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On the regularity of a model non-Newtonian fluid Maxwell, David Aquilla 1997

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ON THE REGULARITY OF A MODEL NON-NEWTONIAN FLUID by DAVID AQUILLA MAXWELL BMath. (Joint Honours Pure and Applied Mathematics), University of Waterloo, 1995 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Mathematics We accept this thesis as conforming to the required standard THE UNIVERSITY OF/BRITISH COLUMBIA August 1997 © David Aquilla Maxwell, 1997 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representa-tives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mathematics The University of British Columbia Vancouver, Canada Date A^^? 1 HI 7 Abstract Existence and regularity of steady and unsteady solutions of a PDE describing the mo-tion of a prototypical incompressible fluid with shear dependent viscosity are studied. The regularity theory is approached by studying the associated elliptic operator. A summary of the classical technique of difference quotients applied to non-linear el-liptic systems is given by applying it to the elliptic system associated with a vector Burgers-like system. Interior regularity is proved for a general class of Stokes-like elliptic operators using a new solenoidal test function that permits the application dif-ference quotient methods to systems with a divergence free constraint. Existence for steady solutions of the incompressible fluid PDE is proven; interior regularity follows immediately from regularity of the Stokes-like elliptic system. Existence and interior regularity for time dependent solutions are proven. ii Table of Contents Abstract ii Table of Contents iii List of Figures iv Acknowledgement v Chapter 1. Introduction 1 1.1 Notation 6 Chapter 2. Weak Solutions of the Stokes-like System 8 2.1 Existence and Uniqueness 8 2.2 Existence of a Pressure 14 Chapter 3. Existence of Second Derivatives for a Model System 16 3.1 Difference Quotients 17 3.2 Interior Regularity 19 3.3 Boundary Regularity 26 3.4 A Simple Application 32 Chapter 4. Interior Regularity for the Stokes-Like Problem 34 4.1 Preliminary Lemmas 40 4.2 A New Test Function 45 4.3 Interior Regularity 46 Chapter 5. Applications to non-Newtonian Fluids 56 5.1 The Steady Case 56 5.2 The Unsteady Case 59 Chapter 6. Epilogue 67 6.1 Boundary Regularity 68 6.2 Higher Time Regularity 70 6.3 Higher Space Regularity 72 6.4 Final Words 73 Bibliography 74 Appendix A. A Direct Coercivity Calculation 76 iii List of Figures 4.1 Comparison of Path Integrals Acknowledgement To begin, I would like to thank Professor K. Hare for introducing me to the beauty of functional analysis. My training under her left me well prepared for my later study of partial differential equations. Professors J. Necas and J. Malek helped me by reading an early draft of my thesis. I would like to thank them for their comments. I would also like to thank Professor M. Ruzicka who spotted an oversight in a draft of Chapter 4. While writing my thesis, I often turned to the books of Professor G.P. Galdi. These references were invaluable to me, as they must also be for many other Navier-Stokes students and researchers. I would like to thank him for providing this resource to our community. The most important influence on my work has been that of my supervisor, Professor John Heywood. Nearly all I have learned about the Navier-Stokes equations I learned from him and his writings. It is no accident that his works permeate the bibliography. His influence exists here also in that I have tried to give the simplest, least general-ized, approach to the solution of problems. His example in this matter provided a good role model for me; my thesis is less clumsy and complicated for it. I would also like to thank Professor Heywood for having taken a personal interest in ensuring my exposure to active Navier-Stokes researchers. Most significantly, he obtained for me two invitations to European conferences where I presented the ideas of this thesis. I would like to sincerely thank him here here for all he has done for me as his student. Finally, to my wife, Kim Fackler, I cannot express enough my gratitude. She had faith in me when I did not. She believed what she could not understand. She understood sometimes when I could not. To her my efforts and this thesis are dedicated. David Maxwell August, 1997 Vancouver, British Columbia v Chapter 1 Introduction In this thesis we study an incompressible fluid with equation of motion ut + u • Vu = -V7T + 2div((u1 + ^ 2|Du| 2)Du) + f divu = 0 (1.1) u L = 0 where Du is the symmetric part of the gradient of u. For simplicity, we will work on a bounded domain 0 contained in R 3 . Our primary goal is to investigate the regularity of solutions of this equation, and to do this we focus on the regularity of the associated nonlinear Stokes-like system -2div((i/i + ^ 2|Du| 2)Du) = - V T T + f divu = 0 (1.2) ulan = 0-We are motivated to study (1.1) for several reasons. Of course, this is a physical model and exhibits so-called "shear thickening" effects seen in some non-Newtonian fluids. However, this is only a specific case of a general class of fluids with power law type Cauchy stress tensors = 2(i/i +i / 2 |Du| 9 )AjU 1 Chapter 1. Introduction which are themselves a specific case of models with stress tensors satisfying Stoke's hypothesis on the stress-strain relationship, Tij = /i(|Du|,detDu)AjU + / 2(|Du|,detDu) [Du 2]... On the other hand, the specific model is associated with the Smagorinsky turbulence model [Sma63]. For a uniform mesh with mesh size h this model can be written with stress tensor 2^(1+ / i 2 |Du| 2 )AjU which is obviously related to the model studied in this thesis. Most importantly, the reason for addressing a prototypical problem is that we can focus on the real difficul-ties of the regularity theory without worrying about being bogged down in details required for generalization. Since we will not be able to come up with a complete reg-ularity theory in the end, it does not appear to be a large loss to use a specific model with a structure that is particularly amenable to analysis. Indeed, the nagging open questions of regularity for nonlinear elliptic systems are especially clear and troubling for the model (1.2). Since the viscosity of this fluid is (v\ + z^lDul2), we see that the viscosity increases wherever Du does, and we would expect that the larger viscosity would then act to damp Du at those places where it is large. Thus we can imagine this as some sort of self-governing fluid and we would expect solutions to exhibit at least the regularity properties of solutions of the Navier-Stokes equations. In particu-lar, in smooth domains with smooth forces we should be able to prove the existence of classical solutions, at least on some time interval (0, T). The fact of the matter is that regularity theory for nonlinear elliptic and parabolic systems (and so also this thesis) is currently unable to address this question, even without the added complications of the solenoidal constraint considered here. With classical solutions as an end-goal, it seems unnecessary to focus too heavily on generalizations when we cannot even obtain desired answers for the specific case. Chapter 1. Introduction Our hope is that the arguments used in this thesis would be accessible to a reader with a basic functional analysis background as well as some familiarity Sobolev spaces and the spaces of divergence free functions such as can be found in a cursory reading of Chapters II and III of [Gal94]. Whenever possible we have chosen to use a simple argument rather than a more complicated, though perhaps more powerful, one par-ticularly when the underlying principle is the same. For example, in addressing the existence of solutions of (1.2) we have chosen to use the easily proven fact that if u f c converges weakly to u, then ||u|| < limfc ||u fe|| rather than the more general principle of weak lower semicontinuity of convex functionals. The structure of the thesis is as follows. In Chapter 2 we prove the existence and uniqueness of weak solutions of (1.2). We introduce the pressure in this Chapter and point out the surprising features of its apparent irregularity. From this we move on in Chapter 3 to a review of how the classical technique of difference quotients can be applied to the study of the related non-linear "vector-Burgers-like" system - div((i/i + i/ 2|Vu| 2)Vu) = f (1.3) U U = °-We do this for two reasons. First, it provides the reader with an introduction to the application of difference quotient techniques to non-linear systems, particularly those with growth condition different from the Laplacian and Stokes systems (i.e. growth condition (1.5) below with p > 2). More importantly, it illustrates why these tech-niques cannot be applied to systems with solenoidal condition directly and it there-fore provides context for the new results in Chapter 4 which form the center-piece of the thesis. Building on the intuition developed in Chapter 3, we prove in Chapter 4 our fundamental regularity result for (1.2). We show that if f is in L2(Q) then the weak solution of (1.2) has second derivatives locally in L 2 . Actually, we prove more than 3 Chapter 1. Introduction this. If T is a C 1 function mapping R " ™ m to R " ™ m such that for some p > 2 dkiTij(A)BijBki > C l ( l + | A | ^ 2 ) | B | 2 (1.4) |9 w T^(A)|< C 2 ( l + | A r 2 ) (1.5) for all symmetric n dimensional matrices A and B, we prove that weak solutions of the system -divT(Du) = -V?r + f div u = 0 U U = 0 have second derivatives locally in L2. To do this we introduce a new test function that allows difference quotient techniques to be extended to system (1.2) and related systems. We have chosen to violate in this Chapter our avoidance of generality for a couple of reasons. Firstly the arguments to get the general proof are no different from those for the specific case, save for some preliminary lemmas that are well motivated by our work in Chapter 3. So we do little extra work to get the stronger result. Also, the conditions (1.4) and (1.5) are easily seen to be analogous to those of ellipticity and growth standard in the theory of elliptic equations and systems. Thus we see that the result is a genuine extension of the theory of difference quotients to non-linear elliptic systems with solenoidal constraint. When presented in this context, not only is the technique new, but it also provides an extension of the class of stress tensors T for which we have regularity results. Therefore it seems appropriate to exhibit the stronger theorem. In Chapter 5 we apply the interior regularity result of Chapter 4 to steady and unsteady solutions of 1.1. Let Q' be any open subset of with closure contained in fi. In the steady case, we prove the existence of solutions and show that every steady solution with f in L2(Q) has second derivatives in L2(Q!). In the unsteady case we show that if the initial velocity is solenoidal and in L2{0) and if the forcing term f is in L2{Q, x [0,T]) then there exists a unique solution u with second spatial 4 Chapter 1. Introduction derivatives in L2(oy x [0,T]) and that tu has a time derivative in L2(Vt x [0,T]). The proof avoids the technical arguments required for existence proofs of weak solutions. Instead, we are motivated by the existence proof for Navier-Stokes equations [Hey80] wherein the existence of regular solutions is proven directly rather than proving weak solutions exist and then examining their regularity. In the end, though, we do not get particularly regular solutions for (1.1) (and certainly not classical ones) and so we are only motivated by some of the preliminary ideas of [Hey80]. The regularity of the motion of an incompressible fluid evolving according to (1.1) has been studied since the pioneering work of Ladyzhenskaya. For a class of fluids includ-ing those modeled by (1.1) she showed the unique existence of weak solutions, in con-trast to the Navier-Stokes equations for which such a result is not yet known in three dimensions. In the steady case, one can find in the work of Giaquinta and Modica [GM82] results concerning the almost-everywhere Holder continuity of the gradients of solutions of a class of equations similar to (1.2), except these equations have growth property of the type (1.5) with p = 2 only and therefore do not exhibit the difficulties associated with the pressure we shall see later. There is a technique in [MNRR96] for the unsteady case that when applied to the Stokes-like system seems likely to give solutions as well as some form of boundary regularity. However, these calculations have not yet been done and would apply directly only in the case that T were derived from a potential, by which we mean Ty(Du) = c\;F(|Du|2) for some scalar function F. For the unsteady problem we have recent work [MNR93] [BBN94] [MNRR96] in the case of periodic boundary conditions wherein the global existence of regular so-lutions with second derivatives in L2(Q, x [0, T]) are proven for a class of equations including (1.1). For Dirichlet boundary conditions, new difficulties appear, as seen in Chapter 4, and the known results are more sketchy. We have an existence proof of classical solutions for small data from Amann [Ama94]. For large data, there is an ex-5 Chapter 1. Introduction istence proof in [MNR96] of regular solutions, i.e. L2(0, T; W2A/3(Q)). The techniques of our proof are different from those in [MNR96], so we obtain a different perspective on the solutions. For example, our proof makes explicit the property that the second derivatives are in in L2(0, T; W2'2(Q')) for open sets fl' with closure contained in O. We have then made apparent the troubling question of whether singularities of type L5 occur in the second derivatives at the boundary. Also, when take in the context of a generalized stress tensor T, our existence proof extends in some respects the results of [MNR96]. The extension of the results of Chapter 4 to the generalized case is taken up in [Max97]. We should ephasize here, though, that since we do not obtain boundary regularity we do not fully recover the results of [MNR96]. 1.1 Notation We end our introductory remarks by outlining our notation. We will use V to denote the gradient operator and D the symmetric part of the gradient. That is, Du has com-ponents (Du)jj = DijU = \{diUj + djUi). If / is a function defined on square matrices, we will write d^j to mean the derivative with respect to the ith,jth component of its argument. Let {efc}£ = 1 denote the standard basis of R™. For any function / defined on R n we define ^ / ( x ) to be the difference quotient /(x+/te^)~/(x). j n addition to difference quotients, we will have occasion to use a smearing operator ah:S defined by <T/,)S/(X) = / 0 X /(x + thes) dt. In all that follows we will use the usual Einstein summation no-tation with the exception that indices appearing in the operators r and a should not be summed unless this is written explicitly. Therefore, T-hjThju would not denote a difference quotient "Laplacian", but YJ7 T-htjThtjU would. If X is a Banach space, we will use | • | to denote its norm if it is finite dimensional, 1  • 11 6 Chapter 1. Introduction if it is L2, || • | | p if it is LP and || • \ \ x otherwise. Also, we will use the generic notation X* to mean the dual of X. We use (/, v) to denote the value of the functional / taken at v. In the case of L2, we use (•, •) to denote the inner product. We denote by W N , P ( Q . ) the usual Sobolev space with W Q , P ( Q , ) the subspace generated by taking the closure in Wn,p(Q) of (Q), the set of smooth functions with compact support. Let J(fl) denote the subset of C£° ( Q ) that are also divergence free (we will not make distinction between spaces of scalar and vector valued functions as this will always be clear from the context). Let J0"'p(^) denote the closure of J{9) in W Q , p ( Q ) . 1 We will simply write Jp for J°' p and J for J2. While our notation for divergence free spaces is not traditional, it is hopefully both consistent and clear. If Q, is an open subset of R n and 0 < 6 < T we write 0^ and [^<5,r] to denote the time cylinders 0, x [0, T] and x [S, T] respectively. We use the standard notation for Bochner spaces. If / e Lp(0,T;X) then J 0 T \\f\\px dt < oo. We identify LP(Q,T; L"(fi)) with W(Or). Since we will always work on a bounded domain in this document, we need not worry about the troubles that arise from this definition in unbounded domains. 7 Chapter 2 Weak Solutions of the Stokes-like System 2.1 Existence and Uniqueness In this section, we shall concern ourselves with the nonlinear elliptic system of equa-tions defined on Cl, a bounded open set in R", namely -2dj ((1/1 + u2 |Du|2) Diju) = dor + fi diUi = 0 (2.1) If f is in (Jo' 4 )*/ w e define a weak solution of (2.1) to be a function u in Jg'4 such that / 2(i/x + v2 | D u | 2 ) A M & - (f, <f>) (2.2) for all solenoidal <j) in CQ°(CI). It is clear by simple integration by parts that any suf-ficiently smooth solution of (2.1) is also a weak solution. Moreover, for any fixed u e J, 1 ' 4 we obtain from Holder's inequality that for every <f> in J (CI), < 2i/i | | D u | | L 4 | | 0 | | 7 l , 4 + / 2(ui + u2 \Hnf)Dij\xdi(j)j dx - (f, 0} i f ! + 2^ 2 | |Du | | i 4 | |0 | | J i , 4 + | | f | | ( t / o l , 4 ) . | | 0 | | J i . < (2ux | f i | 5 ||Du||L4 + 2^ 2 | |Du|| | 4 + + llfl Chapter 2. Weak Solutions of the Stokes-like System Since solenoidal C£° functions are dense in J01,4, it follows that for fixed u the integral in (2.2) defines a continuous linear functional on <j> e J Q 1 ' 4 . In particular, if u is a weak solution, then / 2(i/! + v2 |Du|2)DijUdjVi fix - (f, v) = 0 (2.3) Jn for all v G J Q 1 ' 4 . To investigate the solubility of (2.2) we will consider the functional on J Q 1 ' 4 defined by ;F(u) = [ vx |Du| 2 + ^ |Du| 4 rix - (f, u) Jn 2 (2.4) That T is well defined on J Q 1 ' 4 is clear since |^(u)|< f ^ |Du| 2 + ^ |Du| 4 rix+|(f,u)| Jn 2 < ^ l | V u | 2 + y |Vu | 4 dx+ | |f| | < | | U | | J M + yIMIjM + | | f | | ( j o l , 4 ) . | | u | | J O M . We now apply the theory of variational integrals to the functional T to obtain the existence of a minimizer for T. Since critical points of T are shown to be solutions of (2.2) we will have in hand the existence of a weak solution. Moreover, by using the simple form of T, we will be able to prove these things without calling directly upon the standard results concerning weak upper semi-continuous functionals. As is usual, we define a critical point of T to be a function u in J Q 4 such that for any v in JQ'4, + sv) | = 0. Since T(u + sv) is just a polynomial in s, we can easily compute the derivative of T(u + sv) with respect to s and evaluate it at s = 0. Doing this yields —T u + sv r |2\ / 2(>i + vi |Du|2)DijuDijV dx - (f, v) Jn / 2{vx + v2 |Du|2)DijudiVj fix - (f, v) (2.5) Jn Chapter 2. Weak Solu tions of the Stokes-like System where we arrived at the last line using the fact that if A is a symmetric matrix and B is an anti-symmetric matrix, AijBij = 0. It its clear from (2.5) that a critical point is weak solution. Suppose T has a minimizer u. We now show that u is a critical point of T, and there-fore a weak solution. If v is any other element of J Q 1 ' 4 , then ^(u + sv) is a fourth order polynomial in s; call it r(s). Since u is a minimizer, r(0) < r(s) for all s. Thus, r'(0) = 0, which can be written as ^^"(u + sv) | = 0. Since v is arbitrary, u is a critical point. Given what we have just seen, to show existence of a weak solution it would be suffi-cient to show existence of a minimizer. This is what we turn to now. First, we will show that T is coercive and bounded below. To do this, we split the domain of T into two regions, one wherein the term (f, u) is dominant and the re-mainder of the space where this term in subordinate. We will need to use a case of the Korn inequality proved, for example, in [Nec66]: for u e Jo'4, there is a constant k(Q) > 0 such that | | u | | ^ ,4 < £;(Q)||Du | | L4. Let us now consider the case where (f, u) is subordinate. Suppose where 7 > 0 will be determined later. Then • l l ^ - i i u i i j M i i f i i ^ . IHIjM-IHIJoM||f||(i7 > 7 | | U | I jM 10 Chapter 2. Weak Solutions of the Stokes-like System Taking 7 = 1 gives us the desired coercivity. Taking 7 = 0 we have shown that F is non-negative when | |u | | 3 r l 4 > ^ U l l f II, M r . Now we find a lower bound when I lul I ji,4 < 2 f c^ I |f 1  (ji>4)* by neglecting the positive terms in T to get jr(u)= f vx |Du| 2 + ^  |Du| 4 dx - (f, u) > - (f, u) > - IHI^II f l l ( jM)* Thus we have obtained that T is bounded below by - ( ^ j f ^ ) " I l f 1  (jM)* • We are now able to find a minimizer for J7. Since T is bounded below, there exists a sequence u" of terms in J 0 M such that T(un) ->• inf v G Ji,4 T{y). By coercivity, the sequence is bounded in J 1 / 4 and so converges weakly in J 1 / 4 to some u. We now show that .F(u) is the minimal value for T'. Recall from elementary functional analysis that if u n converges weakly to u in a Ba-nachspace S, then liminfn^ oo | | u n | | B > | | U | | B . Since | |Du | | L 4 < ||u||7|i,4 < /c(ft)||Du||L4, as cited before, we can take |\D • \\L4 as the norm on JQ 1' 4. Since {un} converges weakly to u in J 1 / 4 we have limint^ oo | | D u n | | L 4 > | |Du | | z ,4. Also, since f2 is bounded and u" converges weakly to u in JQ 1' 4, it also converges weakly to u in j]'2. We can easily see by a couple of integrations by parts that | |Du|\L2 = | |u| ^ 1 , 2 . So, lim inf^ oo | |Du™| | L 2 > | | D u | | L 2 . Using these fact together with the weak continuity of linear functionals it 11 Chapter 2. Weak Solutions of the Stokes-like System follows that vex1-4 inf .F(v) = lim F(un) = lim f vx | D u n | 2 + ^ | D u n | 4 dx - (f, un> n~>°° Jn 2 = lim / vx | D u n | 2 + — |Du"| 4 dx - (f, u) n^°° Jn 2 > vx lim inf | |Du n | | £ 2 + ^ lim inf ||Du"||L dx - (f, u) > ^ | | D u | | 2 2 + ^ | | D u | | l 4 fix-<f,u) = ^(u). Thus u is in fact a minimizer for our functional and a weak solution of (2.2). The uniqueness of weak solutions follow from the convexity and smoothness of the functional T. We can show this in an elementary way using finite dimensional argu-ments. Suppose u 1 and u 2 are two distinct weak solutions of (2.2). Then T(sul + (1 - s)u2) is a fourth order polynomial in s, call it r(s), with derivatives r'(s) = / 2 i / i ( s u 1 + (1 - s) u2) Dijiu1 - u2) + Jn + 2u2 |D (su1 + (1 - s) u2) | 2 Dij (su1 + (1 - s) u2) A ^ u 1 - u2) dx -- ( f ^ - u 2 ) ) (2.6) and r"(s) = [ 2vy |D (u1 - u2) | 2 + Au2 | A ; ( S U 1 + (1 - s) u2) Aj (u x - u 2)| 2 + Jn + 2v2 |D (su1 + (1 - s) u2) | 2 J D ^ 1 - u 2)| 2 fix > 0. (2.7) 12 Chapter 2. Weak Solutions of the Stokes-like System From equation (2.6) using (2.3) we obtain r'(0) = f 2u1Diju2Dij(u1 - u2) + 2v2 | D u 2 f A j U 2 A j ( u x - u2) fix - (f, (u1 - u2)) Jn = 0, and r ' ( l ) = / 2 I / I A J U 1 A J ( U 1 - u2) + 2v2 (Du 1! 2 A ^ A j i V - u2) fix - (f, (u1 - u2)> Jn = 0. By the Mean Value Theorem, then, there is an s0 in (0,1) with r"(s0) = 0. This con-tradicts the positivity of r" as shown in (2.7). Thus there can be at most one weak solution. The results of this and the previous section can be summarized in the fol-lowing theorem. Theorem 2.1 Let Vibe a bounded open subset o/R" and f be in (J01,4)*(Q). Then there exists a unique u € J Q 1 ' 4 such that: / 2(ui + v2 |Du | 2 )£)j jUjDjjvdx = (f, v ) Jn for all v G J Q 1 ' 4 . Moreover, there is a constant c(Q) such that ||u||JoM<c(n) ( ^ l l f l l ^ ) * ) 3 -The final estimate of the theorem is easily derived: u22k(n)\\u\\4.1A < 2^ 2 | |Du | | l4 < / 2{y\ + v2 |Du|2)DjjuDijU fix Jn — I 2(u\ + v2 |Du|2)DijudiUj fix Jn = <f,u) 13 Chapter 2. Weak Solutions of the Stokes-like System 2.2 Existence of a Pressure Until now, we have not discussed the pressure which appears in (2.1) but not in the weak formulation. However, using standard results from Navier-Stokes theory we can show a reformulation of the weak formulation in which it appears. Given any f € (Jo'4)* w e have, since J 1 ,' 4 is a closed subspace of W Q 1 ' 4 with the same norm, that f\\(ji,*Y = sup — — (f,u) : SUp By the Hahn-Banach theorem there exists f 6 ( W ^ 4 ) * such that its restriction to J 1 / is just f. Let u be the weak solution of (2.2) for a given f and let f be an extension as described above. Then the linear functional C on WQ'A defined by £(v) = J 2(ux + u2 \Du\2)DijUdjVt dx - , is continuous. From (2.2) we see that £(v) = 0 on J Q 1 ' 4 - It is well known, see for example [Gal94], that every continuous linear functional on W Q , A vanishing on J 1 / 4 has a representation of the form / 7T (V • v) dx Jn for some function 7r e such that JQ IT dx = 0. So we are assured that there exists a function n G L% such that / 2(ui + v2 |Du|2)DijudjVi + 7T (V • v) dx - (f, v) = 0 (2.8) 14 Chapter 2. Weak Solutions of the Stokes-like System for all v e W Q ' \ At this point we should point out a significant difference between weak solutions of the Stokes system proper and the nonlinear Stokes-like system considered here. The weak solution of the Stokes system is found in the space W01,2(ft) which is a larger space than the one containing solutions of the nonlinear system, JQ 1' 4. This phenomenon is what allows us, for example, to show unique weak solutions of the non-steady system (1.1). However, the pressure for the Stokes system lies in L2(Q) as opposed to the pressure appearing in the nonlinear system which lies in l4(fi). Thus the pressure in the nonlinear system apparently comes from a less regular space. All of this arises from balancing terms in the weak formulation. If u is an an arbitrary element of JQ' 4(Q), then 2(vx + v2 |Du| 2 )AjU is in Ls. So we cannot expect that 7r with which it is balanced lies in any smaller space. This balancing also allows us.to have right hand sides f to be in (Jo'4)* which is in a bigger space than the one containing right hand sides for the Stokes system, (Jo'2)*- Indeed, since we normally specify f to obtain u, we can see that the less regular pressure is related to the fact that we are able to specify a less regular right hand side. The structure of the PDE determines spaces in which both solutions and legitimate right hand sides can be found. The less regular space that contains the pressure becomes significant later on when we attempt to prove regularity for solutions of system (2.1). It turns out that a natural way to prove regularity for the Stokes system [SS73] relies heavily that the pressure does not live in any worse a space than I?. Therefore, we will not be able to extend these ideas to the nonlinear Stokes-like system. Indeed, in the end we will obtain regularity only by avoiding the pressure completely. However, before we do this, we first study how classical techniques are applied to a model system similar to the nonlinear Stokes-like system. 15 Chapter 3 Existence of Second Derivatives for a Model System In this Chapter we will examine the system: -dj {(ux + v2 |Vu| 2) djUi) = fi (3.1) u l a n = 0 It is obviously similar to that studied in Chapter 2 , and we will use it as a model to investigate how to prove regularity for the nonlinear Stokes-like system (2.1). There are two differences between the model, which we shall call the nonlinear Poisson-like system, and the nonlinear Stokes-like system. Firstly, we have relaxed the constraint that solutions be solenoidal. This, in turn, increases the set of test functions to work with in the variational formulation. Secondly, we have replaced occurrences of the deformation tensor with the gradient tensor. This is necessary to preserve the nature of the nonlinearity; we want it to force solutions to be in W 0 1 ' 4 , just as in Chapter 2 . By similar techniques as used in Chapter 2 we can show that if f e (Wo'4) * there exists a unique weak solution u e W Q 1 ' 4 of (3.1), i.e. a function u e WQ'A that satisfies / {yi + v21 Vu| 2) diUjdi^j dx = (f, </>) (3.2) Jn for all cj) G C Q ° ( 0 ) . We have the following estimate for the size of such a solution: I H I < 4 < ( ^ l | f | l K T ) 3 . (3.3) 16 Chapter 3. Existence of Second Derivatives for a Model System We now want to show that we have some greater regularity of the solution if we assume some greater regularity from f. In particular, we will assume that f e L 2 . The techniques and results in this Chapter are not new. For example, the ideas we are about to use can be found in standard texts: for elliptic equations in [LU68] and for elliptic systems in [Nec83]. Actually, the structure of the system allows us to treat the system in the same fashion as an elliptic equation. We use the structure further to prove results directly with the idea of motivating the generalizations to come in Chapter 4 . Moreover, the intrinsic calculations are more complicated for the Stokes-like system and it will be useful to see the preliminary ideas in an isolated context. We will show in this Chapter how the theory of difference quotients can be used to attack nonlinear problems. By seeing how the standard theory works in this case, we will also be able to see how its direct application fails in the case of the nonlinear Stokes-like system and therefore why the results of Chapter 4 are interesting. 3.1 Difference Quotients It seems reasonable at this point to give a very brief recollection of the fundamentals of difference quotients. The proofs of these results are well known and can be found, for example, in [Gia93]. Given a function g on R 3 , let ThjTn(g) denote the quantity g(x + hem) - g(x) h If g is defined on our bounded set 0, we extend for simplicity g by 0 to R 3 to define the difference quotient in this case. Naturally the difference quotient in some sense approximates the derivative of a function. This intuition can be made rigorous as is seen by the following Lemma. Lemma 3.1 If g £ Wl'p(fl) and Q' is an open set with closure contained in Q then there is a 17 Chapter 3. Existence of Second Derivatives for a Model System constant c(0, Q') such that Th,mg\\LP(W) < c(Q,Q')\\dmg\\Lp(n')-for all h such that h < d(dfl', <9fi). An upper bound for c(fl, 0') is the number of cubes with The converse of this Lemma is also true and is indeed more useful for our purposes. Lemma 3.2 If g e 1/(0) and fi' is an open set with closure contained in Q and \\Th,m9\\p < & for some fixed k and for all h < d(dCl', <9Q) then g e Wl>p(Q!). Moreover, \\dmg\\P < k and Th,mg converges strongly in LP(Q') to dmg. Derivatives and difference quotients commute. Indeed, if d(dfl', dQ) > 2h then and djTh,mg = Th^mdjg. The difference quotient also enjoys some properties analogous to those of derivatives. We have a rule for "integration by parts". If g G Lp(Cl) and / G U' ( Q ) and either g or / have compact support then length of ft. d(dn',dri) required to cover 0', and therefore it depends only in this way on the regularity 2 for h sufficiently small. We also have a "Leibniz rule", Th,m{fg(x)) = y(x)r/,)m/(x) + /(x + hem)rhtmg(x). 18 Chapter 3. Existence of Second Derivatives for a Model System 3.2 Interior Regularity Before we start the calculation to derive interior regularity, let us first motivate it with the a priori estimate that underlies it. Let x 0 be an interior point of fl and let 77 be a cut-off function in a neighbourhood of x0, so 77 € Co°(£2), 77(x) e [0,1] for all x e fl, and there exists a neighbourhood 0' of x 0 such that 77(x) = 1 in this neighbourhood. Assuming u is a smooth function, we can use dk (r]2dku.)as a test function in (3.1). Doing this yields, after expanding some derivatives, / fidk{rj2dkUi) fix = - / {y(Vu)didkUj + u2drdkusdrus) (2r]dir]dkuj + r)2didkUj) fix Jn Jn and therefore / r)2u(Vv.)didkUjdidkUj dx = — / rj2u2drdkusdrusdidkUjdiUj dx — Jn Jn — I 2riu{Vv)didkUjdir}dkUj dx — Jn — I 2u2T]drdkUsdrusdiUjdir]dkUj dx — Jn — [ fidk{v2dkUi)dx (3.4) in where u(Vu) = u x + u2\Vu|2. Applying Holder's inequality in the above yields easily / ri^iVxijdidkUjdidkUjdx < c(^) [||77f||2 + | | | V T 7 | ^ ( V U ) V U | | 2 1 . (3.5) in L J Since the left hand of (3.5) side bounds the second derivatives of u, i.e. i/i||?72V2u|| < J n r)2v(Vu)didkUjdidkUj dx, we have the desired a priori estimate. The key part of this calculation involved the ellipticity of the system which appeared via / r]2dk (u(Vu)diUj) didkUj dx = r]2u(Vu)didkUjdidkUj dx + Jn Jn + / r)2v2drdkusdrusdidkUjdiUjdx Jn > / r]2u(V\x)didkUjdidkUj dx. (3.6) in 19 Chapter 3. Existence of Second Derivatives for a Model System We now want to use the ideas of the a priori estimate in the difference quotient context. Let 0" be an open set with compact support in 0 such that supp(?7) and supp(?7(x-/iem) are contained in 0" for every h < h0 for some h0 > 0. If h is sufficiently small then ?7277i,m(u) G Wo1'4 and so is r_ /j i m(772r/ l ) m(u)) and it is therefore a legitimate test function. Letting <f> = T_htTn (rfr^m (u)) in (3.2) we obtain / fiT-h,m {v2Th,m dx. = / (ux + V2 \ Vu| 2) d i l t j d j T _ A i T O {j)2Th,m (Uj)) dx The real trick now is to see that the elliptic properties observed for derivatives persist in some sense for difference quotients. We want a bound for the right hand side of (3.7) similar to (3.6). The trick can be found in [LU68] for a scalar quasilinear elliptic equation and it applies here also. We are able to write the quantity v\ + v2 |Vu| 2) diUj) di (v2Th.,m {UJ)) dx (3.7) Th,m ( ( " 1 + ^2 |Vu| 2) diUj) = Thtm (v (Vu) diUj) in a convenient form. For almost every x, T~h,m 1 C d r (u (Vu) diUj) = - / jr- (u (rVu(x + hem) + (1 - r) Vu(x))) • h Jo dr L • (di (ruj(x + hem) (1 - r) Uj(x))) dr vi + T ( I V U ( X + hem)\ + drus(x + hem)drus(x) + + |Vu(x)|2)j<M j 7+ + dkU[ (x + hem) (x + hem) + ]rdkui (x) d^j (x + hem) + + ^dkui (x + nem) diUj (x) + <9fcw/ (x) diUj (x) r A > m (dfcuj) With this explicit expression, we are able to write some estimates (boundedness and 20 Chapter 3. Existence of Second Derivatives for a Model System coercivity) for the bilinear form aikji(h, m). For the coercivity estimate, we have aijjkitk&j = ("l + y (|Vu(x + hem)| 2+a ru s(x + /iem)a ru s(x) + |Vu(x)|2)) |£ | 2 + + ^ [dkUi (x + hem) (x + hem) + dkut (x) 8^ (x + hem) + ^ r i V u f x 4- h p „ ) l 2 4- i V i i f x ^ l 2 ) ^ l £ l 2 > ( 1^ + ( |Vu(  + / iem | 2 + | u( )|2 ) + 13 2 + y (x + hem) ikif + (dkut (x) fH) 2) > ^ h + ^ 2 (|Vu(x + hem)f + |Vu(x)|2)) | £ | 2 . For the boundedness estimate, we have a>ikjitk&j = ( i^ + ^ (|Vu(x + /iem)|2 + «9rus(x + /ieM)<3r7Js(x) + |Vu(x)|2)) |£| + ^ (x + hem) diU3 (x + hem) + ^it/ (x) (x + hem) + dkut (x) (x))£ H &j < {u, + | (|Vu(x + hem)\2 + |Vu(x)|2)) |C| 2 + v2 (dkv,[ (x + /ieTO) diUj (x + /iero) + (x) (x)) < ^ 1 + | ( | V u ( x + / iem)| 2 + |Vu(x)|2) + u2 (|Vu(x + hem)\2 + |Vu(x)|2)) |£ | 2 < \ (vx + "2 (|Vu(x + hem)\2 + |Vu(x)|2)) HI 2. Since we will encounter this expression several times, let us define Hum = vi + »2 (|Vu(x + hem)\2 + |Vu(x)| 2). In summary, then, we have shown, that ; U , m l £ | 2 < a^kidi < \^m\i\2- (3.9) 2 + 21 Chapter 3. Existence of Second Derivatives for a Model System Now we use these estimates to analyse (3.7). From (3.7) and (3.8) we obtain / fiT-h,m(V2Th,m(Ui)) = - <h^kTh,m 9% (?72T)*,m (Uj)) dx Jn Jn = ~ V2aij™idkTh,m (ui) di ( r A , m (UJ)) dx -Jn ahi(kidkTh,m (ui) 2r]diT] (Thjm (UJ)) dx, so / V2aij™idkTh,,m (ut) di (Th%m (UJ)) dx = - / fjT-h^r^T^miuj)) dx -Jn Jn ~ / aijkldkTh,m («/) ZvdiV (Th,m (%)) dx. Jn (3.10) We now want to estimate from above both terms on the right hand side. Let us start with the first of the two: / fjT-h,m{rjiThtm{ui))dx.= / / j[r_ / l ! m(77)77T f t ! m(« j)-r-7?(x-/^em)r_/ l ! m(r7r f e i m(u J))]dx Jn Jn < \\vf\\L^\\r-h,m(v)rh,m{a)\\L2 + + ||/77(x- /iem)||L2||r_h,m(77TA)m(u))||i2 < WVIWLA\r-h,m{v)Th,m(n)||L2 + -J^-1|r/(x + hem)f\\2L2 + + Vie'\\T-h,m(rrh,m(u))\\h- (3.11) At this point we invoke Lemma 3.1 to bound the last term here. From Lemma 3.1 we see that there exists c{Q, 0") such that for all v in W0M(ft"), ( V ) | | L P ) < c(0,Q")||Vv||L P. 2 2 Chapter 3. Existence of Second Derivatives for a Model System From this we obtain \T-h,m(VTh,m{u))\\l2 < c(Q,Q")\\V{rprh,m{a) I I 2 < c(fl,fi") (||V(r;)r,,m(u)||2L2 + l l ^ V r ^ u ) ! ^ ) 2 < c(fi,«") f||V(77)-r^m(u)|j|2 + - | v^77Vr f t , m (u ) | | 2 L 2 < c(0,O/') ^ i V T y r ^ u H 2 , + ^ H V ^ T / V T ^ U ) ! ! ! ^ (3.12) Combining (3.11) and (3.12), letting e = e'c(Q, Q") we obtain the final estimate for this term, / fjT-h,r Jn {v rhjm(uj)) fix < c(Q, fi") [(l + -^j 11(77 + r7(x + hem))f\\l2 + + (1 + i/ie)|(V/7 + Tft^r/j^uHls +c||v/Aih,m»7VrAim(u)||£2. (3.13) Now we turn to the second term on the right hand side of (3.10). It is not hard to show using the form of ah,m and the Cauchy-Schwarz inequality that h,m c ^ I h,m (• (• \ 2 / h,m \ (3.14) Indeed, a.h'm is a positive definite symmetric bilinear form on the vector space of n by n matrices, so (3.14) is just an assertion of Cauchy-Schwarz. Using (3.14) in the second 23 Chapter 3. Existence of Second Derivatives for a Model System term on the right hand side of (3.10) we obtain J S7 • (fijMTh,m {Uj) Th>m {ui) djTjdk-nj 2 d> aij™idkTh'm vdiV (Th,m (UJ)) fix / 2 e f 2 h, ZdkT^mUldiTh^Uj fix c + / ahijkiTh,m (UJ) rKm (ut) djj]dkr]fix i n < e|IV^ft,m»/Vr/, i m(u)||| (2 + + - / A*/i,m|V77|2|rft)mu|2fix. e i n (3.15) On the other hand, we can bound below the left hand side of (3.10) using (3.9) to get ^\\y/p^rjVThtm(u)1||2 < J V2a£j™idk (rh,m (uj)) di (r f t,m fix. (3.16) Letting e = 1/24 we can combine the lower estimate (3.16) with the upper estimates (3.13) and (3.15) for equation (3.10) to get | 2 f i x + | | ( 77+r / (x - f - / i e m ) ) / | | 2 2 + L i n + 1  (V77 + Th,mr))TKrnu\||2 . (3.17) The right hand side of (3.17) is uniformly bounded in h. Indeed, by the absolute con-tinuity of the integral we see ph<m converges strongly in L? to poiTn = v\ 4- 2i/ 2 |Vu| 2, which we will call \i. From Lemma 3.2 we know T ^ U converges strongly in L 4 to <9mu. Finally, from the smoothness of 77 we know T/i imu converges uniformly to dkr]. From the boundedness of convergent sequences follows the boundedness of the right hand side of (3.17). Since ||Vrh i m(lO||L2(fy)|| < —\\y/lJ.h,mVVTh,m{u)\\L2, (3.18) ^1 we see from the uniform bound in h of the right hand side of (3.17) that the left hand side of (3.18) is uniformly bounded in h. From Lemma 3.2, then, we assert that dmdiU 24 Chapter 3. Existence of Second Derivatives for a Model System exists in L2(Q') for each i. Since m is arbitrary, u G W2,2(Q!), and since x0 is arbitrary, we have u G W2'2(Q!) for any C C fi. We would now like to get an estimate for the interior derivatives similar to that of the a priori estimate (3.5). To do this we note that from (3.17) and the uniform boundedness of the right hand side of (3.17), the sequence ^/Jlh^rfs/'^^(u) converges weakly to some limit. From the strong convergence of \Xh,m to p. in L2 and the strong convergence of ?)Vrd]m(u) to 77V<9(u) in L2 we see that the weak limit must be y/jJrjVdmU. Indeed, if <j> is smooth we have from the aforementioned strong convergence lim / JJIh^rjdiT^rnUjfy dx = y/Jk)didmUj(f>j dx. h^°Ju Jn The result follows from the density of smooth functions in L2. Since the norm of the weak limit is less than the limit infimum of the norms of the limiting terms we conclude that \\VfJ-V^Th,mn\\l2 < l iminfc(Q',fi , u{) / \Vr]\2 nh,m\Th,mu\2 dx + + ||(77+r/(x + hem))f\\2L2 + + ||(V?7 + Th,mv)rh,mn\\2L2 . (3.19) From the strong convergence of r f c ] 7 nu to dmu in L 4 and the strong convergence of ^h,m to fi in L2 as well as the uniform convergence of 77(x + hem) and r^mV to 77 and <9m77 respectively we can take the limit in the right hand side of (3.19) to conclude HvW?r3mu||!2 <c(fi ' in >^)[/ ' |V77|2/x|amu|2dx+||?7f||2-2l. (3.20) Indeed this is almost what we obtained in the a priori estimate (3.5). We would want to remove the dependence of the constant on Q and express the interior estimate only in terms of the cut-off function 77. This can be done as we will see in Chapter 4, but we need not do this for now for our purposes. 25 Chapter 3. Existence of Second Derivatives for a Model System 3.3 Boundary Regularity Although we will not be able to prove boundary regularity for the Stokes-like problem (2.1), we will include boundary regularity for the Poisson-like system to complete the ideas of this Chapter. We will assume for this section that dQ, is of class C 1 ' 1 . Let x 0 be a point on dQ, so there exists a C1'1 invertible function G mapping B^ to 0', where Q' is the intersection of some neighbourhood of x 0 and Q. If v is a function on 0!, let v* denote the pull back of v, so v*(y) = v(G(y)). Since G is C 1 , 1 , if v G WlA{Vt') then v* G Wl'4(B+). Let J be the Jacobian of G, so V„u* = V x u J . We define U to be (Vyu*) J 1 . Already this section will be notationally clumsy, so we will introduce A : B to mean AijBij hoping to make some of the calculations more clear. Let us suppose that $ G W0lA(B^), and let 0(x) = $(G - 1(x)). Using this <f> in (3.2) and changing variables we obtain Jn = [ | det J | (vx + u2 |U| 2) U : V ^ J - 1 dy. (3.21) We can now use techniques similar to those in the previous section. As before we take rj to be a cut-off function with support in BR such that rj\y+hem also has support in BR. 26 Chapter 3. Existence of Second Derivatives for a Model System Letting $ = T^htm(r]2Th,m(u*)) in the above where 1 < m < n we obtain / aijM K m U ) ^ . {rh,mV)kl rf dy = J B R f 43 M)e (Vu*(y + te^-r^J"1),, r?2 riy -+ J B + L V ' v . - I \ detJ\fiT-htm(r)2Th,mu*) dy-- / | det J|i/(U)U : (r h , m u*| ^ 2V77T_/l,rnJ-1) r/dy -- / |detJ |KU)U: U M Vu*| r ^ J " 1 W dy. (3.22) Here we have used the notation a>ijM = f I det J(y + r/iTA ) f ny)| [i/(U + r/irA,mU)<5a!<5jj + io u2(U + rhT^mU^jilJ + r/ir A > m U) w ] dr (3.23) and Jo , m Since | det J | is bounded both from above and away from zero, we have the same estimates as the previous section for ah,m, namely (3.24) where this time l*h,m = v\ + v2 (|U(y + hem)\2 + |U | 2 ) . 27 Chapter 3. Existence of Second Derivatives for a Model System As well as a'1'"1 we can also estimate Fh'm. For any e WQl'A(B\) we have, using the fact that | det J | is bounded away from zero and that J is Lipschitz, / F^m9ijdy< j / \dm\detJ(y+rhThtTny)\\u(V+rhThtmlJ)\lJ+rhThtrnlJ\\g\drdy J B + J B + J O < ^ f uhjm ( U | y + / I + |U| 2) dy + ef < J g | 2 d y . (3.25) J B R J B £ In the right hand side of (3.22) we now have six terms to estimate. Let us make the estimates for each term in order denoting each integral by a Roman numeral from I to VI. For the first we have HI • f +^i(^rnU)lj(rh,mU)klV2dy + J B R + ; f A M ( V u * U e ™ W 1 ) . . ( V u * | y + , e m Th>mJ- 0 V2dy >kl <e[ 4;a^U). . (r , , m U) H r 7 2 dy+^ / ^ JB+ « The second term is estimated similarly. For it we have Vu* y+he„ rfdy. (3.26) mi J B + < t f a^kl(rh,m\J)i](rh,mlJ)klr)2dy+C-^ [ ^ , m | ^ u ' | 2 dy. (3.27) From our estimate (3.25) for F / l , m we can bound the third term by mil //ir \n2 ( V u * U e m ^ J " 1 ) . . - (v^ J-1) J B + L V ' V + c(9fi) / j U / , , ^ ' Vu* y+/ie„ + |U| 2) 7?2riy+ dy + e iiKmr)2 \Th>m (Vu*J _ 1 ) | 2 dy. JBt (3.28) 28 Chapter 3. Existence of Second Derivatives for a Model System The fourth term is estimated as before in (3.13): We arrive at |IV| / | det 3\fiT-h,m {v2Th,mu*) dy < c{dQ) l + —)f |detJ||f| 2fiy+(l + ^e)riy f | det J | |Vu*| 2 dy JBt JB+ + e / Vh,mV2 l^.mVu*!2 dy. JBt + (3.29) Since ThtTnVu* = ThjTn(U)J - V u * | y + f t e m ThtTn(3 *) J we can estimate the last term with e / Vh,mV2 KmVu*! 2 < 2e / nh>mr]2 \TKmV\2 dy + c(dQ)e / / i f t , m 7? 2 | V u * | y + / i e m dy. J B + J B + J B + (3.30) Combining (3.29) and (3.30) yields our final estimate for the fourth term: + f I det J | / 2 dy + (l + *ic) / \Vu*\2dy V e z / l / JBt JBt |IV| < c{dQ) + + c(dn)e / pht„ JBt Vu* y+hev dy + 2e / (3.31) Using Holder's inequality and the fact that | det J[ is bounded away from zero and that the derivatives of J - 1 are in L°° we easily estimate the fifth term by |V| = / |detJ|i/(U)U: (r f c i T nu*| h e m V77r A , m J x)r?dy J B + V ' [ u(U)V : UT , 2 dy + / i/(U) l ^ u * ^ dy JBt JBt < c{dQ) (3.32) The last term is mildly more tricky. Since ThjTnVu* — Th>m(U)J - V u * | y + / l 6 m we have |VI| < / |det J|i/(U)U : (rh,m V u * | ^ r ^ J " 1 ) rf dy J B + V ' - [ |detJ|i/(U)U : U ^ d y + c(dfi)e / u(U)\Vu*| V dy 6 JB+R JBt + e [ i/(U) r , , m (U | , m ) % 2 d y . + (3.33) 29 Chap ter 3. Exis tence of Second Deriva fives for a Model Sys tern We want to be able to absorb this last term into the left hand side. Changing variables we have / + K U ) | ^ > r a ( U | y _ f c e m ) | V d y = / + K U | y + f c e m ) K m ( U ) | V | y + f t e m dy < / K U | y + h e J k , m (U) | 2 {r,2 + (hsupVv)2) dy J B + < / Ph,m\Th,m{^J)\2r)2dy + J B + + C(v) / l*h,m( JBt u y+he„ + |U|2) dy. (3.34) Combining (3.33) with (3.34) we arrive at our last estimate for the sixth term, |VI| < - / |detJ>(U)U : Vr]2 dy + c(dfl)e [ v(U)\Vu*\V dy + E J B + J B + JBt Ul y+he„ dy + | U |2 ^ dy + e J +ph,m\Th,m(U)\2V2dy. (3.35) We now combine the lower bound for the left hand side given in (3.24) with the upper estimates in (3.26), (3.27), (3.28), (3.31), (3.32) and (3.35), absorbing terms on the right similar to those on the left. The remainder of the proof continues as in the interior case to get that U has square integrable derivatives in each of the directions 1 < m < n and / v(U) \dmV\2 dy < JBt c(dCl, ui) [ u(\J) |Vu*| 2 dy + f U(U) | U | 2 dy + f |detJ||f| JBt JBt JBt dy . (3.36) We now use the fact that J has determinant bounded above and below to get |U| has bounds by c(dQ)\Vu*\ < |U | < c(dQ)\Vu* 30 Chapter 3. Existence of Second Derivatives for a Model System to get / i/(Vu*) |<9mVu*|2 dy < c{dQ,ul) f v{Vu*) |Vu*| 2 dy + f | det J| |f | 2 dy {337) We have thus shown the existence of and given estimates for dmdku* for 1 < m < n, 1 < k < n. We now turn to obtaining estimates for dndnu*. We do this by controlling it in terms of the other second partial derivatives that we already know. Let us denote the entries of the matrix J - 1 by JXK Then, the partial differential equa-tion in the flattened coordinates reads a,-(| det 3\PkJlk{ux + v2dqu*mJ«rdsu*mJsr)diu*) = | det J\f*. By interior regularity, this equation holds almost everywhere in B^. We are only in-terested in terms involving two derivatives with respect to yn. So, we move all other terms to the right hand side and denote collectively the right hand side by F. Note that all of these terms can be controlled by our previous estimates. We arrive at the equation jnrjnr^ + U2dku*sJkldmU*sJml) + 2v2JnS Jnk dtU* Jlk dmU* J™ = Ft Thus we are lead to consider the invertibility of the matrix 2v2 Jns Jnkdiu* Jlkdmu* Jms 13 Jnr Jnr{ux + u2dku*sJkldmu*Jml) This symmetric matrix exists almost everywhere since JnrJnr > 0 (J is invertible). It is also positive definite since 2v2JnsJnkdiu*Jlkdmu*Jms _ 2v2\Jnkdsu*JskZj\2 5 i j ^ j + J™JnT{vx + v2dku*Jkldmu*sJml)^3' = + JnrJnr{ux + u2dku*sJkldmu*Jml) >l£|2 31 Chapter 3. Existence of Second Derivatives for a Model System Moreover, since 2u2Jns Jnkdiu* Jlkdmu* J Jnr jnr ( N +v2dkU* Jkldmu* Jml) <2, 2u2JnsJnkdlu*Jlkdmu*Jr> JnrJnr(vx + u2dku*sJkldmu*Jml) We have thus established that the matrix is invertible with inverse uniformly bounded in Bft. Therefore [ r?v(Vu*)\dndnu*\2 dy < c{dtt) f —]— \F\W dy JB+ JB+ nVu*) < c(dfl, ui) [ u{Vu)\Wu*\2 dy+ f |detJ||f| 2dy J B + JBt (3.38) Combining (3.37) and (3.38) together with the interior estimate (3.20) and a simple covering estimate based on the compactness of 0 we have our final global estimate | |MVu))^V 2u|U 2 (n) < c(Q,ui) [||f|||2(n) + | | H V u ) ^ V u | | 2 L 2 ( n ) ] (3.39) 3.4 A Simple Application Let us now use our L 2 regularity to prove the existence of W 2 ' 2 solutions of the sta-tionary vector-Burgers-like system -dj [(ui + zy2|Vu|2) djUi] + UjdjUi = /, (3.40) where f G L 2 . We define the operator T T from WQ'4 to WQ'4 by ^i(v) = u where u is the unique weak solution of -dj [(ui + u2\\Vu\2) djUi] = f i - tVjdjVi. Notice that by Sobolev's inequality, in three dimensions v is continuous, so VjdjVi G L 2 . Thus T T is well defined. From the estimate (3.3) together with the fact that u — v w, 1,4 < j !/(Vu)Vu : V(u - v) - i/(Vv)Vv : V(u - v) dx (3.41) 32 Chapter 3. Existence of Second Derivatives for a Model System we easily obtain the continuity of this map uniformly in t from W Q ' A to W 0 1 ' 4 . Inequality (3.41)) follows from arguments used in Corollary 4.1 of Chapter 4. For interest and motivation, we have also proven it directly in Appendix A. Since f and VjdjVi are both in L2, the W 2 , 2 estimate (3.39) together with the compact embedding of W 2 > 2 n W 0 1 , 4 in W 0 1 , 4 imply that Tt is compact. Moreover, when t = 0, we have a unique fixed point of the map, namely the solution of the system studied in the previous sections of this Chapter. Therefore, to apply the Leray-Schauder fixed point theorem, we need only obtain an a priori estimate of the WQ1'4 norm of a fixed point for each t. However, if u is a fixed point then we have using u as a test function in the weak formulation of (3.40) z^HVull4^ < / v(Vu)diUjdiUj dx Jn = / fjUj dx + t UjdjUiUi dx Jn Jn <||f | | L 2 | |Vu | | L4 |Q | i+tc(Q)| |Vu|| 3 L 4 |r2|i < c(Q, v2) 4 4 \t\\b+f + ^ HVu||44 2 where we have used the Poincare inequality to get this estimate. This gives us the desired a priori estimate (independent of t in [0,1]) 1  Vu| |£ 4 < c(Q, u2) [| |f 1 1 J 2 + l] . (3.42) Therefore, we may conclude by the Leray-Schauder fixed point theorem that there exists a fixed point in W 2 ' 2 n WQ'A to each operator Tt and, in particular, there exists at least one solution in the same class to the system (3.40). We now turn to applying the ideas of this Chapter to the Stokes-like system (2.1). 33 Chapter 4 Interior Regularity for the Stokes-Like Problem We shall now concentrate on the interior regularity of weak solutions of system (1.2). As mentioned in the Introduction, we will do this in a more generalized setting. We study weak solutions of the system -divT(Du) = -Vvr + f divu = 0 (4.1) u | a n = 0. where T is a C 1 function mapping R " ™ m to R " ^ m such that for some p > 2 duTijMBijBu > C l ( l + | A r 2 ) | B | 2 (4.2) |^|Ty(A)|<C2(l + | A r 2 ) (4.3) for all symmetric n dimensional matrices A and B . Let us first ensure that these condi-tions are satisfied by the Stokes-like system (2.1). In this case, T ( A ) = (2^i-r-2z/2| A | 2 ) A , so dkiTij(A) = {2ux + 2v2\A\2)dikSjl + 4v2AklAi:i. Therefore dkiTij(A)BkiBij = (2i/! + 2 J / 2 | A | 2 ) | B | 2 + Av2AklBklAi3Bl3 (4.4) > m.\xv{2vx,2u2){\ + | A | 2 ) | B | 2 (4.5) 34 Chap ter 4. In tenor Regularity tor the Stokes-Like Problem and i^T^(A) | - 1(2^  + 2z/ 2|A| 2)|Ef + AutAuAyl (4.6) < max(2j/i, 6i/2)(l + |A| 2). (4.7) Thus our stress tensor is of the type considered. Indeed, the conditions (4.2) and (4.3) are essentially those of ellipticity with a natural growth condition. Let f be in (W^y. Then we say that u is a weak solution of (1.2) if is in WQl'p and satisfies / T-j(Du)Dijcj) dx.=<f,<f>> (4.8) Jn for all (f) in J (and therefore by continuity for all 0 in W0l'p(Q)). From the properties (4.2) and (4.3) of T together with the Fundamental Theorem of Calculus it is well known [MNRR96] that T also satisfies 7^(A) • Aij > c(cup)(l + | A r 2 ) | A | 2 , (4.9) |7ij(A)| < c(c2,n)(l + | A | P _ 2 ) | A | , (4.10) (^•(A) - Ti^B)) • (A - B)ij > C l | A - B | 2 . (4.11) From these properties, it is not difficult to show, for example, with standard Galerkin techniques and monotone operator theory that weak solutions of (1.2) exist. Moreover, from property (4.11) we see that weak solutions are unique. These results (although not always the methods used to prove them) are analogous to those determined for the Stokes-like system in Chapter 2. Aside for these cursory comments on existence we ignore this topic for the remainder of this thesis keeping in mind its scope and hoping to focus on the regularity issues. We now consider the central difficulty presented by the regularity theory of equation (4.8), namely that the weak pressure is a priori in a less regular space than L 2(0), as 35 Chapter 4. Interior Regularity for the Stokes-Like Problem seen for the Stokes-like system in Chapter 2. Indeed, if / is in L2(Ci), the functional C defined on v in W 0 1 , P by £(v) = / Tij-fDuOAiCv) - f • vdx Jn is linear and bounded on W 0 L , P and vanishes on J 0 L , P , so it follows from standard Navier-Stokes theory [Gal94] that there exists a function -K in Lploc(Q) such that f Tij{Du)Dij(<l>)-f-<f>dx= [ 7rdiv(/)dx (4.12) Jn Jn for all 0 in J . Here lies the problem. Although the WQ'P weak solution u has an initially known higher degree of regularity than, for example, W 0 1 ' 2 weak solutions of the Stokes system, the corresponding pressure arising from basic Navier-Stokes theory potentially lies in a less regular space: V' instead of L2. Proceeding formally, if f] is a cut off function corresponding to interior regularity and we use ifdidiU as a test function in (4.12), as we did for the a priori estimate of Chapter 3, we must bound from above a term of the form -KT] (V77 • didiu) dx. in terms of the L2 norm of the second derivatives of u. Just applying Holder's inequal-ity would then create a integral such as f |Vn|27T2dx Jn which is not controlled by knowing 7r is in Lp'oc(fl). This problem does not arise in the case of periodic boundary conditions since Au is a solenoidal test function. We only encounter this problem when we attempt to localize our test function, which is necessary for Dirichlet boundary conditions. Hence there are two obvious lines of attack. Either use the fact that f is more regular than just being in ( W Q ' P ) * to prove something stronger about the pressure or find a 36 L Chapter 4. Interior Regularity for the Stokes-Like Problem solenoidal test function that avoids dealing with the pressure. The approach for the system (4.1) has been to prove, essentially, that the pressure lies in L2(Q) [MNR96]. To do this, however, one assumes that T is derived from a potential (so 7^ - = f3yF(|Du|2) for some scalar valued function F). The proof then continues by approximating the potentials from which T is derived with potentials exhibiting linear growth in |Du| 2 at infinity and then taking limits. Until now, a solenoidal test function approach has not been used and in doing so we eliminate an assumption on the form of T, although at the cost of not yet being able to prove boundary regularity. To motivate our method, let us recall the solenoidal test function approach to interior regularity for the Stokes operator. We will work formally assuming that u is smooth. Following [Lad69] we let 77 be a cut-off function corresponding to interior regularity and use the the fact that the curl of a vector field is solenoidal to set </» = curl(r?2 curl u) in (4.8). Then, distributing the curl operation in 4>, we see For the Stokes system, Ty (Du) = Aj(u) and we quickly arrive after some integrations by parts and applications of Holder's inequality to the desired estimate Perhaps not surprisingly, the same test-function <j> yields a similar a priori estimate for the general case. This time, we will be more explicit since the a priori estimate so obtained will be the sharpest estimate we shall see and should be the goal of our later calculations. For our test function we set 4> = ?y2(-Au + V(div u)) + V772 x curl u = -rr 2 Au 4- 277 (V77 x curl u). (4.13) (4.14) 4>i = —r)2Aui + djT)2diUj — djTj2djUi (4.15) 37 Chapter 4. Interior Regularity for the Stokes-Like Problem which is easily seen to correspond with (4.13) in the three dimensional case and avoids having to reinterpret the curl and cross-product in higher dimensional settings. In-deed, Heywood used this variant of (4.13) in [Hey76] in discussing interior regularity of the Stokes operator. Again, we assume u to be smooth in the following computa-tions. Using (4.15) as the test function in (4.8) and integrating by parts we get f r)2djTkl(Du)dkdjuldx= f ^(T H (Du))f3j(^ U j ) -in Jn - di{Tkl(JDv))dkri2djUi + T ^ D u ) ^ rix - / dk(Tkl{VvL))dj{rj2dluj) - dj(Tkl(Du))dkr12djul + rix. in (4.16) From the estimate (4.2) we obtain immediately / r)2djTkl(Du)dkdjuldx= / rj2dmn (Tw(Du)) Dmn{dj\i)Dkl(dju) rix in in > C L V f r?2(l + i D u l ^ l d p u p r i x . (4.17) j J n Now we need to estimate from above the three integrals on the right hand side of (4.16). For the first we have using (4.3) and Holder's inequality, rix / dkTkldjrl2diujdx. = / dmn (Tw(Du)) Dmn{dku)2r]djr]diuj Jn Jn V / c(c2, n)(l + |Du|p-2)77|r3fcDu| IV?y||Du| fix k J n J2 [ c(c2,n)(l + |Du|p-2)r7|aj-Du||Vr7||Du|dx • Jn < [ c(ci,C2,n)(l + |Du| p - 2 ) |Vr/ | 2 |Du| 2 + in + e(ci) V / c(l + |Du| p- 2)r 7 2 | (3 iDu| 2 rix. (4.18) „• Jn < k j 38 Chapter 4. Interior Regularity for the Stokes-Like Problem For the second we have the similar estimate, also using (4.3) and Holder's inequality, Jn TudtrfdjUi dx < / \dmnTki('Du)\\Dmndju\2r]\dkr]\\djui\dx Jn <J2 c(c2,n)(l +|Du| p- 2)|ajDu|77|V77||Du|dx 3 J N < f c(c1 )c2,n)(l + |Dur 2 ) |V7 ? | 2 |Du| 2 dx + Jn + ^ e ( C l ) f 772(l + |Du| p- 2)|a iDu| 2dx. 3 J N For the last estimate we have by simple application of Holder's inequality, (4.19) / fifc Jn dx <c( (4.20) / -rffiAiii + fidjri2(diUj - djUi) dx Jn :(ci) / (r72|f|2 + | V T 7 J 2 | D U | 2 ) dx + e(Cl) f v2\Au\2 dx Jn Jn We would like to bound the last term in (4.20) in terms of derivatives of Du so as to incorporate it into the left hand side. To do this, we note that for smooth solenoidal functions v, j ?]2|Dv|2 dx = ^ j rj2(diVjdiVj + djVidiVj) dx Jn 2 j n > f ^ 2 |Vv | 2 -4 |V77 | 2 | v | 2 dx. Jn 4 Thus, / n 2 |Vv| 2 dx < 16 f |V7]|2|v|2 + r?2|Dv|2dx. (4.21) Jn Jn Since smooth solenoidal functions are dense in JQ1,2 and since the integrals in (4.21) are continuous on this space, the result holds for all v in J Q 1 ' 2 , and we will have occasion to use this fact later. Thus, letting v = djU in (4.21) and summing on j we find that for smooth solutions, V / 772(l + |Du | p- 2)|a jDu| 2dx< 3 J N c(cl) / 772|f|2dx + c(cl,c2,n) f \ V?y|2 [|Vu|2 + |Du|p] dx. (4.22) Jn Jn 39 Chapter 4. Interior Regularity for the Stokes-Like Problem Our goal now is to find a way to incorporate these ideas into a proof. To do this, we will first need to take care of some preliminaries. 4.1 Preliminary Lemmas To begin, we consider the smearing operator o defined in the introduction. We now prove a simple lemma showing that its action on Sobolev spaces is well behaved. Lemma 4.1 Let v e Wl'p(Vl) and let Q' be an open set with closure contained in Cl with d(fl', dfl) > h. Then the function ^h,mv(x) = / v(x + them)dt Jo is in W1J"(Cl') with diO-h,mv{x) = I div(x + them)dt. (4.23) Jo Moreover, | | c / i , m v | | w i . p ( n ' ) < | | v | | w i . p ( n » ) (4.24) where Q" is any open set such that Q D Q" D CI' and d(Cl', dfl") > h. Proof: Let us first show that ohimv has generalized first derivatives. If ijj is in C0y3(Cl'), then {ahtrnv) dii> dx = / / v(x + them)dii) dt dx. Jn Jo By Fubini's theorem, together with the fact that v(x + them) is bounded uniformly in t in W1'P(Q'), we are able to change the order of integration. We also change variables to get / (o-h,mv)dlipdx= / / v(y)diip{y - them) dydt. Jn Jo Jn 40 Chapter 4. Interior Regularity for the Stokes-Like Problem Now we use Fubini again to change the order of integration and use the fact that ip is smooth to exchange the order of integration and differentiation to obtain / ((Jhtmv) diijj dx = / v(y)di / ip(y - them) dtdy. Jn Jn Jo Since v has generalized derivatives, we can integrate by parts and change variables a final time to arrive at the desired equation / (<Th,mv) diipdx- - / div(x + them) dtip dx. Jn1 Jn' Jo Since ip is arbitrary in Co°(f2') we conclude that diGh,mv exists and is fQl div(x+them) dt. To show that diak,mV is in LP we use Jensen's inequality and Fubini. It follows that / \diO-htTnv\pdx < / / \div(x + them)\pdtdx Jn' Jn' Jo < \div(x + them)\pdxdt Jo Jn' < [ \div(x)\pdQ, Jn" where Q," is any set satisfying the properties assumed. • Our regularity proof will use as before method of difference quotients. In the course of our calculations, we will have occasion to estimate from below the integral i \tA + (1 - t)B\q dt. o This useful estimate is claimed but not proven in [LU68]. We present its proof here since it has a nice, geometrically motivated, argument and gives us the opportunity to put a lone figure in this thesis. If we are looking for a lower bound for f \tA + (1 -t)B\q dt. Jo '0 we can equate this integral with an average value of the function \y\q taken along the line from A to B. Given a fixed B, it seems reasonable the lowest path integral of all 41 Chap ter 4. In terior Regularity for the Stokes-Like Problem Large Path Average Figure 4.1: Comparison of Path Integrals points C on the same level set as A is the one that passes through 0 (see Figure 4.1). Since this point is on the line that connects B and 0, the argument reduces to the one dimensional case, which is easy to prove. Lemma 4.2 Let A and B be any two matrices in R n x n . Then for all q > 0, [ \tA + {l-t)B\Qdt>c(q)(\A\q + \B\'1). (4.25) Jo Proof: Since the integral on the left hand side of (4.25) is symmetric in A and B by change of variables , we may assume without loss of generality that |B| = 5\A\ for some 5 in [0,1]. Since |*A + (1 - t)B\ > \(\tA\ - |(1 - i)B|)| = |(*(5|B|-(1-*)|B|)| = |*(1 + <J) - 1||B| 42 Chapter 4. Interior Regularity for the Stokes-Like Problem for all t in [0,1], and since yq is non-decreasing for all q > 0 and y > 0, it follows that |*A + (1 - t)B\q > \t(l + 5)- l\q\B\q. Integrating this expression in t yields / \tA+{l-t)B\qdt> f \t(l + 5) - l\g\B\"dt Jo Jo • — f -i+s = I B I 9 - ^ - / II -w\qdw 1 r1 - / (1 - w)qdw 2 Jo > \B\q - i i 2 * 4(1+1) <IAI,+ IBI'>' which proves estimate (4.25) with c(q) = 1/(4(^  + 1)). • A Corollary to Lemma 4.2 is an improvement of estimate (4.11). We will use this improvement in our study of the regularity of steady solutions of (1.1). Corollory 4.1 Let T satisfy (4.2) for somep > 2. Then (Ttf(A) - Tij{B))(A - B)ij > C l | A - B | 2 + c(cup)\A - B|". (4.26) Proof: From the Fundamental Theorem of Calculus and (4.2) it is easy to see that (TijiA) - Tij(B))(A - B)ij = [ drsTi3{tA + (1 - t)B)(A - B) r s (A - B){j dt Jo > [ ci ( l + |tA + ( l - i ) B | p - 2 ) d t | A - B | 2 . Jo From Lemma (4.2) we conclude (T^(A) - Ti j(B))(A - B)tj > C l | A - B | 2 + c ( C l ) P ) (|A|"-2 + | B r 2 ) |A - B | 2 > c i | A - B | 2 + c (c i ,p) |A-B| p . 43 Chapter 4. Interior Regularity for the Stokes-Like Problem • The primary application of Lemma 4.2 is to take advantage of the ellipticity property (4.2) of T in a difference quotient setting. In our final preliminary Lemma we prove this classical [LU68] coercivity estimate together with a growth estimate. These are analogous to the properties proven for the bilinear form ah'm of Chapter 3. Lemma 4.3 Let T satisfy the ellipticity property (4.2) and the growth property (4.3) for some p>2. Then for every u in Jo'p(fl) and every B in R"*™m and for almost every x in Cl, r f c i S(31j(Du(x)))rMA ju > c(c 1 > P)(l + \Bu\p-2 + | D u | £ 2 H E J | T M D u | 2 (4.27) 7fc,,(Ty(Du(x)))By| < c(c2,n,p)(l + | D u £ ; 2 e s + |Dur 2 |)K .Du||B|. (4.28) and Proof: For the bound (4.27) we use the fact that for almost every x, jTij(Du(x)) = \ f ^ ( t / i r M ( D u ( x ) ) + Du(x)) dt (4.29) Hence, contracting (4.29) with Th>sDij\x we obtain from (4.2) ,.(Ty(Du(>)) >' f ci(l + \(t D u | x + , e a + (1 - r)Du)|p"2) d*K.Du Jo From inequality (4.25) of Lemma 4.2 we obtain immediately (4.27). 44 Chapter 4. Interior Regularity for the Stokes-Like Problem For the second estimate (4.28) we use the growth assumption (4.3) together with the property that yv~l is convex for every p > 2 to find that for almost every x I T ^ C T ^ D U ) ) ^ ! / dklTi:i(thThtSDu + Du)JD f c /(r / l ) Su) dtBi3 Jo <c(c2,n) [ ( l + | ( i D u | x + ^ + ( l - i ) D u ) | p _ 2 ) ^ | r M D u | | B | Jo < c(c2,n,p) (1 + | D < ; 2 e < + | D u r 2 ) | r M D u | | B | . • 4.2 A New Test Function Our next goal is to find a solenoidal test function to use in the difference quotient con-text to arrive at interior regularity. We are motivated by the form of the test function (4.15). Therefore, let us write <f>i = -rf T-hjThjUi + dj-rfVij - djrf^ji (4.30) 3 where \I> is an unknown tensor to be determined by the solenoidal constraint. Specifi-cally, ^ must satisfy -diV2 Yl T-h,jTh,jUi + drfdj^ji - drfdj^ij = 0. 3 This could be solved, as in the test function (4.15), if one could exhibit a tensor \I> such that dj^ij — 0 and djVji = YJj T-h^h^i- However, in the support of rj, we can write 45 Chapter 4. Interior Regularity for the Stokes-Like Problem the difference quotient "Laplacian" as j j 0 = r _ / i j / djiii(x + htej) dt . ' Jo = T-h jdj / Ut(x + htej) dt = ^ dJT-h,jah,jUi (4.31) where used Lemma 4.1 applied to the Wl'p functions itj. We define Wji = T-hj / Ui(x + htej)dt = T-hj<jhjUi. (4.32) io Then, applying Lemma 4.1 again and using the fact that u is solenoidal, we get di^ji — 0. Therefore ^  is the desired tensor and <f>i = [-^T-hjThjUi + djri2T-htiah,iUj - djT]2T-h!Jah>jUi\ (4.33) j is the test function we seek. 4.3 Interior Regularity We now use the test function found in the previous section to prove our central result. Theorem 4.1 Let 0 be a bounded domain and u be a weak solution of (1.2) where f is in (W01'p(fl))* n L2l0C(Q) and T satisfies properties (4.2) and (4.3). Then u is in W^(Q) and satisfies Jn dx < | V 2 u | 2 + | D u | p - 2 ^ | 0 j D u | j c(Cl,c2,n)^J 772|f|2rix + J \Vr]\2 [|Vu|2 + |Du|p] dx^j (4.34) for every smooth interior cut-off function 77. 46 Chapter 4. Interior Regularity for the Stokes-Like Problem Proof: We divide the proof into two parts. First we show that u has second derivatives. However, we will be constrained in our use of integration by parts when we show the second derivatives exist and will not obtain the sharper estimate (4.34). In the second part we use the existence of second derivatives to obtain (4.34). Part 1. Introducing the test function </> defined by (4.33) into the weak formulation (4.8) we obtain Y / V2ThjTkl(Du)ThJDkludx= / ( - ^ T H ( D u ) | x + f t Thd(r,2)rKjDkln jJn Jn j 3 + Y Tki(Du.)dk(v2)r-h,jrhjUi 3 , + Tkl(Du)dk(djr]2(yji-yij)) = I + II + III + IV (4.35) where we have used the standard properties of difference quotients that can be found in [Gia93] to "integrate by parts" and to distribute a difference quotient over a product. We will bound the left hand side of (4.35) from below and the right hand side from above to get an estimate for the difference quotients of Du. At this point it might seem sufficient to simply cite our work from Chapter 3 to claim the final estimate. However, many calculations are different from those that appear in the standard theory. We have introduced new terms in our test function of the type T-hjOv^-u and we must be careful to introduce approximate second derivatives of the form r ^ D u only and not more general terms rhtkdiU. Therefore, let us find bounds for (4.35) from above and below. Since we will encounter the weight (1 + | D u | p _ 2 + IDu^^ ) several times, p we will denote it as phj. Of course, nnj converges strongly in Z>-2 as h goes to 0 to (1 + 2|Du| p _ 2) which we will simply call p. It will also be convenient to use tth to 47 Chapter 4. Interior Regularity for the Stokes-Like Problem denote (Jsupp(r h ) J77) |Jsupp(77) which approximates supp(??) in the limit as h goes to 0. We use the estimate (4.25) from Lemma 4.2 to bound the left hand side of (4.35) from below by J2 [ V2rhjTkl(Du)rhdDkludx>c(cup)y] [r?2(l + |Du| p - 2 + | D u £ ; 2 )|r f c j-Du| 2dx = c (c i , p )V] / / / A j | r h jDu| 2 dx. (4.36) We now turn to maximizing each of the integrals on the right-hand side of (4.35). Using the bound (4.10) for T we see Jn j V c ( c 2 , n ) / (1 + | D u | p ; 2 ) |Du | a ; + f t e . |r / l , i(r/ 2) | |r / l JDu|rix Y]c(c2,n) / /i / l, j|Du| I + / j e j|r / l J(r7 2)||r / l i :,Du|(ix. (4.37) j J n From the "Leibniz-rule" for different quotients, Th,jV2 = rh,jV (v + V\x+hej) • < < Since 77 is smooth we see r)\x+he = r)(x) + JQh jtr)(x + tej) dt < 77(x) + hc(Vr]) and therefore \ThAv2)\<c{Vv)(ri + h). (4.38) On the other hand, h\rhjDu\ < |Du| 4- | D u | x + / i e . . (4.39) 48 Chapter 4. Interior Regularity for the Stokes-Like Problem We combine (4.38) and (4.39) in (4.37) and apply Young's inequality to get |I| <c(ci,C2,p,n,V77) V / /xfcj-(|Du|2 + | D u | 2 + h )dx + + e{cup) J2 / ^ V j l ^ j D u p d x (4.40) 3 J N where we have written explicitly the dependence of e on the parameters appearing in the constant in (4.36) to make clear that e need only be small enough to merge into this term, along with other terms of this type appearing in the later computations. To estimate the second integral, we must "integrate by parts" with one of the differ-ence quotients. Doing this we have, employing the bound (4.28) of Lemma (4.3) along with (4.10) and Young's inequality, mi / Th!J(TkiCDu)dkri2)Th!Juidx 3 J Q Y Jn [ThATkl(Vu))dkv2 + T H (Du) | s + f c e . ThtJdk(v2)] T h j U l dx < c(c2,p,n)y2 / ^h,j\rhj'Du\r]\\/r]\\Thdu\dx 3 J Q + c(c2, AtojlDulx+ZMy V ( 7 7 2 ) | T A j i t | dx 3 j Q h <c(C l,c 2,p,n,Vr?,V 277) V / ^ j ( | D u | 2 + |Du| 2 + \ThJu\2) dx 3 J u » + e(ci,p) V / r)2p,hj\ThJD\i\2dx.. (4.41) The third integral breaks up naturally into two parts, which we will call Ilia and Illb. For Ilia we have applying Holder's inequality |IIIa| - / TH(Du)(afcaj-r?2)(*iJ - tfy) dx Jn < c(V2ry) j J S U D |T(Du)|p' + |^|pdx. (4.42) 49 Chapter 4. Interior Regularity for the Stokes-Like Problem From (4.10) we have the estimate |T(Du)|p' < c(c 2 ,p,n)(l + |Du|p). Also, we can estimate |^ | p < c(n) ^ \T-h>jcrhiju\p to arrive at our final bound for Ilia, |IIIa| < c(V2r7) f c(c2,p, n)(l + |Du|p) + c(n) ^ K h t j a h J u \ p dx <c(V277) f c (c 2 ,p ; n)( l + |Du|p) + c(n ,supp(r ? ) ,Q)^|Va / l J u| p rix (4.43) J SUVD(V) where we have used the fact that there exists a constant c(supp(r7), Q) independent of h such that for all g in W01,p(f2), / \ThJg\pdx<c(supp(r]),Q) \djg\p dx. isuppfn) Jsuon(n) For second part of integral III we have IIIb = J2 f {djr]2Tki(T>n) - dir]2Tkj(T)u))T^hjdkah^ui dx 3 J Q We integrate the difference quotient by parts to get Illb = Illbi + Illbii where |IIIbi| = / djt^ThjTki(Du)dkahdui - diT]2ThdTkj(Du)dkah!Jui) dx and |IIIbii| J T w (Du) | x + f c e . {Thjdjrj^dkahjui - T f e j (Du) | x + / i e j (ThJdiif)dkahijui . Using estimate (4.28) in Lemma 4.3 together with Holder's inequality we easily see X / ^r7 2 (T A j-T w (Du)9 f c a h j-ujdx <V"c(c 2 ,p,n) / fihJr)\VT]\\Thj'Du\\Vah!Ju\ dx. (4.44) The second term in Illbi requires a bit more care before we can apply Lemma 4.3 directly. However, we may rewrite it using the Kronecker delta before applying (4.28) to get Y] / dir]2(ThJTkmCDu)SmjdkahtjUidx <Y]c(c 2 ,p,n) / /j,hJr]\Wr]\\ThijDu\\'v7ohdu\ dx. j Jn • Jn (4.45) 50 Chapter 4. Interior Regularity for the Stokes-Like Problem Combining (4.44) and (4.45) together with Young's inequality we have the estimate for Illbi |IIIbi| < c(cuc2,p,n)Y] / phJ\VT]\2\Vohju\2+ e(cup)y2 / ^VftjKjDu|2dx. (4.46) On the other hand, Illbii is easy to estimate as it has no singular terms. We may apply Holder's inequality together with (4.10) to get llllbii] < 2>(c2,p,n, V2V) f 1 + | D u | p + \Vvhdu\pdx (4.47) Our last term to estimate is IV, which can be written | IV |= J2 f ftfT-hjThjUi + Y f fidrf^ij-Vji) • 3 J Q 3 J U For the latter term we obtain from Holder's inequality T [ fidtf^ij - dx < v c(n) f r? 2 |/ | 2 + \Vri\2\Thj<Thdu\2 dx (4.48) 3 J N 3 J N For the former, the estimate is completely standard from the theory of difference quo-tients, such as done in Chapter 3, and we have / fiV2T-hjThjUidx. < c(ci,p,supp(77),£2) / ?7 2 | / | 2 dx + • Jn Jn + e(c 1 ) P ) Y f r y X ^ u f d x (4.49) 3 J I I We would like to bound the last term in (4.49) in terms of difference quotients of D u so as to incorporate it into (4.36). To do this, we use (4.21) in (4.49) just as in the a priori estimate to conclude V ] / fiTJ2T-hjThjUidx < c(Ci,p,SUpp(77),ft) / T]2\f\2 dx • Jn Jn + c(ci ,p) Y J\Vv\2\Th,M2 dx + e(ci,p) JQ V2\rhjVu\2. (4.50) 51 Chapter 4. Interior Regularity for the Stokes-Like Problem Now we may combine (4.40), (4.41), (4.43), (4.46), (4.47), (4.48), and (4.50) together with the lower bound (4.36) to arrive at the estimate for the difference quotients Hh,j\ThJ'Du\2 dx < c(ci,p,n, Vr?, V2r7, supp(7?), Q) / r/2|f | 2 rix + j in [ in Yl / \VahJu\2 + |V<7 f cju| p + phJ [|Du|2 + | D u | 2 + f t + Ir^-ul2 + \VahJu\ dx (4.51) Using the absolute continuity of the integral together with the strong LP convergence of r/i^u to dku and the LP boundedness of ohjdj\i proven in Lemma 4.1 we see that the right hand side of (4.51) is uniformly bounded from above as h —> 0. Since the left-hand side of (4.51) bounds fn r? 2|r/ ljDu| 2 rix from above, and since 77 is an arbi-trary interior cutoff function,we conclude that for each j, djDu e Lf0C(Q). Employing (4.21) we conclude further that u e W^(tt). Actually, u is even more regular than this. Since rr2ii,h^\rh^Du^ is bounded in L 1 ( Q ) we conclude that some subsequence of rjy/phjThjDkiu converges weakly in L2(Q). From the strong convergence of y/Phj in L2 to \i and the strong convergence in L2 of Th,jDkiu to djDkiu we see that the weak limit must be r]^/]ldjDkiu and therefore Vy^djDklu e L2(Q). (4.52) However, the bounds given in (4.51) depend on the second derivatives of 77 among other things and are therefore not as sharp as the the desired estimate (4.34). We now use (4.52) to obtain (4.34). Part 2. From (4.52) together with the growth estimate (4.3) it follows that ^ ( D u ) is in LPloc(Vl). From this, we are able to integrate the weak formulation (4.8) by parts to conclude that - f djTij{Bu)(l)idx= [ fifadx. (4.53) in in 52 Chapter 4. Interior Regularity for the Stokes-Like Problem for all cj) in J and by continuity for all <j> in Jp(0') every open fl' with closure in Q. Consider the test function 0 e jy01,2(Q) defined by <t>i = Y [-TfdjThjUi + djr]2ThjUj - djT]2ThijUi] j = Y1 [di i^hjUi)] + djifr^mj (4.54) j An easy computation shows that (f> is also divergence free and lies in LP(Q) and there-fore is a legitimate test-function for (4.53). Indeed a simple mollification argument shows that it is in the closure in LP(Q,) of J. Therefore, introducing it in (4.53), we see Y f dkTki{-Du)dj (v2rh,jUi) dx = Y] [ ^ ( D U ^ T ^ U * dx + f dx (4.55) j, Jn • Jn Jn We would like to integrate both derivatives by parts in the left hand side of (4.55). This is justified since Tjj(Du) is in W^p> and r]2Th,3Ui is in Wr]'p, and the result is / diTij(Du)dkr]2Th,juk - d3Tki(Du)ThjUidkrj2 dx Jn (4.56) We now take the limit as h tends to zero. None of the terms on the right hand side present any difficulty but we must be careful with the term on the left. Since •ndjTki(Du) is in Lp' and since we have established r]dkThjUi converges strongly only in I? we cannot immediately use continuity arguments. However, we may write each integrand as —d3Tki{T>\i)r]y/JldkThjjui. Y / V2djTki(Du)dkThtjUidx = Y 53 Chapter 4. Interior Regularity for the Stokes-Like Problem From (4.3) we see that -^ |^r f c i (Du) | = -^\dTaTki{T>XL)djDrau\ 1 + | D u | p - 2 < c(c2)rj — \djT>u\ v ft < c(c2)r)y/ji\djDu.\ (4.57) which implies j^=<9jTfcj(Du) is in L?. From the weak convergence in L? of rjy/p^dkThjUi jUi to rj^/pZdkdjUi and the strong convergence of y/pnj to ^fp in L 2 we see r]y/]2dkTh, converges weakly in L? to r)^fpdkdjUi. Therefore, we are justified in taking this limit on the left-hand side of (4.56) and we arrive at / r]2djTki(Du)dkdjuidx= / diTij('Du)dkr)2djuk - d^T^Du^-u^r, 2 rix in in + fi {-V29jdjUi + djrfdiUj - djrfdjUi) . in (4.58) Now we estimate from above and below similar to when we when we proved the exis-tence of second derivatives. However, the estimates are cleaner and easier. The lower bound follows directly from (4.2) and we have (using the fact that T is symmetric) / r)2djTki('Du)dkdjUidx= / 7]2d3Tki(T>-a)djDki\xdx in in > d V ( 772(1 + |Du| p- 2)|r3 iDu| 2 rix. (4.59) j J n For the first integral on the right-hand side of (4.58) we use (4.3) together with Young's inequality to obtain / d^iji^dk^dj^ - dj^ktiD^djUidkrr2 dx < in c( C l , c 2 ,n ) f ( l + |Du | p - 2 ) |Vr7 | 2 |Du| 2 r ix + i n + e ( C l ) V [ 772(1 + |Du|p-2)|<9pu|2 rix. (4.60) 54 Chapter 4. Interior Regularity for the Stokes-Like Problem The second integral on the right-hand side of (4.58) requires an application of Holder's inequality together with (4.21) to get / fi {—rfdjdjUi + djT]2diUj — djT]2djUi) dx < Jn c(d) / r]2\i\2dx + c(cun) f |Vr7 | 2 |Vu| 2 f ix + Jn Jn + e(d)J2 f r72(l + |Du|p- 2)|r3 jDu| 2dx. (4.61) , Jn Combining (4.59), (4.60) and (4.61) together with a final application of (4.21) gives us the estimate / Jn |V 2 ui 2 + ^ ( l + |Dur2)|apu| 3 <c(C l) f rf |f|2 dx + c(C l,c 2,n) / | V r 7 | 2 | [ | V U | 2 + (1 + |Dur2)Du|2] dx Jn Jn from which (4.34) follows easily. • 55 Chapter 5 Applications to non-Newtonian Fluids In this Chapter we apply Theorem 4.1 to the steady and unsteady motion of an incom-pressible fluid with equation of motion (1.1). We return our attention to the specific model (1.1) in which the stress tensor T can be written Tij = 2(is1 + v2\Du\2)Diju. 5.1 The Steady Case Our first object of study is the steady case —dj(2(vi + f 2 |Du| 2)Ajii) = —UjdjUi — + fi diUi = 0 u | a n = 0. (5.1) Since the weak solutions of the Stokes-like system (2.1) come from such a regular space, W Q L ' A ( Q ) , the inertial term poses little problem for the regularity theory. Indeed, we define a weak solution of (5.1) to be a function u in J Q A ( Q . ) such that / + u2\D\i\2)DijVidi(f>j dx = / -UjdjUifa + fcfadx (5.2) Jn Jn for all (j> in J. Its regularity follows directly from Theorem 4.1. Theorem 5.1 Let Clbea bounded open subset o/R 3. Iff is in L2(Q) then every weak solution 56 Chapter 5. Applications to non-Newtonian Fluids 1 ^ of (5.2) is in W?£{Q) and there exists a pressure TT in W , 0 ' /(fi) such that u and TT satisfy (5.1) almost everywhere in fi. Proof: If a weak solution u exists, then from Sobolev embedding of W 0 1 , 4 ( f i ) into C ( f i ) we see that u is in L ° ° ( f i ) an therefore the term UjdjUi is in L 4 ( f i ) and hence also L 2 ( f i ) . Therefore u is a weak solution of system (2.1) with right hand side Fi = -UjdjUi + fi in L 2 ( f i ) . From 4.1 we see then that u e W, 2 ' 2 (f i ) . From Section 2.2 we recall that there exists a pressure TT in LX ( f i ) such that for all <f> in ( f i ) , / 2(i/i 4 - v2 |Du|2)Dijudjfa + 7T (V • 0) - Fi</>; fix = 0 Integrating this equation by parts we obtain / [d,{2(vi + i/2 |Du|2)Diju) + Fi] <pidx= / TT (V • 0) fix. (5.3) in in I ^ Therefore, 7 r is in W /o'c3 (fi) and -dj(2(vi + v2 |Du|2)Dyu) = - f 9 j 7 r 4 - -UjdjUi + fi almost everywhere. • Therefore, to complete the theory we need only show that weak solutions exist. We do this along the same lines as can be done for stationary solutions of the Navier-Stokes equations using the Leray-Schauder fixed point theorem. Our proof is different from that used at the end of Chapter 3 for solutions of the stationary Vector Burgers-like equation since we do not have a boundary proof and therefore cannot use the compact embedding of W 2 ' 2 ( f i ) into W ^ f i ) . 57 Chapter 5. Applications to non-Newtonian Fluids Theorem 5.2 There exist weak solutions of (5.1). Proof: We show solutions of (5.2) exist from the unique solvability of (2.1) and the Leray-Schauder fixed point theorem. Indeed, consider the function Tt defined by the map that takes v in W01,P(Q) to the unique u in W0l,p(Q) that solves / 2(ui + i/2|Du|2)DijuDijcf) dx = (-VjdjVi + fi) fa dx. Jn Jn for all (j) in Wr]'p(Q). We show now that this map is compact, so let us suppose that {vr} is a sequence in W 0 1 , P ( Q ) that converges weakly in WQ'p(Q) and therefore strongly in L 4 ( f i ) . Let u r = "^(v7"). Then it follows from (5.2) and an integration by parts that f (2(i/x + i / 2 | D u r | 2 ) A J u r A J ( u r - us) - 2(i/i + J / 2 | D u 8 | 2 ) £ > y u * A i ( u r - us)) dx = Jn f "K - *,W(«r - vt) - v°{dM - vt))(vl - vi) dx (5.4) Jn From (5.4) together with Corollary 4.1 (or Appendix A), Korn's inequality [ N e c 6 6 ] , and Holder's inequality we see c ( Q ) M V ( u r - us)\\l + u2\\V(ur - us)\\t) < | | V v r | | | K " v s | | 2 + c(fi, i/i)||v*||2 x >||v r - v s | | 2 which, recalling Sobolev embedding, shows from the uniform boundedness of ||v r | | 2 and | |v r | | 0 0 and the strong convergence of {vr} in L 4 (0 ) that {ur} is Cauchy in WQlA and therefore also the desired compactness. To apply the Leray-Schauder principle we need only show now that any solution of u = AJF(u) is bounded uniformly for A G [0,1]. However, since / n UjdjUiUi vanishes we obtain from (4.9) and Korn's inequality the estimate I I V w H ^ c ^ l l f H ^ p j . 58 Chapter 5. Applications to non-Newtonian Fluids which implies the existence of solutions of (5.2). • 5.2 The Unsteady Case We now turn our attention to showing the existence of regular solutions of (1.1). Theorem 5.3 Let ttbe a bounded domain in R 3 , let f be in L2(0, T; L2(Q)), let u 0 be in J. Then there exist functions u and -K that satisfy (1.1) almost everywhere in Q x [0, T] with \\u(t) - u 0 | | -» 0 as t ->• 0+ Moreover, u e L4(0, T; J 0 M ) and v^u e W 1 > 2(fi x [0, T]) n L2(0, T; W2'2(0')/o^ every open set Q! with closure contained in Q,. Let {ak} be a basis for J01,4(fl) that we will take to be orthogonal in J(Q) . To begin, we construct Galerkin solutions in the usual way. We let u f c = 2~2j=i <^,k^ where the functions c°,k satisfy the ordinary differential equation d k j / ' k = - E cl,kcm'k(*1 • V a m , a0 - (Dlmuk, Dlma>) + (f, a,-) (5.5) l,m=l with initial condition ^(O) = (a-7, u0). By the standard theory of ordinary differential equations, we find that the cj'k exist on some interval [0, Tk]. Multiplying (5.5) by c?'k and summing on j we obtain in the same way as for the Navier-Stokes equations the energy estimate for the Galerkin approximations ^ | | u f c | | 2 + (2 [ux + u2\Duk\2) £>yu f c ,Dyu f c) = (f,ufc). (5.6) From Gronwall's inequality and (4.9) we arrive in the usual way at ||ufc(t)||2+ f f 2u1\Buk\2 + 2u2\Bu\Uxds<B1 (5.7) Jo Jn 59 Chapter 5. Applications to non-Newtonian Fluids where Bx is finite and depends only on ||u 0 | | and | |f | U2(o)7\;z,2(n))- Therefore, we can extend the interval of existence of the cj'k to any interval [0,T]. We shall now work with some arbitrary but fixed T > 0. The second estimate, roughly based on an a priori estimate found in [Hey93], follows from multiplying (5.5) by ^cj'k and summing on j to get, using u to denote ftu, (2 {ux + i/2|Du*|2) Dlmuk,Dlmuk) + ^||u f c | | 2 < ||u f e • Vu f c | | 2 + ||f||2. (5.8) We would now like to bound the inertial term using the first estimate (5.7) and apply Gronwall's inequality to (5.8). First we show that (2 (vx + i/ 2 |Du f c | 2) Dlmuk, D;mufc) is a derivative. Indeed, 2{D l mu k , Dimiik) = | | | D u f e | | 2 and 2(|Du| 2AmU f e, Dlmuk) = i | | | D u | | 4 and therefore (2 (Ul + u2\T>uk\2) Dlmuk,Dlmuk) = jt ^| |Du f c | | 2 + ^ | | D u f c | | 4 = |^ (Du f c ) . (5.9) Multiplying (5.8) by t we obtain using (5.9), 1 ^ <Du'>] +1 d [ i^(Du f c )]+^| | u f c | | 2 < i | | u f c - V u f c | | 2 + t||f||2 +J-(Dufc(0)). (5.10) We now want to bound the convective term in such a way that we can apply Gron-wall's inequality. As in the stationary case, we can use W1'4 control to our advantage here. From Sobolev's inequality we have ||u f c • Vu f c | | < | |Vu f c | | | |Vu f e | |3. From the boundedness of and Korn's inequality it follows that ||u*- Vu f c | | < c(0)||Du||4. (5.11) Thus, from Gronwall's inequality applied to (5.10) combined with (5.11) we see ^ 1 | | D u f c | | 2 + t^UDu*!!^ f s\\uk(s)|\\ds. < £2(ft,Bi) (5.12) Jo 60 Chapter 5. Applications to non-Newtonian Fluids We now turn to the convergence properties of the sequence uk. As is usual, we shall speak of convergence of subsequences of ufc as convergence of the sequence itself. Since our estimates do not give us control in advance of the second derivatives, we are not able to to prove convergence properties as simply as in the case of the spectral-Galerkin method for the Navier-Stokes equations. On the other hand, since we have more information than is traditional for weak solutions, we are able to prove the esti-mates more easily than those for weak solutions. From (5.7) we have / Jn \u"\zdxdt < TBU nT so ufc converges weakly in L2(flT) to some u. From bounds (5.7) and (5.12) we have / Jn 1 |uT + |uT + |VuT < (1 + T)Br + -B2 nx[6,T] b and so ufc converges weakly in W 1 , 2 ^ ^ ) and therefore strongly in L 2 ( Q , \ S , T ] ) to u. Furthermore, / |u-ufe|2= / |u-ufc|2 + / |u -u f e | 2 JnT Jnx[Q,s] Jnx[5,r] < 2 l l u l lL 2 (nx [o , 1 5] ) + 2 l l u / c | l L 2 ( n x [ o , T ] ) + / |u-ufc|2 Jnx[s,T] < 2||u||L2(nx[0)4]) + 2 r J 5 1 + f \u-uk\2. (5.13) Jnx[8,T] Taking limits in k we have using the strong convergence of uk on 0 x [5,T] that limsupfc^ oo ||ufc — u l l ^ ^ j < c2||u||i2(nX[0)(j]) + 5B\. Since 5 is arbitrary we have shown that {uk} converges strongly in L 2 ( Q T ) to u. From the bound (5.12) we see that for every t > 0, {uk(t)} converges weakly in W01,A(Cl) to some v(c) and thus strongly in L4(tt) and L 2 ( f i ) . Let us show that v(s) = u|t_g. In particular, since u is in W L , 2 ( Q x [8, T]), it has trace values u(i) in L 2 ( 0 ) for each t > 0. We need to show that u(t) = v(t). However, if w is 61 Chapter 5. Applications to non-Newtonian Fluids any function in W 0 1 , 2 (f i ) , then for t2 > h > 0 we have / i n (h - * i ) ( t i » ( x , t 2 ) - tt*(x,t2))iUt(x) fix = (< - tx)(ui - iik)wi(x) + (ui - uh)wi(x) dxdt. (5.14) In. ' x [ i i , « 2 ] Thus, by weak convergence of uk to ii in L? (fi x [i :, £2]) and by weak convergence of u f c to u in L2(Q, x [ti, t2}) we have uk(t2) converges weakly in L2(Q) to u(i2). Since uk(t2) converges strongly in L2(Q) to v(£ 2 ), it must be that v(£ 2) and u(t2) are the same. Since t2 is arbitrary, we have v(i) is in fact the trace value of u at t. Now we turn to showing that u is a solution of the PDE. First we deal with the con-vective term. If 0 is a smooth function, f f uk • V u ^ - u - Wu(j)dxdt< [ [ \uk - u||Vufe||0| + |u||V(u f c - u)||0| dxdt Jo Jn Jo Jn (5.15) By strong convergence of {ufc} to u in L2(ttT) and by weak convergence of {Vufe} to V u in L2(VtT) we see that uk • Vu f c converges weakly in L2(CtT) to u • Vu. Next we deal with the viscosity term. Since {Dufe} remains bounded in L 4 (fi T ) we have 2 {yx + i^lDu^ 2) Du remains bounded in L3(Q,T), and so converges weakly to some tensor T in L^(ilT). We now show T = 2 {v\ + f 2 |Du| 2) Du via a simple mono-tonicity technique. If 4>l is any function of the form 2 ^ = i bj(t)a3 with bj continuous in t then (5.5) and (4.11) imply for k > I [ [ (uk + uk • Vu f c - f) • <j)1 dx dt = - [ [2 (vx 4- v2 |Du f c |2) Dij^DijCJ)1 dx dt. Js Jn Js Jn Taking the limit in k using the weak convergence of uk to u, the weak convergence of ufc . yufc t o u . y u ^ a n ( j t n e w e a ] < convergence of T(Du f c) to T we see [ [ (u + u • Vu - f) • (j)1 dx dt = - [ [ TijDirf1 dx dt. (5.16) Js Jn Js Jn 62 Chapter 5. Applications to non-Newtonian Fluids In particular, for 4>l = ul we have r-T f [ (u + u • Vu - f) • u' dx dt = - [ [ TijDijU1 dx Js Jn Js Jn dt. Taking the limit in I using the strong convergence of u' to u in L 2 ( £ I T ) and the weak convergence of Vu' to V u in L 4 ( f i r ) and the fact that fQ u • V u • u dx vanishes we see f [ (ii + u • Vu - f) • u' dx dt = - [ [ T^D^u1 dx dt. (5.17) Js Jn Js Jn From (4.9) we have f [ [2 ( i / i + u2\Buk\2) DijUk — 2 ( i / i + u2\D(j)l\2)Dij(j)1] ^-(u* - <f>1) dxdt > 0 Js Jn r-T i Jand therefore, from (5.5), [ [ -2 ( i / i + i A j | D 0 ' | 2 ) A i ( i i f e - (j}) + (u f e + u • V u - f>< - (ufe - /)u f c dxdt > 0. Js Jn Taking the limit as before (now also using the weak convergence of u f c and the strong convergence of u in L 2 ( Q , x [S, T}) to handle the u f cu term) and using (5.17) we see [ [ (Tij - 2 + i / 2 |D<£f ) )Aj (u - <t>1) dxdt > 0 Js Jn and by density of functions of the type 4>l in L4{6, T; J 0 1 , 4 (f i)) and continuity of the previous integral on <f>1 in this space we see f [{Tij - 2 + i / 2 |D0 ' | 2 ) )Aj (u - 4>) dxdt > 0 (5.18) for all (j) in L4(^, T; J 0 1 , 4 ( f i ) ) . Letting 0 = u - eaJ and taking the limit in epsilon we see [ [ {Tij - 2 ( i / i + z/2|Du|2) Diju)Dij{u -4>)dxdt = Q Js Jn fT ' JSince the above is true also for <f> = u + eaJ we determine that, in fact, 's Jn [ [ {Tj - 2 (vx + ^ 2|Du| 2) DijUjDijia?) = 0. (5.19) ./<J . / f i63 Chapter 5. Applications to non-Newtonian Fluids Let us now show that u satisfies the initial condition. From (5.5) we see (u*(t),a'(t)) = (u0,a'(0)) + f [ -uk • Vu* • a' - 2 {ux + i/2|Du*|2) D^D^a1 + i-aldxdt Jo Jn Taking the limit in k, using now also the strong convergence in L2(Q) of uk to u we obtain (u(t),a'(t)) = (uo, a'(0)) + / / - u • Vu • a' - 2 (i/x + J / 2 | D U | 2 ) DyiiAja' + f • a' dx dt. Therefore u(c) converges to u(0) weakly in L 2 ( f i ) . However, from the energy inequal-ity (5.6) we see that ||ufc(r)||2 tends to ||ufe(0)||2 uniformly in t and therefore ||u(£)|| 2 tends to | |u 0 | | 2 . Thus we have established that u(t) converges strongly to u 0 as t —> 0 +. We are now in a position to show that u has second spatial derivatives. Although we were not able to control these in taking the limit of the u f c, we have ample control over all other aspects of the equation to make this possible. In particular, if £ is any continuous function, then letting cf)1 = £(t)al(x) in (5.16) we see, using (5.19), '8 Now since f £(t) [ 2 (i/i + u2 |Du|2) D^uD^ + (u + u • V u - f) • a' dx dt = 0. Js Jn / 2(ui + ^ 2|Du| 2) DijuD^a3' 4- (u + u • V u - f) • aj dx Jn is in L5(r5, T) and continuous functions are dense in L4(5, T) , the Rietz-Ritz represen-tation theorem implies f Ti:j(Du)Di:jaj - - / (u + u • V u - f) • a> dx Jn Jn at almost every t in [5, T]. Repeating this process for each of the countably many functions a3', using the fact that a countable collection of zero measure sets has zero 64 Chapter 5. Applications to non-Newtonian Fluids measure, we see that (5.2) holds for every basis function a? for almost every t. Since ii + u • V u - f is in L 2 ( f i ) for almost every t we obtain immediately from Theorem 4.1 and the fact that 6 is arbitrary that at almost every t > 0, u is in W^(Q). Moreover, at almost every t, the estimate (4.34) from 4.1 holds. However, we are not quite in a posi-tion to claim that u e L2(5, T; W2'2(Cl')) since we have not determined that | | u | | w 2 , 2 ( n ' ) is measurable. However, since u l l z , 2 ( n ' ) < c(fi',fi) [£||f||^ + t||u||2 + * | |u-Vu| | | + t | | V u | | l + *||Du||3 (5.20) we get from the Dominated Convergence Theorem and the almost everywhere con-vergence of «||r h ) m?7 2Vu|| to i||<9m??2Vu|| that y/iu e L2(0,T; W 2 ' 2(fi ')) for every open set fi' with closure contained in fi. Finally, we obtain in the usual way from (5.16) and 4 the regularity of u the existence of a TT with Vir e Lf0C(Q) such that u and TT satisfy (1.1) almost everywhere. • We complete our discussion of the time-dependent case by mentioning that our solu-tion is unique.1 Indeed, let v and u be two such solutions and let w be their difference. Then we see that w satisfies I d , , , , 2 2dt" 11 + 2 / (yi + f 2 |Du | 2 )L>M - (ui + zy2|Dv|2)Dvdx = - / WiDijUWj dx. Jn Jn in Ll. Applying estimate (4.11) we see then that at almost every time 1 d 2dt < - WiDiji Jn IwjI + 2i/i||Dw|| — / WiDijUWj dx.. 1 Actually, it is not hard to show that our solutions are also weak solutions in the sense of La-dyzhenskaya and are therefore unique by her arguments. 65 Chapter 5. Applications to non-Newtonian Fluids We now use Holder's inequality and a Sobolev inequality to get at almost every time ^7-||w||2 + 2*/1||Dw||2 < c(n)||Vw||i||w||3||Du|| < ^i||Vw|| 2 + c(Q, i/i)||Du||4||w||2. Thus we find ^ | | w | | 2 + ^i||Dw||2 < c(fi ,^i)| |Du|| 4 | |w|| 2 and using the fact that w converges strongly in L2 to 0 as t tends to 0 we apply Gron-wall's inequality and find that | |w| | = 0 at almost every time. We have proven Theorem 5.4 The solutions of (1.1) with the regularity of those found in Theorem 5.3 are unique. 66 Chapter 6 Epilogue We have presented in this thesis an approach for studying interior regularity of elliptic and parabolic systems with solenoidal constraint. Unfortunately, we have left many questions unanswered. What we have shown is that if f is in L 2 ( f i x [0, T]) and u 0 is in J , then there exists a solution of u t + u • Vu = - V T T + 2div((i/i + I A , | D U | 2 ) D U ) 4- f div u = 0 (6.1) ulan = 0 ult=o = uo such that for every open set fi' with closure contained in fi u has second derivatives in L 2 ( f i ' x [0, T]) and that tu has a time derivative in L 2 ( f i x [0, T]). Thus we naturally ask whether we can extend these results up to smooth portions of the boundary, and whether higher regularity of u can be obtained in time and space. We hope with these final words to briefly outline some of the difficulties involved in extending our results to answer these questions. 67 Chapter 6. Epilogue 6.1 Boundary Regularity We note from the method of proof in Theorem 5.3 that a boundary regularity proof (e.g. u in W 2 > 2 ( V L ) ) of the Stokes-like system -2dj ( ( ^ i + u2 |Du|2) Diju) = di-K + ^ diUi = 0 (6.2) u | a n = 0. would translate immediately to boundary regularity of (6.1). The central difficulty for proving boundary regularity of(6.2) is the same as that of interior regularity, namely that we need more regularity from the pressure than that which arises naturally for weak solutions, as seen in Chapter 4. We avoided this difficulty in the interior case by using a solenoidal test function. However, we do not have even for the Stokes system a solenoidal test function approach for boundary regularity. Since such an approach to boundary regularity has not yet been obtained for the well studied Stokes system, it seems that seems that without a truly remarkable result we must work with the pressure. Recall that the troublesome term to estimate is 7rr7 (V77 • didiu) dx. So it would be sufficient to show that TT is in L 2 ( f i ) . From Theorem 4.1 we deduce that the pressure is in W ^ 3 ( Q ) as well as in (fi). Such a result by itself would certainly be useless in proving something about TT up to the boundary since one can imagine many functions that are smooth in a domain but have a singularity of arbitrary growth at a boundary point (e.g. l/|x — x 0 | 9 for x 0 a boundary point of fi). However, we also obtain from Theorem 4.1 a growth estimate which could say something about TT up to the boundary. Let us take for the sake of exposition fi to be the cube Q in R 3 defined by {x : < 1, \x2\ < 1,0 < xz < 2}. Let & be a cut-off function on [0,2] that is 1 on [8,1]. It is easy to show that we can find such functions such that |Vf,$| < | for 68 Chapter 6. Epilogue some fixed constant c. Let f (xi, x2) be an arbitrary cut-off function on the unit ball of R 2 . Then 77a(x) defined by ((xx, x2)£s(x3) is a cut-off function on Q and if we define Qs{* eQ : x3> 5} we obtain from the growth estimate of Theorem 4.1 / | V 7 r | i d x < c(cuc2,n) ( [ K D u ) 2 + r7 2 |f | 2rix+ f \Vrjs\2 [|Vu|2 + |Du|4] rix) •^Qa \JQ6 JQS J <c(Cl,c2,n)( f ^(Du)2 + | f | 2 dx+4 / [|Vu|2 + |Du|4] r i x V \JQS 6 JQI\Q6 / All terms on the right hand side are uniformly bounded in 5 except 0 JQi\Qs Therefore, limr52 / | V 7 r | i d x = 0. Jo.* We are lead to ask, then, what sort of boundary singularity in TT might arise in growth of this type. One might hope that this restricts TT to be in a smaller space than lh (Q). However, consider the function TT singular given by TT = —A _ . Then a simple calcula-tion shows 1 singular^ dx = 0[ 'Qs for every e > 0. / VltsingularV ^ X = o( — ) . Since l / | x | ( " e) isn't even in LX for e small, we see that our restriction does not, in this approach, offer useful information at the boundary. We are not entirely without hope, however. We note that we do not even need that TT lies in L2. If 7 r / ^ ( D u ) is in L2, then we can estimate this term by f TTT] (Vr; • dAu) dx < c(e) f | V r ? ! 2 - ^ - d x + e / 772i/(Du)|r3ir9iu|2 dx, i n i n f(Du) Jn which would be sufficient to incorporate the final term into the lower bound obtained by ellipticity. Unfortunately, an estimate of this type would be hard to obtain (even 69 Chapter 6. Epilogue though it is weaker that an L2 estimate) since it involves a very strong pairing between the growth of TT and Vu. 6.2 Higher Time Regularity To get study higher time regularity one might expect to proceed as in the Navier-Stokes equations and take derivatives of (6.1) to obtain a sequence of inequalities to use in Gronwall's inequality and thereby control growth of higher time derivatives. For example, this has already been done for a single step higher in [Hey93]. To see how this works, we will assume for simplicity that f is 0. Then taking the derivative of (6.1) with respect to time, multiplying by ii and integrating over fi we see i||u||2 + 2 j (u{Dii + f 2 Du : DiiDu) Dii dx = — f uittDijUUjttdx. 2 Jn Jn By a Sobolev inequality we see UijdiUjUjjdx < c||u||2||Du|| < ^ | | V u | | 2 + c(^)||Du|| 4||u|| 2 which implies J||ii| | 2+ f i/ 1 |Du| 2 + i/ 2(Du:Dii) 2dx<c(i/ 1)||Du|| 4||u|| 2. 2 Jn Modulo computability conditions on the initial data, this is sufficient to apply Gron-wall's inequality. The key here was to find a lower bound for / (uiDu + i/2Du : DiiDu) : Diidx, Jn namely 0. Actually, this result is a consequence of the ellipticity of the associated elliptic operator. Indeed, suppose T satisfies condition (4.2). Then / ^-Tij(Du)Dijudx = [ awTij-(Du)£>wuAiudx Jn dt JQ > C ( C I ) [ |Vi i | 2 + |Du| p - 2 |Du| 2 dx Jn 70 Chapter 6. Epilogue which gives an equivalent estimate. The next natural estimate in the series is to take the derivative of (6.1) with respect to time and then multiply by £Pdj(v(DvL)Dij\i) (where P is the L2 orthogonal projection onto divergence free functions). Integrating this over fi yields, J ^ ( D u ) | | D u | 2 + ^ ( D u ) D u : ^Dudx + J |p | (5 i (V(KDu)A J u)) ) | 2 = f d d - / ujdjui—Pdk(u(Du)Diku) + Ujdjui—Pdk(u(Bu)Diku^ (6.3) Jn dt at The problem here is the left-hand side, not the right. We would like to express the first integral on the left-hand side as a time derivative of some quantity so as to apply Gronwall's inequality. Inspection shows that because of the nonlinearity it isn't a time derivative. Given the form of the first term, we would perhaps expect to get ! i > ( D u ) | D u ^ x which we could make by adding correction terms to (6.3). To do this, though, we would have to find a control for (* d d I* d / - i / (Du)Du : — Diidx = 2 / u2T>u : DiiDu : — Diidx Jndt K ' dt Jn dt which seems elusive. One might avoid the problem generated by the nonlinearity by multiplying by Pdkdku instead of £Pdj(v(Du)Diju). But then the problem shifts to the second term on the right hand side. We obtain jt j | V i i | 2 dx + J P - ^ - ( V H D u ) D i 3 u ) ) ) P ( d k d k U i ) dx = - / u]dJuiP(dkdkUi) + UjdjUiP{dkdkUi) dx (6.4) Jn We would like to show ^P^(a j (V(z/(Du)A ,u)))P(a^ f c u l )dx (6.5) 71 Chapter 6. Epilogue is positive or is bounded from below by a positive term and a term with controlled growth. This is made difficult by the action of the projection on the product. In the case of the Navier-Stokes equations, however, (6.5) reduces to just which is clearly positive and allows a successful application of Gronwall's inequality. Problems like this related the nonlinearity persist for higher derivatives and must be overcome to obtain higher time regularity. Higher space derivatives for nonlinear elliptic systems remain a famous open prob-lem. Because of this notoriety (and thereby implied difficulty), we did not make a serious attempt to tackle higher space regularity. We summarize briefly here starting points for the interested reader. For general elliptic equations, Holder regularity of the first derivatives can be obtained by the famed de Giorgi theorem, which can be found with exposition in [Gia93]. Further regularity follows from a bootstrap argument. For elliptic systems we have some counterexamples, e.g. [Nec75], to show that we cannot expect everywhere regularity for general elliptic systems. However, these counterex-amples do not exclude the possibility that it is possible to prove everywhere regularity for systems with additional structure. In particular, we have the result of Uhlenbeck [Uhl77] that weak solutions of elliptic systems of the form have Holder continuous first derivatives. Notice the similarity between this system and the Poisson-like system considered in Chapter 3. Finally, we remark that in two dimensions the regularity problems are not the same. Indeed, there is a very recent preprint [KMS97] which proves that in two dimensions the stationary version of (6.1) with periodic boundary conditions has classical solutions. 6.3 Higher Space Regularity dj (F(\Vu\2)djUi) = 0 72 Chapter 6. Epilogue 6.4 Final Words Despite what we are unable to prove, we must keep in mind what has been accom-plished. We have shown a new method for obtain interior regularity of elliptic systems with solenoidal constraint. Moreover, we have extended the class of systems for which this is possible. We have shown how interior regularity for the elliptic system allows us to prove (and clarify) interior regularity for the model parabolic system of inter-est. We have done all this using straight-forward, classically motivated, techniques. Therefore, our failure to find classical solutions should not be viewed too harshly. Our hope is that by presenting our shortcomings the reader might be inclined to think about this problem and help drive our knowledge toward the truth. 73 Bibliography [Ama94] H. Amann, Stability of the rest state of a viscous incompressible fluid, Arch. Rat. Mech. Anal. 126 (1994), 231-242. [BBN94] H. Bellout, R Bloom, and J. Necas, Young measure-valued solutions for non-nezvtonian incompressable fluids, Comm. in PDE 19 (1994), 1763-1803. [Gal94] G. R Galdi, An introduction to the mathematical theory of the Navier-Stokes equations, vol. 1, Springer-Verlag, 1994. [Gia93] M. Giaquinta, Introduction to regularity theory for nonlinear elliptic systems, Birkhauser, Basel, 1993. [GM82] M. Giaquinta and G. Modica, On linear systems of the type of the stationary navier-stokes system, J. fur'Math. 330 (1982), 173-214. [Hey76] J. G. Heywood, On uniqueness questions in the theory of viscous flow, Acta Math. 136 (1976), 61-102. [Hey80] J.G. Heywood, The navier-stokes equations: on the existence, regularity and decay of solutions, Indiana Univ. Math. J. 5 (1980), 639-681. [Hey93] J.G. Heywood, Remarks on the possible global regularity of solutions of the three-dimensional navier-stokes equations, Progress in Theoretical and Com-putational Fluid Mechanics, Pitman Research Notes in Mathematics, no. 308, Longman Group, 1993. [KMS97] P. Kaplicky, J. Malek, and J. Stara, Full regularity of weak solutions ot a class of nonlinear fluids in two dimensions - stationary, periodic problems, submitted to Comment. Math. Carol. Univ., 1997. [Lad69] O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, 2 ed., Gordon and Breach, New York, 1969. [LU68] O.A. Ladyzhenskaya and N. N. Ural'tseva, Linear and quasilinear elliptic equations, Academic Press, New York, 1968. [Max97] D. Maxwell, On an elliptic regularity theorem with applications to a class of non-newtownian fluids, to appear, 1997. [MNR93] J. Malek, J. Necas, and M. Ruzicka, On the non-newtonian incompressible fluids, Meth. Math. Model. App. Sci. 3 (1993), 35-63. [MNR96] J. Malek, J. Necas, and M. Ruzicka, On weak solutions ot a class of non-newtonian incompressible fluids bounded three-dimensional domains, the case p>2, submitted to Adv. in Diff. Eq., Preprint SFB 256, No. 481,1996. 74 Bibliography [MNRR96] J. Malek, J. Necas, M. Rokyta, and M. Ruzicka, Weak and measure-valued solutions to evolutionary partial differential equations, Applied Mathematics and Mathematical Computation, vol. 13, Chapman and Hall, 1996. [Nec66] J. Necas, Sur les normes equivalentes dans W~£(Q,) et sur la coercivite des formes formellement positives, Les presses de l'Universite de Montreal, 1966. [Nec75] J. Necas, Example of an irregular solutiosn ot a nonlinear elliptic system with analytic coefficients and conditions for regularity, Theory of Non-Linear Oper-ators, Abhandluing Akad. der Wissen. der DDR, Proc. of a Summer School held in Berlin, 1975. [Nec83] J. Necas, Introduction to the theory of nonlinear elliptic equations, Teubner-Texte zur Mathematik, no. 52, Teubner Verlagsgesellschaft, 1983. [Sma63] J. S. Smagorinski, General circularion model of the atmosphere, Mon. Weather Rev. 91 (1963), 99-164. [S§73] V.A. Solonnikov and V.E. Scadilov, On a boundary value problem for a station-ary system of Navier-Stokes equations, Trudy Mat. Inst. Steklov 125 (1973), 186-199. [Uhl77] K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math. 138 (1977), 219-240. 75 Appendix A A Direct Coercivity Calculation We wish to show directly that for two functions u and v in W Q ' 4 ( Q ) , / + ^ 2 |Vu| 2 )V^u - (vx + ^|Vv| 2 )Vyv) V 0-(u - v) dx > c||V(u - v) in This calculation inspired Corollary 4.1 and is very easy to follow. Thus this calculation is included here for both completeness and interest. Lemma A . l Let u and v be in W 0 1 , A ( Q ) , I (( l^ + Z A , | V u | 2 ) V ^ U - K + ^|Vv| 2 )Vyv) V y ( u - V ) d x > ^ 11 V(u - v) |\\. Jn 4 Proof: Let u(Vu) denote (ux + u2\Vu|2), let w = u - v and let V u : Vv denote V^uV^v. Then / ^(Vu)Vu : V(w)-zy(Vv)Vv : Vwdx = (A.l) Jn = / (i/(Vu) - v(Vv)) Vu : Vw + j/(Vv)Vw : Vw dx Jn = / v2 (|Vu| 2 - |Vv| 2) Vu : Vw + z;(Vv)|Vw|2dx Jn = / u2V(u + v) : V w V u : Vw + z^(Vv)|Vw|2 dx Jn = / J / 2 ( V U : V W ) 2 + Z A , ( V U : Vw)(Vv: Vw) + u(Vv)|Vw| 2. Jn (A.2) 76 Appendix A. A Direct Coercivity Calculation Now, noticing that expression / n ^(Vu)Vu : V(u - v) - ^(Vv)Vv : V(u - v) rix is symmetric in u and v, since / i/(Vu)Vu : V(u - v) - i/(Vv)Vv : V(u - v) rix = in / z/(Vu)|Vu|2 + z/(Vv)|Vv|2 - (i/(Vu) + i/(Vv))Vu : V v i x , we conclude from (A.2) that / z^(Vu)Vu : V(u - v) - i/(Vv)Vv : V(u - v) rix = Jn / i / 2 (Vv : Vw) 2 + J / 2 ( V U : Vw)(Vv : Vw) + u(Vu)|Vw| 2. (A.3) Jn Averaging (A.2) and (A.3) we see then that / i/(Vu)Vu : Vw - i/(Vv)Vv : VwrJx = Jn = ~k I (Vv : V w ) 2 + 2(Vv : Vw)(Vu : Vw) + (Vu : V w ) 2 rfx + 2 Jn + J (^ + ^ ( | V v | 2 + |Vu| 2)) | V w | 2 i x = f v2 (V(u + v) : Vw) 2 rix + f (ux + ^ (|Vv| 2 + |Vu| 2)) |Vw| 2 rix in in ^ 4 ' - f^l + jlVw|2)lVwl2rfx > l^|Vw||t (A.4) • Since nothing in the previous proof relied on the fact that we used V u as opposed to Du, we obtain a similar result in this case. Lemma A .2 Let u and v be in J01,4(Q). Then, I ((i/i + ^ 2 |Du| 2 )AjU - {yx + i/ 2 |Dv| 2)A;v) Aj(u - v) rix > ^ | |D(u - v) in 4 77 

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