Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

A sharp inequality for Poisson's equation in arbitrary domains and its applications to Burgers' equation Xie, Wenzheng 1991

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1991_A1 X53.pdf [ 1.76MB ]
Metadata
JSON: 831-1.0080353.json
JSON-LD: 831-1.0080353-ld.json
RDF/XML (Pretty): 831-1.0080353-rdf.xml
RDF/JSON: 831-1.0080353-rdf.json
Turtle: 831-1.0080353-turtle.txt
N-Triples: 831-1.0080353-rdf-ntriples.txt
Original Record: 831-1.0080353-source.json
Full Text
831-1.0080353-fulltext.txt
Citation
831-1.0080353.ris

Full Text

A S H A R P I N E Q U A L I T Y F O R POISSON'S E Q U A T I O N IN A R B I T R A R Y D O M A I N S A N D ITS A P P L I C A T I O N S T O B U R G E R S ' E Q U A T I O N By Wenzheng Xie B. Sc., Zhongshan University, 1982 M. Sc., Fudan University, 1985 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y in T H E F A C U L T Y O F G R A D U A T E STUDIES M A T H E M A T I C S We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F BRITISH C O L U M B I A March 1991 © Wenzheng Xie, 1991 In presenting this thesis in partial fulfilment 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 representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada Date DE-6 (2/88) Abstract Let fl be an arbitrary open set in M3 . Let || • || denote the L2(fi) norm, and let is shown to be the best possible. Previously, inequalities of this type were proven only for bounded smooth domains or convex domains, with constants depending on the regularity of the boundary. A new method is employed to obtain this sharp inequality. The key idea is to es-timate the maximum value of the quotient \u(x)\/ || Vu|| x/ 2 || AuH 1 / 2 , where the point x is fixed, and the function u varies in the span of a finite number of eigenfunctions of the Laplacian. This method admits generalizations to other elliptic operators and other domains. The inequality is applied to study the initial-boundary value problem for Burgers' equation: tained. Adapting and refining known theory for Navier-Stokes equations, the exis-tence and uniqueness of bounded smooth solutions are established. As corollaries of the inequality and its proof, pointwise bounds are given for eigen-functions of the Laplacian in terms of the corresponding eigenvalues in two- and three-dimensional domains. Ho(£l) denote the completion of Co°(fi) in the Dirichlet norm ||V-||. The pointwise bound n Table of Contents Abstract ii Acknowledgements iv 1 A sharp inequality for Poisson's equation 1 1.1 Introduction and the main result 1 1.2 Proof of the inequality for bounded smooth domains 3 1.3 Proof of the inequality for arbitrary domains 6 1.4 Proof that the constant in the inequality is optimal 7 1.5 Corollaries 8 1.6 Pointwise bounds for eigenfunctions of the Laplacian 9 2 Application to Burgers' equation 12 2.1 The main result 12 2.2 Preliminaries 14 2.3 Galerkin approximations 15 2.4 Main estimates 16 2.5 Further estimates 18 2.6 Proof of the theorem 23 Bibliography 25 iii Acknowledgements It is a pleasure to thank my advisor, Professor John G. Heywood, for suggesting the topic for this research, and for his very helpful advice and encouragement. I would also like to thank Professor Lon Rosen for helpful discussions about eigenfunction expansions. iv Chapter 1 A sharp inequality for Poisson's equation 1.1 Introduction and the main result In this chapter we establish the following Theorem 1 Let ft be an arbitrary open set in 2R3 . For all u £ HQ(CI) with Au £ L2(Vl), there holds sup \u\ < WVuf'2 H A u l l 1 / 2 . (1.1) Cl V27T The constant \/yj2~Tr is the best possible. Throughout this thesis, || • || denotes the L2(Vl) norm. The gradient V and the Laplacian A are understood in the distributional sense. The homogeneous Sobolev space HQ(Q,) is denned to be the completion of Cfffo) in the Dirichlet norm ||V-1| . Inequalities of this type are used in the study of nonlinear partial differential equations (see [4, p. 299], [10, p. 12]). For bounded domains with smooth boundaries, an inequality of the form of (1.1), but with a constant depending on the domain, can be obtained by combining the Sobolev inequality sup |u| < c||u | | ^ ( n ) |H|J / 2 2 ( n ) , (1.2) with the Poincare inequality H<c||Vu||, (1.3) and the a priori estimate < C I | A « | | • (1.4) 1 Chapter 1. A sharp inequality for Poisson's equation 2 The inequality (1.2) has been proven for domains that satisfy a weak cone condition [1]. The estimate (1.4) has been proven for domains with C 1 ' 1 boundaries or convex domains (see [6]). However, simple examples given in [2] show that (1.4) fails to hold for domains with reentrant angles. For bounded domains with possibly nonsmooth boundaries, the pointwise bound sup|u| < c||Au|| (1.5) n is known [8], with a constant c depending on the domain. In comparison, Inequality (1.1) has a smaller exponent on | | A u | | . For some applications, it is crucial that this exponent is less than one (see the remark in Section 2.4). In Section 1.5, as a corollary of the inequality (1.1), we give a bound for the constant c. Our proof of the inequality (1.1) is independent of such Sobolev inequalities and a priori estimates for elliptic equations, and of the various methods that are used in proving them. The key idea in the proof is to estimate the maximum value of the quotient \u(x)\/ | | V u H 1 / 2 || A u H 1 / 2 , where the point x is fixed, and the function u varies in the span of a finite number of eigenfunctions of the Laplacian. The method can be generalized to other elliptic operators and to other domains. Some of these generalizations will be given by the author in separate papers. That an inequality of the form (1.1) should be valid for arbitrary open sets was suggested to the author by Professor J. G. Heywood. He conjectured that an anal-ogous inequality also holds for the Stokes operator, and can be combined with the methods of [3] and [4] to obtain a regularity theory for the Navier-Stokes equations in arbitrary open sets. Partial results toward the proof of the analogous inequality for the Stokes operator have been obtained by the author. An existence theorem for smooth solutions of Burgers' equation based on (1.1) and the methods of [4] and [3] is given in Chapter 2. Chapter 1. A sharp inequality for Poisson's equation 3 1.2 Proof of the inequality for bounded smooth domains In this section, we assume that fl is bounded, with a C°° boundary 9f2. It is well known that the eigenfunctions of —A can be chosen to form a complete orthonor-mal basis for L2(f2). Let <f)n denote the eigenfunctions, and An the corresponding eigenvalues. Then <f>n £ C°° (f l ) , An > 0, and they satisfy —A<^>n = \n(f>n , <l>n\dQ = 0, ||^ n|| = l , (ra = l,2, •••), / <t>i<f>j dx = Q, {i^j). Our proof of (1.1) has three steps. Step 1. We first consider functions of the form m U( X) = X) Cn<t>n{x) , where c\, • • •, cm are real numbers. We have m m n—1 n = l Hence, for any y € f l , (1.6) Vn=l / \ n = l Let y and m be fixed. Then this quotient is a smooth and homogeneous function of (ci,---,cm) in JRm\{0) . Hence it attains its maximum value at some point (ci, • • •, cm), i.e., when the function is u = Y^=\ ^n<i>n • This maximum value is greater than zero, by the well-known fact that <f>i(y) ^ 0 . Differentiating l o g i i v . ' n L i i 5 5 l 0 i " 2 { y ) ' \loe 1 1 V "" 2 - 5 l o g l | A u | 1 2 Chapter 1. A sharp inequality for Poisson's equation 4 with respect to cn at the critical point, we get 2<Mv) Anc„ A 2c n = 0 , u(y) ||Vu||2 ||Au|p for n = l , - - - , m . Letting p, = || A£i||2/|| Vu||2 , we obtain ^(y) v -/i + A B ~ | | A « | P W Hence ^ / 2 ^ ( y ) \ 2 / u(y) V ^ 2 2 u (^y)_ BtiU + A j l l|Afi|p ; ^ " » ||Afi||» Therefore the maximum value of the quotient (1.6) is u\y) \\Au\ |Vu||||Au|| ||Vfi| I | A U | | 2 - 4 ^ U + AJ • ( L 7 ) Step 2. To bound the right hand side of (1.7), we use the Green function for the Helmholtz equation and its eigenfunction expansion. Let y and u be fixed as above, and let e-y/t\x-y\ s(x) = h—f- <L8) 47r|x — y\ It is easy to verify that Ag(x) = pg(x) for all x ^ y . Let h(x) satisfy Ah = ph, h\dQ — g\au . There is a unique solution h 6 C°°(f l ) , since p > 0 and the domain is bounded and smooth. The function h(x) attains its minimum value at some point i i 6 fl. If xy G dft, then h(xi) = g(x\) > 0 . If x\ 6 fl, then Ah(xi) > 0 , and hence M x i ) = (1/A4)A/I(XI) > 0 . Therefore, we always have h(x) > 0 in fl. Let G = g—h. Then AG = fiG in fl\{y}, G\dU = 0 . We obtain G(x) > 0 in fl\{j/} similarly. Hence we have 0 < G(x) < g(x) for all x 6 ^\{y} , and therefore Chapter 1. A sharp inequality for Poisson's equation 5 Let J l j C O be a ball centered at y with radius e. For any n > 1 , we have / 0„AC7 - GA<j>n) dx= f U n ^ ~ dS, Ju\n€ Jd(n\nc) \ ou ou J by Green's formula, where u is the outward normal to the boundary. Hence / (p + K)G(f>n dx = - f (<f>n^r ~ G ^ 1 ) dS , JQ\QC Jr=€ \ Or Or J where r = |x — y\. Since G has the same type of singularity as l/47rr, by letting e —• 0, we obtain (/* + AN) / G<f>ndx = <j>n{y). (i-io) JQ Therefore, by Parseval's equality, \ G2dx = J2( G<f>ndx) = £ 2 00 '*»(y)V ^ + Any This, together with (1.9), provide a bound for the maximum value (1.7): u2(y) < Ay/u I G2 dx < = ^~ ||Vu||||Au|| - v W n ~ 2TT' Thus we have proven Inequality (1.1) for all functions of the form u = cn<f>n • Step 3. Now, let u be any function in HQ(VI) with Ait 6 L2(Vt). Since Cl is bounded, we have HQ(CI) C L2(Cl) by virtue of the Poincare inequality. Hence we have the expansion u = Y^i^^n in L2{Ci), where cn — fnu<f)ndx. Let um = Yl™=i cn<f>n • Integrating by parts, we have / V « • V u m dx = — uAum dx — y2 Ancn / ucj>n dx = \nc2 = ||Vum||2 . J a J Q n = l J n n = l Hence we get ||Vum|| < ||Vu|j by using the Schwarz inequality. Similarly, from / A « m A u rfi = — A„cn / <f>n Au dx — — Anc„ / uAcj)ndx Jn ~i Jn ~. Jn 7 1 = 1 7 1 = 1 m = £ « = ||Aur "m||2 j n = l Chapter 1. A sharp inequality for Poisson's equation 6 we get ||Aum|| < ||Au||. Therefore, by the result of Step 2, we obtain sup |um|2 < ± - ||Vum|| ||Aum|| < -J- ||Vu|| ||Au||. By a well-known interior regularity theorem for elliptic equations, Au £ L2(tt) implies u £ H2oc{fl), which in turn implies that u £ C ( A ) , by a well-known Sobolev imbedding theorem. Now, if (1.1) were not true, then there would be some x0 £ fi such that \u(x0)\2 > ||Vu|| ||Au|| > sup |um|2 , Z 7 T n • which is obviously contradictory to the fact that limT n_ ).0 0 \\um — u\\ = 0. This completes the proof of (1.1) for bounded smooth domains. 1.3 Proof of the inequality for arbitrary domains Let fi be an arbitrary open set in IR?, and suppose that u £ H^ft) and Au £ L2(fl). We can choose a sequence of bounded domains fln with smooth boundaries, such that fl\ C O2 C • • •, and IJSLi An = fi • For each n > 1, there exists a unique un £ /z~o(^ n) such that / Vun-Vvdx=f Vu-Vvdx, V v £ H^(fln), (1.11) by the Riesz theorem. We get || Vun||£2(o,n) < ||Vit|| by letting v = un and using the Schwarz inequality. Integrating by parts on the right hand side of (1.11), we obtain / Vun-Vvdx = -f (Au)vdx, V v £ #o(A„), and hence Aun = Au\un . Therefore, by the result of Step 3, we have sup K|2 < i - ||Vu n|| L 2 ( f 2 n ) ||Atxn||L2(n„) < i - ||V«|| ||Au||. (1.12) Setting un equal to zero in fl\fln , we get un £ H^fl) • From (1.11) we have lim / Vun-Vvdx= j Vu-Vvdx, V v £ CZ°(fl). Chapter 1. A sharp inequality for Poisson's equation 7 This and ||Vwn|| < ||Vu|| imply that lim^oo un = u in H^ft) . Therefore, by the inequality ||w||z,6(n) < c||Vt>|| (see [6, p. 10] for a simple proof giving c — \/4&), we have linin^oo \\un — « | | L 6 ( O . ) = 0. This and (1.12) imply (1.1) by reasoning similarly as in Step 3 above. 1.4 Proof that the constant in the inequality is optimal We first consider the case fl = IR3 . Define u(x) = f(r) = < 1 - c -r = 0, , r > 0 , where r = |x|. The function u is continuous, with a maximum value u(0) = 1. We notice that U/ATT is equal to the difference between the fundamental solution for the Laplace equation, and that of the Helmholtz equation ((1.8), with u = 1). Hence we immediately obtain Au — —e~r/r in the distributional sense. We have Au £ L2(fl) since |Au|2 dx = jT° ) 4?rr2 cir = 2?r. Integrating by parts (which is easily justified), we obtain f IT-, ,i i f * , f°° 1 ~ e~r e _ r / Vu di — — I uAudx = / JJR* 7JR3 Jo r 4 7 r r 2 dr = 2ir Hence, the equality in (1.1) actually occurs for the function u . To show that u £ HQ(]R3) , we modify the function u to define a sequence of functions. Let / ' denote df/dr. For each n > 1, let /'(1/n) un(x) 2n f'(n)2 4/(n) 0, ( n V - l) , 0 < r < 1/n, 1/n < r < n, , n < r < rn = n rn < r < oo. 2/(n) Chapter 1. A sharp inequality for Poisson's equation 8 It is easily seen that un G CQ(M3) and that un is piecewise C 2 . By explicit calcu-lation, we find that Jim | | V ( u B - u)\\L2(E?) = 0. Hence u G HQ(M3) . (It is interesting to note that u g" HQ(1R3) since u g" L2(M3)). By explicit calculation, we also find that lim u„(0) = 1, lim ||A(u„ - u)\\L2tR3) = 0. Hence ^ H V U n l l ^ H . ) ||Au n ||$ R 3 ) l l V u l l ^ , | | A U | | ^ K 3 ) V 7 ^ ' We now consider an arbitrary open set fi. Let \x — xo\ < e be a ball contained in it. For each n > 1, define vn(x) = un(er~1(a; — x0)). Then u n vanishes outside the ball. Clearly, we have vn G Cl(fi) and A v n G L 2 ( f i ) , for all n = 1,2, • • • . It is easy to verify that Vn(xo) _ Un(0)  \\Vvn\\^2\\Avn\\^2 - | | V u n | | # w ) | | A u n | | ^ 3 ) ' Noticing (1.13), we conclude that the constant l/y/2n in Inequality (1.1) cannot be improved, for any given domain. This completes the proof of our theorem. 1.5 Corollaries In this section, we give several immediate corollaries of Theorem 1. Let fl denote an arbitrary open set in JR3 , except in Corollary 4. Note that the constants in the corollaries are not claimed to be the best possible, except for the special case stated in Corollary 1. Corollary 1 / / u G H^{fl) a n d A u G L 2 ( f i ) , then sup|u|<^=(||Vu||+ | | A u | | ) . The equality occurs for some functions in the case fl = M3 . Chapter 1. A sharp inequality for Poisson's equation 9 Corollary 2 If u £ HQ(Q) and Au £ L2(ft), then u satisfies sup |u|< -)= Hull1/4 || A u f / 4 . n V27T Corollary 3 // u £ H^(Ct) and Au £ Z2(fi), tfien 8 u p H < - L = ( H + 3 | | A u | | ) . n 4v27r Corollary 4 Let fl 6e an open set in Ft3 such that the Poincare inequality \\u\\ <7||Vu||, Vu£ J ff 0 1 (A) (1.14) holds. Then, for all u £ HQ(Q,) with Au £ L2{Vi), there holds s^ P 1^ 1 < 7^||Au||. Proof of the corollaries. Corollary 1 follows from Inequality (1.1) directly and the example given in Section 1.4. Corollary 2 follows from (1.1) by using ||Vu||2 = - / uAudx < ||u|| ||Au||. (1.15) Corollary 3 follows from Corollary 2 and Young's inequality. Corollary 4 follows from (1.1), since we have ||Vu|| < 7 ||Au|| from (1.15) and (1.14). It is easy to show that Theorem 1 and the corollaries are also valid for vector-valued or complex-valued functions. 1.6 Pointwise bounds for eigenfunctions of the Laplacian As a special case of Corollary 2, we have Theorem 2 Let fl be an arbitrary open set in M3 . If A > 0 and <f> satisfy -A<f> = \<j>, teH^n), p>|| = i , then A 3 / 4 sup |^ | < Q V27T Chapter 1. A sharp inequality for Poisson's equation 10 That is, we have a pointwise bound for any eigenfunction <f> of the Laplacian, depending only on the corresponding eigenvalue A . As in Corollary 2, the constant here is not optimal. We give a better constant in the theorem below. Theorem 3 Let fl be a bounded open set in M3 with a smooth boundary. If A > 0 and (j) satisfy -A<t> = \<f>, 4>£Hl0(fl), U\\ = \ , then T w s \ / ! ( f ) 3 / 4 - ( 1 1 6 ) Proof. For any y 6 fl and any p > 0, as the equality (1.10), we have (f>(y) = (u + \) I G<j>dx. Jn B y the Schwarz inequality and (1.9), we have \Hy)\ = f> + A) / G<f>a Jn < ( , + A ) ( / n C 7 ^ x ) 1 / 2 ( / f l ^ ^ 1 / 2 < ^ + A)(L^ 2^) 1 / 2 (p + A) ( — ^ 1/2 87ry/jlJ The right hand side attains a minimum value when p = A/3 . Letting p, = A/3 , we obtain (1.16). The following theorem is an analogue of Theorem 3 in two dimensions. Theorem 4 Let fl be a bounded open set in M2 with a smooth boundary. If A > 0 and (f> satisfy -A<f> = \<!>, teH^fi), U\\ = i, then supM<J- . (1.17) fi V 7T Chapter 1. A sharp inequality for Poisson's equation 11 Proof. In two dimensions, the fundamental solution corresponding to (1.8) is = -^Ko(yMx - y\) > where KQ is a modified Bessel function. We have Hence, similar to the proof of Theorem 3, we have The right hand side attains a minimum value when u = A. Letting \i = A , we obtain (1.17). Since for special domains, the eigenvalues and eigenfunctions are explicitly known in terms of special functions, these pointwise bounds can be used to derive inequalities for the special functions. Chapter 2 Application to Burgers' equation 2 . 1 The main result In this chapter, we apply the inequality (1.1) to study the following problem for the three-dimensional Burgers' equation: du _ — + u • Vu - Au, at u(t) G Hl(Cl), (2.1) u(0) = UQ . Here, the spatial domain 0 is an arbitrary open set in IR3 , and the initial vector field «o is given in Hl(Vl) = H^Cl)3. The Burgers' equation is studied for its analogy with the Navier-Stokes equations. We establish the following theorem by methods which will carry over immediately to the Navier-Stokes equations, if we are successful in proving the analogue of (1.1) for the Stokes operator. Hereafter, we use Dt to denote the partial derivative with respect -to the time variable (t or s). Theorem 5 Let 0 be an arbitrary open set in M3 . Let u0 G HQ(CI) be given. Let T _ 256TT2 27||Vw0||4 ' Then there exists a unique vector field u such that u G C°°(f2 x (0, T)) n C°°((0, T), Xoo(O)), u - uo G C ([0, T), Hl(il)) n C~ ((0, T), Hl(Q)) , A w G C°°((0, T), L2(ft)), 12 Chapter 2. Application to Burgers' equation 13 and satisfies Problem (2.1). The solution also satisfies the estimates: \\u(t) - wol2 < tF(t), tk+i/2 ||£)*+iu|| + \\DfuWoo + tk \\VDku\\ + tk+1'2 ||ADku\\ < F(t), k > 0, j[f(||A«||3 + |Mll, + | | A « | | 2 ) d 8 < F ( t ) l J\2k (\\Dk+lu\\2 + s-^ \\Dktu\\lo + s'1 \\VDku\\2 + ||A£>M|2) ds < F(t), k > 1, for all t £ (0, T) , where the F(t) denotes appropriate continuous functions on [0, T) that can be obtained explicitly in terms of k and ||V«o||, independently of f l . Theorem 5 is an adaptation and refinement of known existence theorems for the Navier-Stokes equations, based on a differential inequality for ||Vu(r)|| , and its ana-logue for Galerkin approximations. The method originated with Prodi [9], who used it to prove the existence of generalized solutions in bounded domains. Hey wood [3] introduced a further infinite sequence of differential inequalities, to obtain classically smooth solutions. He also extended the method to unbounded domains. Heywood and Rannacher [4] developed the method further through use of weight functions de-pendent on the time variable to give more precise estimates as t —> 0 + . All of these developments are incorporated into the existence theorems given here. The principal innovation here is that the nonlinear term is now estimated in a new way, using the inequality (1.1), to give results that are not only sharper but also valid in arbitrary domains. The energy estimate basic to many works on the Navier-Stokes equations is not valid for the three-dimensional Burgers' equation. Theorem 5 is independent of it. Observe that the solution can have an infinite X2(fl)-norm in unbounded domains. We point out that unlike the Navier-Stokes equations, there is a maximum prin-ciple for solutions of Burgers' equation. An existence theorem for Burgers' equation based on the maximum principle was given by Kiselev and Ladyzhenskaya [5]. Incor-porating it with Theorem 5, the solution can be continued globally in time. Chapter 2. Application to Burgers' equation 14 2.2 Preliminaries In this section we list some lemmas that will be used later. Notations. We use boldface symbols to denote three-dimensional vector-valued functions and their spaces. We use || • ||p to denote the LP(D.) or Lp(tt) norm. When p = 2, we simply use || • || to denote the norm, and use (•, •) to denote the inner product. We use || • ||oo to denote the supn | • | norm. The vector version of Theorem 1. Let fl be an arbitrary open set in JB? . If Proof. The inequality (2.2) is obtained by simply applying (1.1) to each component: u € iT 0(fl) and Aw 6 L2 (fl), then 1 (2.2) The constant 1 /y/2~w is the best possible. 3 oo It is obvious that the constant remains optimal. Holder's inequalities. If p, q > 1 and l /p+ 1/q = 1, then (2.3) If p, q, r > 1 and 1/p + l/q + l/r = 1, then dx < (J | /| p <k) 1 / P ( / \g\qdxy/q (J \h\rdx) l / r These are well known. We will use the case p = 6, q = 2, r = 3: (2.4) Chapter 2. Application to Burgers' equation 15 Sobolev inequalities. For all u G C 0 x 3(iR 3), there hold H i e < c||V«||, (2.5) ||u||3 < cHwll^l lVwII 1 / 2 . (2.6) A simple proof of (2.5) can be found in [6, p. 10], with c = \/48. Letting p = 4/3,? = 4 and / = g = |u|3/2 in (2.3), we obtain \3/4 / » \ 1/4 J \u\3dx < (J \u\2dx^j (J \u\*dx^ Combining this with (2.5), we obtain (2.6). Young's inequality. If a,6,p, q > 0 and l/p + l/q = I, then ap bq a f e <^_ + L . (2.7) p q 2.3 Galerkin approximations Similar to Section 1.2, we first assume that fl is a bounded open set in M3, with a C°° boundary d f l . The vector-valued eigenfunctions of —A can be chosen to form a complete orthonormal basis for £ 2 ( f l ) . Let <pn denote the eigenfunctions, and An the corresponding eigenvalues. Then <pn 6 C°°(f l ) , An > 0, and they satisfy -A<pn = An<£n, <pn\att = 0, ||<pn|| = l , (n = l,2,-.-), (</>,, <p,) = 0, (i^j). These eigenfunctions and eigenvalues can be obtained immediately from their scalar counterparts. We seek Galerkin approximations in the form TO «m(M) = 53C(*)*n(*) n-1 where the cm(£) are smooth functions of t. The advantage of using this form is that dum/dt and Aum are also linear combinations of the first m eigenfunctions. Let Chapter 2. Application to Burgers' equation 16 um satisfy — -Aum+um-Vum,<pn\ = 0, (2.8) ( u m ( - , 0 ) - « o , cj>n) = 0, (n = l ,2 , - . . ,m) , (2.9) i.e., j m - C ( t ) = - A n C ( i ) - E ^ - ^ ^ n W W , d * . J = i (2.10) C (0 ) = ( « o , < £ „ ) , ( n = l , 2 , - - - , r o ) . To find a time interval on which the solution exists, we need a priori estimates. Hereafter we suppress the superscript m. From (2.8) we obtain (Dk+1u-ADku + Dk(u-Vu), v) =0, V v G spanf^, • • •,<pm} . (2.11) In particular, we can take v = D\A^u. Here i,j,k>0. 2.4 Main estimates Let h = 0 and let v = Aw in (2.11). We get I|||W||2+ ||Au||2 = ( u - V u , A«) (2.12) < M o o ||Vti|| ||Ati|| (2.13) < -j= ||V«||3/2 ||A«||3/2 (2.14) < ^ ( a - 6 | | V W f + 3a2||A«||2) (2.15) = C7e||V«||6 + e||Aw||2. (2.16) We obtain (2.12) by integration by parts; (2.13) by using the Schwarz inequality; (2.14) by using (2.2); (2.15) for any a > 0, by using Young's inequality (2.7); and (2.16) by letting _ 3a2 _ a~6 27 e — —;= , C, — 4v/27' ' 4 ^ 10247r2e3' Hence ~l|V«H2 + (l-e)||Au||2<C7£||V«f. Chapter 2. Application to Burgers' equation 17 Let ¥>(*)= ||V«(*)U2 4-2(l - e ) f \\Au(s)r ds . Jo Then, when 0 < e < 1, we have d -<p(t)<2C^(t), V(0)= ||V«(0)||2. Comparing this with j $ { t ) = 2C^\t), $(0) = ||V«o||2, and noticing that we have ||Vu(0)||2 < ||VM 0|| 2 from (2.9), we obtain ^ ) < $ W = (||V W o||- 4-4C7 £ i ) - 1 / 2 , ||Vtt(t)||: where + 2(1 - e) f \\Au(s)\\2ds < . l | V " ° 1 1 2 , (0 < t < e3T) J o yj\ - t/e3T 256TT 2 27||Vu0|h Letting e = 1 we obtain , 2 ^ I I V U O H 2 l | V w W " 7rf / r ' ( 0 - t < r ) - ( 2 - 1 7 ) Letting e = ^Jt/T , we obtain / * ||Aw||2^ < l | V " ° 1 1 2 (0 < t < T). (2.18) 2(l-fi/T)y/l-y/t/T Since ||Vu(*)||2 = £ m = 1 An|cm(f)|2, the a priori estimate (2.17) ensures that the solution of the o.d.e. system (2.10), hence the Galerkin approximation, exists on the interval [0, T), with T independent of m and Cl. Remark. If the inequality (1.5) is used instead of (1.1), one would obtain \ft\\Vu\\2+ ||A«||2<c||Vu||||AU||2, Chapter 2. Application to Burgers' equation 18 which does not lead to a bound for ||Vw||, unless one restricts the initial values to those satisfying c||Vtt0|| < 1 • 2.5 Further estimates To prove the smoothness of the solution, we need further a priori estimates. We first prove some differential inequalities. We use c to denote a constant that does not depend on m or fi, but possibly depends on k . The actual value of c may change at each occurrence. Lemma 1 For 0 < t < oo, there hold \\Dku\\2 < cHA^-^H 2 + c £ IIVA*-1-'"!!2, * > 1, (2-19) i=0 ft\\Dku\\*+ \\WDku\\2 < c(||V«||4 + l)||DM|2 +c£ \\VDiu\\2 ||VZ?tfc-,"«Ha , * > 1, (2.20) | ||VA*«||a + ||AAfc«Ha < c(||Vu||* + H|L) ||VD*«||2 + * £ WD>\\lo IIV^-^H2 , k > 1, (2.21) i=l HAfc«l|2oo < c||V2?*u||||AD*u||, k>0. (2.22) Proof of (2.19). Letting v = Dk+1 u in (2.11) and using the Schwarz inequality, we get \\Dk+1u\\ < \\ADku- Dk(u- V«)||. Hence \\Dk+1u\\2 < 2 ||AAfc«||2 + 2 \\Dk(u • Vu)\\2. (2.23) We have k Dk(u • Vu) = £ cD\u • VZ>*-'ii, (2.24) t=0 Chapter 2. Application to Burgers' equation 19 by the Leibniz formula, hence \\Dk(u • Vu)|| < cjz \\D\u\U \\VDkru\\. (2.25) «'=o Using this in (2.23) and replacing k by k — 1, (2.19) is obtained. Proof of (2.20). Letting v = Dku in (2.11) we get \jt\\Dktu\? + \\VD'u\\2 = -(Dl(u.Vu),Dku) (2.26) < c ^ l p i t t l l f l l l V D f - u l l l l A ^ l l a (2.27) j'=0 < l|VZ?;u|| ||Vflf-'u|| \\Dku\\^\\VDku\\^ (2.28) i=0 < i ||V£>*«||3 + c ( ||V«||4 + l) \\Dktu\\2 + c £ II VD\uf || V^-wH2 . (2.29) i=i We obtain (2.26) by integration by parts; (2.27) by using Leibniz's formula (2.24) and Holder's inequality (2.4); (2.28) by using Sobolev inequalities (2.5) and (2.6); (2.29) by using Young's inequalities. Hence (2.20) is proved. Proof of (2.21). Letting v = AD*u in (2.11) we get ld_ 2dt \j\\VDktu\\2+\\&Dku\\2 = (Dk(u.Vu),ADku) < \\\ADku\\* + c\\Dk(u.Vu)\\\ (2.30) From (2.25), we have \\Dk(u.Vu)f < i||ADNH2 + c||V«|H|VAfc«||2 k-i +cE||JD>||Ll|VAfc-tw||2, (2-31) i=0 since IPMIL liv«||2 < ^ ||VAfc«ll \\&Dku\\ ||W||2 < i||AAfc«ll2 + c||vw||4||VAfc«ir, Chapter 2. Application to Burgers' equation 20 by using (2.2) and (2.7). The inequality (2.21) follows from (2.30) and (2.31). Proof of (2.22). This is directly obtained by applying (2.2) to the function u. Now, differential inequalities (2.20) and (2.21) cannot be integrated directly, be-cause we do not have initial values at t = 0. To overcome this difficulty, we will follow the method of Hey wood and Rannacher [4], introducing the weight functions Lemma 2 Suppose ip, xp, ct, /3 E C 1(0,T) are all non-negative and satisfy -£ + tp < oap + P, (0 < t < T). Suppose also that where n is a positive integer and F\,F2,F3 £ C[0,T) . Then we have where F4 = (F 3 + nFi) exp F2 £ (7[0, T). Proof. We have d — ( t » + tnif> < atn(p + tN(3 + nt71'1^. dt For 0 < e < t < T, let $£(r) = tn(p(t) + f snr^(s) ds . Then dt Hence $e(<) < e"cp(e) + / ' exp (- f' a dr) [snp + n s n " V ds exp ads < [eV(e) + F3(t) + nF^t)] exp F2(t) Chapter 2. Application to Burgers' equation 21 for t<t<T. The existence of / sn 1ip(s)ds implies that Jo liminfeTV(<0 = 0, £—0+ r W ' completing the proof of the lemma. Now, corresponding to Lemma 1, we prove the following estimates. Lemma 3 For 0 < t < T, there hold /V1"2 ll/J>||2d5 < F(t), i>l, (2.32) Jo i 2 * ' " 1 II D\u\\2 + /V'"1 \\VD\ufds < Fit), i>l, (2.33) Jo r2 i ||VD;u||2 + [*s2i\\AD\u\\2ds < F(t), z>0, (2.34) Jo f s^WDiuWlds < F(t), i>0, (2.35) Jo where the F(t) denotes generically a continuous and increasing function on [0, T) depending only on i and ||Vwo|| > independent of fl and m . Proof. We obtain (2.34-0) from (2.17) and (2.18), and then obtain (2.35-0) by using (2.22-0). From (2.34-0) and (2.35-0), we see that the coefficients in the differential inequal-ities (2.20) and (2.21) are integrable, i.e., they satisfy the condition set for a in Lemma 2. We proceed by mathematical induction. Let k > 1 and assume that (2.32)-(2.35) are true for i < k — 1 . From the assumption and (2.19) we have f s2k-2\\Dku\\2ds < c f s^WAD^ufds Jo Jo +cE f {s2k~2-2t llVZ^-'wH2) (s2' ||L>||2) ds i=0 J o < F(t). Chapter 2. Application to Burgers' equation 22 Hence we obtain (2.32-k). This technique of appropriately distributing the weight functions will be repeatedly used below without further comment. With (2.32-k) and the assumption, we can apply Lemma 2 to the differential inequality (2.20) to obtain (2.33-k), which in turn enables us to apply Lemma 2 again to the differential inequality (2.21) and obtain (2.34-k). Finally, we obtain (2.35-k) from (2.22). This completes the proof. Lemma 4 For 0 < t < T, there holds t2i+1 \\ADiu\\2 < t2i+1/2\\Dlu\\l < Proof. Letting v = ADku in (2.11) we get ||ADku\\2 < 2 \\Dk+1u\f + 2\\Dk(u • Vu)\\2. Using (2.31) we obtain ||A£>M|2 < c\\Dk+1u\\2 + c\\Vu\\4\\VDku\\2 +cE\\Diu\\l\\VDtiu\\2. (2.38) »=o Using the estimates obtained in Lemma 3, by mathematical induction on (2.38) and (2.22), the proof is completed. 2.6 Proof of the theorem With the estimates given in Lemma 3 and Lemma 4, we can follow the argument in [3] to prove the existence and regularity of the solution of Problem (2.1), as asserted in Theorem 5. The solution in the bounded smooth domain is obtained as a limit of a subsequence of the Galerkin approximations. Given an arbitrary domain, we can choose a sequence of bounded smooth subdomains expanding to the domain, and F(t), i > 0 , Fit), » > 0 , (2.36) (2.37) Chapter 2. Application to Burgers' equation 23 solve the problem in each subdomain with properly chosen initial data, and take a subsequence of these solutions which converge to the solution in the given domain. The estimates carry over in the above mentioned processes of taking limits. Thus, we need only prove the uniqueness of the solution. Suppose v is another solution. Let w = v — u. Then dw — + u • Vto + w Vu + w • Vw = Aw , (2.39) and Km||V«»(t)|| = 0. (2.40) Multiplying (2.39) with Aw and integrating over 0, we get ij4||Vu>||2 + ||Au>||2 = (u • Vw + w • Vu + w • Vw , Aw). We have {u-Vw,Aw) < Halloo [|Vw|| ||Aw|| < ^WAwr + cWuWlWVwf, (w-Vu,Aw) < \\w\loo \\Vu\\ \\Aw\\ < clival 1/ 21|V«|| ||Aw||3/2 < i||A«,||2 + c||VW||4||V«;||2, (w-Vw, Aw) < I I I O I I ^ H V I O I I I I A I D I I < C||V«J||3/2||AU.||3/2 < | | | A « | | A + C||Vtc||6, using the inequality (2.2) and Young's inequality. Hence dt Since ||VH| 2 <c(H| 2 0+ l|Vu||4) ||V«,||2 +c||V«;| j f (NIL + liv«||4) ds < F(t) Chapter 2. Application to Burgers' equation 24 and we have (2.40), it is easy to prove that || Vu>(<)|| = 0 for all 0 < t < T. Hence w(t) = 0. R e m a r k 1: If a bounded portion of the boundary of 0 is Cm , then D*[u(x,t) is uniformly Cm up to that portion of the boundary, for all k . R e m a r k 2: If we consider Burgers' equation with a "viscosity coefficient" v. du — + u • V u = i/Att, then T should be multiplied by v3 . If we consider nonhomogeneous boundary values and an external force term, then we can prove a similar theorem of local existence and uniqueness of the solution, with T depending on the given data. Bibliography [1] R. A . A D A M S A N D J . J . F O U R N I E R , Cone conditions and properties of Sobolev spaces, J. Math. Anal. Appl. 61 (1977), 713-734. [2] P . G R I S V A R D , Elliptic Problems in Nonsmooth Domains, Monographs and Stud-ies in Mathematics 24, Pitman Publishing Inc., Boston, 1985. [3] J . G . H E Y W O O D , The Navier-Stokes equations: on the existence, regularity and decay of solutions, Indiana Univ. Math. J. 29 (1980), 639-681. [4] J. G . H E Y W O O D A N D R. R A N N A C H E R , Finite element approximation of the nonstationary Navier-Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization, SIAM J. Numer. Anal. 19 (1982), 275-311. [5] A. A. K l S E L E V A N D O. A. L A D Y Z H E N S K A Y A , On the existence and uniqueness of solutions of the non-stationary problems for flows of non- compressible fluids, AMS Translations Series 2, 24, 79-106. [6] O. A. L A D Y Z H E N S K A Y A , The Mathematical Theorey of Viscous Incompressible Flow, Second Edition, Gordon and Breach, New York, 1969. [7] O. A. L A D Y Z H E N S K A Y A , The Boundary Value Problems of Mathematical f Physics, Applied Mathematical Sciences 49, Springer-Verlag, New York, 1985. [8] J. N E C A S , Les Methodes Directes en Theorie des Equations Elliptiques, Masson, Paris, 1967. [9] G . P R O D I , Theoremi di tipo locale per il sistema di Navier-Stokes e stabilita della soluzioni stazionarie, Rend. Sem. Mat. Univ. Padova, 32 (1962), 374-397. [10] R. T E M A M , Navier-Stokes Equations and Nonlinear Functional Analysis, SIAM, Philadelphia, 1983. 25 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0080353/manifest

Comment

Related Items