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Uniqueness theory for compressible flows Ravindran, S. S. 1991

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UNIQUENESS THEORY FOR COMPRESSIBLE FLOWS By S.S.RAVINDRAN B. Sc.(Hons.), The University of Jaffna, Sri Lanka, 1987 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF M A S T E R O F SCIENCE in T H E FACULTY OF G R A D U A T E STUDIES D E P A R T M E N T OF MATHEMATICS We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA January 1991 © S.S.RAVINDRAN, 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 a a * 0 tA«rch 11 DE-6 (2/88) ABSTRACT This thesis investigates questions of uniqueness in the theory of Compressible flow. First, various uniqueness theorems for compressible flow are reviewed in an expository manner. Roughly, these the-orems state that fluid motion in a bounded region fi = fi(t) is uniquely determined by its initial data together along with certain boundary conditions. Next, this analysis is extended to magnetohydrody-namic flows and uniqueness theorems are given for a variety of possible cases. The basic question in all these theorems is the determination of appropriate boundary conditions. The proofs are by energy estimates. ii Table of Contents A B S T R A C T ii A C K N O W L E D G E M E N T S v 1 INTRODUCTION 1 1.1 VISCOUS COMPRESSIBLE FLUIDS 2 1.2 INVISCID COMPRESSIBLE FLUIDS 4 1.3 INVISCID HEAT CONDUCTIVE COMPRESSIBLE FLUIDS 5 1.4 VISCOUS COMPRESSIBLE MHD 6 1.5 INVISCID COMPRESSIBLE MHD 8 1.6 INVISCID HEAT CONDUCTIVE COMPRESSIBLE MHD 9 2 UNIQUENESS OF VISCOUS COMPRESSIBLE FLUID MOTIONS 11 2.1 AUXILIARY EQUATIONS 11 2.2 SERRIN'S FUNDAMENTAL UNIQUENESS THEOREM FOR VISCOUS FLUIDS . . 14 2.3 UNIQUENESS THEOREM FOR THE CASE ft > ft > 0, C > 0, K = 0 27 2.4 UNIQUENESS THEOREM FOR THE CASE y. - 0, C > Ci > 0, K = 0 29 2.5 UNIQUENESS THEOREM FOR THE CASE ^ = 0, C > Ci > 0, K > 7ci > 0 34 3 UNIQUENESS OF INVISCID COMPRESSIBLE FLUID MOTIONS 37 3.1 UNIQUENESS THEOREM FOR INVISID COMPRESSIBLE FLUIDS 37 3.2 UNIQUENESS THEOREM FOR INVISCID HEAT CONDUCTIVE FLUIDS 42 4 UNIQUENESS FOR COMPRESSIBLE M A G N E T O H Y D R O D Y N A M I C S 47 4.1 FUNDAMENTAL UNIQUENESS THEOREM FOR VISCOUS MHD 47 4.2 UNIQUENESS THEOREM FOR THE CASE u > ft > 0, C > 0, K = 0, rj > 0 . . . . 52 4.3 UNIQUENESS THEOREM FOR THE CASE p = 0, C > Ci > 0, ic > m > 0, fj > 0 . 53 4.4 UNIQUENESS THEOREM FOR THE CASE \i = 0, C > Ci > 0, K = 0, rj > 0 . . . . 55 4.5 UNIQUENESS THEOREM FOR INVISCID COMPRESSIBLE MHD 57 iii 4.6 UNIQUENESS THEOREM FOR INVISCID HEAT CONDUCTIVE MHD Bibliography iv A C K N O W L E D G E M E N T S I would like to express my sincere gratitude to Professor John Heywood, my thesis supervisor, for introducing me to the Navier-Stokes theory and suggesting the topic for this thesis. His advice and encouragement was very valuable and greatly appreciated. I am grateful to Professor Anton Bui for his valuable time in reading this thesis. I am also grateful for financial support from the Department of Mathematics, University of British Columbia, in the form of Teaching Assistantship, and from the Natural Science and Engineering Research Council of Canada, grant 5-84150. v Chapter 1 INTRODUCTION One of the central problems in the mathematical theory of fluid dynamics is the determination of whether certain initial boundary value problems are well posed. That is, do solutions exist, are they unique, and do they depend continuously on the data. In this thesis we study questions of uniqueness for the equations of compressible flow. The principal objective of such a study is to find appropriate boundary conditions, which together with prescribed initial values for the velocity, temperature, density and magnetic field, will uniquely determine the subsequent motion. Our methods and results in this thesis stem from a number of important prior works. In 1953 GRAFFI [6] proved a uniqueness theorem for compressible barotropic fluids. These are fluids for which the pressure P depends only on the density p. His proof was by classical energy methods. In 1959 SERRIN [15] extended this uniqueness theory to general compressible flows. He dealt with each thermodynamic case except that in which the coefficients of viscosity satisfy the Stokes relation 3A + 2/i = 0, and the coefficient of thermal conductivity tt is zero. In 1981 VALLI [17] gave a still more general development of uniqueness theorems for compressible flow which includes this exceptional case. For magnetohydrodynamic flows, uniqueness theorems have been proved under various hypothesis by NARDINI [11], FERRARI [3], and KANWAL [9]. The methods and results of Kanwal are largely analogous to those given by Serrin in the non-magnetic case. In Chapter 2 we give an expository presentation of VALLI's proof of uniqueness for viscous non-magnetic fluids. In chapter 3 we present SERRIN's theorem for non-viscous non-magnetic fluids. In the last Chapter we extend this analysis to magnetohydrodynamic flows, following mainly the works of Kanwal. Throughout this thesis we allow for the possibility that the viscous coefficients ft and A, the thermal conductivity K, and the coefficient of specific heat at constant volume Cv, are not constants, merely assuming that they are non negative functions of the density p, the absolute temperature 6, and the velocity field v. The region occupied by fluid is assumed to be a three dimensional bounded domain 1 Chapter!. INTRODUCTION 2 fi = fi(t). We suppose that the boundary of fi is sufficiently regular to allow use of the divergence theorem and transport equation. 1.1 VISCOUS COMPRESSIBLE FLUIDS The motion of a fluid is governed by the following standard equations of fluid dynamics. Let v, P, p, b be the velocity, pressure, density and external force respectively. Let D be the deformation tensor Dij — l(t>,j + Vjti), and £ the bulk viscosity ( = 3 * | 2 / i . The stress tensor T is introduced below. We denote specific internal energy by E, the heat flux vector by q, the temperature by 6, and the heat supply per unit mass per unit time by r. The coefficient of specific heat at constant volume is c„ = Below, Dt = gf^ -, v< = ^ r , Pg = etc.. The fundamental governing equations are the following. Equation of Motion: 'D\ Di - b = div T , T = - P I + \divv I + 2uD or or 5v at + (v • V)v - b = — V P + V • (2/iD) + div(\div\l) ^ + (v • V)v - b Equation of continuity: Energy Equation: DE 3 E = - V P + £[2? t ( / i l>*v) + Dk(»Vvk)) + V[(C - ^)divv ]. (1.1) g + p ^ v = 0. p— = T : D - div q + pr, q = -KV6, E = CV9 (1.2) or or C V P dt = -Pdiv\ + V • (kV0) + pr + [Xdivvl + 2/iD] : D _ + v . V * (1.3) Chapter 1. INTRODUCTION 3 In these equations the symbols have their usual meanings, in particular, (divT), = ]T Q^-Tji , I = Identity matrix, T : D = J^Ttf A>, and the material derivative ~ is a differential operator defined by _ = - + rod. The reader unfamiliar with equations (1.1)-(1.3) may find a complete derivation and explanation in the memoir of TRUESDELL[16] or in SERRIN[14]. To equations (1.1)-(1.3) must be added certain equations of state relating the thermodynamic variables p, P, E and 9. For this purpose we shall take P = P(M), E = E(p,B) as the defining relations. Here P(p, 6) and E(p, 9) are given twice differentiable functions ofthe variables p and 9, whose explicit form depends of course on the particular fluid in question. We suppose that 0 < Ci < Cv(p,9) < C7. Moreover the solution is assumed to be such that o<pi<p(t,x) V x G n(«), vt e [O,T]. In posing an initial boundary value problem, we always prescribe initial values for the velocity, density and temperature. Initial conditions: Plt=o = Po, v| t = 0 = v0, ^|t=0 = ^ 0-As mentioned before, our main objective is to find appropriate boundary conditions. Let n denote the outward unit normal to fi, and G the outward normal velocity of dQ. Then U=vn-G is the normal speed of the particles at the boundary relative to the boundary. In the various theorems of Chapter 2.1 we will consider the possibility of prescribing various combination of the following Boundary Conditions. Boundary conditions: At all points of the boundary dQ, and at all instants of time, the velocity vector v is given. Furthermore, Chapter 1. INTRODVCTION 4 (a) At all points where U < 0, the absolute temperature and the density are prescribed. (b) At all points where U > 0, the absolute temperature is given. (c) At all points where (7 = 0, one of the following conditions is assumed: (cj) The temperature is prescribed. (C2) The heat flux q • n is prescribed. (03) The heat flux is assumed to be proportional to the difference between the fluid temperature 8 and a given wall temperature t90, that is, q n = k(6-e0), * > 0. 1.2 INVISCID COMPRESSIBLE FLUIDS The first law of thermodynamics DQ = DE + PD(^) can be expressed with the help of the equation of state P = pRB as DQ = DE + pR6D{-), P where Q denotes the amount of heat and R the gas constant. Let S denote the specific entropy DS = 2f-\rcver,ibu- Then we obtain DS = ~ - Dp. (1.4) 0 p Assuming that the fluid is devoid of heat conduction as well as viscosity, the thermodynamic equation of state is more conveniently assumed to be given in the form P = P(p, 5), where P(p, S) is a continuously differentiable function satisfying dp) s The Energy equation (1.3) becomes p—— = —Pdivv. H dt Using the equation (1.4) we can express the above equation as H Dt p Dt With the help of the continuity equation (1.2), the above equation can be written as Chapter 1. INTRODUCTION 5 Then the reduced system of governing equations for this case take the form: D v p^^pb-gradP, (1.5) ^ + ^ v = 0 , (1.6) Dt ^ = 0. (1.7) The initial and boundary conditions to be considered are the following: Initial conditions: p|t=o = Po, v | t = o = v 0 , S\i=o — So-Boundary Conditions: At all points of the boundary dQ, and at all instants of time, the normal velocity component v • n is given. Furthermore, at all points where U < 0, the velocity, density, and entropy are assigned. 1.3 INVISCID H E A T C O N D U C T I V E COMPRESSIBLE FLUIDS The differential equations for inviscid heat conductive fluids are obtained by specializing (1.1)-(1.3) to the forms p-^T - ph- gradP, (1.8) Dt ^+pdivv = 0, (1.9) DE — — P div v — divq + pr. (110) As in the treatment of viscous fluids, we shall consider only the equations of state P = P{p,6) E = CV6, where Cv = Cv(p,B) is a positive and continuously differentiable function. It will also be assumed that the pressure-density-temperature relation is twice differentiable and satisfies the inequality ( d p ) s > > °" 5 / S The initial and boundary conditions to be considered are the following: Initial conditions: p|t=o = Po, v| 1 =o = vo, |^t=0 = 0^-Chapter 1. INTRODUCTION 6 Boundary conditions: At all points of the boundary dQ, and at all instants of time, the normal velocity component v • n is given. Furthermore (a) At all points where U < 0, the velocity, density, and tempreature, are assigned. (b) At all points where U > 0, the temperature is given. (c) At all points where [7 = 0, one of the following conditions is assumed: (ci) The teperature is prescribed. (02) The heat flux q n is prescribed. (03) The heat flux is assumed to be proportional to the difference between the fluid temperature 8 and a given wall temperature 9o, that is, q-n=fc(0-0 o), k>0. 1.4 VISCOUS COMPRESSIBLE M H D The equations of magnetohydrodynamics, in the absence of Maxwell's displacement currents, are the following; see [1]: cur/H = 47rj, divH = 0, (1.11) ATI C U r / E = -A i . (^ ) , (1.12) j = «T(E + / i e vxH) , (1.13) ^ + pdivx = 0, (1.14) D\ p— = pb + divT + ne5 x H, T = —PI + Xdiv vl + 2p.D, (1.15) where H stands for the magnetic field, j the current density, E the electric intensity, v the velocity vector, T the stress tensor, p the density, P the pressure, and b the applied force. The quantities /i« and cr are the coefficients of magnetic permiability and electrical conductivity, respectively. Again, D is the deformation tensor, I is the identity matrix and the quantities A and ft are the coefficients of viscosity. In a compressible fluid, the equations (1.11)-(1.15) have to be supplemented by the equation of energy. If E is the specific internal energy of a fluid element, q is the heat flux vector, and r is the heat supply per unit mass per unit time, then the required equation is DE — = T : D - divq + pr, q=-ngrad8, (1.16) Chapter 1. INTRODUCTION 7 where 6 stands for the temperature and K is the thermal conductivity. Here it is assumed that the heating effect due to the flow of currents is negligible compared to that of heat conduction and viscosity, and it has therefore been ignored. In addition the equation of state relating the thermodynamic variables p, P, E, and 6 is assumed to be of the form: p = p(P,ey, E = E{P,$). The functions P and E are assumed to be twice differentiable. We also assume E = Cv0 and C„ > 0. The quantity j x H entering the equation (1.15) can be split into two parts: j x H = -grad(-) + V ; = -grad(-) + dtv-—. (1.17) If is uniform, the equations (1.11) and (1.13) imply curia = 4*<7(E + >iev x H). Taking the cross product of V with this equation and using (1.12), we obtain V x (V x H) = 4ir/ie<r[V x (v x H) - (1.18) Since divH = 0, the vector identities V x V x H = VV • H - V • V H and V x (v x H) = H • Vv - HV • v - v • V H + W • H reduce to V x V x H = A H and V x (v x H) = H • gradv - Hdivv - v • gradH. With the help of these identities the equation (1.18) can be expressed as ^5 - -Hdivv+ H-gradv+ r)AH, (1.19) dt where rj = (47r/!er7)-1. In what follows we use only the equations (1.17) and (1.19) for H. The other electromagnetic variables become auxiliary quantities, whose values can be obtained from equations (1.11) and (1.13). Chapter 1. INTRODUCTION 8 The constant fj is called the magnetic diffusivity; since p e > 0 and a > 0, we have fj > 0. Here we will consider the case of a finite conductivity o ^ oo, so that fj is a positive constant. Furthermore, it will be assumed that p is positive in the closure of fi. The Governing equations are DH ^ =-Hdu>v + H ffradv-r rjAH, (1.20) Dt d«vH = 0, (1.21) § + ^  = 0, (1.22) />-r— = pb 4- difT — grad—— + div ——, (1.23) Dt 87T 47T p ^ = T : D - divq + pr. (1.24) Initial conditions are given as follows: Initial conditions: p|t=o = Po, v| t =o = vo, 0\t=o = 6o, H| < =o = Hrj. In Chapter 4.1, we will consider the possibility of prescribing various combinations of the following boundary conditions: Boundary conditions: At all points ofthe boundary dQ, and at all instants of time, the velocity v and the magnetic field H are given. Furthermore, (a) At all points where U < 0, the absolute temperature and the density are prescribed. (b) At all points where U > 0, the absolute temperature is given. (c) At all points where U — 0, one of the following conditions is assumed: (cj) The teperature is prescribed. ( C 2 ) The heat flux q n is prescribed. (c3) The heat flux is assumed to be proportional to the difference between the fluid temperature 6 and a given wall temperature 60, that is, q-n = *(0-0o), * > 0. 1.5 INVISCID COMPRESSIBLE M H D Assuming that the fluid is devoid of heat conduction as well as viscosity, the governing equations (1.20)-(1.24) reduce to the form: ^5- = -Hdiwv + H • gradv + r)AH, (1.25) Dt Chapter J. INTRODUCTION 9 divH = 0, (1.26) j£-rPdivv = 0, (1.27) p-RT7 = Ph-9radP-grad— + div — , (1.28) Dt off 4 t f = »• <»•») The initial and boundary conditions to be considered are the following: Initial conditions: p|t=o = Po> v | < = 0 = v 0 , S\i=o = So, H|t=o = Ho-Boundary conditions: At all points of the boundary dQ, and at all instants of time, the normal velocity component v n and the magnetic field H are given. Futhermore, at all points where U < 0, the velocity, density and entropy are given. 1.6 INVISCID H E A T C O N D U C T I V E COMPRESSIBLE M H D The differential equations for inviscid heat conductive compressible magnetohydrodynamics are obtained by reduceing (1.20)-(1.24) to the form: =±L = -Hdivv + H • gradv + rjAH, (1.30) divH = 0, (1.31) ?l + pdivv = 0, (1.32) p—— = pb — gradP — grad— h div——, (1.33) y Dt H y " 8tt 4tt v ' £)£• p-^- = —Pdivv — divq + pr. (1.34) The initial and boundary conditions to be considered are the following: Initial conditions: p\t=o = po, v|,=o = v 0 , 0|«=o = 0o, H| 1 =o = H 0 . Boundary conditions: At all points of the boundary dQ, and at all instants of time, the normal velocity component v • n and the magnetic field H are given. Furthermore, (a) At all points where U < 0, the velocity, density and temperature are given. (b) At all points where U > 0, the temperature is given. Chapter I. INTRODUCTION 10 (c) At all point* where U = 0, one of the following conditions is assumed: (ci) The temperature is prescribed. (C2) The heat flux q n is prescribed. (C3) The heat flux is assumed to be proportional to the difference between the fluid temperature 9 and a given wall temperature 90, that is, q • n = k(9 - 90), k>0. Chapter 2 UNIQUENESS OF VISCOUS COMPRESSIBLE FLUID MOTIONS 2.1 AUXILIARY EQUATIONS Let (p,f?,v) and (p, 6,x) be two solutions of (1 -l)-( 1.3) and set n = p - p, y = 6 — 6, u = v -Below, a tilde over P, Cv, fi, C or K means that these functions are evaluated for the second solution 6, v. Since both flows satisfy the equation of motion, we obtain by subtraction: pv, - pv, + p(v • V)v - p(y • Vv) - ph + ph + (P,Vp + PeV6) - (PpVp + 3 3 3 3 + P*V0) - Dk(^il) + Dk(uVvk) - Y, Dk{-uDkv) + £ Di(pDtv) -1 = 1 Jt = l 1 = 1 1 = 1 ~ V [ ( C - | ^ ) V-v] + V [ ( C - | / i ) V-v] = 0, pv, - pv, + pv, - pv, -f- (p(v • V)v - p(v • V)v) + (p(v • V)v - p(v • V)v) + + (p(v • V)v - p(v • V)v) - b[p - p] + P„Vp - P pVp + P,Vp - P„Vp + P* V0 -3 3 3 - P,VB - PeV9 -^2Dk(pVvk) + ^ Dk(pVvk) - Y Dk(i*Vvk) + 1=1 1=1 1=1 3 3 3 3 3 + £Dk{ftVvk) - Dk(-uDkv) + ^Dk(p\Dkv) - £Dk(jiDkv) + 53Dk(uDkv) -i=i i=i 1=1 1=1 i=i - V[(C - | / i)V • v] + V[«~ - |/i)V • v] - V[(< - |/2)V • v] + V[(C - | / i)V • v] = 0, pu, + nv, + p(u • V)v + p(v • V)u + n(v • V)v - br/ + P pVn + (Pp - Pp)Vp + 3 3 3 + P,(VT) + (Pt - P«)V0 - 53 A( / iV«*) - J3z?t(/J - /i)Vi>* - 53 WDku) -1=1 1=1 1=1 - 53 Di (p - p)D tv - V[C - | / i ) V • u] - V[(C - IA - C + • v] = 0, 1=1 p[u, + (v • V)u + u • Vv] + ?,Vt) + (Pp - P^Vp + P ( V 7 + (P, - P,)V« + 3 3 3 4- n[v1 + v V v - b ] - 5 3 D l ( p V u l ) - 5 3 D i ( ( / i - / i ) V v i ) - 5 3 D t ( / i D i u ) -i=i i=i i=i 11 Chapter 2. UNIQUENESS OF VISCOUS COMPRESSIBLE FLUID MOTIONS 12 3 - ^ A ( p - ^ ) Z ? I V - V [ ( C - | / i ) V u ] - V [ ( C - | / i - C + ^ ) V - v ] = 0. (2.1) Also, since both flows satisfy the equation of continuity, we have by subtraction: - -j-) + (v • Vp - v • Vp) + (v • Vp - v • Vp) + (pdivv — pdivv) + (pdivv — pdivv) = 0, di dt ^7 + v • Vn + u • Vp + rjdivv + pdivu = 0. (2.2) at Turning next to the Energy equation, we have by subtraction: (ptf, - p0«) + (p0, - p8t) + (pv • ve - pv • ve) + (pv • ve - pv • ve) + (pv • V A -0 0 0 - 0 - pv • V0) + (—Pedivv - -r-Pedivv) + (^-P6divv - -^-Pgdivv) + Ct, Cv Cv 6 - 6 6 6 1 1 + (—Pedivv - —Pedivv) + (—Pgdivv - — Pgdivv) - (-^-rp - -^rp) -3 3 - (-Lrp- -^ rp) - (-^^DJkDj) - i £ ^ ( k D ^ ) ) -3 3 3 - ( J - E ^ ( * ^ * ) - p r E - (?r E ^ ( « ^ 0 ) -3 3 3 iL/^ i,k=i i,*=i Z C v i,t=i 3 - ^ E ( 0 * « ' + A V ) V ( ; t ( < - 5 * ) ( ^ " ijtTi t'" - (4-(C ~ |A)(^v)2 - i-(C - |p-)(*vv)2) - (i-(C - !/0(<fivv)> -- ^(C-|/i)(^v)2) = 0, 7 - 1 1 -P7t + "0t + Pu • V0 + pv • V 7 + nv • V0+ -J-Pedivv + 0(- — )Pedivv + Cv Cv t^ ti 3 + - P,)<K»v + -^(ditm) - -L, - (4- - i - ) r p - 4- £ Dk(kDky) -1 3 1 1 3 - 3 - iry£Dk(k-K)Dke-(7r--L),£Dk(KDke)-Jir + A S * + c " *=i G " u " t=i Z O w i,*=i + Dtv< + A v l ) . ( A u ' ' + Dtuk) - E + Divk? ~ f (7T " 2CV ^ C„ Chapter 2. UNIQUENESS OF VISCOUS COMPRESSIBLE FLUID MOTIONS 13 3 u i,k=i C" 6 - i ( C - ^ - < + f ' / i ) ( ^ ^ (2-3) Transport equation: jtJpFdx = J2i(pF)dx + j>pFGds, (2.4) where G is the outward normal speed of the boundary of fi, and F(x, y, z, t) is a continuously differen-tiable function defined for (x,y,z) € fi('). f° r * ' n s o m e timt interval. Below, we use the notation (f.g)= / f-gdx. Jn From the transport equation (2.4) and the divergence theorem, one obtains g(pu,u) = j J ^ d x + f^pJGds - J [^ -(P"2) + div(pu2\)]dx + j> pu2(G - v • n)ds = / P[(w2)« + v • ^(u2)] + u2(P< + pV • v + v • Vp)dx - * pu2Uds, Jn Jan where U = (v • n — G). Using equation (1.2) we have / u2(p, + pV • v + v • Vp)dx = 0. Jn Thus i.e. -^(pu, u) = / p[2u • u, + 2[(v • V)u] • u]dx - / pu2Uds U l Jn Jan = 2(pu,,u) + 2(p(v V)u,u)- * pu2Uds, Jan I^(pu,u)+ ^^Pu2Uds = (put,u) + (p(v-V)u,u). (2.5) Chapter 2. UNIQUENESS OF VISCOUS COMPRESSIBLE FLUID MOTIONS 14 2.2 SERRIN'S F U N D A M E N T A L UNIQUENESS T H E O R E M FOR VISCOUS FLUIDS We now state the uniqueness theorem for viscous compressible fluids under the initial and boundary conditions of §1.1. The solutions under consideration are assumed to be sufficiently smooth that all the norms used in estimating them are finite. Theorem 2.1 Let the coefficients of viscosity and heat conduction satisfy V- > £i > 0, C > 0, K > ^ > 0. Then there can be at most one solution of the equations (1.1), (1.2) and (1.8) satisfying the following boundary conditions: (a) At all points of dQ, v ts given. (b) At all points where U < 0, p and 6 are given. (c) Ai all points where U > 0, 6 is given. Proof. Taking the scalar product in L2(Q) of (2.1) with u, and using (2.5), one obtains (pu.u) + J I pu2Uds + (p(uV)v,u) + (PfiVritu)+((Pp-Pp)Vp,u) + (PsV-f,vi) + * Jen ID_ 2Dt iJa+ ((Pg -P«)V0 ,u) + (r;vt,u) + (r,(vV)v,u)-(7 ?b,u)-3 3 3 - (£ Dk(pDku), u) - ( £ Dk([Jl,- p)Dkv), u) - . ( £ Dk(pVul),u) -i = l t=l i = l - (T Dk[(~u - M)V A u) - (V[(C - U)V • u], u) - (V([C - c -- |(p-p)]d^v),u) = 0. (2.6) The next step is to estimate the various terms which appear on the righthand side of equation (2.6). For this purpose it is convenient to restrict our attention to some fixed time interval 0 < t < T. Let C be the generic notation for an upper bound; the number C will thus not be the same in each estimate, though it would be possible to determine its size at any stage. Below, we set ||/||x,tP = L, where L is the Lipschitz constant for f over Q, and 3 Q~ ||£>v(t)||oo H sup max 53 | ^ - | . o i f axi Also |v||oo = sup max |t>,| n • Chapter 2. UNIQUENESS OF VISCOUS COMPRESSIBLE FLUID MOTIONS 15 First, |(p(u-V)v,u)|<||Vv(0 | |oo / pJdx. Jn Next, |((P, - PP)Vp,u)\ < || VHIocll^lU.P / M(l i? I + I 7 \)dx Jn < C\\Vp\\J\Pp\\Lip f (u2 + r,2 + y2)dx Jn < CHVplloollPplli,-,- / (pu2 + pr,2 + p72)dx P Jn < C\\Vp\U\Pp\\Lip-. /(pu2-r-pn2 + pT 2)dx Pi Jn = C || Vp Hood Pp \\Lip [(pu, u) + (pr,, r,) + (p7,7)j-Here the basic tools aTe the inequality 2ab < a2 + b2 , and the assumption that o < pi <p(i,x), Vx e vi e [o,r]. Next, I {{P, - Pe)V9, u)| < || P, \\LiP\\ Vff IU / (I i, I I u | + | 7 | | u |)(fx < C\\P, \\uP\\ V 0 Hoc [(pu, u) + (pr,, r,) + (p7,7)]. Next, K ^ . u ) ! < Hv.lleo / I r? I | u | dx Jn < C || V, ||oo [(PU.U)+(/)!»; I?)]. Next, | (u(v.V)v ,u) | < IIVvlloollvHoo / h i | U I dx Jn < C,||Vv||00||v||00[(pu,u)+(pr,,r,)]. Next, |(nb,u)| < Hblloe / M|u|dx Jn < C || b ||oo [(pn,r,)-r(pu,u)]. Next, - j PpVrjudx = - <£ PPm nds+ I (VP,) ur/dx + / P,r/V udx. Jn Jen Jn Jn Chapter 2. UNIQUENESS OF VISCOUS COMPRESSIBLE FLUID MOTIONS 16 Taking the boundary conditions into consideration, we get - / P„Vn udx = I PppVpur)dx + f PpeV§ • undx + f P ^ V udx Jn Jn Jn Jn < C || P„ lull Vp | U / (pu2 + pr,*)dx + C || P„ Hooll Vt? ||e • ^ (pu2 + pr,7)dx + C || Pp\\l j pr,* dx + |(dnm, divu) < C[\\ PPP Hooll Vp lU + || ^ HoollVfflloo + HPpHL] • • [(pu, u) + (pj, J) + (pr), rf)] + |(dttm, dttm). Next, - / P«V7.udx = / (VP s)-7udx+ / P$ydivudx- 6 Pgyuds. Jn Jn Jn Jdn Since the boundary term vanishes in view of the boundary conditions, we obtain - / PeVy.udx = / PpjVp •U7a?x+ / P,eV6 -uydx + / Pgydivudx Jn Jn Jn Jn < q i ^ i u i i v p i u / ( p U 2 + p72)dx + Jn + C || Pgg Hooll V0 Hoo / (P72 + P« 2) dx + Jn + C\\Pg\\lJj<py7)dx+C-{divu, divu) < C[||P„||oo||Vfl|oo + llA.HooHVtflloo + HAUL] • • [(pu, u) + (p7,7) + {pr), rf)] -f ^(divu, divu)]. Next, Y] / Dk{fiDku)udx = J2 f ftDkUitiiUkds - / fiDkUiDkUidx < -^(Vu.vu) , where again the boundary term vanishes. Next 3 , 3 \Y\ I Dk[(ji-n)Dkv]udx\ = IV, <f & - n)DkViUinkds - V / (fi - ii)DkvxDkxiidx\, where also the boundary term vanishes. Thus 3 Chapter 2. UNIQUENESS OF VISCOUS COMPRESSIBLE FLUID MOTIONS 17 < CH/ilLupllDvlU 53 / (|D tu| |u| + |£>fcu| \t>\ + + \Dku\\y\)dx Next, 3 < i 3 < < C\\n\\l,p\\Vv\\l j{pv? W +py2)dx + £(Vu, Vu). 3 3 3 £ / A^Vu' judx = £ / (JDl/i)(Vul)udx + 5 ; / pDk(Vuk)udx t^J" tTiJn t?iJn 3 3 53 / \Dkp\\Vul\\u\dx + Y^ ( pV{Dkv.k)udx 3 r 3 r 5 3 / Pt/i| |Vu l | |u|dx + 53 f pDkukUinids-H[Jn t=iJ™ 3 r 3 r - 5 / {Vp)Dkukudx - 53 / frDkv.kV udx. Again, since the boundary term vanishes, we get 3 3 3 f Dk(pVuk)udx < C Y ] / |D tA| 2 |u| 2dx+ / | V u f d x - f (Vp)divu udx-— I pdivu divudx Jn < C / | V / i | 2 | u | 2 c f x + £ ( V u , V u ) + | /(ilpVp + ileVe + Jn 6 Jn + p\\V\)divu udx\ — (jldivu, divu) < C ((|pp|2|Vp|2+|AS|2|V/i|2+|Av|2|Vv|2)|U|2dx-r Jn + £(Vu, Vu) + C(\\pfi\\WVp\\l + + Further, 3 + I IMVI IL I IVVH^) / pv?dx + e(divu, divu) - {pdivu, divu) Jn < O(HA^II5O I I^I5O + HA»H2oll^»ll2o -i- Hwll2oH^M5o) / />«2 Jn + -(Vu, Vu) + - ( V • u, V • u) - (fidivu, divu). 6 2 dvk , , A f , d v k dui ^JnDk[(p-p)Vvk]udx\ = l E / j A - M ) ^ ^ Taking the boundary term into consideration, we get 3 , 3 k = lJU i,k = lJ" Chapter 2. UNIQUENESS OF VISCOUS COMPRESSIBLE FLUID MOTIONS < q M M | V v | | r o £ / I^-KIUI+M + ITDA < C|MlLpH V v l l~ / (pu7 + pr,7 + P77)dx + ./n + |(Vu,Vu). Moreover, ^ V [ ( C - ^ ) V u ] u r f x = j f ^ C - I ^ V - u u i n i d s - ^ C - l / i K V - u j I ^ A c . Since the boundary term vanishes, we obtain / V[(C-|A)V-u]-udx = - / C(V.u)2dx+^ / M V u ) 2 d x Jn * Jn <> ./n < | ( A V - u , V - u ) - C i ( V . u , V - u ) . Turning finally to the last term of (2.6), we obtain integrating by parts / [ V [ C - | / i - C + ! / ' ] V - v ] - u d x = j> [C-_|/i_C + |^V . v « I - i . ( r f*-y [ C - | l u - C -2 + r/i]V • vV • udx. 0 Taking the boundary term into consideration and estimating the righthand side, we get ^ [ V [ C - ^ - C + ^ ] V v ] u d x = jf((-(*)(Vv)(Vu)rfx + + | j f ( £ - A « ) ( V - v ) ( V - u ) d x < CHCllLpllVvH^ / (pu7 + pr,7 + p72)dx + + £(divu, «fi«u) + cii^HLpiivviiL Jn(Pu 2 + + prr2 + py7)dx + |(dit)u, divu) = CaiCHLoll Vv||l , + II^HLpl|Vv||L)((pu, u) + + (P7, 7) + (M> V)) + ^(dirm, divu). When these estimates are inserted into (2.6) there results lR.{pu^) + \j>^pu7Uds < - ^ 1 ( V u ) V u ) + £ ( V u , V u ) + C ( p X l | V p | | 2 0 + i an + l l ^ l ! L l | V ^ + ll/iv||20O||Vv||20O)(pu>u) + £(Vu,Vu) + Chapter 2. UNIQUENESS OF VISCOUS COMPRESSIBLE FLUID MOTIONS 19 + §(V u,V -u) - (pdivu,divu) + §(Vu, Vu) + + |(jiV • u, V • u) + CdlCllLpllVvH^ + H/illi^ llVvll^ ) • • ((pu, u) + (py, T) + (PI, l)) + |(dt'vu, divu) + + C[\\P,\\l + H^ lloollV.plloo + HPp.lloollVfflloo + H A U L + + I IPMIIOO I I^HOO + H l^lccllVplloolKpu, u) + (pr,, IJ) + + (P7.7)] + HVvllooC^ i.u) + C||PP|Ut>||Vp||00[(/HiJu) + + (/>»),»?) + (P7.7)] " C i + C||P#||L.>||V0||oo • • [(pu, u) + (pr,, n) + (P7,7)] + (||v,||oo + ||Vv||oo||v||oo + + HblUJKpu, u) + (pr,, r,)) + C\\C\\hp\\V • v\\l({pu,u) + (pr,, n) + + (py,7)) + |(Vu,Vu). Thus ~(pu,») + 5 jf P u ^ d s < Ci(0[(A«i,») + (W.»?) + (7,7)] " Ai(Vu, Vu) -- ipx(V • u, V • u) - Ci(V • u, V • u) + |(V • u, V • u) + + £(Vu,Vu), (2.7) where Ci(t) = CWJUVplll + \\pe\ll\MWl + H/ivllLllVvIlL + HCllLpllVvIlL + + IHhpW^Wl + \\Pp\\l + HP^ IIcoHVplloo + HP^ IIoollVfllloo + \\P$\\lo + + ||PM|U||Vff||eo + HP^ IIeollVplU + HVvlleo + \\Pphip II Vp||cc + + HP.IU.-pllVtflloe + llv.Hoo + IIVvllooHvlleo + ||b|U + ||C||L>I|V • v||2TO]. Taking the scalar product in L2(Q) of (2.2) with n, we obtain (pr,t, n) + (pv • Vr?, n) + (pu • Vp, n) + (pndt'vv, n) + (p7divu, n) = 0. We obtain the following estimate for n in the same way as we obtained the equation (2.5) for u. We get Inserting this into the preceding equation, we get \TZ(P^TI) + \ / pr,7Uds-r(pu-Vp,ri) + (pr)divv,r,) + (p2divu,r,) = 0. (2.8) 2 Dt 2 JM Chapter 2. UNIQUENESS OF VISCOUS COMPRESSIBLE FLUID MOTIONS 20 Estimating the terms of (2.8) in the usual way, we get the inequality 52£(/"».»») < -l£npT]2Uds + C 'I V ' P 1'°° Jn(P*2 + P<l')d*+ II divv\\eoJpntdx + + C\\p\\l JnPr,7dx+£-(divu,divu) = C[\\Vp\\M + \\divy\\00 + \\p\\l)[(pn,u) + {pr}lrl))- \ I Pn7Uds + 1 Jan + § ( d i v u , d i v u ) . 6 Thus IJfiiPl* f ) ^ C 2 (0[(PU- u ) + (pi?, ij)] + | ( d « v u , d i v u ) - i jf p^C/ds, (2.9) where CaWsClHVplloo + H d.vv HOC + IIPHL]. Turning next to the equation (2.3), and taking its scalar product in L7(Q) with y, and using the version of (2.5) corresponding to 7, one obtains 5-K7(P7.7) + 5 / py7Uds+{pu-V§,y) + (ti§uy) + (tjv • V0,7) + (4-7^divv, 7)+ + ( ( " i )0P*divv, 7) + ( i (P, - Pedivv, 7) + ( £ P«divu, 7) -3 3 " (7T X>*(*£ >*7)>7) " <7T " * W ] , 7 ) -^» t=i ^ i=i 3 3 - ( ( ^ - 7 ^ ) E ^ ( « 0 * * ) , 7 ) M ; ^ £ (Ab€"> A C * + A « f c ) -• (A^*' + A « * ) , 7 ) " + A « * ) S , 7 ) " 3 - - 7 ^ ) E (£>*"'' + A « * ) 2 , T ) - ( i ( C - h)(divy + d.vv)d.vu,T) -- ( i ( C - \v - C + \n){divv)\ y) - (4- - i-)(C - fp.)(d.vv)2,7) -- ( ^ , 7 ) - ( ( i - ^ ) r p , 7 ) = 0 . (2.10) Now we have the following estimates valid for 0 < t < T: First, / pu • V6ydx < \\V0\\oo I {pu7 + py7)dx. Jn Jn Chapter 2. UNIQUENESS OF VISCOUS COMPRESSIBLE FLUID MOTIONS 21 Next, Next, Next, Next, Next, Next, / ijtvr<fc< Halloo / (pJ + P72)dx. Jn Jn f rrv • Vdydx < HvlUIIV^IU / W + P72)d*-Jn Jn f ±i*P,divvdx < ||P#||eo||d»HI~ / py7dx. Jn Cv Jn f(i-~ 7r)9P,div*i4x < CIICIIiolltflloollAlUllrfiuvlloe / {P77 W ) * Jn Cv i^v Jn j ^{Pe-P*)divv7dx<\\6\\x\\Pe\\Lip\\div^\x f (py2 + pr)7)dx. Jn Jn f ^rPedivuydx<\\e\\l\\Pe\\l [ py7dx + £-{divu,divu). Jn Jn £ Next, 3 3 J2 / ( ± y ) D k { k D k y ) d x = T, <f j-{kDky)nkds - I'{kDky)Dk{^)dx j~i Jn Cv Jen Cv Jn Cv = f{kDky)Dk(^r)dx+ I £ & d . £rj Jn Cv Jan Cv on = " E / -^{Dky)k{Dky)dx + J2 I l~(DkCv)k{Dky)dx + Jan Cv on = - ^ ( V T , V 7 ) + X : / (fy{CVtP)(Vp)k{Dky)dx + + E / ^z{Cv,t)(V9)k(DkJ)dx+i JTj Jn C-f Jen C„ on < - g ( V 7 , v T ) + [^||c„,,||LI|vtf||20||*||20 + Jon Cv °n = - g r ( ^ 7 , v 7 ) + [^llc.,,||Lllvfl||Ll|s|lL + Chapter 2. UNIQUENESS OF VISCOUS COMPRESSIBLE FLUID MOTIONS 22 + ^IICvJ|Ll|Vp1|20oPllco][(PT,7) + (pu1u)-r(pr,,n)] + 2 Jan C„ Next, 3 f 7TDk[(k-K)Dke)dx = ^l(*-*)(Dke)nk]ds-Y; [ (k-K)(Dk0) Next, Dk(Z-)dx = ~ E I(Z-K)(Dk9)Dk(-2r)dx+ I Uk-n)^-yds ^Jn Cv J&nCv on 3 3 = " E [{*-K)Dk8Dky±-dx + T I(k-K)(Dke)y i=i - ' n C v " in 3 3 = - E / (* - *) w ) ( D t T ) 4 - d x + E / ( * - « ) w f r • t = 1 i n C u k=lJn 3 • ( i ) ( C „ , p ) V p d x + 2 / ( k - K ) ( D k 0 ) y ( ± ) . 1 = 1 ^ n • (CVte)V8dx+ <( ^-(k-K)^-yds Jan Cv on < CIWIlLpllV^Jloo2 / (P7 2 + pr? + Pu2)dx + Jn + e(Vy, Vy) + / ±(k - K)^-yds + C||«|U i p||V0|| o o||C„, p||c Jan C„ on • HVplloo / (pu2 + pr,2 + py2)dx + C\\K\\LIP • Jn • ||V0||o o||C„1,|U||Vfl||o o / (pu2+pr,2 + pT2)<ix Jn = C\\K\\lip\\W\\l[(ry, y) + (pu, u) + (pr,,,)] + + C||K|U,>||V0||Oo(||a i,||oo||V^|| o o + ||C,Bl#||oo||Vff||00) • • [(py, y) + (P«- u) + (pr,, r,)] + |(V7, v7) + + 4 -r - ( K - K)^-yds. JenCv 'dn 3 3 ^ I -±-{Cv - Cv)Dk{KDk9)ydx = ^ I 7L-(C«-Cv){DkK){Dke)ydx + ^_jin Gt,Ot; i=i ^ OJ,OD Chapter 2. UNIQUENESS OF VISCOUS COMPRESSIBLE FLUID MOTIONS 2 + 3 , , + E / TrrrVv - CMDIW* < (c\\cV\\L,P\\Kfi\u\Vp\\x\\ve\\x) / (Pr Jn + /»lja)dx+C||C„||Li.||«#||oo||Vtf||00a. • / (p7a+w2)«/x + C||Cv||Wp||icv||oo||Vv|| Jn • HVtflleo) / (P72 + pr,2)dx + CliailiipH'cllc Jn • \\DHU f (pu2 + py2 + pr,2)dx Jn = C||C„||L.>(IMU|0j0||« + IkplloollVplU • | |V0| | o o -f | |^ | | c o | |V0| | o o 2 + | K | | T O | | V v | | o o • IIV l^oo) f (pu2 + py2 + pr,2)dx. Jn Next, 3 ^2 I -£ - (AbC + AC* + A V + A«* ) (A t i ' + Diuk)ydx 3 < Cll^ lloollVvHoo V (Dlui + Diuh)ydx + i J ^ i J n 3 r + CH/illooll-DvlIco (DkU< + DiUl)ydx < Cm\l\\Vv\\lJ^Py2dx+ | (Vu,Vu) + + q i / i l l L l l V v l ^ ^ ^ d x + ^ V u . V u ) = CHAill^dlVvIlL + flVvH^) / (py2 + pu2 + pr,2)dx + Jn + | ( V u , V u ) . Next, 3 T [ ^(Dkv'-rDiV^ydxKCMu^DyllJ f (pu2 + pr,2 + py2)d> ik=\Jn Jn Next, 3 E / f ( ^ K ^ - C . ) ( A t ' , ' + A^)Vx<C | | / i | | 0 0 | |C < ; |U i p | |Dv | | 0 0 2 / (py2 + pr,2)d> ik~ZiJci CVCV Jn Chapter 2. UNIQUENESS OF VISCOUS COMPRESSIBLE FLUID MOTIONS 24 Next, I -rr-(C — \p)(div\ + divv)(divu)ydx = / -^-£(div\ + divv)(divu)ydx — Jn Cv 3 Jn cv - f — • < C\\C\\l\\Vv\\l I py2dx+£-(divu,divu) + -p(dtvv + divv)(divu)ydx J> Jn + Cg\\l\\Vv\\lJnP72dx+£-(divu,dwu)-r + C||ji||a00||Vv||>e jf py2dx + '-(divu, divu) + + C||p||Ll|Vv||»0 Jnpy2dx+c-(divu,divu) = caiaiL + IIAIlLJOIVvii^ + iivviiL)-(py7 + pn2 + pu7)dx + {j(d»'vu, divu). Next / jr{C-h-C + lri)(divy)2ydx < f i ( ( - ( ) ( ^ v ) V x + Jn Cv »> J Jn Cv 2 / 1 + o / _ /i)(dt't/v)27orx « in C v < q|ClUip||Vv||L / (p72 + p«i2-rpr,2)dx in + CIMMIVvH*, / (py7 + pu2 + pr,7)dx Jn = q|Cv|u<P||Vv||L(||c||L,P + |Mkip) / (PT: in + pu2 + pn2)dx. 2 + Next, / (4- - i )(C - |p)(d.vv)2Tdx = / ~ r ( C „ - C v)C(d«rv) 2 Tdx -in C„ Ct o in CvCv - / 7 J_(Cv-Cv)|/i(d.t;v) 2 7 dx in CVCV « < CIICv lU .p l lC l loo l lVvHL / (py2 + pr,2 + pu2)dx Jn + C||Cv|U<p||H|0o||Vv||2» / (p 7 2 + P« 2 + pr,2)dx in = q i a i M l v v i & a i f l U + IIPIU) / (pr Jn + pr/2 + pu2)dx. ~2 + Chapter 2. UNIQUENESS OF VISCOUS COMPRESSIBLE FLUID MOTIONS 25 Moreover, I ^rnydx < C||r||oo / (P72 + pr)2)dx < CM* / (py2 + pr,2 + pu2)dx. Jn Cv Jn Jn Finally, [(TT-T; )"7>Tdx < / - Cv)rpydx Jn Cv is v Jn LVCV < CIICII^Hrlloo / (P72 + pri2)dx Jn < CIICH^IIrlU / (pj2 + pr,2 + pu2)dx. Jn When these estimates are inserted into (2.10), we obtain ^7&(P7,T) + \ I P72Uds < C3(t)[(pu,u)+(py,y) + (pr,,r,))-£-(Vy,Vy) + It t + s(Vj, Vy) + —(divu, divu) + -(Vu, Vu) + o Z Jan Cv on J8{x Cv on where C3(t) = CHIVfllloo+llfltlleo + llvlUIIVfllU+llAlloolldtWV + ||fl||oo||P.||L.>||*«v||oo + IM&IWI?. + (\\Cv,e\\l\M\\l + \\CvJ\l\\Vp\\l)\\k\\l + + ||«|li.>l|Vfl||L + IlKllwpllVfllUdla.pllcoHVplloo + ||C...||oo||V*||oo) + + ||C„||Lv(||K||oo||i?^||oo + H l^loollVplleoHVfllleo + |M|oo||V<C + + IKHoollVviuiivffiU) + (HcllL + M\l)(\\Vy\\l + \\vnl) + + ||VV|&(||C||L.> + Mtip) + l|Cu|U,-p||Vv||20(||C||oo + IIPIICC) + \\r\ui + \\cv\\Lip)]. Adding inequalities (2.7), (2.9) and (2.11), one obtains i-^[(pu, u) + (pr,, IJ) + (py,7)] < C(t)[(pu, u) + (pr,, r?) + (fry, y)] - £i(Vu, Vu) -- ^ i ( V - u , V u ) - J r ( V T , V 7 ) + e(divu,divu) + o O 2 + c(V 7 , V7) + e(Vu, Vu) - I i [pu2 + 1 Jan + pr,2 + py2)Uds+ i (±)k(p-)yds + Jan Cv on Jan Cv on Chapter 2. UNIQUENESS OF VISCOUS COMPRESSIBLE FLUID MOTIONS 26 where C(t) = Ci(0 + Cs(0 + Cs(0-If £ is chosen as the min(|pi, then we have 5^[(/m,u) + (pi?,»?) + (p7,7)] < C(0[(/m,«)+(p»?,i?)+(p7,7)]-5jf W + + pu2-rPy2)Uds+ <f J-(RdJ.-(-K-K)^-)ds. Jan Cv on On This coimpletes the main estimate. Now we consider the boundary terms in the right hand side of the preceding inequality. Define K= <£ (pu2 + pr,2 + py2)Uds. Jan Then K > 0 if U > 0, and K=0 if U < 0, as in this case v, p, 8 are prescribed at the boundary. Thus K >0 in all cases. The last boundary term vanishes because 8 is prescribed on the boundary. Hence the above inequality becomes -^[(pu, u) + (pn, n) + (pj, y)] < C\(i)[(pu, u) + (pn, n) + (py, y)], 0 < t < T, where C1(t) = 2C(t). Integrating, we obtain [(pu, u) + (pn, n) + (py, y)) < [(pu, u) + (pr,, n) + (py, y)]o exp(- / Ci(t)dt), 0 < t < T. Jo Since (pu,u), (pn, n) and (p7,7) are zero when t=0, it follows that they remain zero throughout the whole time interval in question. But this forces u=7 = n = 0, and so the two flows are identical at least until t=T. Finally, T being arbitary, the flows are identical so long as they exist. Hence one has the uniqueness of the solution, in the class of functions p, 8, v for which Ci(t), C7(t), and Cs(t) belong to L1(0,T). For instance, if the outward normal velocity G is zero, and one sets QT = (0,T) x fi, one has uniqueness in the class: p>0, p€L°°(QT), VpeL2(0,T;L°°(n)), v e H W , vve i 2 (0 ,r ; i~ ( f i ) ) , 8£L00(QT), V8eL2(0,T;L°°(Sl)), D28 e L1(0,T;Loo(Q)), Chapter 2. UNIQUENESS OF VISCOUS COMPRESSIBLE FLUID MOTIONS 27 having assumed that b G ^ ( O . T ; ! 0 0 ^ ) ) , r € I ^ O , T ; p „ e c \ P s e c 1 , C G C 1 , P G C 1 , c e c 1 , « € c 1 . 2.3 UNIQUENESS T H E O R E M FOR T H E CASE p > ft > 0, C > 0, k = 0 Q.E.D In this case the governing equations (1 1.3) reduce to the form dy 8t 3 p[% + (v • V)v - b] = - V P + JT[Dk(pDky) + Dk(pVvk)] + V[(( - \p)divy], (2.12) ^+Pdivy = 0, (2.13) 3 Pljt+y V0] = - ^ ^ A - w v + ±-pr + JL. £ + A , 1 ) 2 + ^-(C - |/i)(A'w)2. (2.14) With the initial conditions v|,=o = v 0, p|t=o = Po, 6\t=0 = 6o, we have the following result for this problem. Theorem 2.2 Lei the coefficients of viscosity and heat conduction satisfy p>Pi>0, C > 0, K = 0. Then there can be at most one solution of the equations (2.12)-(2.14) satisfying the following boundary conditions: (a) At all points of dCl, v ts given. (b) At all points where U < 0, p and 8 are given. Proof. Since the difference between the present case and the preceding case lies entirely in the energy equation, the inequalities (2.7) and (2.9) holds exactly as before. But in place of (2.11) we obtain the following inequality by taking inner product in L2(fi) of (2.14) with j and using the usual kind of estimates: \D\^P1^ + \ j s n n 2 U d S ~ C6(0[(P".") + ( P ' / . ' ? ) + ( / ' T ' T ) ] + |(rfiwu,rfiwu) + + |(Vu,Vu), (2.15) Chapter 2. UNIQUENESS OF VISCOUS COMPRESSIBLE FLUID MOTIONS 28 where Ct(t) = COIWIU.+ HtfJU + ||v|W|V0||T O + ||P,||oo||dit;v||eo + + \\Cv\\Lip\\0\U\Pt\U\divv\\M + |^||=o||P»|U,p||d«Vv||« + ||tf||Ll|P#HL + + (IICIIL + IIAII'ooXIIVvHL + HVvIlL) + ||Vv||^(||C|UiP + \\u\\Lip) + + IIC.IU.pHVvllLdlCIIco + IMloo) + l|r||co(l + \\Cu\\uP)]. Adding the inequalities (2.7), (2.9) and (2.15) for suitably adjusted e, one obtains ~[(/>u, u) + (pn, n) + (py, 7)] < C(t)[(pu,u) + (pn, n) + (py, y)] - ^ (Vu, Vu) -- \p\(V • u, V • u) + e(divu, divu) + e(Vu, Vu) -- I f [pv.7 + py7 + PV2]Uds, * Jan where C(i) = C2(t) + C4(t) + C6(t) and the non-negative term C(V • u, V • u) in the left side of the inequality has been droped. Choosing t < i 3 J-, we get \ -^[(pu, u) + (pn, n) + (P7, y)) < C(t)[(pu, u) + (pn, n) + (py, y)) - i j*^*2 + + py7)Uds. Define K= I (pn7 + pu7 + py2)Uds. Jan Then ii" > 0 in all cases as p, 8 and v are prescribed when U < 0. Thus we have •f%[{p*,u) + (pr,, i,) + (pT,7)] < Ci(t)[(p*,u) + (pn, n) + (py,y)}, 0 < t < T, where Ci(t) = 2C(t). Integrating the preceding differential inequality, we obtain [(pu,u) + (pij)fj) + (P7,7)]<[(pu,u)+(pij1ij) + (p 7 ,7)]o«p(- / Ci(t)dt), 0 < t < T. Jo Since (pu,u), (pn,n) and (p7,7) are zero when t=0, one has uniqueness of solution, in the class of functions p,0,v for which C?(t),CA(t) and C6(i) belong to Ll(Q,T). For instance if G=0, one has uniqueness in the class: p>0, PeL°°(QT), Vp€L 2(0,T;LT O(fi)), v € I ° ° ( Q t ) , Vv€L 2 (0 ,T;L°°(f i ) ) , 0eL°°(QT), V0€ i 2 ( o , r ; i ° ° ( n ) ) , Chapter 2. UNIQUENESS OF VISCOUS COMPRESSIBLE FLUID MOTIONS 29 having assumed that b 6 ^(O.TjL-fn)), r G L l (0,T;I°°(n)) , PP€C\ P,€C\ Cv G C 1 , p.£C\ ( 6 C 1 , / ceC 1 . Q.E.D. 2.4 UNIQUENESS T H E O R E M FOR T H E CASE n = 0, C > Ci > 0, k = 0 The governing equations in this case reduce from (1.1)-(1.3) to the form P& + ( v • v ) v " b l = ~ v p + V[Cdtt>v], (2.16) ot ^ + pdivv = 0, (2.17) P[% + v • VO] = -±.¥Ldivv+ ~-pv + ^ C ( ^ v ) 2 . (2.18) With the initial conditions v|t=o = v 0 ) p\t=o = po, e\t=Q = e0, the main result for this case is the following. Theorem 2.3 Let the coefficients of viscosity and heat conduction satisfy A* = 0, 0 < Ci < C, « = 0. Then there can be at most one soution of the initial boundary value problem consisting the equations (2.16)-(2.18) and the boundary conditions: (a) At all points of dQ, x n is given. (b) At all points where U < 0, v, p, and 6 are given. Proof. We obtain the folloing equations from the equations (2.16)-(2.18) in the same way as we obtained equations (2.l)-(2.3) in'§2.1. p[ut + (v • V)u + (u • V)v] + PfiVri + (Pp - P„)Vp + PeV-r + (Pe - Pe)V0 + + r , [v t - r (v .V)v-b] -V[CVu]-V[ (C-C)V-v] = 0, (2.19) Chapter 2. UNIQUENESS OF VISCOUS COMPRESSIBLE FLUID MOTIONS 30 r)t + v • Vn + u • Vp + vdivv + pdivu = 0, (2.20) p[jt + v • V7 + U • V6] + n[0, + v • V0] + -l-yPtdiv\ + (4- - -^-)0P»dtt>v+ Cv Cv C v 0 0 1 1 + jr{P« ~ P»)div\ + -zrPgdivu - -^ -C(du>v + dt'trv).diim - ^-(C - C)(d«w) 2 -Cv Cv Cv Cv - ( ^ - - ? i - K ( ^ v ) 2 - i - r n - ( i - - 7 L ) r p = 0. (2.21) Cv t^) Cv Cv Taking the scalar product in £ 2 ( f i ) of (2.12) with u, one obtains (pu,,u) + (p(v • V)u,u) + (pu- Vv,u)-r-(PpVr;,u) + ((P p-P,)Vp,u) + (F8V7,u) + + ((Ps - Pe)V0, u) + (i,v,, u) + (r,(v • V)v, u) - (rjb, u) - (V(CV • u),u) -- (V(C-C)Vv ,u) = 0. Utilizing (2.5), we get ~(pM,n) + 5 / pu2C/dS + (p(uV)v,u)+(P,Vrj,u)+((P,-Pp)Vp,u)-r + (PeVT,u) + ((P, - P«)V0,u) + (i/v,, u) + (ij(v • V)v, u) - (nb,u) -- (V(CV • u), u) - (V((C - <)V.v), u) = 0, i-g(pu,u) = - (p(uV)v ,u) - (P ,Vr , ,u ) - ( (P p -P p )Vp,u) - (P s V7,u) -- ((Pe - Pe)V6, u) - (r,v,,u) - (r,(x • V)v, u) + (nb, u) + (V(CV • u), u) + + ( V [ ( C - C ) V v ] , u ) - i / pv?Vds. (2.22) * Jan Now , / V(CV • u) • udx = V] I ((V u)u,nids -J2 I C V u J ^ d x . Jn ~{Jan ~[ Jn dx> Clearly / a f l (C^ " u)ui«ids = 0 in virtue of the given boundary conditions. Thus / V ( C V u ) u d x = - / C(Vu) 2dx. Jn Jn Most terms on the right side of (2.22) are estimated as in the previous theorem. Additionally, we have the following estimates valid for 0 < t <T: First, from the preceding equality, we get / V(CV • u) • u dx <-Ci / ( V u ) 2 d x , Jn Jn Chapter 2. UNIQUENESS OF VISCOUS COMPRESSIBLE FLUID MOTIONS 31 where the assumtion that 0 < C\ < ( has been employed. Next, / V((C - C)V • v)u dx = * ( C - C ) V v u n d s - / (C - C)V • v V • udx Jn Jan Jn < I / (C-C)(V-v)(V-u)dx| Jn < || C lU.pll V • v Hoc / |V • u|(|u| + \r,\ + h\)dx Jn < II CHLpHV • r\\l jf (|u|J + n 2 + y*)dx + | ( V • u, V • u) < C|KllL>l|V • vl&Kpu, u) + (pr,, n) + (py, y)) + | ( V • u, V • u), where t is an arbitary positive number to be fixed later. When the above estimates are inserted into (2.22) there results i-^(pu,u) + Ci(dt'«u,dtwu) < C4(t)[(pu,u) + (pri,ri) + (py,y)]+e;(div\i,divu)-- \ I pv?Uds, (2.23) 2 Jan where C<(t) = C[|| Vv(i) lU + || P„ \\Uf\\ Vp IU + || Pe \\Lip\\ V6 lU +UV.HOO + + ||Vv||ool|v||oo+ || b Heo + || C |U.-.|| V • V lU + || P p l l»+ II P PP 11 oo 11 Vp ||oc + + II Pp* Hooll Ve lU + II Pe II2 + II Ptp 11.11 vp lu + || pM lull Ve ||ro]. Taking the scalar product in L2(fi) of (2.21) with 7, we obtain (Pit,7) + (pv Vy,y) + (puV6,y) + (ri§t,y) + (Tiv •Ve,y)-r(-lryPedivv,y) + + ( (4 - - i )0P^«t;v,7) + (^-(A-P«)d«t ,v,7)- ( -(-?-^ ( f t - t , U i T )_ Cu Cv Cv Cu - (4-C(^v + divv).divu,7) - ( i - (C - C)(dt'vv)2,7) -- ( ( i - ^)C(^v) 2 ,7 ) - ( i r „ , 7 ) - ( ( i - ^-)rp,7) = 0. (2.24) Cu Cu Cu ^« We also obtain the following equation for 7 in the same way as we obtained the equation (2.5) for u. We get Inserting this into (2.24) and estimating the terms, we get 5757(P7,7) + \i Pi'uds < || ve | U / (P« 7 + PT2)*** + c\\et IU / (pn 2 + Dt i Jon Jn Jn Chapter 2. UNIQUENESS OF VISCOUS COMPRESSIBLE FLUID MOTIONS 32 + p72)rfx + C||*||M||V0||oo [ w + py^dx-r Jn + C || Pe Hooll divv ||oo [ pj7dx + Jn + C\\CV \\LiP\\ 6 Hooll P» Hooll dtttf Hoc / (PT2 + P"2)<*x + Jn + C || 6 Hooll |U<p ||di«v||oo / (P77 + Pr)7)dx + Jn + C || *||»oll*||L J PJ7dx + j(divu, divu) + + C || CHLdl Vv||»o+ II Vv\\l)JnP72dx-r ^(divu,divu) + + C||C|U,P||Vv||L / (P72 + prf + Pu2)dx + Jn + C H C H l . - p I I C I I O O I I V v H ' , /(py2 + pu7 + pr,7)dx + Jn + C||r||oo / (pri2 + P72)dx + C \\ Cv \\Lip \\r\U f (p77 + Jn Jn + PV2)dx = C[\\Ve\\oo + Halloo + llvllooHVtflloo + llAllooH^vHoc + + ||a||L l p||0||oo||A||oo||^v||M + ||ff||oo||P.||i.ip||df«v||oo + + m\l\\P'\\l + IIC||L(IIVv||L + ||Vv||L) + HCIUvllVvii 2, + + liaiU.pllClloollVvH^ + ||r||oo(l + ||C,|U,p)][(pu,u)-r + (PV, V) + (P7,7)] + divu). Thus, we obtain \§;(rr, 7) < Ct(t)[(pu, u) + (pr,, 17) + (py, 7)] + |(A"wu, divu) ~\j P72Uds, (2.25) where C»(t) = CHIV^ Iloo + Halloo + HvlloellV l^loo + ||PS||co||d«t>v||oo + + IICvlUipllfflloollAHoolldt-fvHoo + C||0||oo||^||Lip||^v||oo||0||oo2||P»||oo2 + + ||CHoo(||Vv||M2 + ||Vv||oo2) + ||ClU,p||Vv||oo2 + ||C,||L,p||C||oo||Vv||TO2 + + i M M i + l i a i M ] . Adding inequalities (2.23), (2.9) and (2.25), we obtain ~[(putu) + (pr,,r,) + (P7.7)] < C(t)[(pu,u) + (pr,,»?) + (/>7.7)] + Ci(V • u, V • u) Chapter 2. UNIQUENESS OF VISCOUS COMPRESSIBLE FLUID MOTIONS 33 \ l [pu2 + pr,2 + py2]Uds + 1 Jan + + e(divu, divu), where C(t) = C2(t) + C4(t) + Cs(t). If e is chosen small enough, for instance e = ^ , then we have I D - 0 - — [(pu,u) + (pr,,r,) + (p7,7)] < C(<)[(pu,u) + (P'?,'?) + (P7>T)] ~ - \ l (pu2 + pr,2 + PJ2)Uds. 1 Jan This completes the main estimate. It is necessary now to consider the last term, the integral in braces. Define K = I Jas Then K > 0 if U > 0, and K = 0 if U < 0, as in this case v, p, 9 are prescribed at the boundary. Thus K > 0 in all cases. Hence, we have -^[(pu, u) + (py, y) + (pr,, r,)] < C^OKpu, u) + (py, y) + (pr,, r,)}, 0 < t < T, where Ci(<) = 2C(t). This differential inequality has the integrating factor exp(— Ci(l)dt), and the solution (pu2 + pr,2 + py2)Uds. / n [(pu, u) + (pr,, r,) + (py, y)] < [(pu, u) + (pr,, r,) + (py, y)]0exp(- I deft), 0 < t < T. Jo '0 Since (pu,u), (pn, r,) and (py, y) are zero when t=0, one has uniqueness ofthe solution, in the class of functions p, 9, v for which C2(t), C^(t), and Cs(t) belong to Ll(0,T). For instance if G=0, one has uniqueness in the class: p>0, P6L°°(QT), VpeL:(0,T;L°°(n)), veI ° ° ( Q T ) , V v 6 L 2 ( 0 , T ; L » ( f ! ) ) , 9eL°°(QT), V9 £L1(0,T;LDO(Q)), having assumed that beL^o^L^W), rei'to.Tinn)), Chapter 2. UNIQUENESS OF VISCOUS COMPRESSIBLE FLUID MOTIONS 34 pP£C\ Peec\ c „ e c \ pec\ c e c \ k g C 1 . Q.E.D. 2.5 UNIQUENESS T H E O R E M FOR T H E CASE p = 0, C > Ci > 0, /c > m > 0 The governining equations in this case take the form ^ + pdiw = 0, (2.27) p[ft + v • v ^ = ~c~%divv + ^T,d^kD"9) + ^-/r + c~vadivv)2- ( 2 2 8 ) With the initial conditions v|<=o = v 0, p\t=o = Po, 6\t=o = 00, we obtain the following result for this problem. Theorem 2.4 Let the coefficients of viscosity and heat conduction satisfy P = 0, C > Ci > 0, K > ^ > 0. Then there can be at most one solution of the equations (2.26)-(2.28) satisfying the following boundary conditions: (a) At all points of dQ, v n ts given. (b) At all points where U < 0, v, p and 6 are given. (c) At all points where U > 0, 6 is given. Proof. Since the difference between the present case and the preceding case lies entirly in the energy equation, the inequalities (2.9) and (2.13) hold exactly as in the preceding section. In place of (2.25) we obtain the following inequality from (2.28) applying the methods developed in the preceding sections: ^(P7,7)+ \j'BnP'^Uds - C7{t)[(pu,u) + (pr,,*) + (py,y)] + e(divu,divu) + + e ( V 7 , V 7 ) - H py2Uds+i 4-* p-yds - £ - ( V 7 , V 7 ) + Jan Jan Cv on a + / -*)!%**, (2-29) Jan Cv dn Chapter 2. UNIQUENESS OF VISCOUS COMPRESSIBLE FLUID MOTIONS 35 where C7(t) = C7[||V5||oo + Halloo + llvlloollV l^leo + HAlloolldl-Wvlloo + + IIC.IlL.-plMloollAlloollrftfvHoo + I I ^ I I C O I I ^ K L . P I I ^ V I I O C + ||0||L||P,||L + + (\\cvA\l\\v'e\\l + liev.pllLllVpiiL)!!^!2.!!^!!.-,!^^!?. + + ||*IU.-pHVtf||0o(||CulP||oeHVp||00 + IICW.^ MoollVfl'lloo) + + IICiUipdkiiccllD^iu + iKlwiVpiwivfliioo + ||^||oo||Vfl||>0 + + IMIoollVviwivoiu) + HCHLdlVviiL + \\vv\\l) + I K I U . ' P I I V V H ^ + + liaiUipHCHoellVvIlL+ 11^00(1 + 11^11 )^]. Adding the inequalities (2.23), (2.9) and (2.29), one obtains ^-j^[(pu, u) + (py, y) + (pr), n)) < C(t)[(pu,u) + (pr), r,) + (py, y)} - (i(divu, divu) -- TT(V7, V7) + t(divu, divu) + e(V7, V7) -C2 - I f W + PV2 - py7]Uds + * Jan + <t> —K-^-yds+f — ( K - K)—yds, JanCv dn J8(x cv dn where C(t) = CA(t) + C2(t) + C7(t). Choosing e < min^!,^-), and utilizing the fact that the terms integrated on dQ in the above inequality are non positive, one obtains -^[(pu, u) + (pi,, ij) + (py,7)] < C2(<)[(pu, u) + (pr,, 1,) + (py, y)}, 0 < t < T, where C2(0 = 2C(t). Hence one can argue as before, obtaining uniqueness in the class (for G=0): p>0, p£L°°(QT), V p e L ^ O . T i L 0 0 ^ ) ) , V G I ° ° ( Q T ) , V v 6 l 2 ( 0 , T ! L » ( n ) ) , eeL°°(QT), Ve £L2(0,T;L°°(Q)), D20eL1(Q,T;Leo(Q)), Chapter 2. UNIQUENESS OF VISCOUS COMPRESSIBLE FLUID MOTIONS having assumed that b € L1 (0,T; L°°(fi)), r € ^(0,T\2 ,~(n)), pPec\ peec\ c v e c \ Chapter 3 UNIQUENESS OF INVISCID COMPRESSIBLE FLUID MOTIONS 3.1 UNIQUENESS T H E O R E M FOR INVISID COMPRESSIBLE FLUIDS In this section we state the main theorem for inviscid compressible fluids under the initial and boundary conditions of §1.2. The solutions under consideration are assumed to be sufficiently smooth that all the norms used in estimating them are finite. Theorem 3.1 Lei the coefficients of viscosity and heat conduction satisfy X = p. = K = 0. Suppose the equation of state P = P(p,S) is of class for all p and S, and suppose that _ o. s Then there can be ai most one solution ofthe equations (1.5)-(1.7) satisfying the following boundary conditions: (a) At all points of the boundary dQ and at all instants of time, the normal velocity component v n is given. (b) Ai all points where U < 0 the velocity v, the density p and the entropy S are assigned. Moreover, if for some solution, U = v • n — G > \J(^fj)s on a subset dQ* of the boundary, then the boundary conditions on dQ* are redundant with respect to the given data; thai is, the conditions assigned on the remainder of the boundary, together with the given initial values, uniquely determine the solution. Proof. Let (\,p,S) and (v,p,S) be two solutions satisfying the above mentioned conditions and equations and set n = p — p, u = v — v, f3 = S — S. Since both flows satisfy the equation of motion (1.5), we have by subtraction: du d\ p{-£ + v • Vu + u • Vv) + n(-^ + v • Vv) = nb - VP', + + pn Vv = nb - VP'. (3.1) 37 Chapter 3. UNIQUENESS OF INVISCID COMPRESSIBLE FLUID MOTIONS 38 Forming the scalar product of this equation with the vector u, we get PD1^\u^ + VUDT + P U V 7 U = V U H ~ U'VF"' With the help of the equation of continuity, the transport equation can easily be rewritten as Using the transport formula (3.2), one obtains J l-pu*dx = - J [ M | ~ - " b) + pu • Vv • u + u • VP'jdx - f l-pv?Uds. (3.3) Since both flows satisfy the equation of continuity, one obtains by subtraction: ^ + u • Vp + ndivv + pdivu = 0. (3.4) Turning next to the energy equation (1.7), we have by subtraction: ^ + u-V5 = 0. (3.5) Forming the scalar product of this equation with p(i, we obtain Using the transport formula (3.2), one obtains / \ppdx=-l \p(PUds- j fJpu-VSdx. (3.6) u t Jn £ Jan1 Jn Now, we derive an identity for the term / n u • VP'dx in (3.3). First, / u • VP'cfx = / u (PpVr, + (pp _ p p ) V p + P,V/? + (P, - P,)VS)dx Jn Jn = - / VPP • undx - / Ppndivudx + / u • (Pp - Pp)Vpdx -Jn Jn Jn - I VP, • u/3dx — I P,0divudx+ f u(P, - P,)VSdx + Jn Jn Jn + I (Ppr,u + P,pu)ds. (3.7) Jan From equation (3.4), one obtains divu - + - u • Vp+ -div\. (3.8) p Dt p p Chapter 3. UNIQUENESS OF INVISCID COMPRESSIBLE FLUID MOTIONS (3 Inserting this into the equation (3.7), we get / u-VP'dx = - ( PppVpur,dx- f PP,VS ur,dx + f ( ~ + - u - V p + Jn Jn Jn Jn P Dt p + -divv)Ppr)dx + I u(Pp - Pp)Vpdx - f P,pVpu0dx-P Jn Jn - / P..VSupdx + / P,j3{-^ + -u • Vp + ^divv)dx + Jn Jn P Dt p p + / u(P, - P,)VSdx + I {Ppqn.n + P.0u • n)ds. Jn Jan Next, we derive the following identities. First, noting the identity fl~ Dn, f D .Ppr)\. [ Ppr? Dp , fri2DPp, we can use the transport equation (3.2), to obtain JnP plDt DtJn 2P fan 2 p Jn " p* Dt Ja2p Dt Similarly, we obtain JnP Dt Jn Dt p7 Jn p Dt Jn P Dt Jn p2 Di = ° f * l h d x + l ^ U d s + f l u V s P , d x - [ ^ d x -Dt Jn P Jan P Jn P Jn P Dt j 2r,(3P, Dp Jn P Di When these identities are inserted into (3.9), we get the desired result: j u • VP'dx = - J PppVp • undx - J PP,VS • undx + ^ J ^jfdx + Jen 2 P Jn P7 Dt Jn 2p Di + [ -u VpPPT}dx + / ?-divvPpdx + j u (Pp - Pp)Vpdx -Jn P Jn P Jn - j P.pVpu0dx- f P,.VSupdx+£- I ^-dx + Jn Jn Dt Jn p + / !*P.Ud.+ [Hu.VSP.dx- l ^ d x -Jan P Jn P Jn P LH - [MPtE£dx+ [ M u . Vpdx + / ^divtdx + Jn P Dt Jn p Jn P + / u(P, -P.)VSdx-r I (Ppr)un +P,pun)ds. (3. Jn Jen Chapter 3. UNIQUENESS OF INVISCID COMPRESSIBLE FLUID MOTIONS 40 Substituting this formula into (3.3), we get ^(pu,u), = - ( - ) (^-b) ,u) - ( /m.Vv,u) + |p w Vp.u 1 ) <Jx + - 4 [(^Sr-U+ U) + (Ppr,un + P,/3u-n))ds-Jan IP P Jn P7 dt y n2p Dt Jnp - I ?-divvPpdx- j u(Pp- Pp)Vpdx + f P,pVpupdx + Jn P Jn Jn + [p..VSuPdx- f VvSP.dx + / ^ d x + / M p , Jn JnP Jn P D t Jn P - f M u - V p d x - / ^-divvdx- f u ( P , - P.)VSdx. Jn P Jn P Jn Rearranging the terms and estimating the right hand side in the usual way, we get 7i7 / lp[u2 + ^r>2 + ^rVP)dx < Dy(t)l(^) + (PV,ri) + (P0,0)]- O-H) Dt Jn 2 pl p* - I l[pU(n7+L\ri7 + ^rl0) + 2(Ppi1 + P,l3)u-n]ds, Jan 2 P P %dx-Dt where D1(t) = C f l l ^ H o o + l lbHoo + I W I o o l l V v H o o + HApIUII^IIOO + H ^ . l l o c l l V S I I o c + + l l P p l l o o l l ^ H o e + H a l l o o + l l A l l o o H V p l l o o + ||d.«v|l«»l|Ppl|oo + + ||Pp||Lip||VH|oc + l l A p l U I I V p l l o o + I I P . . H 0 0 I I V S H 0 0 + ||V5||oo||P.||oo + + H a l l o o + l l A l l e o l l ^ H o o + H P . I I o o l l V p l l o e + | | P . ||oo \\div ?\\co + + ||P.|U,>||V5||TO]. Recalling that Pp > 0, we choose <<r > P , 2p7 With this choice of M, let us multiply (3.6) by M and add it to (3.11). Then we get I pLdx<D2(t) [ p(r,7 + u7 + 02)dx - I I [pUL + 2(Ppr, + PJ)xi-n}ds, Dt Jn Jn 1 Jan where £ » 2 ( 0 = A ( 0 + l | V S | k Chapter 3. UNIQUENESS OF INVISCID COMPRESSIBLE FLUID MOTIONS 41 and p o p p L = u 2 + ^ r , 2 + illtf + Af/?2 > t/2 + ^ ( » ? 2 + /?2) > 0. p^  p'' zp' Now define K = pf/L + 2{Ppr) + P,f3)u • n, on dfi. Then it follows from the boundary conditions that K > 0 at all points of dQ. Moreover K > 0 on the set dQ* irrespective of the boundary conditions. For on dQ* we have U > \fP~p, so that either K > py/FpL + 2u(P,»? + P,0) or K > pyfprpL-2u(Ppr,+ P,f3). But p P/p ? Hence the required inequality follows. Since K > 0, the resulting boundary integral is non negative, and we obtain / pldx < D2{t) I p{r? + v? + p)dx -C^ Jn Jn < D2(t) J p[u* + far? + p))dx, where D2(t) = D2(<)[l + 2||4-||00]. With the help of (3.12), we get - ^ ^ p L d x < D2(t)J^pLdx, This differential inequality has the integrating factor exp~ Jo D,(-i^d\ and the solution / pLdx < [ f pLdx]0 exp- 5 , { , ) d i , 0 < t < T. Jn Jn Chapter 3. UNIQUENESS OF INVISCID COMPRESSIBLE FLUID MOTIONS 42 It follows from the initial condition that Jn pLdx = 0, and then using (3.12) that u = n = j3 — 0. Hence one can obtain uniqueness, in the class (for G=0): p>0, p(EL°°(QT), V p e l^O.r;! 0 0^)), ^ € Lr(0, T; I°°(fi)), veL°°(Qr), V v e l ^ C r ; ! 0 0 ^ ) ) , ^ G L ^ O . T ; ! 0 0 ^ ) ) , having assumed that beL\0,T;L°°(Q)), P„ P.eC1. Q.E.D 3.2 UNIQUENESS T H E O R E M FOR INVISCID HEAT C O N D U C T I V E FLUIDS The initial and boundary conditions of §1.3 being understood, we have the following uniqueness theorem for non-viscous heat conducting fluids. Theorem 3.2 Let the coefficients of viscosity and heat conduction satisfy X = p. = 0, K > ki > 0. Suppose ( f ) . > ° Then there can be at most one solution to the initial boundary value problem corresponding to the equations (1.8)-(1.10) the and boundary conditions below: (a) At all points of the boundary dQ, and at all instants of time, the normal velocity component v • n ts given. (b) Ai all points where U < 0, the velocity, density, and temperature are assigned. (c) At all points where U > 0, the temperature is given. (d) Ai all points where (7 = 0, one of the conditions listed below is assumed: (d\) The temperature is prescribed. (d2) The heat flux q • n is prescribed. (^ 3) The flux is assumed to be proportional to the difference between the fluid temperature 8 and a given wall temperature 80, that is, q • n = k(8 — 80), with k > 0. Chapter 3. UNIQUENESS OF INVISCID COMPRESSIBLE FLUID MOTIONS 43 Moreover, if for some solution of this problem we have U = v • n - G > yj(^)e on a subset dQ* of the boundary, then the boundary conditions on dQ* are partly superfluous with respect to the given data; that is, the conditions assigned on the remainder of the boundary, together with the temperature on dQ* and the given initial conditions, uniquely determine the solution. Proof. Let (v, p,6) and (v, p, 6) be two solutions ofthe problem under discussion and set 77 = p—p, u = v — v, P = S — S. Applying the same methods developed in the preceding section, one obtains from the equation (1.8) J \pu*dx = - j [ M § j r - b) + pu • Vv • u + u • VPldx - jf ^ \pu*Uds. (3.12) The energy equation (1.10) yields by subtraction PlU + + P u • v# + -jr-lPtdivv + (-J- - -^)6Pedivy + ~ ( P # - Pe)divv + Dt Dt Cv Cv C„ Cv 3 3 + ±P0divu - ^Y^Dk(kDk7) - J - Dk[{k - *)Dk6) -v cv fc=1 cv k = 1 3 " (^-^-)EDk(KDke)-^rri-(^-±)rp = 0. Cv ° f t = 1 Cv Cv Multiplying the preceding equation by 7 and integrating, we get 55? i / 7 * * * + \ fdnP-r2Uds+J^puV6dx +Jnr,^ydx+ (3.13) / —yPgdivvydx + I y(-l- - ^r)epediv\dx + Jn Cv Jn Cv c « I ~r{Pe - Pe)div\dx-r- I -^-Pgdivuydx-Jn Cv Jn Cv r 1 3 r 3 - / 4-^2Dk(kDky)ydx- / ^-£>t[(* - K)Dk6]dx -3 ~ / T t i - ^ E ^ ^ ^ x - / -J-rn 7 «ix- / 7 (-J- -in Cv c « ~ in Cv in Cv - J-)rpdx = 0. O v Now we derive an identity for the difficult term J n u • VP'dx in (3.13). First, [ u-VP'dx = / u(P, - Pp)Vpudx + / uP,Vr>dx+ / (P, - P,)V0udx + in Jn in in + + L PfVyudx Chapter 3. UNIQUENESS OF INVISCID COMPRESSIBLE FLUID MOTIONS 44 = ((?(,- PP)Vpudx + / (Pf - P,)V0udx + * P„nu • nds -Jn Jn Jen - I u VPpr,dx - I Ppndivudx. Jn Jn Inserting the equation (3.8) into the preceding equation, one obtains [ u-VP'dx = / (P„ - P,)Vp • udx + / (Pt-Pt)Veudx-+ & Ppr,unds-Jn Jn Jn Jan - I uPppVpr,dx- / uP„,V0ndx+ / PllPldx + Jn Jn Jn P Dt + f P£lLu.Vpdx+ [ ^-divydx. Jn P Jn P Inserting the identity [ PrfDr,, f D.Ppr,\ [ DPP r,7 [ Ppr,7 Dp Jn — Dld* = L ' Di ( " V )dX ~ Jn~DTy> + Jn~prDtdx' into the preceding formula and then applying the transport formula (3.2), we obtain / u VP'dx = [ (Pp-Pp)Vpudx+ [ (Pe - P9)V0udx+ S PpVunds+ (3.14) Jn Jn Jn Jan + R f ^ d x U Z^uds- f ^ p x + [ ^ ! d x + DtJu 2p Van 2p Jn Dt 2p Jn p* + f ^ u Vpdx + / ^-divydx - / u PppVpr,dx - / uP,,V0ijdx. Jn P Jn P Jn Jn Substituting (3.15) into (3.13), we obtain - ^ ^ J p ( u 2 + ^ 2 ) d x = -Jhu(^-b) + puVyu}dx-Jn(Pp-Pp)Vpudx-- I {Po- Pe)V0udx - I \[pU(u7 + ^  + Jn Jan 1 P _ / 5 ^ V p d x - / &-divvdx + I uPppVpr,dx + Jn P Jn P Jn + / uPpeV6r,dx. (3 15) Jn Estimating the right hand side of this equation, one obtains 757 / \P^7 + ^ )dx < D4(t) I [pu7 + pr,7 + PJ7]dx -Dt Jn 2 pl Jn - I I[pL/(u2 + 5^!)-r2P,» 7u.n]d f i, (3.16) Jan1 P Chapter 3. UNIQUENESS OF INVISCID COMPRESSIBLE FLUID MOTIONS 45 where D*{t) =- c n i ^ i u + iibiu + H P I I C O I I V V I U + iiPpiu.piivpiu + DP + Hftlli.-pHVfflloo + H ^ I U + \\Pp\\oo + IIPpHoellVplloo + + IIPplloolldlwHoo + H-PpplloollVplloo + IIP^IUIIWHoo]. Now inserting the identity M97P9 _ _ 6yPe _ into (3.14), and then applying the divergence theorem and the usual estimates, we get the inequality \D\J r?ix+\j p 7 2 U d s * D S { T ) J [Pu2 + P17 + P72}dx + C-2 Jan Cv 0 n + Z-(k-K)^--p-P$u.n]ds, (3.17) C v on Cv where D5(t) = CrjIplloollWIU + H-^lleo + llAllooll^vHoo + HCIUiplMloollAllooHd.-I'vlU + + llffllooll^llxvll^vlloo + IWIoollVflIU + + lltflUHP.plloollVplloo + + IMIeollPMlloollVtflloo + ||0||T O||P*||oo||d J o c l l V p l U + ||ff||oo||P.||eo||C I ) l,|| 0 0 • • Halloo + | |G\ ,«IILI|V0| |LPIIL + l lc u ,p | |Ll |Vp| |col | s | |L + IMIi,>l|v*||L + + ||«||L,>||V0|| o o(||CV i P|| o o||Vp||T O + IIC^IIoollWIU) + ||C I )|U,p(||/c|| 0 0||Z?'fi|| 0 0 + + IMUIVpl leo l lWHoo + ll/cHooHVflllL + \K Hoc || V v | U | V0||TO ) + ||r||oo + + IIC.IU.-plHloollrlloo]. Adding the inequalities (3.17) and (3.18), one obtains 7S7 / U " 2 + 4 ^ + S)dx $ I W " 2 +> ^  + T2)]d* + (* - ? M ( V % V 7 ) + 7n * P Jn Ci JanCvdn cv on Jan^ P2 + 72)ds + 2P,IJU • n + 2P« pfu • n]ds, Chapter 3. UNIQUENESS OF INVISCID COMPRESSIBLE FLUID MOTIONS 46 for 0 < t < 7, where D€(t) = D4(<) + D6(t). Now define K = C/(u2+ 5^! + 7 2 ) + 2PpTjun + 2P,^ 7 -un, on dQ. Clearly £ > 0 and f . k dj 7 , rt / a n C ^ " C / 5" in virtue of the given boundary conditions. Moreover, on dQ*, K = U(p2u7 + ^ j ! ) + 2Ppr,u • n and r.2 />2 so that K > 0 on dfi* irrespective of boundary conditions. Thus we have < D7(t)N + (e - ^-)(V7tV7), o < t < r , where and / ^ ( ^ + ^ ! + 7 2 ) ( f x £>7(0 = AWU + 2||^||oo]. Choosing e = ^- and solving the differential inequality leads to the identical vanishing of u, t], and 7. Hence one has uniqueness in the class (for G=0): p>0, peL°° (Qr) , Vp€l a (0,T;I~(n)), ^ £ ^(0, 7; L°°(n)), V G H W , VvGl 2 (0 ,T;L°°( f i ) ) , ^ G ^ (O, T; L°°(fi)), 0 € l ° ° ( Q r ) , V0€ L 2(0,T;L°°(n)), £>20 G LHO.T; ! 0 0 ^) ) , under the assumptions b G ^ (O.r ; ! 0 0 ^)) , r G I^O, 7; L°°(Q)). Q.E.D Chapter 4 UNIQUENESS FOR COMPRESSIBLE M A G N E T O H Y D R O D Y N A M I C S 4.1 F U N D A M E N T A L UNIQUENESS T H E O R E M FOR VISCOUS M H D In this section, we state the main uniqueness theorem for viscous compressible magnetohydrodynamics under the conditions of §1.4. The solutions under consideration are assumed to be sufficiently smooth that all the norms used in estimating them are finite. Theorem 4.1 Let the coefficients of viscosity, heat conduction and the magnetic diffusivity satisfy the inequalities p > 7 J j > 0 , C > 0. k > Tci > 0, r?>0. Then there can be at most one solution of the equations (1.20)-(l.24) satisfying the following boundary conditions: At all potnts of the boundary dQ, and ai all instants of time, the velocity v and magnetic field H are given. Futhermore, (a) At all points where U < 0, the absolute temperature and density are prescribed. (b) At all points where U > 0, ihe absolute temperature is given. (c) Ai all points where [7 = 0, one of the following conditions is assumed: (cj) The temperature is prescribed. (c2) The heat flux q • n is prescribed. (cz) The heat flux is assumed to be proporiinal to ihe difference between the fluid temperature 6 and a given temperature 6o, that is, q n = k{0 - 0Q), with k>0. Proof. The difference between the present problem and the problem considered in §2.2 lies entirely in the momentum equation and the equation for H. Therefore, we make considerable use of the equations and the inequalities derived in the preceding chapters. Let (v,H,p,0) and (v,H,p, 0) be two solutions of the problem under discussion, and set H' = H - H, u = v — v, n = p — p, y = 0 — 6. Since both 47 Chapter 4. UNIQUENESS FOR COMPRESSIBLE MAGNETOHYDRODYNAMICS 48 fields satisfy the equation (1.20), we have by subtraction: DH' ~ " —— = - u • V H + Hdt'uv - Hrft'jjv + Hdi'uv - Hdivv - H Vv + H Vv -Dt - H • Vv -f- H • Vv + r/AH', = - u V H - Hdtvu - H'divv + H • Vu + H' • Vv + fj AH'. (4.1) Multiplying the above equation by pH', we obtain p—(-#'*) = -pn • (VH) • H' - pH • H'divu - pH'7divv + + pH • (Vu) • H' + pH' • (Vv) • H' + fjpH' • AH'. (4.2) Inserting the identities pH • (Vu) • H' = div{p(H' • u)H} - (u • H')(H • Vp) - pH • (VH') • u and pH' • AH' = dt'u{p(VH') • H'} - Vp • (VH') • H' - pVH' : VH' . into the equation (4.2), we get p ^ ( i ^ ' 2 ) = -pu • (VH) • H' - pH • H'divu - pH'7div\ + pH' • (Vv) • H' -- (H' • u)(H • Vp) - pH(VH') • u - rj(Vp) • (VH') • H' - fjpVH' : V H ' + + div{p(H' • u)H) + p(VH') • H'. (4.3) If we make use of the transport formula (3.2) and the divergence theorem, then the integration of equation (4.3) yields -5- / \PH'7dx = - I pu (VH) • H'dx - [ pH H'divudx - f pH'7divvdx + Dt Jn 2 j n Jn Jn + / pH' • (Vv) • H'dx - / (H' • u)(H • Vp)dx - / pH (VH') • udx -Jn Jn Jn - / r/Vp • (VH') • H'dx - / rjpVH': VH'dx - / \pH'7Uds. (4.4) Jn Jn Jan2 Since both flows satisfy equation (1.23), we obtain by subtraction: p[u, + (v • V)u + u • Vv] + P,Vr, + (Pp - P,)Vp + PeVy + (Pg - P»)V0 + (4.5) + r,[v, + v • Vv - b] - £D k(flVu l) - Y Dk({p - u)Vvk) - £Dk(fiDku) -k k k - Y W - ^ v - VK< - ?P)V • u] - V[(C - U - < + | /i)V • v] - R' = 0, Chapter 4. UNIQUENESS FOR COMPRESSIBLE MAGNETOHYDRODYNAMICS 49 where R' = -VH2 ,. , H H , V i / 2 .. , H H . R' = — + dtv(-) + — — - dxv(—~), Sir 4ir B3T 4JT 1 f > s dHk 1 ^ - 6Hi 1 A 6Hk 1 A dHi Therefore and R' = - i - ( V H ' ) • H - -J-(VH) • H' + -^-H • (VH') + -^-H' • (VH) 4?r 4TT Air 4ir u R' = - — {pu • (VH') H + pu (VH) H' - pH {VH')u - pH' • (VH) • u}. (4.6) 4TT Taking the innerproduct in L2(Q) of (4.5) with u and inserting (4.6) into the resulting equation, one obtains in the usual way i-^(pu,u) + i^p U 2 C/d S +(p(uV)v,u)-r(P p Vn,u)+(P p -P,)Vp,u) + + (PgV7, u) + {(Pg - Pg)V6, u) + (IJV,, u) + (n(v • V)v, u) - (nb, u) -3 3 - (^ Dk(pDku),u) - (^ Dk{[p - p]Dkv,u) - (53 Dk(pVuk),u) -k = l *=1 i 3 1 - (53 Dk[(p - p)Vvk), u) - (V([C -C-UP- P)]divv), u) -- (V[(C-^)V-u],u)+-L / pu • (VH') • Hdx + i - / pu • (VH) • H'dx -6 4TT Jn \n Jn - ^- f pH • (VH') • udx - / pH' • (VH) • udx = 0. (4.7) 4T Jn Jn Estimating the terms in (4.4), (2.8), (2.10) and (4.7), we obtain the following integral inequalities: / \PH'2dx < £,(<) / p(u2 + tf'2)dx + e(V-u,V-u) + Dt Jn i Jn + £ ( V H ' , V H ' ) - r ] p i ( V H ' , V H ' ) - / \PH'2Udsy (4.8) Jen * where Ei(t) = CHIVHIU + ||H|| 2 T O + lldtvvHoo + ||Vv||T O + + IIHIIoollVplU-r-HVpIlL], Chapter 4. UNIQUENESS FOR COMPRESSIBLE MAGNETOHYDRODYNAMICS 50 where where and where £ I bl7dx < E,{t) I ptf + « 2)dx + e(V • u, V • u) - J / pn'Uds, (4.9) Dt Jn 2 Jn 2 J6n ^ ( O B C D i V p i u + lldivviu + iHlL], RJ^pu'dx < £ 3 (<) p(u2 + ^  + T 2 ) ^ - /I,(Vu, Vu) - | ( V • u, V • u) + + e(Vu,Vu) + e ( V - u , V u ) - / ^-pu2Uds + e(VH', VH'), (4.10) Jan 2 E3(t) = c[\\,ip\\l\\vp\\l + \\te\\l\W\\l + ||A»ll?»l|Vv||20 + HcllLpllwiiL + + IWlLpllVvU*, + \\Pp\\l + llPpplloollVplloo + llPp.lloollVfllU + \\p,\\l + + I I P M I I O O H V ^ I I O O + H^lloollVplloo + IIVvHoo + ||Pp||iip||VH|oo + + llftlUo-IIWHoo + Hv.lloo + IIVvllooHvlloo + ||b|ioo + IICIlLpllV.vHL + + IIHHL + HVHlleo], jLJJ-py'dx < E<(t) p(v? + n2 + 7 2)dx - g ( V 7 , V 7 ) + e(Vu, Vu) + + e(V u, V u ) + £ ( V 7 , V 7 ) - i / py2Uds+I -Lkp-yds + 2 yen Jan Cv on + / ^{k-K^yds, (4.11) Jan Cv on EA(t) = C[||V5i|oc + Halloo + llvHeoHVSlloo + H l^looMrftt/vlloo + . + ||aiU,>||ff||0o||P,|UI|d««v||00 + ||tf||oo||P.||L.>||dft,v||eo + IMILII^ III, + + (\\c«A\U\*~e\\l, + lia.pllLllvpiiDpHL + | |« | |L P | | v<C + + ||*||Lip||Vfl|| 0 0(||CW lp||oo||Vp|| o e + llC^IWIVelU) + ||C„||L,>(||K||oo|Paff||eo + + IlKplUlVplUIVflllco + ||lC,||oo||Vfl||L + ||K„||oo||Vv||00||Vff||00) + + (IICIIL + IIAIlSoXIIVviiL + HVviiL) + HVvi^diciUip + IHU.>) + + ||c„||i.1>||Vv||a0(||Clloo + ll/illoo) + IWI~(i + liaiU,p)]. Chapter 4. UNIQUENESS FOR COMPRESSIBLE MAGNETOHYDRODYNAMICS 51 Adding the inequalities (4.8), (4.9), (4.10) and (4.11), one obtains Dt jn lp(H'2 + ^ + u * + T 2 ) d x ^ E*W J P(u7 + r? + 7J + H'2)dx -- 5 l ( V 7 , V 7 ) - ^ ( V u , V u ) - ^ - ( V . u , V . u ) + -I- e (V.u ,V-u) + E(VH',VH') + £(Vu,Vu) + + e ( V T , V 7 ) - ( ^ V H ' , V H ' ) , where Es(t) = Ei(t) + E2(t) + E3(t) + E4(t). The surface integral in the above inequality vanishes due to the boundary condition. Choosing c < min(r7pi, ^-), we obtain Integrating, we get where p < E,(t)L. L < L0exp(- I E5(t)dt), Jo L= I p(H'2 + u 2 + r?2 + y*)dx. Jn Since Lo=0, L=0 in the interval 0 < t < T. This forces u=H'=77=7=0 in 0 < t < T. Finally, since T is arbitary, they vanishes as long as they exist. For instance, if G=0, one has uniqueness in the class: p > 0, p G L°°(QT), Vp G L2(0, T\ I°°(fi)), v G l ° ° ( Q r ) , VveI2(0 ,Tii»(fi)), 6 G L°°(QT), V6 G I2(0, T; I°°(fi)), V H G L°°(QT), H G I2(0, T; I°°(n)), having assumed that b G L ^ T ; ! 0 0 ^ ) ) , r G Ll(Q,T; L°°(Q)), Pp, Pe, Cv G C 1 , H, C, K € C 1 . Q.E.D. Chapter 4. UNIQUENESS FOR COMPRESSIBLE MAGNETOHYDRODYNAMICS 52 4.2 UNIQUENESS T H E O R E M FOR T H E CASE p. > ft > 0, C > 0, K = 0, IJ > 0 The governing equations for this case can be reduced from (1.20)-(1.24) to the form a 3 o iii HH ^ + ( v V ) v - b ] = -VP- rg[Z) i(pI) lv) + Dl(/iVt;i)]+V[(C-^/i)d.-H-<?rad—+div—, (4.12) ^ + pdivv = 0, (4.13) g p [ ?J + v . V0] = ~ ^ d i w + ±pr+J^Yl (Dkv< + A . 1 ) 2 + £r(C ~ |^)(^v) 2 , (4.14) ^ = -Hdiuv + H • gradv + rjAU. (4.15) at With the initial conditions v| ( =o = v 0, H|t=o = H 0 , p\t=o = po, 6\t-o = 80, we have the following result for this problem. Theorem 4.2 Lei ihe coefficients of viscosity, the heat conductivity and the magnetic diffusiviiy satisfy the inequalities P- > A<i > 0- C > 0, K = 0, rj > 0. TAen i/iere can be almost one solution of the equations (4-12)-(4-15) satisfying the following boundary conditions: (a) Ai all points of dQ, v and H are prescribed. (b) At all points where U < 0, p and 8 are prescribed. Proof. Since the difference between this case and the preceding case lies entirely in the energy equation, the inequlities (4.8), (4.9) and (4.10) hold as before. On the other hand, the energy equation is exactly the same as it is in §2.3. Hence the inequality (2.15) holds here as well. Adding the inequalities (2.15), (4.8), (4.9) and (4.10), one obtains ^ J\p(u2 + r,2 + T 2 + H'*)dx < Ml[(pUjU)+(pr}>rl) + (p1,y) + (pH',H')}-- l<fW + PV7 + P"7]Vds + * Jan + e(V • u, V • u) + c(Vu, Vu) + e(VH', VH') -- f)px (VH', VH') - ft (Vu, Vu) - ^-(V • u, V • u), Chapter 4. UNIQUENESS FOR COMPRESSIBLE MAGNETOHYDRODYNAMICS 53 where E6(t) = Ei{t) + E2{t) + E3(t) + C6(0-Choosing t < min(^-, f)pi) and solving the resulting differential inequality, we obtain uniqueness in the class (for G=0): p>0, peL°°(QT), Vp6L 2 (0,T;L°°(n)) , veI°°(Qr), Vv€l 2 (0,r;L°°(f i )) , • 6eL°°(QT), v J e i ' ^ r i i * ^ ) ) , VHeI°°(Qr), H € L2(0,T;L°°(fi)), having assumed that b e i ^ o . T ; ! 0 0 ^ ) ) , rei 1 (0 ) r ;Lo o (n ) ) > . Pp, Pe, Cv G C 1, Q.E.D. 4.3 UNIQUENESS T H E O R E M FOR T H E CASE p = 0, C > Ci > 0, « > K i > 0, 5? > 0 The governing equations for this case take the form PllZ- + ( v • V ) v " b) = " V P + VlCA'uv] - ffrad^ + d u S , (4.16) OT 87T 47T ^ + pdivv = 0, (4.17) 3 P[% + v • ™ ] = - ^ ^ d i « v + i - g Dk(KDk0) + ^-pr + ^ C ( * « v ) 2 , (4.18) ^ =-Hdfrv + H-ffradv + rJAH. (4.19) dt With the initial conditions v|t=o = vo, H, =o = Ho, p|t=o = Po, 0|<=o = ^ o, we have the following result. Chapter 4. UNIQUENESS FOR COMPRESSIBLE MAGNETOHYDRODYNAMICS 54 Theorem 4.3 Let the coefficients of viscosity, heat conductivity and magnetic diffusivity satisfy the inequalities Then there can be atmost one solution of the equations (J.16)-(J.19) satisfying the the following boundary conditions: (a) Ai all points of dQ, v n and H are prescribed. (b) At all points where U < 0, v, p and 6 are prescribed. (c) At all points where U > 0, 6 is given. Proof. Because ofthe similarity between Theorem 4.3 and Theorems 2.5 and 4.1, the inequalities (4.8), (4.9) and (2.29) hold exactly as before. For the equation of motion we obtain the following inequality from (4.16) in the usual way: /J = 0, C > C i > 0. « > / c i > 0 , rj>0. - C i ( V u , V u ) + e(VH',VH') (4.20) where f?7(0 = C4(0 + ||H||i, + | |VH| | . Adding the inequalities (2.29), (4.8) and (4.20), and choosing e < min(£i, ^,fjpi), one obtains where E6(t) = Eiit) + E3(t) + E7(t) + C7(t). Finally, solving this differential inequality, one obtains uniqueness in the class (for G=0): P>0, peL°°(QT), VpGl 2 (0 ,T;L~(n)), V€I°°(QT), V v 6 L 2 ( 0 , T ; L»( f i ) ) , 0 € L°°(QT), VO G I2(0,T; L°°(fi)), V H G L°°(QT), H G I 2(0, T\ i ° ° ( n ) ) D26 £Ll{b,T;L°°{Q)), Chapter 4. UNIQUENESS FOR COMPRESSIBLE MAGNETOHYDRODYNAMICS 55 having assumed that b € L\0, T; r £ I l (0, T; I°°(n)), Pp i Rf i Cv € C 1 , Q.E.D. 4.4 UNIQUENESS T H E O R E M FOR T H E CASE p. = 0, C > C i > 0, * = 0, r? > 0 The governing equations in this case take the form />I?7 + (v • V)v - b] = - V P + V[C<«H - grad^- + d n ^ , (4.21) ^ + pdivv = 0, (4.22) p[g + v . V<?) = ~ ^ d i w + ±-pr + ^ C ( ^ v ) 2 , (4.23) ^ = -Hdt'vv + H-pradv + rjAH. (4.24) dt With the initial conditions v|« =o = v0, H| t =o = Ho, p\t=o = Po, #|t=o = o^, we have the following result for this case. Theorem 4.4 Let the coefficients of viscosity, heat conductivity and magnetic diffusivity satisfy the inequalities M = 0, C > C i >0, K = 0, j j>0. Then there can be at most one solution of ihe equations (4-21)-(4-24) satisfying the following boundary conditions: (a) Ai all points of dQ, v • n and H are prescribed. (b) At all points where U < 0, v, p and 6 are prescribed. Proof. The difference between this problem and the problem considered in §2.4 lies entirely in the momentum equation and the equation for H. Hence the inequalities (2.25) and (2.9) hold as before. For Chapter 4. UNIQUENESS FOR COMPRESSIBLE MAGNETOHYDRODYNAMICS 56 the equation (2.24), the inequality (4.8) holds as it does in §4.1. Finally, for the momentum equation we derive the following inequality from (4.21) in the usual manner: £-J^lpu2dx < Es(t) JnP(u> + rl* + 7*)dx+e-(V.u,V•u) + + e(VH',VH')-Ci(V-u,V-u), (4.25) where ^ ( O s C ^ O + IIHHL + IIVHlloo. Adding the inequalities (2.9), (2.25), (4.8) and (4.25), we obtain ^JJ-p^ + y'+tf + H'^dx < Bl0(t)J p{u2 + ^ + ^i2 + H'2)dx-Cl^•n,V•u)-- i jMVH', VH') + e(V • u, V • u) + e(VH'.VH'), where Eio(t) = C2(i) + C5(t) -1- £i(<) + E9(t). Choosing t < min(£i, r)p\) and solving the resulting inequality, one obtains uniqueness in the class (for G=0): p>0, p€ i~(Qr) , Vp6L2(0,T;L°°(Q)), vGL°°(Qr), Vv6l 2 (0 ,T;L~(fi)), 8-€Lee(QT), V0€l 2(O ,T;L°°(n)), V H € L°°(QT), H € I2(0, T; I~(fi)), having assumed that b G L ' iCr; !"^) ) , r G I^O, T\ L°°(fi)), P p, P$, Cv G C 1 p, C, " G C 1 . Q.E.D. Chapter 4. UNIQUENESS FOR COMPRESSIBLE MAGNETOHYDRODYNAMICS 57 4.5 UNIQUENESS T H E O R E M FOR INVISCID COMPRESSIBLE M H D We now state the uniqueness theorem for inviscid compressible magnetohydrodynamics under the initial and boundary conditions of §1.5. The solutions under consideration are assumed to be sufficiently smooth that all the norms used in estimating them are finite. Theorem 4 .5 Let the coefficients of viscosity, heat conductivity and magnetic diffusiviiy satisfy X = y = K = 0, rj> 0. Suppose thai ihe equation of state P=P(p,S) is of class C^ for all p and S, and suppose that Then there can be at most one solution io the initial boundary value problem described in §7 .5 . Moreover, if for some solution U = v n - G > y/P^ on a subset dQ* of ihe boundary, then the boundary conditions on dQ* are redundant with respect to the given data; that is, the conditions assigned on ihe remainder of the boundary, together with the given initial values, uniquely determine the solution. Proof. Let (v,p,S) and (v,p, S, H) be two solutions satisfying the above mentioned conditions and equations and set H' = H — H, u = v — v, n = p — p, j3 = S — S. Since both flows satisfy the equation (1.28), we have by subtraction: />(§? + v - V u + u V v ) + » ; ( ^ - r v V v ) = T 7 b - V P ' - ^ - ( V H ' ) H -at ot • 4T - ^ ( V H ) H ' + - ^ H ( V H ' ) + -^H'-(VH). 4ir 4ir 4TT Multiplication by u gives P§j(\»2) + p u - V v u + » ? u ^ = n u b - u V P ' - i ; ; { p u ( V H ' ) H + p u ( V H ) H ' -- p H ( V H ' ) u - p H ' ( V H ) u ) . Using the transport formula (3.3), one obtains / \pu*dx = - [ n (u^- -b )dx- f pu V v u d x - / u -VP'dx- I \pu2Uds-Dt 2 Jn Di Jn Jn Jan 2 - - W pu (VH') Hdx - — f pu (VH) • H'dx + ± - [ pH (VH') • udx + 4TT Jn 4TT Jn 4ir Jn + — / pH' (VH) • udx. (4.26) 4* Jn Chapter 4. UNIQUENESS FOR COMPRESSIBLE MAGNET0HYDR0DYNAM1CS 58 Inserting (3.10) into (4.26), we obtain j^J ^pu2dx = - ( n ( ^ - b ) , u ) - ( p u V v , u ) + j P„Vp • undx + J P „ V S • undx -+ ft* „ „ * _ /n ^ + /n - /n I. -- / ^divvPpdx- f u • (P, - P,)Vpdx + / P„Vpu/?dx + Jn P Jn Jn + / P. .V5u/Hx - / V vSP.dx + / ^ d x + / * P Jn Jn P Jn P 0 1 Jn - j ^ u - V p d x - / L\Pldivvdx- [ u(P, - P,)VSdx -Jn P Jn P Jn - T - / pu • (VH') • H<fx - -L / pu (VH) • H'dx + 4?r J n 4?r y n -I- — / pH (VH') • udx + - i - / pH' • (VH) • udx. 4* Jn An Jn Rearranging the terms and estimating the right hand side, one obtains ^ / \PW + h«2 + 2^vfldx < G1(t)[(pu,U) + (pn,n) + (p^,^)-r(pH',H')]-Dt Jn 2 p1 pl - i i[p^("2+Pf'?2 + 24«/?) + 2(P,n + Jan 2 p2 pJ + P,/?)un]dS + e(VH',VH'), (4.27) where d ( 0 = C l l l ^ l l e o + l lb l loo + l l p l l o o l l V v l U + 11^11001^00 + + ||Pp,||oo||VS||oo + llAllooll J l l o o + H a l l o o + llAlloollVplloo + + H d i v v l l o o l l P p l l o o + I I P p l U . p H V p l l o o + HPp . l lool lVpl loo + + l|P»||oo | |V5||oo + I I V 5 I U I P . I U + H ^ H o o + IIP.IIooll^Hoo + + l lAllooHVplloo + llP.HoollditrvHoo + ||P.||z.ip||V5||eo + + ||H||L + ||VH||T O]. Turning next to the equation (4.4), the second term on the right hand side can be rewritten with the help of the identity f pH H'divudx = I pUH'uknkds- / H H'u Vpdx - / pu • (VH) • H'dx -Jn Jen Jn Jn Chapter 4. UNIQUENESS FOR COMPRESSIBLE MAGNETOHYDRODYNAMICS 59 - / pu • (VH') • Hdx. Jn Thus the equation (4.4) becomes / lpH'2dx = - I pu (VH) • H'dx - I pH H'uknkds + / H H'u Vpdx + Di Jn* Jn Jan Jn + j pu (VH) H'dx + [ pu (VH') • Hdx - / pK'7divvdx + in Jn Jn + f pH' (Vv) • H'dx - / (H u)(H • Vp)dx - / pH (VH') • udx -Jn Jn Jn - f fjVp- (VH') • H'dx - / rjpVH' : VH'dx - / \PH'2Uds. Jn Jn Jan * Estimating the terms appearing on the right hand side, we obtain ILJJ-pH'2dx < G2(t)[(pu,u) + (pH',H')] + | ( V H ' , V H ' ) -- ^ ( V H ' , V H ' ) - i / pH'2Uds, (4.28) * Jan where G2{t) = COIVHIU + IIHIU || V p l U + HVHHoo + ||H|& + \\divvU*, + + HVHHoo + HVplloollHlloo + Choosing M = max1<x(-2^- + jps), and multiplying (3.7) by M and adding it to (4.27), we obtain -%{\PW + + ^r"/3 + MP}) < G 3 ( 0 / PW + u2 + /?2 + H,2)dx -Dt 2 p-1 pl Jn - l-i[p(u2+Pjr]2-r^-r)0 + M02)Uds + 2 Jan P P + 2(P> + P,/?)un]ds-r|(VH',VH'), ( 4. 29) where G3(t) = G1(t) + ||VS||oc. Adding the inequalities (4.28) and (4.29), we get -g- / pLdx < G<(t) I p(r)2 + u2 + 02 + H'2)dx-Di Jn Jn - I I [pUL + 2(P> + P,p)u • n)ds + 2 Jan + (e-ifa)(VH\VH'), (4.30) Chapter 4. UNIQUENESS FOR COMPRESSIBLE MAGNETOHYDRODYNAMICS 60 where P op P P > u2 + H' 2+pz(r,7+p)>0 (4.31) and G4(t) = G3(t) + G2(t). Now define K = pf/L -(- 2(P,,n + AP*)U • n, on «9£2. Then it follows from the boundary conditions that K > 0 at all points of dQ. Moreover K > 0 on the set 5Q* irrespective of the boundary conditions. For on dQ* we have U > \fP~P, so that either K>pyf?pL + 2u(P pn + P,/?) or * >P\fp~PL-2u(Ppr,+ P,l3). Hence the required inequality follows. Since K > 0 the boundary integral in (4.30) is non negative. Thus if we choose t < fjp'i, the inequality (4.30) reduces to / pLdx < G4(t) I p(v7 + tz2 + /?2 + #'2)dx. Using (4.31), we get where §iJnpLdx < G*(t) J pLdx, G;(0 = G?4(0[I + 2 | | £ | | O O ] . Solving this differential inequality and then using the initial condition, we obtain L = 0. Finally, using (4.31), we obtain u=n=/?=H'=0. Hence one obtains uniqueness in the class (for G=0): p>0, peL">(QT), VP€L7(0,T;L°°(Q)), ^ei'to.Tinn)), v e i ° ° ( Q T ) , V v 6 I1(0,7,;I-oo(n)). ^ € L1(p,T;L°°(Q)), SeL°°(QT), VS£Ll(0,T;L°°(Q)), H € I2(0,T;L°°(Q)), VH G Lx(0,T;I°°(n)), Chapter 4. UNIQUENESS FOR COMPRESSIBLE MAGNETOHYDRODYNAMICS 61 having assumed that bGLHO.T;!, 0 0^)), P„,P.eC\ Q.E.D. 4.6 UNIQUENESS T H E O R E M FOR INVISCID HEAT C O N D U C T I V E M H D Here we state the uniqueness theorem for inviscid heat conductive compressible magnetohydrodynamics. Theorem 4.6 Let the coefficients of viscosity, heat conductivity and magnetic diffusivity satisfy X = p = 0, K > *i > 0, fj > 0. Suppose > 0. {dP)t Then there can be dt most one solution of the equations (l.S0)-(I.S4) satisfying the boundary conditions described in §1.6. Moreover, if for some solution of this problem we have U = v.n — G > \ J { ^ ) i o n a subset dQ* of the boundary, then the boundary conditions on dQ* are partly superfluous with respect to the given data; that is, the conditions assigned on the remainder of the boundary, together with the temperature on dQ* and the given initial conditions uniquely determine the solution. Proof. Let (v,p,0,H) and (v,p,0,H) be two solutions ofthe problem under discussion and set H' = H - H , u = v - v , n = p —p, 7 = 0 — 0. Applying the same methods developed in the preceding section, one obtains from the equation (1.33) §ij\p^dx = - ^ [ n u ( ^ - b ) + pu V v . u + u . V P l d x - £ J p u 2 l / d s -- -i- / pu (VH') • Hdx --j- f pu • (VH) • H'dx + 4ir Jn 4TT Jn + / p H ( V H ' ) u d x + - ^ - / p H ' ( V H ) u d x . (4.32) 47r Jn 4TT Jn Inserting (3.15) into (4.32), we obtain S/>u2 + T' 2 ) d X = - / n M ^ - b ) + pu.Vv.u]dx-- / (Pp - Pp)Vp udx - / (Pe - P«)V0 • udx -Jn Jn Chapter 4. UNIQUENESS FOR COMPRESSIBLE MAGNETO-HYDRODYNAMICS 62 _ / Pf!lix _ / £ ^ d x - / ^ d i . v d x + Jn P2 Jn P Jn P + / nuP^Vpdx + I nuP„,V0dx - — f pu (VH') • Hdx -Jn Jn 4ir Jn - I pu (VH) • H'dx + -J- / p¥L • (VH') • udx + 47T Jn 4n Jn + ^- f pH' • (VH) • udx. (4.33) 4 i r Jn Estimating the right hand side of this equation, we get P2 Di Sn ^ P("2 + "7" 2 ) < i X " G s ( 0 L P { U 7 + ^  + T' + H ' 2 ) d x ' - I hpU(u2+Z£) + 2Ppr,un]ds + Jan2 P + |(VH',VH') , (4.34) where G5(t) = C t H ^ l l o o + H b l U + I IPIIOOIIVVIIC + U P p l U i p H V p l l o o + ~ DP , + H^IU.pH^lloo -I- l l ^^ l loo -I" Halloo "I- I I ^ H o o l l V p l l o o + + ||J>,||oo||<tt«v||oo + H P p p l l o o l l V p j I o o + I I P ^ H o o l l V f l l l o e + | |H| | 2 r o + | |VH|| r o]. Adding the inequalities (4.34), (3.18) and (4.28), one obtains 757 / + ^ + ? + H")dx * G«(<) / + "2 + T2 + H'2)dx + Dt Jn 2 pl Jn + ( e _^-) (V 7 ,V7) + ( £ - ^ i ) ( V H ' , V H ' ) + / . k dy y ,. . 50. , JatiCvdn Cv d n Jan 2 P* + 2 P » 7 f u n ] d s , for 0 < f < T, where G6(<) = G5(i) + D5(i) + G2(<)- Now define ^ = f / ( u 2 - r ^ ! + 72 + tf'2) + 2 Pp"un + 2P 9 ^ 7 -un, on 5fi. p Cu Clearly # > 0 and f k dy y .60 , . JanCvdn r dn Chapter 4. UNIQUENESS FOR COMPRESSIBLE MAGNETOHYDRODYNAMICS 63 in virtue of the given boundary conditions. Moreover, on dQ*, K = U(P7u2 + + H'2) + 2P,nu • n and |2P ,nu -n |<yp; (pV+ P i^) , so that K > 0 on dQ* irrespective ofthe boundary conditions. Thus, we have the inequality ^ < G7(t)J + (e- g ) ( V 7 , V 7 ) + (c - W)(VH',VH'), where Jil2 and G 7 ( t ) = G6(t)[l + | | £ | | 0 O ] . Choosing e < min(^-,f}pi) and solving the resulting differential inequality leads to the identical van-ishing of u, n, f and H'. Hence one has uniqueness in the class (for G=0): p>0, P<LL°°{QT), Vp€L 2 (0 ,T;L°°(f i ) ) , ^ € L'^T; L°°(Q)), v e i ° ° ( Q T ) , V v e i 2 ( o , T ; L ~ ( f i ) ) , ^ e i ^ o . T j L - c n ) ) , 0 e I ° ° ( Q T ) , V0€ L ^ O . T j L 0 0 ^ ) ) , H € I2(0,T;L°°(n)), V H e l H O T ; ! 0 0 ^ ) ) , under the assumptions Cv, Pp, Ps, K € Cl, b € ^ ( O . r j X 0 0 ^ ) ) , r 6 . ^ ( O . T i L 0 0 ^ ) ) . Q.E.D. Bibliography [1] COWLING, T .G. : Magnetohydrodynamics. New York: Interscience Publishers 1957. [2] DOLIDZE, D.E.: Doklady Akadernii Nauk SSSR. 96, 437-439 (1954). [3] FERRARI, I.: Su un teorema di unicita per le equazioni dell' idromagnetismo Atti del Seminario Matematico e Fisico dell' Universita di Modena 9, 205-217 (1960). [4] FRIEDRICHS, K., k H .LEWY: Uber die Eindeutigkeit und das Abhongigkeitsgebiet der Losungen beim Anfangswertproblem linearer hyperbolischer Differentialgleichungen. Math. An-nalen 98, 192-204 (1927). [5] FOA, E.: Sull' impiego dell' analisi dimensionale nello studio del moto turbolento L'Industria 43, 426-429 (1929). [6] GRAFFI , D.: II teorema di unicita nella dinamica dei fluidi compressibili. J. Rational Mech. Anal. 2, 99-106 (1953). [7] H A D A M A R D , J. : Sur l'integrale residuelle. Bull. Societe Mathematique de France 28, 69-90 (1900). [8] H E Y W O O D , J.G.: On the uniqueness questions in the theory of viscous flow, Acta Math. 136(1976), 61-102. [9] K A N W A L R.P.: Uniqueness of magnetohydrodynamic flows. Arch. Rational Mech. Anal. 4, 335-340 (1960). [10] MISES, R.V.: Mathematical theory of compressible fluid flow. New York: Academic Press 1958. [11] NARDINI, R.: Due teoremi di unicita nella teoria delle onde magneto-idrodinamiche. Rend. Sem. Mat. Univ. Padova 21, 303-315 (1952). [12] ORR, W. McF.: The stability or instability of the steady motions of a liquid. Part II A viscous liquid. Proc. Royal Irish Acadamy (A) 27, 69-138 (1907). [13] REYNOLDS, O.: On the dynamical theory of incompressible viscous fliuds and the determina-tion ofthe criterion. Philosophical Transactions Royal Society of London (A) 186, 123-164 (1895). [14] SERRIN, J.B.: Mathematical principles of classical fluid mechanics. Handbuch der Physik, Bd. 8. Berlin-Gottingen-Heidelberg: Springer- Verlag, 1959. [15] SERRIN, J.B.: On the uniqueness of compressible fluid motions. Arch. Rational Mech.Anal. 3, 271-288 (1959). [16] T R U E S D E L L , C : The mechanical foundations of elastisity and fluid dynamics, J. Rational Mech. k Analysis, vol.1, 1952. [17] A L B E R T O VALLI: Uniqueness theorems for compressible viscous fluids, especially when the stokes relation holds. Bollettino U.M.I., Analisi Funzionale e Applicazioni, Serie v, Vol. XVII-C, N.1-1981. 64 Bibliography 65 [18] Z A R E M B A , S.: Sopra un teorema d'unicita relative- alia equazione delle onde spheriche. Rendi-conti, Accademia Nazionale dei Lincei (5) 24, 904- 908 (1915). 

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