ASYMPTOTICS TOWARD THE PLANAR RAREFACTION WAVE FOR VISCOUS CONSERVATION LAW IN TWO SPACE DIMENSIONS

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1 TANSACTIONS OF THE AMEICAN MATHEMATICAL SOCIETY Volume 35, Number 3, Pages S -9947(999-4 Article electronically publishe on September, 1999 ASYMPTOTICS TOWAD THE PLANA AEFACTION WAVE FO VISCOUS CONSEVATION LAW IN TWO SPACE DIMENSIONS MASATAKA NISHIKAWA AND KENJI NISHIHAA Abstract. This paper is concerne with the asymptotic behavior of the solution towar the planar rarefaction wave r( x t connecting u + an u for the scalar viscous conservation law in two space imensions. We assume that the initial ata u (x, y tens to constant states u ± as x ±, respectively. Then, the convergence rate to r( x ofthesolutionu(t, x, y is investigate t without the smallness conitions of u + u an the initial isturbance. The proof is given by elementary L -energy metho. 1. Introuction We consier the Cauchy problem for the scalar viscous conservation law in two space imensions: (1.1 (1. u t + f(u x + g(u y = µ u, (t, x, y +, u(,x,y=u (x, y, where f an g are smooth functions, an µ is a positive constant. We assume that f is convex, i.e., (1.3 f (u α> for u, an that the initial ata is asymptotically constant: (1.4 u (x, y u ± as x ± for any fixe y, where u ± are constants satisfying u <u +. The asymptotic behavior as t of the solution is closely relate to that of the iemann problem for the corresponing hyperbolic conservation law in one space imension: (1.5 (1.6 r t + f(r x =, (t, x ( 1,, { r( 1,x=r(x u for x<, for x>. u + eceive by the eitors July 8, 1996 an, in revise form, October 14, Mathematics Subject Classification. Primary 35L65, 35L67, 76L5. Key wors an phrases. Nonlinear stable, viscous conservation law, planar rarefaction wave, L -energy metho. 13 c 1999 American Mathematical Society

2 14 MASATAKA NISHIKAWA AND KENJI NISHIHAA The entropy solution r(t, x of(1.5, (1.6 is given by u for x<f (u (t +1, r(t, x = (f 1 x (1.7 ( t+1 for f (u (t +1 x f (u + (t +1, for f (u + (t +1<x. u + The function (t, x, y r(t, x is calle the planar rarefaction wave. In a one imensional case, the asymptotic behaviors of solutions were originally investigate by Il in an Oleinik [3]. Harabetian [1] obtaine the convergence rate towar the rarefaction wave. Hattori an Nishihara [] showe more precise behaviors of the solution for the Burgers equation, employing the Hopf-Cole transformation. See also [5], [6], [7], [8], [1]. In a two imensional case, Xin [9] has first investigate the stability of the planar rarefaction wave. Ito [4] has recently shown the convergence rate towar the planar rarefaction wave. In both papers, the smallness of initial isturbance is essentially assume. In [4], the rarefaction wave is also assume to be weak. Our main purpose in this paper is to show that the solution u(t, x, y asymptotically behaves as r(t, x with the same rate as that in [4] without smallness conitions, which improves their results. Denote + = {(x, y ; x>}, ={(x, y ; x<}an D =( x, y. Then, our main theorem is as follows. Theorem 1. Suppose that u (x, y u ± L (± L1 (± an Dα u (x, y H 1 (, α =1. Then the problem (1.1,(1. has a unique global solution u(t, x, y satisfying (1.8 sup u(t,,y r(t, L ( x C(1 + t 1 4 log( + t, y where C is a positive constant epening on u. Our plan in this paper is as follows. In the next section, we construct a smooth rarefaction wave, which is ifferent from that in [4], an reformulate our problem. In the last two sections, we give the proofs of theorems for the reformulate problems.. Smooth approximation an preliminaries We first introuce the function w(t, x as a solution to the problem: (.1 w t + w w x =µ w xx, (t, x ( 1,, (. w( 1,x= r (x f (r(x. The Hopf-Cole transformation gives the information of the properties of w. Using w(t, x, we efine the smooth rarefaction wave w(t, x as (.3 w(t, x =(f 1 ( w(t, x t. Accoring to (1.3, w(t, x satisfies (.4 w t + f(w x = µw xx + µ f (w f (w w x, (t, x +, (.5 w(,x=w (x f ( w(,x. The properties of the smooth rarefaction wave w(t, x are state in the following lemma. From now on, we enote several constants by C or c without confusion.

3 ASYMPTOTICS TOWAD THE PLANA AEFACTION WAVES 15 Lemma 1 (Hattori an Nishihara []. The smooth rarefaction wave w(t, x given by (.3 satisfies the following properties: (i w(t, x u ± Cexp( c x, (ii w x (t, x >, (iii w x (t, L p ( (1 + t 1+ 1 p, wxx (t, L p ( (1 + t 1, (iv w(t, r(t, L p ( C(1 + t p 1 p. Since there is a forcing term f (w f (w w x in the equation (.4, we further introuce the smooth rarefaction wave U(t, x approximate to w, which satisfies (.6 (.7 U t + f(u x = U xx, (t, x +, U(,x=U (x f ( w(,x. The monotonicity in x of U(t, x was obtaine by Xin [9], which is important in the a priori estimates in 4. Lemma (Xin [9]. Suppose that U (x is monotonically increasing: (.8 t U (x >, x. Then, the solution U(t, x of (.6,(.7 satisfies (.9 Thus, setting x U(t, x >, (t, x +. u(t, x, y r(t, x ={w(t, x r(t, x} + {U(t, x w(t, x} + {u(t, x, y U(t, x} {w(t, x r(t, x} + v(t, x+v(t, x, y, we have reache two reformulate problems: (.1 (.11 an (.1 (.13 v t + {f(w + v f(w} x = µv xx µ f (w f (w w x, (t, x +, v(,x=u (x w(,x v (x, V t + {f(u + V f(u} x + g(u + V y = µ V, (t, x, y +, V (,x,y=u (x, y U (x V (x, y. The perturbations v an V satisfy the following theorems, respectively. Theorem (Decay estimate. Suppose that v H ( L 1 (. Then the problem (.1,(.11 has a unique global solution v(t, x satisfying an v C ([, ; H ( C ([, ; L 1 (, v x L (,T;H (, (.14 v(t, L ( C(1 + t 1 4 log( + t.

4 16 MASATAKA NISHIKAWA AND KENJI NISHIHAA Theorem 3 (Decay estimate. Suppose that V H ( L 1 (. Then, the problem (.1,(.13 has a unique global solution V (t, x, y satisfying V C ([, ; H (, V L (, ; H (, an (.15 sup V (t,,y L ( x C(1 + t 3 4. y Theorem, Theorem 3 an Lemma 1 (iv yiel the esire estimate (1.8. In the next two sections, we evote ourselves to the proofs of Theorems an 3, respectively. 3. Decay estimates for the perturbation v We begin with the Cauchy problem v t + {f(w + v f(w} x = µv xx µ f (w (3.1 f (w w x, (t, x +, (3. v(,x=u(,x w(,x v (x. We shall show that the problem (3.1,(3. has a unique global solution in the solution space X(,, where X M (,T= ψ ψ C ([,T]; H (, ψ x L (,T;H ( an sup ψ(t, H M. [,T ] In what follows, we often abbreviate the omain of H (, etc. Proposition 1 (Local existence. Suppose that v H (. For any M >, there exists a positive constant T epening on M such that if v H M, then the problem (3.1, (3. has a unique solution v(t, x X M (,T. Proposition 1 can be prove in a stanar way. So we omit the proof. Next, we show a priori estimates of v. Proposition (A priori estimate. Suppose that v is a solution of (3.1,(3. in X M (,T for positive constants T an M. Then there exists a positive constant C such that t t (3.3 v(t H + w x v xτ + v x (τ H τ C ( v H +1. Proof. Multiplying (3.1 by v, wehave (3.4 1 v x + v{f(w + v f(w} x x + µ vx t x = v f (w f (w w x x. The secon term of (3.4 is estimate by the following: v{f(w + v f(w} x x = v x {f(w + v f(w}x [ ( w+v ] (3.5 = f(yy f(wv + {f(w + v f(w f (wv}w x x w x α w x v x.

5 ASYMPTOTICS TOWAD THE PLANA AEFACTION WAVES 17 The right han sie is estimate as follows: v f (w f (w w xx C wx v x α w x v x + C w 4 xx. 3 Integrating (3.4 over [,t] an using Lemma 1 (iii, we get t t (3.6 v(t + w x v xτ + v x (τ τ C ( v +1. Here an later, by we enote the L -norm in or without confusions. Next, we erive the higher orer estimates. Multiplying (3.1 by ( v xx, we have 1 v t xx v xx {f(w + v f(w} x x + µ vxxx (3.7 f (w = µ v xx f (w w x x. The right-han sie is estimate as f (w v xx f (w w x x 1 4 v xx + C w x 4 L 4. The secon term of (3.7 is estimate as v xx {f(w + v f(w} x x µ { } 4 v xx + C v x (t + w x v x. Here, the maximum principle for a parabolic equation has been employe. Hence, we have (3.8 v x (t + t v xx (τ τ C ( v H Differentiating (3.1 twice in x, an multiplying it by v xx,wehave 1 t v xx(t + v xx {f(w + v f(w} xxx x + µ v xxx (t ( f (w = µ v xx f (w w x x, xx which yiels (3.9 v xx (t + t Thus, the proof of Proposition 3 is complete. v xxx (τ τ C ( v H +1. Combining Proposition 1 with Proposition, we obtain the global result. Theorem 4 (Global existence. Suppose that v (x H (. Then the problem (3.1,(3. has a unique global solution v(t, x satisfying an the estimate (3.3. v C ([, ; H (, v x L (, ; H (, In orer to obtain the ecay orer of v, we further assume that v L 1 (.

6 18 MASATAKA NISHIKAWA AND KENJI NISHIHAA Lemma 3. Suppose that v L 1 ( H (. Then the solution v(t, x also satisfies (3.1 v(t L 1 v L 1 +C 1 log(1 + t, where C 1 is a constant epening on u + u. Proof. The L 1 -estimate (3.1 of v can be prove by the same metho as that in [4]. So we omit the proof. Theorem 5 (Decay estimate. Suppose that v H ( L 1 (. Then, for any <ε< 1, the solution v(t, x of (3.1,(3. satisfies (3.11 (1 + t k+ 1 +ε k xv(t + t (1 + τ k+ 1 +ε ( CI k (1 + t ε ρ k (t, k =,1, w x k x v(τ x + k x v x(τ τ (3.1 (1 + t +ε x v(t + t (1 + τ +ε ( CI (1 + t ε ρ (t, w x xv(τ x + xv x (τ τ where an I k =( v L 1 + v H k +1, k =,1, ρ =log ( + t, ρ 1 =log 1 ( + t, I =( v L 1 + v H , ρ =log 6 ( + t. emark. The estimate (3.11 with k =shows (.14 in Theorem. Proof. The proof is similar to one in Ito [4]. However, the smooth rarefaction wave w(t, x in [4] is ifferent from ours an its estimates are one for the linearize equation aroun w(t, x. Hence, we give the outline of the proof. First, we show (3.11 with k =. From (3.4 an Lemma 1 (iii, we have (3.13 t v(t + Multiplying (3.13 by (1 + t 1 +ε,wehave (3.14 w x v x + v x (t C(1 + t. t {(1 + t 1 +ε v(t } +(1+t 1 +ε ( C{(1 + t 1 +ε v(t +(1+t 3 +ε }. By the Gagliaro-Nirenberg inequality w x v x + v x (t (3.15 v(t C v(t 4 3 L 1 ( v x(t 3,

7 ASYMPTOTICS TOWAD THE PLANA AEFACTION WAVES 19 we obtain ( t {(1 + t 1 +ε v(t } +(1+t 1 +ε w x v x + v x (t C{(1 + t 1 +ε v(t 4 3 L 1 v x (t 3 +(1+t 3 +ε } 1 (1 + t 1 +ε v x (t + C{(1 + t 1+ε v(t L 1 +(1+t 3 +ε } 1 (1 + t 1 +ε v x (t + C{(1 + t 1+ε ( v L 1 + C 1 log (1 + t+(1+t 3 +ε }; that is, (3.16 ( t {(1 + t 1 +ε v(t } +(1+t 1 +ε w x v x + v x (t C{(1 + t 1+ε ( v L 1 + C 1 log (1 + t+(1+t 3 +ε }. Integrating (3.16 over [,t] in t, we obtain (3.11 with k =. Next, we erive (3.11 with k = 1. From (3.7, we have 1 (3.17 t v x(t v xx {f(w + v f(w} x x + µ v xx (t C(1 + t 3. Here v xx {f(w + v f(w} x x = 1 [ ] 1 f (w + vw x v xx + v3 x v xx {f (w + v f (w}w x x. Hence, ue to (1.3, we have t v x(t + α w x vxx + v xx (t { } (3.18 C v xx v w x x + v x 3 x +(1+t 3 1 { } v xx(t + C wx v x + v x (t 3 L 3 +(1+t 3. Multiplying (3.18 by (1 + t 3 +ε,wehave t {(1 + t 3 +ε v x (t } + α(1 + t 3 +ε w x vx x +(1+t3 +ε v xx (t (3.19 C {(1 + t 1 +ε v x (t +(1+t 3 +ε wxv x +(1 + t 3 +ε v x (t 3 +ε} L 3 +(1+t 3. Noting that (1 + t 3 +ε wx v x (1 + t 3 +ε w x (t L w x v x C(1 + t 1 +ε w x v x,

8 11 MASATAKA NISHIKAWA AND KENJI NISHIHAA an making use of (3.11 with k = an the Gagliaro-Nirenberg inequality (3. we obtain (1 + t 3 +ε vx (t + { C I (1 + t ε ρ + { C I (1 + t ε ρ + v x (t 3 L 3 ( C v xx(t 7 4 L ( v(t 5 4 L (, t t t (1 + τ 3 (α +ε (1 + τ 3 +ε v(τ 1 L ( τ } w x v x x + v xx(τ τ (1 + τ 3 +ε (I (1 + τ 1 ρ 5 τ which yiels (3.11 with k = 1. Finally, multiply (3.9 by (1 + t +ε an use (3.11. After several calculations, we can obtain the esire estimate (3.1. Though the etails are omitte, we cannot multiply (3.9 by (1 + t 5 +ε in our metho. Because we have the ecay orer w xx (t = O(t, not O(t 5 (cf. Ito [4]. Thus the proof is complete. 4. Decay estimates for the perturbation V In this section, we consier the Cauchy problem in two space imension: (4.1 V t + {f(u + V f(u} x + g(u + V y = V, (4. V (,x,y=v (x, y u (x, y U (x. The solution space is X M (,T= ψ ψ C ([,T]; H (, ψ L (,T;H ( an sup ψ(t,, H M, [,T ] with T>. Then we have Proposition 3 (Local existence. Suppose that V H (. For any M >, there exists a positive constant T epening on M such that if V H M, then the problem (4.1, (4. has a unique solution V (t, x, y X M (,T. Proposition 3 can be prove in a stanar way. So we omit the proof. Next, we show a priori estimates of V. Proposition 4 (A priori estimate. Suppose that V is a solution of (4.1,(4. in X M (,T for positive constants T an M. Then there exists a positive constant C 1 epening on V such that t t (4.3 V (t H + U x V xyτ + V (τ H τ C 1 V H. Proof. Multiplying (4.1 by V an integrating the resultant equation over,we have 1 t V (t + V {f(u + V f(u} x xy (4.4 + Vg(U +V y xy + µ V (t =. },

9 ASYMPTOTICS TOWAD THE PLANA AEFACTION WAVES 111 The secon an thir terms are, respectively, estimate as follows: V {f(u + V f(u} x xy = V x {f(u + V f(u}xy [ ( U+V = f(yy f(uv (4.5 U x + {f(u + V f(u f (UV }U x ] xy α U x V xy. Since U is inepenent of y, Vg(U+V y xy = V y g(u + V xy ( U+V (4.6 = g(ξξ xy =. y Using (4.5 an (4.6, we have the basic estimate t t (4.7 V (t + U x V xyτ + V (τ τ C V. The estimates of the erivatives in x, y of V can be obtaine similarly to those in Proposition. We omit the etails. The combination of Propositions 3 an 4 gives the global result. Theorem 6 (Global existence. Suppose that V (x H (. Then the problem (4.1,(4. has a unique global solution V (t, x, y satisfying V C ([, ; H (, V L (, ; H (. an the estimate (4.3. We now show the ecay estimates on V. As in Lemma 3, the following L 1 - estimate plays an important roll. Lemma 4 (Ito [4]. Suppose further, in Theorem 6, that V L 1 (. Then the solution V (t, x, y also satisfies (4.8 V (t L 1 ( V L 1 (. Applying Lemma 4, we have the following theorem. Theorem 7 (Decay estimate. Suppose that V (x, y H ( L 1 ( an let V (t, x, y be the solution of (4.1,(4.. Then, for any ε>, there exists a constant C>such that the following ecay estimates hol: (4.9 (1 + t 1+ε V (t t ( + (1 + τ 1+ε U x V (τ xy + V (τ τ C(1 + t ε ( V L 1 + V, U

10 11 MASATAKA NISHIKAWA AND KENJI NISHIHAA (4.1 (1 + t ε V x (t t ( + (1 + τ ε U x V x (τ xy + V x (τ τ C(1 + t ε log 4 ( + t( V L 1 + V H 1, (4.11 (1 + t +ε V y (t t ( + (1 + τ +ε U x V y (τ xy + V y (τ τ C(1 + t ε ( V L 1 + V H 1, (4.1 (1 + t ε V xx (t + t (1 + τ ε ( U x V xx (τ xy + V xx (τ C(1 + t ε log 8 ( + t( V L 1 + V H, τ (4.13 (1 + t 3 8 +ε V xy (t t ( + (1 + τ 3 8 +ε U x V xy (τ xy + V xy (τ τ C(1 + t ε log 8 ( + t( V L 1 + V H, (4.14 (1 + t 3+ε V yy (t t ( + (1 + τ 3+ε U x V yy (τ xy + V yy (τ τ C(1 + t ε ( V L 1 + V H. emark. From (4.9 an (4.11, the estimate (.15 in Theorem 3 is obtaine as sup V (t,,y C V(t,, V y (t,, y C(1 + t 1 1 = C(1 + t 3. Proof. From (4.4 (4.6, we get 1 t V (t + 1 (4.15 α U x V xy + µ V (t. Multiplying (4.15 by (1 + t 1+ε,wehave (4.16 t {(1 + t1+ε V (t } +(1+t 1+ε By the Gagliaro-Nirenberg inequality ( α U x V xy + V (t (1 + ε(1 + t ε V (t. V (t L ( C V (t L 1 ( V (t L (,

11 ASYMPTOTICS TOWAD THE PLANA AEFACTION WAVES 113 we obtain (4.17 t {(1 + t1+ε V (t } +(1+t 1+ε C(1 + t 1+ε V (t (1 + t ε 1 V (t L 1 ( ( α U x V xy + V (t (1 + t 1+ε V (t + C(1 + t ε 1 V (t L 1 ( (1 + t 1+ε V (t + C(1 + t ε 1 V L 1 (. Integrating (4.17 over [,t]int,weobtain(4.9. Next, we estimate V y an V x. First, multiplying y (4.1 by V y,wehave 1 t V y(t + V y {f(u + V f(u} xy xy ( V y g(u + V yy xy + µ V y (t =. The integration by parts gives: The secon an thir terms of (4.18 = 1 f (U + V U x Vy xy + 1 {f (U + V V x Vy + g (U + V Vy 3 }xy. Hence, 1 t V y(t + α U x Vy xy + µ V y(t (4.19 C ( V y 3 + V x V y xy. Since C V x V y xy C sup V x (t, x, y V y (t, x, L ( x y y C V x (t, x, 1 L ( V y xy(t, x, 1 L ( V y y(t, x, L ( x y µ 4 V xy(t L ( + C V x (t, x, 3 L ( V y y(t, x, 8 3 L ( x y µ 4 V xy(t L ( + C V x (t, x, 3 L ( V (t, x, 4 3 y L ( V y yy(t, x, 4 3 L ( x y µ 4 V y(t L ( + C V x (t, x, L ( V (t, x, y 4 L ( x y µ 4 V y(t L ( + C sup V (t, x, 4 L ( V y x(t L ( x µ 4 V y(t L ( + C V (t L ( y V x(t 4 L (

12 114 MASATAKA NISHIKAWA AND KENJI NISHIHAA an C V y 3 xy µ 4 V y(t L ( + C V (t L ( V x(t L ( V y(t L (, we have 1 t V y(t + α U x Vy xy + µ V y(t (4. C V (t L ( V x(t L ( ( V x(t L ( + V y(t L (. Noting that V (t L ( C(1 + t 1, we multiply (4. by (1 + t +ε an integrating it over [,t]toobtain(4.11. Secon, multiplying x (4.1 by V x. Then, after similar calculations to the above, we have 1 t V x(t + α U x V x xy + µ V x(t (4.1 C V (t L ( V x(t L ( ( V x(t L ( + V y(t L ( + C U x (t L U x V xy. Since U x (t L ( w x (t L (+ v x (t L ( C(1 + t 7 8 log 4 ( + t by virtue of Theorem 5, we can multiply (4.1 by (1+t ε,not(1+t +ε,toobtain (4.1. The estimates (4.1 (4.14 for the secon erivatives of V are obtaine by more complicate calculations than those for the first erivatives. We omit the etails. Thus the proof of Theorem 7 is complete. Acknowlegment The work of the secon author was supporte in part by Wasea University Grant for Special esearch Project 96A-199. eferences [1] E. Harabetian, arefactions an large time behavior for parabolic equations an monotone schemes, Comm. Math. Phys. 114 (1988, M 89:3584 [] Y.HattorianK.Nishihara,A note on the stability of the rarefaction wave of the Burgers equation, Japan J. Inust. Appl. Math. 8 (1991, M 91k:357 [3] A. M. Il in an O. A. Oleinik, Asymptotic behavior of solutions of the Cauchy problem for certain quasilinear equations for large time (ussian, Mat. Sb. 51 (196, M :11 [4] K. Ito, Asymptotic ecay towar the planar rarefaction waves of solutions for viscous conservation laws in several space imensions, Math. Moels Methos Appl. Sci. 6 (1996, M 97e:3516 [5] A. Matsumura an K. Nishihara, Asymptotics towar the rarefaction waves of the solutions of a one-imensional moel system for compressible viscous gas, Japan J. Appl. Math. 3 (1986, M 88e:35173 [6] A. Matsumura an K. Nishihara, Global stability of the rarefaction wave of a oneimensional moel system for compressible viscous gas, Comm. Math. Phys. 144 (199, M 93:7656 [7] A. Matsumura an K. Nishihara, Asymptotics towar the rarefaction wave of the solutions of Burgers equation with nonlinear egenerate viscosity, Nonlinear Analysis 3 (1994, M 95m:35168 [8] A. Szeppesy an K. Zumbrun, Stability of rarefaction waves in viscous meia, Arch. ational Mech. Anal. 133 (1996, M 97a:3519

13 ASYMPTOTICS TOWAD THE PLANA AEFACTION WAVES 115 [9] Z. P. Xin, Asymptotic stability of planar rarefaction waves for viscous conservation laws in several imensions, Trans. Amer. Math. Soc. 319 (199, M 9j:35138 [1] Z. P. Xin, Asymptotic stability of rarefaction waves for viscous hyperbolic conservation laws the two-moe case, J. Differential Equations 78 (1989, M 9h:35155 Department of Mathematics, School of Science an Engineering, Wasea University, Okubo, Shinjuku, Tokyo 169, Japan aress: masataka@mn.wasea.ac.jp School of Political Science an Economics, Wasea University Tokyo, 169-5, Japan aress: kenji@mn.wasea.ac.jp