Asymptotic Behaviour Of Solutions Of Matrix Burgers Equation

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1 Applied Mathematics E-Notes, 727, c ISSN Available free at mirror sites of amen/ Asymptotic Behaviour Of Solutions Of Matrix Burgers Equation Kayyunnapara Thomas Joseph Received 13 July 26 Abstract In this paper, we study the large time behaviour of solutions of initial value problem for a system of parabolic equations which can be written as Matrix Burgers equation. This work generalizes a result of Hopf 195 for the Burgers equation. 1 Introduction In this paper we consider a system of parabolic partial differential equations for the unknown variables u 1,u 1,u 2,..., u N of the form u j t j u i u j i+1 x = 2 u j xx,j =1, 2,..., N 1 i=1 in <x<, t>, with initial conditions at time t =, u j x, = u j x,j =1, 2,..., N. 2 Here > is the viscosity coefficient. The system 1 is called Matrix Burgers equation because it can be written as A t A2 x = 2 A xx. 3 where A =A i,j isan N lower triangular matrix with the i, j-th entry A i,j takes values from {u 1,u 2,..., u N }: A i,j =,j >i,a i+k,i = u k+1,k=1, 2,..., N i, i =1, 2,..., N. The system 1 has a rich mathematical structure and solutions of special cases of this system were studied previously by many authors in the contexts of vanishing viscosity and large time behaviours. The singularity in the solutions of 1 as goes to zero increases as N increases. Mathematics Subject Classifications: 35B4, 35L6. School of Mathematics, TIFR, Homi Bhabha Road, Mumbai 45, India. 186

2 K. T. Joseph 187 When N = 1 the equation 1 is the Burgers equation, u t +1/2u 2 x = 2 u xx. Hopf [2] used the Hopf-Cole transformation u = v x /v to reduce it into the one dimensional heat equation v t = 2 v xx and solved the initial value problem for the Burgers equation explicitly. Letting go to in the formula, he derived an explicit formula for the entropy solution for the inviscid Burgers equation u t +1/2u 2 x = with bounded measurable initial data. For = in 1, the standard theory of Lax [7] does not apply when N>1. Joseph [3] studied the cases N = 2 to construct solution for the Riemann problem for the system of conservation laws u 1 t +1/2u 2 1 x =, u 2 t +u 1 u 2 x =, whose solution does not belong to the L class but may contain δ measures as well. When N = 3, the system 1 is more singular than the cases N = 1, 2 in the passage to the it as goes to zero and was studied in [4] and solutions were constructed in the algebra of generalized functions of Colombeau [1]. Hopf [2], analyzed another aspect of solution of 1 for the case N = 1, that is the asymptotic behaviour of solutions of Burgers equation for large time with >. This result was generalized for the case N = 3 in [6]. In this paper we extend Hopf s work [2] on the large time behaviour of solutions of the Burgers equation to the solutions of the system 1 with initial data 2. For notational convenience we drop the explicit dependence of on the solutions. 2 Explicit Solution and Asymptotic Form An explicit formula for the solution of the initial value problem for 1 and 2 was obtained by Joseph and Vasudeva Murthy [5], using a generalized form of Hopf-Cole transformation. To give this explicit formula we need some notations. Let us consider the polynomial px =a 1 x + a 2 x a N 1 x N 1 of degree N 1 with real coefficients a 1,a 2,..., a N 1. We define E j and L j as coefficients of x j 1 in the power series expansion around x = of the functions exppx and log1 + px respectively. More precisely d j E j+1 a 1,a 2,..., a j = 1 j! dx j [epx ] x=,j =, 1, 2,... 4 and are polynomial of a 1,a 2,..., a j,a j, of degree j. For example the first four coefficients are given by E 1 =1,E 2 a 1 =a 1,E 3 a 1,a 2 =a 2 + a12 2!,E 4a 1,a 2,a 3 =a 3 + a 2 a a 1 3!,E 5a 1,a 2,a 3,a 4 =a 4 + a 3 a 1 + a2a1 2 2! + a a1 4 4!. Similarly d j L j+1 a 1,a 2,..., a j = 1 j! dx [log1 + px] x=,j =1, 2,... 5 j Here again it follows from 5 that L j+1 is a polynomial of a 1,a 2,..., a j of degree j. For example the first four coefficients are given by L 2 a 1 =a 1,L 3 a 1,a 2 =a 2 a !,L 4a 1,a 2,a 3 =a 3 a 1 a 2 + a1 3,L 5a 1,a 2,a 3,a 4 =a 4 a 1 a 3 + a 2 1 a 2 a2 For j =1, 2,...N, let us denote w j x = x 2 2 a u j ydy, 6

3 188 Matrix Burgers Equation E j x =E j w2 y, w 3y,..., w jy where E j is defined by 4. Define the functions v j,j =1, 2,..., N by 1 v j x, t = 2πt 1/2 R 1 E j ye 1 7 [w1y+ x y2 2t ] dy. 8 Since E 1 x =1,v 1 x, t >. Assume that u j x, j =1, 2, N be measurable functions on R 1 with w j defined by 6 satisfy w j x = o x 2, as x. Then it was proved in [5], that u j defined by u 1 x, t = xv 1 x, t v 1 x, t u j x, t = x L j v2 x, t v 1 x, t, v 3x, t v 1 x, t,... v jx, t v 1 x, t 9,j =2, 3,..., N 1 is a classical solution to 1 and 2. The aim of this paper is to study the asymptotic behaviour of this solution as t tends to infinity. We introduce the variable ξ = x/ t and a function which appear in the asymptotic form of the solution, namely V j ξ = E w2 j + E w2 j,..., wj,..., wj e w 1 e w 1 ξ e y2 /2 dy e y2 /2 dy. ξ 11 Since E 1 =1,V 1 ξ >. The main result of the paper is the following. THEOREM. Assume that the initial data u j x,j =1, 2,..., N are integrable, then the solution u j x, t,j =1, 2,..., N of 1 and 2 given by 9 and 1 has the following asymptotic behaviour as t tends to infinity: t/.u1 x, t = d dξ logv 1ξ 12 t/.uj x, t = d V2 ξ L j dξ V 1 ξ, V 3ξ V 1 ξ,..., V jξ,j =2, 3,..., N, 13 V 1 ξ uniformly with respect to the variable ξ = x t. PROOF. To prove this theorem, we write the formula 1 for the solution in a convenient way, namely j 1 v2 u j x, t = k L j,..., v j x v k+1 v k+1 x v 1 v 1 v 1 v 1 v 1 v 1 k=1 where k L j denotes derivative of L j a 1,..., a j with respect to the k-th variable a k. Note that typical terms in these expressions are of the form v j x, t = 1 2πt 14 E j ye 1 x y2 [w1y+ 2t ] dy 15

4 K. T. Joseph 189 and its partial derivative with respect to x, where E j x and w 1 x has finite it as x tends to infinity because of the integrability assumption on u j. Keeping the variable ξ = x/ t fixed, we make a change of variable z = tξ y t and renaming z as y, weget v j x, t = 1 2π E j tξ ye [ w 1 tξ y +y 2 /2] dy. 16 Now we split the integral in 16 in the following fashion. 2πvj x, t = ξ δ E j tξ ye [ 1 w1 tξ y+y 2 /2] dy + ξ+δ E j tξ ye [ 1 w1 tξ y+y 2 /2] dy + ξ+δ ξ δ E j tξ ye [ 1 w1 tξ y+y 2 /2] dy. 17 Now we fix δ> and study each of these integrals as t tends to infinity, we get ξ δ ξ+δ E j tξ ye [ 1 wj tξ y+y 2 /2] dy = e w j E j ξ δ e y2 /2 dy, E j tξ ye [ 1 wj tξ y+y 2 /2] dy = e w j E j e y2 /2 dy, ξ+δ ξ+δ sup E j tξ ye 1 [wj tξ y+y 2 /2] dy = Oδ, ξ δ uniformly with respect to ξ. Now first let t tends to infinity and then δ tends to, in 17 we get ξ 2π.vj x, t =e w 1 E j e y2 /2 dy+e w 1 E j This it is valid uniformly for ξ R 1 and for the x-derivative we have 2πt. x v j x, t =E j e w 1 ξ e y2 /2 dy. 18 E j e w 1 e ξ2 /2. 19 In this analysis, we have used the fact that u j,j =1, 2,..., N are integrable and hence w j,w j,e j,e j exists. Indeed w j = E 1 = 1, and for j =2, 3,..., N u j ydy, w j = E j = E j w2 E j = E j w2 u j ydy 2,..., wj,..., wj 21

5 19 Matrix Burgers Equation Since V 1 ξ > from 11,18,19,2 and 21 we get, for j =1, 2,..., N v j x, t v 1 x, t = V jξ V 1 ξ, 22 x v j x, t.t = V jξ v 1 x, t V 1 ξ, 23 uniformly with respect to ξ. Here V j ξ means derivative of V 1 ξ with respect to ξ. From 9, 14, 22 and 23, we get t/u1 x, t = V 1ξ V 1 ξ, t/.uj x, t = j 1 k=1 kl j = d L V2ξ dξ j V2ξ V 1ξ, V3ξ V 1ξ V, V3ξ 1ξ V 1ξ Vjξ,..., V 1ξ,..., Vj ξ V 1ξ V k+1 ξ V 1ξ Vk+1ξ V 1ξ. V 1 ξ V 1ξ The proof of the theorem is complete. ACKNOWLEDGMENTS. This work is supported by a grant number from the Indo-French Centre for the promotion of advanced Research, IFCPAR Centre Franco-Indien pour la promotion de la Recherche Avancee, CEFIPRA, New Delhi. References [1] J. F. Colombeau, New Generalized Functions and Multiplication of Distributions, North Holland, Amsterdam, [2] E. Hopf, The partial differential equation u t +uu x = u xx, Comm. Pure Appl.Math., 3195, [3] K. T. Joseph, A Riemann problem whose viscosity solution contain δ-measures, Asym.Anal., 71993, [4] K. T. Joseph, Explicit generalized solutions to a system of conservation laws, Proc. Indian Acad. Sci. Math. Sci., , [5] K. T. Joseph and A. S. Vasudeva Murthy, Hopf-Cole transformation to some systems of partial differential equations, Nonlinear Diff. Equ. Appl., [6] K. T. Joseph, Asymptotic behaviour of solutions of a system of viscous conservation laws, Appl. Math. E-Notes, 525, [7] P. D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math., 11957,

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