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Un développement asymptotique pour la solution du problème de Riemann généralisé. (An asymptotic expansion for the solution of the generalized Riemann problem). (French) Zbl 0619.35074
Consider nonlinear hyperbolic systems of conservation laws with a source term $(1)\quad \partial_ tu+\partial_ xf(u,x,t)=g(u,x,t),\quad u(x,t)\in {\mathbb{R}}^ P,\quad x\in {\mathbb{R}},\quad t>0.$ Here, f and $$g: {\mathbb{R}}^ p\times {\mathbb{R}}\times {\mathbb{R}}_+\to {\mathbb{R}}^ P$$ are given smooth functions. The generalized Riemann problem is as follows: given two smooth functions $$u_ L: {\mathbb{R}}_-\to {\mathbb{R}}^ P$$ and $$u_ R: {\mathbb{R}}_+\to {\mathbb{R}}^ P$$, find an entropy weak solution of (1) both with the following piecewise smooth initial data: $(2)\quad u(x,0)=u_ L(x)\quad for\quad x<0,\quad u_ R(x)\quad for\quad x>0.$ For the solution of this problem (1)$$\cdot (2)$$, we propose an asymptotic expansion of the form $(3)\quad u(x,t)=\sum_{k\in {\mathbb{N}}}u^ k(\frac{x}{t})t^ k,$ and we derive explicit formulas for the functions $$u^ k: {\mathbb{R}}\to {\mathbb{R}}^ P$$. The function $$u^ 0$$ is the well-known solution of the classical Riemann problem $\partial_ tu^ 0+\partial_ xf(u^ 0,0,0)=0;\quad u^ 0(x,0)=u_ L(0)\quad for\quad x<0,\quad u_ R(0)\quad for\quad x>0.$ The main property of our construction of the expansion (3) is as follows: each function $$u^ k$$ for $$k\geq 1$$ is (as made by P. Lax for $$k=0)$$ characterized by $$(p+1)$$ constant vectors of $${\mathbb{R}}^ P$$ and algebraic linear equations are provided to determine these vectors.
Approximate solutions of (1)$$\cdot (2)$$ are defined from (3) and bounds of error are obtained. The details of the derivation will be published in Ann. Inst. Henri Poincaré, Nonlinear Analysis (1988).

MSC:
 35L65 Hyperbolic conservation laws 35C20 Asymptotic expansions of solutions to PDEs