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Existence of solutions for a class of hyperbolic systems of conservation laws in several space dimensions. (English) Zbl 1061.35048
The authors study the Cauchy problem for the system of conservation laws $\partial_t u+\sum_{\alpha=1}^m \partial_{x_\alpha} f_\alpha(| u| )u=0, \quad u=(u_1,\ldots,u_n)\in {\mathbb R}^n,$ with initial condition $$u(0,\cdot)=\bar u$$. As was recently shown by Bressan, this problem can be ill-posed for general initial data $$\bar u\in L^\infty$$. The authors consider the case $$\bar u\in BV_{loc}({\mathbb R}^n)$$ and prove existence of the entropy solution. The radial part $$\rho=| u|$$ is required to be a Kruzhkov’s entropy solution for the scalar conservation law $\partial_t\rho+\sum_{\alpha=1}^m \partial_{x_\alpha} f_\alpha(\rho)\rho=0$ with corresponding initial data $$\rho(0,\cdot)=| \bar u|$$, while the angular part $$\theta=u/\rho$$ must satisfy the linear transport equation $\partial_t\rho\theta+\sum_{\alpha=1}^m \partial_{x_\alpha} f_\alpha(\rho)\rho\theta=0$ with coefficients $$(\rho,f_\alpha(\rho)\rho)\in BV_{loc}$$. The authors use the theory recently developed by the first author [Invent. Math. 158, No. 2, 227–260 (2004; Zbl 1075.35087)] to prove existence of weak solution $$\theta$$ of the transport equation, which satisfies the key condition $$| \theta| =1$$, thus proving existence of the entropy solution $$u=\rho\theta$$ of the original problem. The paper also contains some remarks about conditions of uniqueness and stability of entropy solutions.

##### MSC:
 35L65 Hyperbolic conservation laws 35L45 Initial value problems for first-order hyperbolic systems 35F10 Initial value problems for linear first-order PDEs
Zbl 1075.35087
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