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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.

35L65 Hyperbolic conservation laws
35L45 Initial value problems for first-order hyperbolic systems
35F10 Initial value problems for linear first-order PDEs
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