Well-posedness for hyperbolic systems of conservation laws with large BV data. (English) Zbl 1065.35189
The author studies the Cauchy problem for a strictly hyperbolic \(n\times n\) system of conservation laws \(u_t+f(u)_x=0\). It is required that each characteristic field of this system is either genuinely nonlinear or linearly degenerate. The initial data \(u(0,x)=\bar u(x)\) is assumed to have bounded but possibly large total variation. Under some linearized stability condition the author utilizes the wave-front-tracking algorithm and proves existence and uniqueness of a (local in time) BV-solution. Moreover, the Lyapunov functional is constructed, which yields existence of a Lipschitz continuous flow of solutions. The last section contains some applications to the system of gas dynamics.
Reviewer: Evgeniy Panov (Novgorod)
MSC:
| 35L65 | Hyperbolic conservation laws |
| 35L45 | Initial value problems for first-order hyperbolic systems |
| 76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |
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