On the \(L^1\) well posedness of systems of conservation laws near solutions containing two large shocks
The authors consider the Cauchy problem for a strictly hyperbolic \(n\times n\) system of conservation laws \[ u_t+f(u)_x=0, \;u(0,x)=\bar u(x). \] Each characteristic field is assumed to be either linearly degenerate or genuinely nonlinear. It is known that for the initial data \(\bar u\in (L^1\cap BV)({\mathbb R},{\mathbb R}^n)\) with suitably small total variation the entropy solutions of the Cauchy problem exist and constitute a semigroup which is Lipschitz continuous with respect to time and initial data. Well-posedness of the Cauchy problem for arbitrarily large initial data remains an open problem. The authors investigate this problem in the model case when the initial function is close to the Riemann data \((u_0^-,u_0^+)\) and the corresponding Riemann problem is solved by two large shocks. They discuss existence of global admissible solutions to the Cauchy problem, and stability of this solutions under small perturbations of its initial data. The so-called finiteness and stability conditions are formulated. It is shown that the first condition guarantees existence of admissible solution when \(\bar u\) stays close to the Riemann data, while the second one is essential for the stability, yielding the existence of a Lipschitz semigroup of entropy solutions. For construction of solutions the authors make use of the wave front tracking algorithm.
scientific article; zbMATH DE number 1744187
scientific article