Well-posedness of the global entropy solution to the Cauchy problem of a hyperbolic conservation laws with relaxation (Q1433353)

The authors study the Cauchy problem for a hyperbolic conservation laws with relaxation \(u_t+\sigma_x=0\), \((\sigma-f(u))_t+(\sigma-\mu f(u))/\delta=0\), where \(\delta>0\) and \(\mu\in (0,1)\) are constants. The initial functions \(u(0,x)\), \(\sigma(0,x)\) are generally discontinuous. The function \(f(u)\) is not supposed to satisfy any convexity or monotonicity conditions. The authors introduce a notion of an entropy solution using the condition similar to the Kruzhkov entropy inequality known for scalar conservation laws. Under some regularity assumptions they prove existence and uniqueness of a global entropy solution to the problem under consideration.