Global existence for hyperbolic-parabolic systems with large periodic initial data (Q1979111)

This paper studies the solvability of the Cauchy problem for the one-dimensional hyperbolic-parabolic system \(u_t+f(u)_x=(B(u)u_x)_x\), \(u\in \mathbb{R}^n\) where the data are assumed to be periodic. For \(n=1\) Ladyzhenskaya, Solonnikov and Ural'tseva proved in 1988 the existence of unique smooth global solution for any \(L^{\infty }\) periodic initial data. In this paper, the author proves that assuming the appropriate growth conditions on \(f\), the existence of an entropy \(E\) of the system, and under natural properties of the matrix \(B\) and entropy \(E\), the system in question has for locally \(L^2\) periodical initial condition a global solution in a weak sense. The conditions under which the solution is unique are specified. The result is obtained by the method of combining the local existence result and suitable a~priori estimates. The result is then applied to the Keyfitz-Kranzer system. The paper is clearly written and the result is interesting.