The semigroup generated by a Temple class system with large data

Abstract

We consider the Cauchy problem $$ u_t+\big[F(u)\big]_x=0,\qquad u(0,x)=\bar u(x)\tag{*} $$ for a nonlinear $n\times n$ system of conservation laws with coinciding shock and rarefaction curves. Assuming the existence of a coordinates system made of Riemann invariants, we prove the existence of a weak solution of $(*)$ that depends in a Lipschitz continuous way on the initial data, in the class of functions with arbitrarily large but bounded total variation.