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We consider the Riemann problem composed of two shocks for the 1D Eulersystem. We show that the Riemann solution with two shocks is stable and uniquein the class of weak inviscid limits of solutions to the Navier-Stokesequations with initial data with bounded energy. This work extends to the caseof two shocks a previous result of the authors in the case of a single shock.It is based on the method of weighted relative entropy with shifts known as$a$-contraction theory. A major difficulty due to the method is that verylittle control is available on the shifts. A modification of the constructionof the shifts is needed to ensure that the two shock waves are well separated,at the level of the Navier-Stokes system, even when subjected to largeperturbations. This work put the foundations needed to consider a large familyof interacting waves. It is a key result in the program to solve theBianchini-Bressan conjecture, that is the inviscid limit of solutions to theNavier-Stokes equation to the unique BV solution of the Euler equation, in thecase of small BV initial values.

Well-posedness of the Riemann problem with two shocks for the isentropic Euler system in a class of vanishing physical viscosity limits