Relative energy and weak-strong uniqueness of the two-phase viscoelastic phase separation model

Abstract

The aim of this paper is to analyze a viscoelastic phase separation model. We derive a suitable notion of the relative energy taking into account the non-convex nature of the energy law for the viscoelastic phase separation. This allows us to prove the weak-strong uniqueness principle. We will provide the estimates for the full model in two space dimensions. For a reduced model we present the estimates in three space dimensions and derive conditional relative energy estimates.Application of this technique to viscoelastic equations can be found in [26,4] . Maximal dissipative solutions applying the weakstrong uniqueness were analysed in [23] .The aim of this paper is to develop a relative energy concept for the viscoelastic phase separation model. We will further derive the weak-strong-uniqueness principle for the viscoelastic phase separation model. For the two-dimensional case the weak-strong uniqueness result is unconditional. Furthermore, we state the corresponding results in three space dimensions requiring the existence of a global weak solution. Finally, by combination with the results of [4] we derive a conditional weak-strong uniqueness principle for the full model in three space dimensions.