Relative energy and weak–strong uniqueness of a 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 nonconvex 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.

Existence and weak-strong uniqueness for global weak solutions for the viscoelastic phase separation model in three space dimensions

“…A related model is the Navier-Stokes-Korteweg system, the corresponding results can be found in [16]. The relative energy and the corresponding weak-strong uniqueness results for the viscoelastic phase separation model are given in our recent work [3]. The proof is valid in two space dimensions and conditionally valid in three space dimensions by requiring the existence of a suitable weak solution.…”

“…For the standard Lebesgue spaces, L p (Ω) the norm is denoted by • p . We use the standard notation for the Sobolev spaces and introduce the notation V := H 1 0,div (Ω) 3 , H := L 2 div (Ω) 3 . As usual these space are obtained as closure with respect to the L 2 , H 1 norm of the space of C 0,∞ ( Ω) function with zero divergence, respectively.…”

“…In principle, all relevant calculations are already done in [3]. We have to verify several regularity assumptions of the weak solution and compute the necessary regularity of the strong solutions.…”

“…In this work, we aim to generalize our two-dimensional results for a viscoelastic phase separation model from [4,5,3] to three-space dimensions. Furthermore, we will not consider the viscoelastic effects due to non-linear rheology, but focus only on the impact of the additional pressure on the Date: August 3, 2022.…”

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The aim of this work is to prove the global-in-time existence of weak solutions for a viscoelastic phase separation model in three space dimensions. To this end we apply the relative energy concept provided by [3]. We consider the case of regular polynomial-type potentials and positive mobilities, as well as the degenerate case with logarithmic potential and vanishing mobility.

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