Weak-strong uniqueness and energy-variational solutions for a class of viscoelastoplastic fluid models. (English) Zbl 1504.35313
Summary: We study a model for a fluid showing viscoelastic and viscoplastic behavior, which describes the flow in terms of the fluid velocity and a symmetric deviatoric stress tensor. This stress tensor is transported via the Zaremba-Jaumann rate, and it is subject to two dissipation processes: one induced by a nonsmooth convex potential and one by stress diffusion. We show short-time existence of strong solutions as well as their uniqueness in a class of Leray-Hopf-type weak solutions satisfying the tensorial component in the sense of an evolutionary variational inequality. The global-in-time existence of such generalized solutions has been established in a previous work. We further study the limit when stress diffusion vanishes. In this case, the above notion of generalized solutions is no longer suitable, and we introduce the concept of energy-variational solutions, which is based on an inequality for the relative energy. We derive general properties of energy-variational solutions and show their existence by passing to the nondiffusive limit in the relative energy inequality satisfied by generalized solutions for nonzero stress diffusion.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 35Q86 | PDEs in connection with geophysics |
| 35K61 | Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations |
| 76A10 | Viscoelastic fluids |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 35D30 | Weak solutions to PDEs |
| 35D35 | Strong solutions to PDEs |
Keywords:
viscoelastic fluids; viscoplasticity; weak-strong uniqueness; relative energy inequality; nonsmooth potential; vanishing stress diffusionReferences:
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