Weak-strong uniqueness and energy-variational solutions for a class of viscoelastoplastic fluid models
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 b...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2022-10-01
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| Series: | Advances in Nonlinear Analysis |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/anona-2022-0274 |
| _version_ | 1797989414961414144 |
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| author | Eiter Thomas Hopf Katharina Lasarzik Robert |
| author_facet | Eiter Thomas Hopf Katharina Lasarzik Robert |
| author_sort | Eiter Thomas |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-11T08:19:52Z |
| format | Article |
| id | doaj.art-14fe4b2e20ba40ac94d95d6000a52d3f |
| institution | Directory Open Access Journal |
| issn | 2191-950X |
| language | English |
| last_indexed | 2024-04-11T08:19:52Z |
| publishDate | 2022-10-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Advances in Nonlinear Analysis |
| spelling | doaj.art-14fe4b2e20ba40ac94d95d6000a52d3f2022-12-22T04:34:59ZengDe GruyterAdvances in Nonlinear Analysis2191-950X2022-10-0112120521610.1515/anona-2022-0274Weak-strong uniqueness and energy-variational solutions for a class of viscoelastoplastic fluid modelsEiter Thomas0Hopf Katharina1Lasarzik Robert2Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, GermanyWeierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, GermanyWeierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, GermanyWe 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.https://doi.org/10.1515/anona-2022-0274viscoelastic fluidsviscoplasticityweak-strong uniquenessrelative energy inequalitynonsmooth potentialvanishing stress diffusion35k6135q3535q8676a1076d03 |
| spellingShingle | Eiter Thomas Hopf Katharina Lasarzik Robert Weak-strong uniqueness and energy-variational solutions for a class of viscoelastoplastic fluid models Advances in Nonlinear Analysis viscoelastic fluids viscoplasticity weak-strong uniqueness relative energy inequality nonsmooth potential vanishing stress diffusion 35k61 35q35 35q86 76a10 76d03 |
| title | Weak-strong uniqueness and energy-variational solutions for a class of viscoelastoplastic fluid models |
| title_full | Weak-strong uniqueness and energy-variational solutions for a class of viscoelastoplastic fluid models |
| title_fullStr | Weak-strong uniqueness and energy-variational solutions for a class of viscoelastoplastic fluid models |
| title_full_unstemmed | Weak-strong uniqueness and energy-variational solutions for a class of viscoelastoplastic fluid models |
| title_short | Weak-strong uniqueness and energy-variational solutions for a class of viscoelastoplastic fluid models |
| title_sort | weak strong uniqueness and energy variational solutions for a class of viscoelastoplastic fluid models |
| topic | viscoelastic fluids viscoplasticity weak-strong uniqueness relative energy inequality nonsmooth potential vanishing stress diffusion 35k61 35q35 35q86 76a10 76d03 |
| url | https://doi.org/10.1515/anona-2022-0274 |
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