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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2412.08622 |
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| _version_ | 1866909997203456000 |
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| author | Berkemeier, Stefanie Elisabeth |
| author_facet | Berkemeier, Stefanie Elisabeth |
| contents | We study long time behavior of shear-thinning fluid flows in $d \geq 3$ dimensions, driven by additive stochastic forcing of trace class, with power-law indices ranging from $1$ to $ \frac{2d}{d+2}$. We particularly focus on Leray-Hopf solutions, i.e. on analytically weak solutions satisfying energy inequality. Introducing a new kind of energy related functional into the technique of convex integration enables the construction of infinitely many such solutions that are probabilistically strong for a certain initial value. Furthermore, we provide global i time estimates which lead to the existence of infinitely many stationary and even ergodic Leray--Hopf solutions. These results represent the first construction of Leray-Hopf solutions in the framework of stochastic shear-thinning fluids within this range of power-law indices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_08622 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence and Non-Uniqueness of Ergodic Leray-Hopf Solutions to the Stochastic Power-Law Flows Berkemeier, Stefanie Elisabeth Analysis of PDEs We study long time behavior of shear-thinning fluid flows in $d \geq 3$ dimensions, driven by additive stochastic forcing of trace class, with power-law indices ranging from $1$ to $ \frac{2d}{d+2}$. We particularly focus on Leray-Hopf solutions, i.e. on analytically weak solutions satisfying energy inequality. Introducing a new kind of energy related functional into the technique of convex integration enables the construction of infinitely many such solutions that are probabilistically strong for a certain initial value. Furthermore, we provide global i time estimates which lead to the existence of infinitely many stationary and even ergodic Leray--Hopf solutions. These results represent the first construction of Leray-Hopf solutions in the framework of stochastic shear-thinning fluids within this range of power-law indices. |
| title | Existence and Non-Uniqueness of Ergodic Leray-Hopf Solutions to the Stochastic Power-Law Flows |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2412.08622 |