Global-in-time well-posedness for the two-dimensional incompressible Navier-Stokes equations with freely transported viscosity coefficient
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Liao, Xian Zimmermann, Rebekka |
| author_facet | Liao, Xian Zimmermann, Rebekka |
| contents | We establish the global-in-time well-posedness of the two-dimensional incompressible Navier-Stokes equations with freely transported viscosity coefficient, under a scaling-invariant smallness condition on the initial data. The viscosity coefficient is allowed to exhibit large jumps across $W^{2,2+ε}$-interfaces.
The viscous stress tensor $μSu$ is carefully analyzed. Specifically, $(R^\perp\otimes R):(μSu)$, where $R$ denotes the Riesz operator, defines a ``good unknown'' that satisfies time-weighted $H^1$-energy estimates. Combined with tangential regularity, this leads to the $W^{1,2+ε}$-regularity of another ``good unknown'', $(\barτ\otimes n):(μSu)$, where $\barτ$ and $n$ denote the unit tangential and normal vectors of the interfaces, respectively. These results collectively provide a Lipschitz estimate for the velocity field, even in the presence of significant discontinuities in $μ$.
As applications, we investigate the well-posedness of the Boussinesq equations without heat conduction and the density-dependent incompressible Navier-Stokes equations in two spatial dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_06517 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global-in-time well-posedness for the two-dimensional incompressible Navier-Stokes equations with freely transported viscosity coefficient Liao, Xian Zimmermann, Rebekka Analysis of PDEs 35Q30, 76D03 We establish the global-in-time well-posedness of the two-dimensional incompressible Navier-Stokes equations with freely transported viscosity coefficient, under a scaling-invariant smallness condition on the initial data. The viscosity coefficient is allowed to exhibit large jumps across $W^{2,2+ε}$-interfaces. The viscous stress tensor $μSu$ is carefully analyzed. Specifically, $(R^\perp\otimes R):(μSu)$, where $R$ denotes the Riesz operator, defines a ``good unknown'' that satisfies time-weighted $H^1$-energy estimates. Combined with tangential regularity, this leads to the $W^{1,2+ε}$-regularity of another ``good unknown'', $(\barτ\otimes n):(μSu)$, where $\barτ$ and $n$ denote the unit tangential and normal vectors of the interfaces, respectively. These results collectively provide a Lipschitz estimate for the velocity field, even in the presence of significant discontinuities in $μ$. As applications, we investigate the well-posedness of the Boussinesq equations without heat conduction and the density-dependent incompressible Navier-Stokes equations in two spatial dimensions. |
| title | Global-in-time well-posedness for the two-dimensional incompressible Navier-Stokes equations with freely transported viscosity coefficient |
| topic | Analysis of PDEs 35Q30, 76D03 |
| url | https://arxiv.org/abs/2409.06517 |