Free boundary regularity of vacuum states for incompressible viscous flows in unbounded domains
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| Format: | Preprint |
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2023
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| _version_ | 1866909084217769984 |
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| author | Prange, Christophe Tan, Jin |
| author_facet | Prange, Christophe Tan, Jin |
| contents | In the well-known book of Lions [{\em Mathematical topics in fluid mechanics. Incompressible models}, 1996], global existence results of finite energy weak solutions of the inhomogeneous incompressible Navier-Stokes equations (INS) were proved without assuming positive lower bounds on the initial density, hence allowing for vacuum. Uniqueness, regularity and persistence of boundary re\-gularity of density patches were listed as open problems. A breakthrough on Lions' problems was recently made by Danchin and Mucha [The incompressible Navier-Stokes equations in vacuum, {\em Comm. Pure Appl. Math.}, 72 (2019), 1351--1385] in the case where the fluid domain is either bounded or the torus. However, the case of unbounded domains was left open because of the lack of Poincaré-type inequalities. In this paper, we obtain regularity and uniqueness of Lions' weak solutions for (INS) with \emph{only bounded and nonnegative initial density} and additional regularity only assumed for the initial velocity, in the whole-space case $\mathbb R^d$, $d=2$ or $3$. In particular, our result allows us to study the evolution of a vacuum bubble embedded in an incompressible fluid, as well as a patch of a homogeneous fluid embedded in the vacuum, which provides an answer to Lions' question in the whole-space case. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_09288 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Free boundary regularity of vacuum states for incompressible viscous flows in unbounded domains Prange, Christophe Tan, Jin Analysis of PDEs 35A02, 35Q30, 35R35, 76D05, 76D27 In the well-known book of Lions [{\em Mathematical topics in fluid mechanics. Incompressible models}, 1996], global existence results of finite energy weak solutions of the inhomogeneous incompressible Navier-Stokes equations (INS) were proved without assuming positive lower bounds on the initial density, hence allowing for vacuum. Uniqueness, regularity and persistence of boundary re\-gularity of density patches were listed as open problems. A breakthrough on Lions' problems was recently made by Danchin and Mucha [The incompressible Navier-Stokes equations in vacuum, {\em Comm. Pure Appl. Math.}, 72 (2019), 1351--1385] in the case where the fluid domain is either bounded or the torus. However, the case of unbounded domains was left open because of the lack of Poincaré-type inequalities. In this paper, we obtain regularity and uniqueness of Lions' weak solutions for (INS) with \emph{only bounded and nonnegative initial density} and additional regularity only assumed for the initial velocity, in the whole-space case $\mathbb R^d$, $d=2$ or $3$. In particular, our result allows us to study the evolution of a vacuum bubble embedded in an incompressible fluid, as well as a patch of a homogeneous fluid embedded in the vacuum, which provides an answer to Lions' question in the whole-space case. |
| title | Free boundary regularity of vacuum states for incompressible viscous flows in unbounded domains |
| topic | Analysis of PDEs 35A02, 35Q30, 35R35, 76D05, 76D27 |
| url | https://arxiv.org/abs/2310.09288 |