Discontinuous solutions for the Navier-Stokes equations with density-dependent viscosity
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866913504094584832 |
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| author | Zodji, Sagbo Marcel |
| author_facet | Zodji, Sagbo Marcel |
| contents | We prove existence of a unique global-in-time weak solutions of the Navier-Stokes equations that govern the motion of a compressible viscous fluid with density-dependent viscosity in two-dimensional space. The initial velocity belongs to the Sobolev space $H^1(\mathbb{R}^2)$, and the initial fluid density is $α$-Hölder continuous on both sides of a $\mathscr{C}^{1+α}$-regular interface with some geometrical assumption. We prove that this configuration persists over time: the initial interface is transported by the flow to an interface that maintains the same regularity as the initial one.
Our result generalizes previous known of Hoff [21], Hoff and Santos [22] concerning the propagation of regularity for discontinuity surfaces by allowing more general nonlinear pressure law and density-dependent viscosity. Moreover, it supplements the work by Danchin, Fanelli and Paicu [6] with global-in-time well-posedness, even for density-dependent viscosity and we achieve uniqueness in a large space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_07578 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Discontinuous solutions for the Navier-Stokes equations with density-dependent viscosity Zodji, Sagbo Marcel Analysis of PDEs 35R35, 35A02, 35Q30, 76N10 We prove existence of a unique global-in-time weak solutions of the Navier-Stokes equations that govern the motion of a compressible viscous fluid with density-dependent viscosity in two-dimensional space. The initial velocity belongs to the Sobolev space $H^1(\mathbb{R}^2)$, and the initial fluid density is $α$-Hölder continuous on both sides of a $\mathscr{C}^{1+α}$-regular interface with some geometrical assumption. We prove that this configuration persists over time: the initial interface is transported by the flow to an interface that maintains the same regularity as the initial one. Our result generalizes previous known of Hoff [21], Hoff and Santos [22] concerning the propagation of regularity for discontinuity surfaces by allowing more general nonlinear pressure law and density-dependent viscosity. Moreover, it supplements the work by Danchin, Fanelli and Paicu [6] with global-in-time well-posedness, even for density-dependent viscosity and we achieve uniqueness in a large space. |
| title | Discontinuous solutions for the Navier-Stokes equations with density-dependent viscosity |
| topic | Analysis of PDEs 35R35, 35A02, 35Q30, 76N10 |
| url | https://arxiv.org/abs/2312.07578 |