Global well-posedness and self-similar solution of the inhomogeneous Navier-Stokes system
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866912138967121920 |
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| author | Hao, Tiantian Shao, Feng Wei, Dongyi Zhang, Ping Zhang, Zhifei |
| author_facet | Hao, Tiantian Shao, Feng Wei, Dongyi Zhang, Ping Zhang, Zhifei |
| contents | In this paper, we study the global well-posedness of the 3-D inhomogeneous incompressible Navier-Stokes system (INS in short) with initial density $ρ_0$ being discontinuous and initial velocity $u_0$ belonging to some critical space. Firstly, if $ρ_0u_0$ is sufficiently small in the space $\dot{B}^{-1+\frac{3}{p}}_{p,\infty}(\mathbb{R}^3)$ and $ρ_0$ is close enough to a positive constant in $L^\infty$, we establish the global existence of strong solution to (INS) for $3<p<\infty$ and provide the uniqueness of the solution for $3<p<6$. This result corresponds to Cannone-Meyer-Planchon solution of the classical Navier-Stokes system. Furthermore, with the additional assumption that $u_0\in L^2(\mathbb{R}^3)$, we prove the weak-strong uniqueness between Cannone-Meyer-Planchon solution and Lions weak solution of (INS). Finally, we prove the global well-posedness of (INS) with $u_0\in \dot{B}^{\frac{1}{2}}_{2,\infty}(\mathbb{R}^3)$ being small and only an upper bound on the density. This gives the first existence result of the forward self-similar solution for (INS). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_00390 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global well-posedness and self-similar solution of the inhomogeneous Navier-Stokes system Hao, Tiantian Shao, Feng Wei, Dongyi Zhang, Ping Zhang, Zhifei Analysis of PDEs In this paper, we study the global well-posedness of the 3-D inhomogeneous incompressible Navier-Stokes system (INS in short) with initial density $ρ_0$ being discontinuous and initial velocity $u_0$ belonging to some critical space. Firstly, if $ρ_0u_0$ is sufficiently small in the space $\dot{B}^{-1+\frac{3}{p}}_{p,\infty}(\mathbb{R}^3)$ and $ρ_0$ is close enough to a positive constant in $L^\infty$, we establish the global existence of strong solution to (INS) for $3<p<\infty$ and provide the uniqueness of the solution for $3<p<6$. This result corresponds to Cannone-Meyer-Planchon solution of the classical Navier-Stokes system. Furthermore, with the additional assumption that $u_0\in L^2(\mathbb{R}^3)$, we prove the weak-strong uniqueness between Cannone-Meyer-Planchon solution and Lions weak solution of (INS). Finally, we prove the global well-posedness of (INS) with $u_0\in \dot{B}^{\frac{1}{2}}_{2,\infty}(\mathbb{R}^3)$ being small and only an upper bound on the density. This gives the first existence result of the forward self-similar solution for (INS). |
| title | Global well-posedness and self-similar solution of the inhomogeneous Navier-Stokes system |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2412.00390 |