On Liouiville Type Theorem for the 3D Isentropic Navier-Stokes System without D-condition
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914234958348288 |
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| author | Jiu, Quansen Tan, Jie Yan, Zhihong |
| author_facet | Jiu, Quansen Tan, Jie Yan, Zhihong |
| contents | In this paper, we establish Liouville-type theorems for the steady compressible Navier-Stokes system. Assuming a smooth solution \(u \in L^p(\mathbb{R}^3)\), \(3 \le p \le \frac{9}{2}\), with bounded density, one obtains \(u \equiv0\). This generalizes the result of Li-Yu \cite{Li-Yu} by removing the Dirichlet condition \(\int_{\mathbb{R}^3} |\nabla u|^2 \, dx < \infty\). If \(\frac{9}{2} < p < 6\), Liouville-type theorem holds under the additional oscillation condition for momentum \(ρu \in \dot{B}^{\frac{3}{p} - \frac{3}{2}}_{\infty,\infty}(\mathbb{R}^3)\). For the marginal case \(u \in L^6(\mathbb{R}^3)\), the oscillation condition can be replaced by \(ρu \in BMO^{-1}(\mathbb{R}^3)\). We also present results in Morrey-type spaces: \(u \in \dot{M}^{s,6}(\mathbb{R}^3)\) and \(ρu \in \dot{M}_w^{q,3}(\mathbb{R}^3)\) for \(2 \le s \le 6\) and \(\frac{3}{2} < q \le 3\). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_02791 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Liouiville Type Theorem for the 3D Isentropic Navier-Stokes System without D-condition Jiu, Quansen Tan, Jie Yan, Zhihong Analysis of PDEs 35Q30, 76N10 In this paper, we establish Liouville-type theorems for the steady compressible Navier-Stokes system. Assuming a smooth solution \(u \in L^p(\mathbb{R}^3)\), \(3 \le p \le \frac{9}{2}\), with bounded density, one obtains \(u \equiv0\). This generalizes the result of Li-Yu \cite{Li-Yu} by removing the Dirichlet condition \(\int_{\mathbb{R}^3} |\nabla u|^2 \, dx < \infty\). If \(\frac{9}{2} < p < 6\), Liouville-type theorem holds under the additional oscillation condition for momentum \(ρu \in \dot{B}^{\frac{3}{p} - \frac{3}{2}}_{\infty,\infty}(\mathbb{R}^3)\). For the marginal case \(u \in L^6(\mathbb{R}^3)\), the oscillation condition can be replaced by \(ρu \in BMO^{-1}(\mathbb{R}^3)\). We also present results in Morrey-type spaces: \(u \in \dot{M}^{s,6}(\mathbb{R}^3)\) and \(ρu \in \dot{M}_w^{q,3}(\mathbb{R}^3)\) for \(2 \le s \le 6\) and \(\frac{3}{2} < q \le 3\). |
| title | On Liouiville Type Theorem for the 3D Isentropic Navier-Stokes System without D-condition |
| topic | Analysis of PDEs 35Q30, 76N10 |
| url | https://arxiv.org/abs/2601.02791 |