Stability and Decay for the 2D Anisotropic Navier-Stokes Equations with Fractional Horizontal Dissipation on $\mathbb{R}^2$
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910151614660608 |
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| author | Wang, Zhibin Wu, Jiahong Zhu, Ning |
| author_facet | Wang, Zhibin Wu, Jiahong Zhu, Ning |
| contents | The stability problem for the 2D Navier-Stokes equations with dissipation in only one direction on $\mathbb R^2$ is not fully understood. This dissipation is in the intermediate regime between the fully dissipative Navier-Stokes and the inviscid Euler. Navier-Stokes solutions in $\mathbb R^2$ decay algebraically in time while Euler solutions can grow rather rapidly in time. This paper solves the fundamental stability and large-time behavior problem on the anisotropic Navier-Stokes with fractional dissipation $Λ_1^{2s}$ for all $0\leq s<1$. The case $s=1$ corresponds to the standard one directional dissipation $\partial_1^2$. Different techniques are developed to treat different ranges of fractional exponents: $0\leq s\leq \frac34$, $\frac34<s<\frac{11}{12}$, and $\frac{11}{12} \leq s <1$. The final range is the most difficult case, for which we introduce the spatial polynomial $A_2$ weights and exploit the boundedness of Riesz transforms on weighted $L^2$-spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_16110 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Stability and Decay for the 2D Anisotropic Navier-Stokes Equations with Fractional Horizontal Dissipation on $\mathbb{R}^2$ Wang, Zhibin Wu, Jiahong Zhu, Ning Analysis of PDEs The stability problem for the 2D Navier-Stokes equations with dissipation in only one direction on $\mathbb R^2$ is not fully understood. This dissipation is in the intermediate regime between the fully dissipative Navier-Stokes and the inviscid Euler. Navier-Stokes solutions in $\mathbb R^2$ decay algebraically in time while Euler solutions can grow rather rapidly in time. This paper solves the fundamental stability and large-time behavior problem on the anisotropic Navier-Stokes with fractional dissipation $Λ_1^{2s}$ for all $0\leq s<1$. The case $s=1$ corresponds to the standard one directional dissipation $\partial_1^2$. Different techniques are developed to treat different ranges of fractional exponents: $0\leq s\leq \frac34$, $\frac34<s<\frac{11}{12}$, and $\frac{11}{12} \leq s <1$. The final range is the most difficult case, for which we introduce the spatial polynomial $A_2$ weights and exploit the boundedness of Riesz transforms on weighted $L^2$-spaces. |
| title | Stability and Decay for the 2D Anisotropic Navier-Stokes Equations with Fractional Horizontal Dissipation on $\mathbb{R}^2$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2601.16110 |