The optimal time-decay estimates for 2-D inhomogeneous Navier-Stokes equations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929409314783232 |
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| author | Liu, Yanlin |
| author_facet | Liu, Yanlin |
| contents | In this paper, we derive the optimal time-decay estimates for 2-D inhomogeneous Navier-Stokes equations. In particular, we prove that $\|u(t)\|_{\dot{B}^θ_{p,1}({\mathop{\mathbb R\kern 0pt}\nolimits}^2)}={\mathcal O} (t^{\frac1p-\frac32-\fracθ2})$ as $t\rightarrow\infty$ for any $p\in[2,\infty[,~θ\in [0,2]$ if initially $ρ_0u_0\in \dot{B}^{-2}_{2,\infty}({\mathop{\mathbb R\kern 0pt}\nolimits}^2)$. This is optimal even for the classical homogeneous Navier-Stokes equations. Different with Schonbek and Wiegner's Fourier splitting device, our method here seems more direct, and can adapt to many other equations as well. Moreover, our method allows us to work in the $L^p$-based spaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_03752 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The optimal time-decay estimates for 2-D inhomogeneous Navier-Stokes equations Liu, Yanlin Analysis of PDEs In this paper, we derive the optimal time-decay estimates for 2-D inhomogeneous Navier-Stokes equations. In particular, we prove that $\|u(t)\|_{\dot{B}^θ_{p,1}({\mathop{\mathbb R\kern 0pt}\nolimits}^2)}={\mathcal O} (t^{\frac1p-\frac32-\fracθ2})$ as $t\rightarrow\infty$ for any $p\in[2,\infty[,~θ\in [0,2]$ if initially $ρ_0u_0\in \dot{B}^{-2}_{2,\infty}({\mathop{\mathbb R\kern 0pt}\nolimits}^2)$. This is optimal even for the classical homogeneous Navier-Stokes equations. Different with Schonbek and Wiegner's Fourier splitting device, our method here seems more direct, and can adapt to many other equations as well. Moreover, our method allows us to work in the $L^p$-based spaces. |
| title | The optimal time-decay estimates for 2-D inhomogeneous Navier-Stokes equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2407.03752 |