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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.06527 |
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| _version_ | 1866916476620898304 |
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| author | Raees, Faiq Zhao, Weiren |
| author_facet | Raees, Faiq Zhao, Weiren |
| contents | In this paper, we prove the local well-posedness of a scaled anisotropic Navier-Stokes-Maxwell system in a two-dimensional striped domain with a transverse magnetic field around $ (0,0,1)$ in Gevrey-2 class. We also justify the limit from the scaled anisotropic equations to the associated hydrostatic system and obtain the precise convergence rate. Then, we prove the global well-posedness for the system and show that small perturbations near $(0,0,1)$ decay exponentially in time. Finally, we show the optimality of the Gevrey-2 regularity by proving the solution to linearized hydrostatic system around shear flows $(V(y),0,0)=(y(1-y),0,0)$ with some initial data $(ζ, ζ^1)$ grows exponentially. More precisely, for some large parameter $ \lvert k \rvert>M\gg 1 $ corresponding to the frequency in $x$, there exists a solution $ h_k(t,x,y)$ of the system
\begin{equation*}
\begin{cases} \partial_{tt}h_k+\partial_th_k-\partial_{yy}h_k+V(y) \partial_x h_k =0,\\
h_k(0,x,y)=ζ,\quad \partial_th_k(0,x,y)= ζ^1,\\
h_k(t, x,0)=h_k(t, x, 1)=0, \end{cases}
\end{equation*}
such that for any $s\in [0,\frac{1}{2})$ and $t\in [T_k,T_0)$ with $T_{k}\approx |k|^{s-\frac{1}{2}}\to 0$ as $|k|\to \infty$ and some $T_0$ small and independent of $k$, it satisfies
\begin{align*}
\lVert h_k(t) \rVert_{L^2 }\geq C \, e^{\sqrt{|k|}t}( \lVert ζ\rVert_{L^2} + \lVert ζ^1 \rVert_{L^2}),
\end{align*}
for some $C > 0$ independent of $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_06527 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On The Hydrostatic Approximation of Navier-Stokes-Maxwell System with 2D Electronic Fields Raees, Faiq Zhao, Weiren Analysis of PDEs In this paper, we prove the local well-posedness of a scaled anisotropic Navier-Stokes-Maxwell system in a two-dimensional striped domain with a transverse magnetic field around $ (0,0,1)$ in Gevrey-2 class. We also justify the limit from the scaled anisotropic equations to the associated hydrostatic system and obtain the precise convergence rate. Then, we prove the global well-posedness for the system and show that small perturbations near $(0,0,1)$ decay exponentially in time. Finally, we show the optimality of the Gevrey-2 regularity by proving the solution to linearized hydrostatic system around shear flows $(V(y),0,0)=(y(1-y),0,0)$ with some initial data $(ζ, ζ^1)$ grows exponentially. More precisely, for some large parameter $ \lvert k \rvert>M\gg 1 $ corresponding to the frequency in $x$, there exists a solution $ h_k(t,x,y)$ of the system \begin{equation*} \begin{cases} \partial_{tt}h_k+\partial_th_k-\partial_{yy}h_k+V(y) \partial_x h_k =0,\\ h_k(0,x,y)=ζ,\quad \partial_th_k(0,x,y)= ζ^1,\\ h_k(t, x,0)=h_k(t, x, 1)=0, \end{cases} \end{equation*} such that for any $s\in [0,\frac{1}{2})$ and $t\in [T_k,T_0)$ with $T_{k}\approx |k|^{s-\frac{1}{2}}\to 0$ as $|k|\to \infty$ and some $T_0$ small and independent of $k$, it satisfies \begin{align*} \lVert h_k(t) \rVert_{L^2 }\geq C \, e^{\sqrt{|k|}t}( \lVert ζ\rVert_{L^2} + \lVert ζ^1 \rVert_{L^2}), \end{align*} for some $C > 0$ independent of $k$. |
| title | On The Hydrostatic Approximation of Navier-Stokes-Maxwell System with 2D Electronic Fields |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2411.06527 |