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arXiv 2409.12918math.AP
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三维Navier-Stokes方程在$L^3$及邻近空间中的渐近稳定性

Asymptotic stability for the 3D Navier-Stokes equations in $L^3$ and nearby spaces

Zachary Bradshaw, Weinan Wang

AI总结:

本文证明三维Navier-Stokes方程在$L^3$及邻近空间(如Lorentz空间和弱-$L^3$闭包)中围绕小向量场(含Landau解)的渐近稳定性,并指出该性质在弱-$L^3$中一般不成立。

摘要:

我们给出了在弱-$L^3$中小的向量场(包括小的Landau解)周围$L^3$-渐近稳定性的简短证明。我们证明,渐近稳定性同样适用于严格介于$L^3$和弱-$L^3$之间的Lorentz空间范围内的向量场,以及弱-$L^3$中测试函数的闭包中的向量场。为了提供关于此问题的全面视角,我们通过一个反例观察到,Landau解的渐近稳定性通常不能推广到弱-$L^3$。

英文摘要:

We provide a short proof of $L^3$-asymptotic stability around vector fields that are small in weak-$L^3$, including small Landau solutions. We show that asymptotic stability also holds for vector fields in the range of Lorentz spaces strictly between $L^3$ and weak-$L^3$, as well as in the closure of the test functions in weak-$L^3$. To provide a comprehensive perspective on the matter, we observe that asymptotic stability of Landau solutions does not generally extend to weak-$L^3$ via a counterexample.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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