不可压缩纳维 - 斯托克斯方程强解在无穷远处的高阶渐近展开
High Order Asymptotic Expansion at Infinity for Strong Solutions to Incompressible Navier-Stokes Equations
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中文总结 AI 辅助
研究不可压缩纳维 - 斯托克斯方程速度方程与其涡度形式区别，通过推导展开式表明“良好局部化”初值产生的速度方程\(L^p\)强解在无穷远处行为，改进了前人一阶展开。
中文摘要 AI 辅助
我们讨论了全空间中不可压缩纳维 - 斯托克斯方程（速度方程）与其涡度形式之间有趣的区别。表明若初始涡度有高斯界，则在强解的最大寿命内该界会继承。但速度方程不具有此性质。“良好局部化”初值产生的速度方程的\(L^p\)强解在无穷远处的行为类似于拉普拉斯基本解的（\(\geq3\)阶）导数。为此推导了直至最大寿命的精确展开式。这改进了L. Brandolese和F. Vigneron给出的一阶展开。
英文摘要
We discuss an interesting distinction between the incompressible Navier-Stokes equations (the velocity equations) and its vorticity form in whole space. We show that if the initial vorticity has a Gaussian bound then the bound is inherited up to the maximal lifespan of the strong solution. However, it turns out that the velocity equations don not share the same property. In fact, $L^p$-strong solutions to the velocity equations arising from ``well-localized" initial value generally behave at infinity like derivatives (of order $\geq3$) of the fundamental solution of Laplacian. To show this, a clean expansion up to maximal lifespan is derived : \begin{align} u(x,t)=-\nabla\sum_{|α|=0}^{d-1}\frac{(-1)^{|α|}}{α!}\partial^α\partial_{i,j}^2Γ(x)\int_0^t{\rm M}_α^{i,j}(s){\rm d}s+O\big(|x|^{-2d-1}\big)\nonumber \end{align} where ${\rm M}_α^{i,j}(t):=\int_{\mathbb R^d}y^αu^i(y,t)u^j(y,t){\rm d}y$ and $Γ$ is the fundamental solution of Laplacian. This improves the first order expansion given by L. Brandolese and F. Vigneron \cite{BV07}.
