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arXiv 2108.11940math.AP
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$3$ D各向异性Navier-斯托克斯方程解的大时间行为

Large time behavior of solutions to the $3$D anisotropic Navier-Stokes equation

MIkihiro Fujii

摘要:

考虑了具有水平粘性$Δ_{\rm h} u=\partial_1^2 u+\partial_2^2 u$的$3$ D Navier-斯托克斯方程解的大时间行为,并证明了速度场的水平分量的$L^p$衰减率与$2$ D热核的衰减率一致,而垂直分量的衰减率与$3$ D热核的衰减率相似.此外,我们考虑解的渐近展开,发现一部分非线性项影响速度场水平分量的首项,而垂直分量的首项仅由线性解给出。

英文摘要:

We consider the large time behavior of the solution to the $3$D Navier-Stokes equation with horizontal viscosity $Δ_{\rm h} u=\partial_1^2 u+\partial_2^2 u$ and show that the $L^p$ decay rate of the horizontal components of the velocity field coinsides to that of the $2$D heat kernel, while the vertical component decays like the $3$D heat kernel. Moreover, we consider the asymptotic expansion of the solution and find that a portion of the nonlinear term affect the leading term of the horizontal components of the velocity field, whereas the leading term of the vertical component is given by only the linear solution.

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