半空间中Navier-Stokes方程在一类加权Lebesgue空间中的初边值问题
The IBVP for the Navier-Stokes equations in the half-space in a class of weighted Lebesgue spaces
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中文总结 AI 辅助
本文将Stokes系统初边值问题的加权Lebesgue空间框架推广到Navier-Stokes方程，证明了其光滑解的局部存在唯一性，得到L^q估计与速度场空间渐近行为。
中文摘要 AI 辅助
我们研究半空间中Navier-Stokes方程的初边值问题，其初值属于合适的加权Lebesgue空间。更准确地说，我们考虑与一个权函数相关的加权Lebesgue空间，该权函数定义为有限多个固定点距离的幂次乘积。该框架由A. P. Di Feola和V. Pane在之前的研究（J. Math. Anal. Appl. 558 (2026), 130390）中引入，用于研究Stokes系统的初边值问题。本文完成了该分析，同时将Maremonti和Pane（J. Math. Fluid Mech. 27 (2025), Art. 2）针对Navier-Stokes Cauchy问题得到的结果推广到初边值问题。我们证明了光滑解的局部存在性与唯一性，推导出q>n时的L^q估计，以及速度场的空间渐近行为。
英文摘要
We study the initial-boundary value problem for the Navier-Stokes equations in the half-space with initial data belonging to a suitable weighted Lebesgue space. More precisely, we consider a weighted Lebesgue space associated with a weight function defined as a product of powers of the distances from finitely many fixed points. This framework was introduced in a previous work by A. P. Di Feola and V. Pane (J. Math. Anal. Appl. 558 (2026), 130390) for the study of the initial-boundary value problem for the Stokes system. The present paper completes that analysis and, at the same time, generalizes to the initial-boundary value problem the results obtained by Maremonti and Pane (J. Math. Fluid Mech. 27 (2025), Art. 2) for the Navier-Stokes Cauchy problem. We prove the existence (local) and uniqueness of a smooth solution and derive $L^q$-estimates, with $q>n$, together with the spatial asymptotic behavior of the velocity field.
