标题： 二维不可压缩Navier-Stokes方程在类似通道区域上的全局吸引子的正则性 显示英文标题

标题： Regularity of the global attractor for the 2D incompressible Navier-Stokes equations on channel-like domains

摘要： 考虑二维区域上不可压缩Navier-Stokes方程的全局吸引子的正则性。 假设域$\Omega$是一个类似通道的域，即任意有界或无界域，在最初没有任何关于其边界的正则性假设，仅假设在其上满足Poincaré不等式。 系统相空间$H$是光滑无散向量场在$L^2(\Omega)^2$范数下的通常闭包，这些向量场在$\Omega.$中具有紧支集。通过相对于$H^1(\Omega)^2$范数得到的相应空间记为$V.$。假设强迫项属于对偶空间$V'.$。在这种情况下已知全局吸引子存在于相空间$H.$中。本文表明全局吸引子也是$V$中的紧集，并且由于方程的正则化效应，解在$V$范数下对于在$H$中有界初始条件一致地收敛到吸引子。 此外，还证明了如果强迫项属于$D(A^{-s})$，对于某些$0<s\leq 1/2$，其中$A$是Stokes算子，那么全局吸引子在$D(A^{-s+1})$中是紧的。 如果强迫项属于$H,$对应的极限情况$s=0,$，则进一步假设区域是均匀的$\Ccal^{1,1}$光滑区域或有界 Lipschitz 区域，以得到全局吸引子在$D(A).$中是紧的 显示英文摘要

摘要： The regularity of the global attractor of the incompressible Navier-Stokes equations for flows on two-dimensional domains is considered. It is assumed that domain $\Omega$ is a channel-like domain, i.e an arbitrary bounded or unbounded domain, at first without any regularity assumption on its boundary, with the only assumption that the Poincar\'e inequality holds on it. The phase space $H$ for the system is the usual closure in the $L^2(\Omega)^2$ norm of the space of smooth divergent-free vector-fields with compact support in $\Omega.$ The corresponding space obtained as the closure with respect to the $H^1(\Omega)^2$ norm is denoted by $V.$ The forcing term is assumed to belong to dual space $V'.$ It is known in this case that the global attractor exists in the phase space $H.$ It is shown in this work that the global attractor is also a compact set in $V$, and that due to the regularization effect of the equations, the solutions converge to the attractor in the norm of $V$, uniformly for initial conditions bounded in $H$. Moreover, it is shown that if the forcing term belongs to $D(A^{-s})$, for some $0<s\leq 1/2$, where $A$ is the Stokes operator, then the global attractor is compact in $D(A^{-s+1})$. If the forcing term is in $H,$ corresponding to the limit case $s=0,$ then it is further assumed that the domain is either a uniformly $\Ccal^{1,1}$ smooth domain or a bounded Lipschitz domain in order to obtain that the global attractor is compact in $D(A).$