Research on well-posedness for the Navier-Stokes equations

在域Ω R^n中，考虑u∈L^∞（0，T ; L^n（Ω））类中Navier-Stokes方程的弱解u。如果limsup_<t-t_*-0><$u（t）<$^n-<$u（t_*）<$^n_n在t_*∈（0，T）的每一点都很小，则u在Ω^^-×（0，T）上正则.作为应用，我们给出了奇异时间的精确刻画，即，我们证明了如果Navier-Stokes方程的解u在初始时是光滑的，并且在以后的某个时刻T_*<T时失去了正则性，那么要么lim_<t-T_*-0> u（t）= +∞，要么u（t）在L^n（Ω）中以足够大的振幅围绕弱极限w-lim_<t-T_*-0>u（t）振荡.进一步证明了（0，T）上有界变差的弱解u在L^n（Ω）中是正则的.考虑Ω×（0，T）中的非定常Navier-Stokes方程，其中Ω是R^3中的一个区域.本文证明了存在一个绝对常数ε_0，使得在D×（a，B）的任一紧致子集上的时空变量中，具有性质sup_&lt;t∈（a，B）&gt; ≠ Ω且0&lt;a&lt;B&lt;T。作为应用，我们证明了如果弱解u在（x_0，t_0）∈Ω×（0，T）如u（x，t）=o（|x-x_0|当<-1>x→x_0在t_0的某个邻域内在t内一致时，则（x_0，t_0）是u的可去奇点.考虑Serrin类Navier-Stokes方程w ∈L^α（0，∞ ; L^q（Ω）），其中2/α + 3/q = 1，3&lt;q[小于或等于]∞，Ω是R^3中的一般无界区域.我们将证明，尽管来自w的初始扰动和外部扰动都很大，但当ω（t）-w（t）ω_&lt;L^2（Ω）&gt;→0时，每个满足能量不等式的扰动流u都渐近收敛于w，当t→∞时，ω（t）-ω w（t）ω_&lt;L^2（Ω）&gt;=O（t^&lt;-1/2&gt;）.

In a domain Ω⊂R^n, consider a weak solution u of the Navier-Stokes equations in the class u∈L^∞ (0, T ; L^n (Ω)). If lim sup_<t-t_*-0>‖u (t) ‖^n_n-‖u (t_*) ‖^n_n is small at each point of t_*∈ (0, T), then u is regular on Ω^^-× (0, T). As an application, we give a precise characterization of the singular time, i.e., we show that if a solution u of the Navier-Stokes equations is initially smooth and loses its regularity at some later time T_*<T, then either lim sup_<t-T_*-0>‖u (t) ‖_<L^n (Ω) >= +∞, or u (t) oscillates in L^n (Ω) around the weak limit w-lim_<t-T_*-0>u (t) with sufficiently large amplitude. Furthermore, we prove that every weak solution u of bounded variation on (0, T) with values in L^n (Ω) becomes regular.Consider the nonstationary Navier-Stokes equations in Ω× (0, T), where Ω is a domain in R^3. We show that there is an absolute constant ε_0 such that every weak solution u with the property sup_<t∈ (a, b) >‖u (t) ‖^3_W (D) 【less than or equal】ε_0 is necessarily of class C^∞ in the space-time variables on any compact subset of D× (a, b), where D ⊂⊂Ω and 0<a<b<T.As an application, we prove that if the weak solution u behaves around (x_0, t_0) ∈Ω× (0, T) like u (x, t) =o (|x-x_0|^<-1>) as x→x_0 uniformly in t in some neighborhood of t_0, then (x_0, t_0) is a removable singularity of u.Consider weak solutions w of the Navier-Stokes equations in Serrin's classw∈L^α (0, ∞ ; L^q (Ω)) for 2/α + 3/q = 1 with 3<q【less than or equal】∞,where Ω is a general unbounded domain in R^3. We shall show that although the inital and exteral disturbances from w are large, every perturbed flow u with the energy inequality converges asymptotically to w as‖υ (t) -w (t) ‖_<L^2 (Ω) >→0, ‖▽υ(t) -▽w (t) ‖_<L^2 (Ω) >=O (t^<-1/2>) as t→∞.