Research on well-posedness for the Navier-Stokes equations

在区域Ω⊂R^n中，考虑u∈L^∞(0，T；L^n(Ω))中的N-S方程的弱解u。如果LIM sup_&lt；t-t_*-0&gt；‖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划，即，如果N-S方程的解u最初是光滑的，并且在以后的某个时刻T_*&lt；T_*&lt；t-T_*-0&gt；‖u(T)‖_&lt；L^n(Ω)&gt；=+∞，或者u(T)在L^n(Ω)的弱极限w-Lim_&lt；t-T_*-0&gt附近振荡；U(T)具有足够大的幅度。考虑Ω×(0，T)中的非定常N-S方程，其中Ω是R^3中的区域，我们证明了存在一个绝对常数ε_0，使得每一个性质为∈(a，b)&gt；‖u(T)‖^3_W(D)[小于或等于]ε_0在D×(a，b)的任何紧子集上的时空变量中必然是C^∞类，其中D⊂⊂Ω和0&lt；a&lt；b&lt；T。作为应用，我们证明了如果弱解u表现在(x_0，t_0)∈Ω×(0，T)Like u(x，t)=o(|x-x_0|^&lt；-1&gt；T；)当x→x_0在t_0的某个邻域内一致于t时，则(x_0，t_0)是u的一个可去奇点。考虑∈类中的N-S方程的弱解w，其中2/α+3/q=1有3&lt；Q[小于或等于]∞，其中Ω是R^3中的一般无界区域。我们将证明，尽管来自w的初始和外部扰动很大，但每个具有能量不等的扰动流u都渐近收敛到w，‖υ(T)-w(T)‖_&lt；L^2(Ω)&gt；→0，‖▽υ(T)-▽w(T)‖_&lt；L^2(Ω)&gt；=O(t^&lt；-1/2&gt；)as t→∞。

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→∞.