Research on well-posedness for the Navier-Stokes equations

在一个定域Ω∧R^n中，考虑类u∈L^∞（0，T; L^n （Ω））中Navier-Stokes方程的弱解u。如果lim sup_ < t-t_ * 0 >为u (t)为^ n_n -为u (t_ *)识别为^ n_n很小的每一点t_ *识别∈(0,t),然后u是常规Ω^ ^ -×(0,t)。作为一个应用，我们给出了奇异时间的一个精确表征，即我们证明了如果Navier-Stokes方程的解u最初是光滑的，并且在以后的某个时间T_*<T时失去其规律性，那么要么lim sup_< T -T_*-0>‖u (T)‖_<L^n （Ω） >= +∞，要么u (T)在L^n （Ω）中围绕弱极限w-lim_< T -T_*-0>u (T)以足够大的振幅振荡。进一步证明了（0，T）上有界变分的值在L^n （Ω）上的每一个弱解u都是正则解。考虑Ω× （0， T）中的非平稳Navier-Stokes方程，其中Ω是R^3中的定义域。我们证明了存在一个绝对常数ε_0，使得具有sup_<t∈(a, b) >‖u (t)‖^3_W (D)【小于或等于】ε_0性质的每一个弱解u在dx （a, b）的任意紧子集上的时空变量中必然是C^∞类，其中D≠Ω且0<a<b< t。作为一个应用程序中,我们证明如果弱解u的行为(从t_0)∈Ω×(0,T)像u (x, T) = o (| x-x_0 | ^ < 1 >) x→x_0均匀t_0 T在某些社区,然后(x_0 t_0)是一个可移动的奇点的u.Consider navier - stokes方程的弱解w Serrin的classw∈L ^α(0,∞;L ^ q(Ω))2 /α+ 3 / q = 1与3 <问【小于或等于】∞,Ω普遍无限域R ^ 3。我们将证明，尽管来自w的初始和外部扰动很大，但具有能量不等式的每个扰动流u渐近收敛于w为‖υ (t) -w (t)‖_<L^2 （Ω） >→0，‖▽υ(t) -▽w (t)‖_<L^2 （Ω） >=O （t^<-1/2>）→∞。

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→∞.