United theory of existence of global solution and its asymptotic behavior to the nonlinear partial differential equations

1. 外域内Navier-Stokes方程的弱解构造。我们证明了给定非齐次边界数据的局部极弱解的唯一存在性，这些数据属于比通常迹类更大的一类。我们的解满足Serrin条件，该条件蕴涵着缩放不变量2类。Navier-Stokes方程弱解的新正则性判据。证明了在α>1/2的时间区间内，动能为α-霍尔德连续的每一个湍流解必然是正则的。一致C^2类非紧边界无界域上的Helmholtz-Weyl复合。尽管在L^r中没有证明Helmholtz-Weyl分解有效性的反例，但我们引入了L^r的和与交空间，并证明了该空间中的Helmholtz-Weyl分解。作为一个应用程序，我们可以定义Stokes操作符。

1. Constructions of very weak solutions of the Navier-Stokes equations in exterior domains.We show the unique existence of local very weak solutions to the prescribed non-homogeneous boundary data which belong to the larger class than the usual trace class. Our solutions satisfy the Serrin condition which implies the scaling invariant class.2. New regularity criterion on weak solutions of the Navier-Stokes equations.We prove that every turbulent solution which is α-Hoelder continuous in the kinetic energy in the time interval with α>1/2 necessarily regular.3. Helmholtz-Weyl de composition in unbounded domains with non-compact boundaries of uniformly C^2-class.Despite of a counter example of valiclity of the Helmholtz-Weyl decomposition in L^r, we introduce the space of sum and intersection of L^r and prove the Helmholtz-Weyl decomposition in such spaces. As an application, we can define the Stokes operator.