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

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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.