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Our purpose is to formulate an abstract result, motivated by the recent paper [8], allowing to treat the solutions of critical and super-critical equations as limits of solutions to their regularizations. In both cases we are improving the viscosity, making it stronger, solving the obtained regularizations with the use of Dan Henry's technique, then passing to the limit in the improved viscosity term to get a solution of the limit problem. While in case of the critical problems we will just consider a 'bit higher' fractional power of the viscosity term, for super-critical problems we need to use a version of the 'vanishing viscosity technique' that comes back to the considerations of E. Hopf, O.A. Oleinik, P.D. Lax and J.-L. Lions from 1950th. In both cases, the key to that method are the uniform with respect to the parameter estimates of the approximating solutions. The abstract result is illustrated with the Navier-Stokes equation in space dimensions 2 to 4, and with the 2-D quasi-geostrophic equation. Various technical estimates related to that problems and their fractional generalizations are also presented in the paper. [ABSTRACT FROM AUTHOR]

We consider fractional Navier-Stokes equations in a smooth bounded domain Ω ⊂ RN , N ≥ 2. Following the geometric theory of abstract parabolic problems we give the detailed analysis concerning existence, uniqueness, regularization and continuation properties of the solution. For the original Navier-Stokes problem we construct next global solution of the Leray-Hopf type satisfying also Duhamel's integral formula. Focusing finally on the 3-D model with zero external force we estimate a time after which the latter solution regularizes to strong solution. We also estimate a time such that if a local strong solution exists until that time, then it exists for ever. [ABSTRACT FROM AUTHOR]

We consider the Navier–Stokes equation (N-S) in dimensions two and three as limits of the fractional approximations. In 2-D the N-S problem is critical with respect to the standard L2 a priori estimates and we consider its regular approximations with the fractional power operator (-PΔ)1+α, α>0 small, where P is the projector on the space of divergence-free functions. In 3-D different properties of the N-S problem with respect to the standard L2 a priori estimate are obtained and the 3-D regular approximating problem involves fractional power operator (-PΔ)s with s>54. Using Dan Henry’s semigroup approach and the Giga-Miyakawa estimates we construct regular solutions to such approximations. The solutions are global in time, unique, smooth and regularized through the equation in time. Solution to 2-D and 3-D N-S equations are obtained next as a limit of such regular solutions of the approximations. Moreover, since the nonlinearity of the N-S equation is of quadratic type, the solutions corresponding to small initial data and small f are shown to be global in time and regular. [ABSTRACT FROM AUTHOR]

Solvability of Cauchy's problem in [ABSTRACT FROM AUTHOR]