Abstract
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 L^2 a priori estimates and we consider its regular approximations with the fractional power operator (-PDelta )^{1+alpha }, alpha >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 L^2 a priori estimate are obtained and the 3-D regular approximating problem involves fractional power operator (-PDelta )^s with s>frac{5}{4}. 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.
Highlights
Fifteen years ago, in the monograph [3], we were studying a direct generalization of semilinear parabolic equations, namely abstract semilinear equation with sectorial positive operator in the main part
We will obtain and study its solutions constructed as limits of solutions to sub-critical approximations (1.2) when α → 0+ where, in 2-D, the P operator is replaced with its fractional power −(−P )1+α, α > 0 (P is the projector on the space of divergence-free functions; see e.g. [15])
The main part operator A will not control the nonlinearity in that case, unless we find a better a priori estimate
Summary
In the monograph [3], we were studying a direct generalization of semilinear parabolic equations, namely abstract semilinear equation with sectorial positive operator in the main part. Such approach, located inside the semigroup theory, proves its strength and utility in the study of several classical problems; some of them were reported in [3,21] Such technique offers further possible generalizations, first to study the problems, like e.g. Korteweg-de Vries equation and its extensions [7,8,12,13], where the solutions are obtained as a limit of solutions to parabolic regularizations of such equations (the method known as vanishing viscosity technique, originated by Hopf, Oleinik, Lax in 1950th); see [9,10]. There were several tries of replacing the classical N-S equation, or the viscosity term in it, with another equation having better properties of solutions, starting with Leray α -regularization reported in paper [31], see [14]. In this paper we follow the idea of Lions to replace the diffusion term with a stronger one fractal diffusion term
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