Algunos detalles relevantes de las ecuaciones de navier-stokes incompresibles.

Abstract

In the present work we study the incompressible classical Navier-Stokes Cauchy problem (NS problem), which describes the behavior of an incompressible Newtonian fluid with initial velocity u_0 and that is subjected to an external force f. We represent by u and p the velocity of such fluid and its pressure, respectively, which are functions of temporal and spatial variables. When f is in the Bochner space L^2 ((0,T), L^2 (R^3,R^3)) (for some positive T), and u_0 in the Sobolev space H^1(R ^3,R^3) we show in detail that the NS problem admits a unique mild solution (u,p) in the space [L^∞([0,T_0],H^1(R^3,R^3))∩L ^2 ((0,T), Ḣ^2 (R^3,R^3))]× L^2 ((0,T), Ḣ^1 (R^3,R^3)) for some T_0 small enough on the interval (0,T) (in the literature it is known as the Fujita-Kato Theorem). To achieve this, first we develop the appropriate scenario (using harmonic analysis tools such as the Fourier transform) to introduce the Leray Projector, which in turn allow us to establish the notion of a mild solution in order to start the demonstration of the Fujita-Kato theorem. The main idea to get the mild solution of the NS problem is based on a fixed point argument over a suitable Banach space. Moreover, we talk about the regularity and stability (with respect to the data u_0 and f) of this solution.