Luis Adauto Medeiross and Juan Limaco Ferrel

Elliptic Regularization And Navier-Stokes System

Applications of the method of elliptic regularization to the Navier--Stokes system in a bounded domain of $\bold{R}^3$ with moving boundary. We follow the ideas of J. L. Lions [7] and Temam [13]. There is an extensive literature about the existence of weak solutions for the Navier--Stokes system. In the references at the end of this article, one can find a small selection, far from completion, which can lead the reader indications to other references. Starting from the sixties, there appear many results on the Navier-Stokes system in noncylindrical domains, as, for example, Fujita--Sauer [5] and Lions [7]. Among the most employed methods, there was the so called penalty method, which consists in considering a perturbation, by means of a term with a parameter $\ve$, transforming the problem into a cylindrical domain, and passing to limit as $\ve$ tends to zero. The objective of the present article is to study the Navier-Stokes system by a method idealized by Lions in 1983, cf. [7]. It consists in transforming a parabolic problem into a family of elliptic problems indexed by a parameter $\ve>0$. We solve these problems by elliptic methods and try to obtain the solution of the original parabolic problem as a limit as $\ve$ tends to zero. This method is also used, in a certain sense, by Salvi [11] in a different context. For the study of the Navier-Stokes system in the noncylindrical case by another method, we refer to M. Milla Miranda--J. Limaco Ferrel [10]. The abstract aspect of the elliptic regularization can be found in Lions [9], and for the Navier--Stokes system in cylindrical domains see Temam [14], [15] or Tartar [13] and the references therein. We also use a techniques of Fujita-Sauer [5].

Mathematics Subject Classification: 35F30, 35K55.

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