**Luis Adauto Medeiross and Juan Limaco Ferrel**

## Elliptic Regularization And Navier-Stokes System

**abstract:**

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.

**Key words and phrases:**
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