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**Zero viscosity limit for analytic solutions, of the Navier-Stokes equation on a half-space. I: Existence for Euler and Prandtl equations.**
*(English)*
Zbl 0913.35102

This first part of the paper deals with the difficult parallel study of Euler, Prandtl and Navier-Stokes equations. In the introduction, useful also for the second part, the authors present an up to date situation of the problem and expose the plan of the entire paper; in the first part their attention is focused on the Euler and Prandtl equations and in the second part, on the Navier-Stokes equations with respect to the existence of a solution and their inner connections.

The paper contains many theoretical results. The most interesting are: 1) the short-time existence of solutions to Euler and Prandtl equations in a half space, with analytical initial data, 2) the existence of solutions to the unsteady Prandtl system, 3) the connection between these solutions, considered as starting point for the existence of solutions to the Navier-Stokes equations. The main theoretical tool in this analysis is the abstract Cauchy-Kovalewski theorem, but also other mathematical methods are used: functional analysis, operator theory, abstract normed spaces and so on. The results are obtained for bi- and tri-dimensional domains.

[For part II see ibid., 463-491 (1998; reviewed below)].

The paper contains many theoretical results. The most interesting are: 1) the short-time existence of solutions to Euler and Prandtl equations in a half space, with analytical initial data, 2) the existence of solutions to the unsteady Prandtl system, 3) the connection between these solutions, considered as starting point for the existence of solutions to the Navier-Stokes equations. The main theoretical tool in this analysis is the abstract Cauchy-Kovalewski theorem, but also other mathematical methods are used: functional analysis, operator theory, abstract normed spaces and so on. The results are obtained for bi- and tri-dimensional domains.

[For part II see ibid., 463-491 (1998; reviewed below)].

Reviewer: V.Ionescu (Bucureşti)

### MSC:

35Q30 | Navier-Stokes equations |

35Q05 | Euler-Poisson-Darboux equations |

35Q35 | PDEs in connection with fluid mechanics |