# American Institute of Mathematical Sciences

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September  2002, 1(3): 379-415. doi: 10.3934/cpaa.2002.1.379

## In a horizontal layer with free upper surface

 1 Department of Mathematics, University of Ferrara,44100 Ferrara, Italy, Italy

Received  August 2001 Revised  January 2002 Published  June 2002

We propose a new existence proof of global in time solutions of isothermal viscous gases in a layer bounded below by a horizontal plane, and above by a free upper surface, which are periodic in the two horizontal variables. Despite the importance of compressible fluids for physical applications, the problem of uniform in time estimates is scarcely explored. The rest state with a steady distribution of density in a rectangular domain is stable, without restrictions on initial data, in a "weak" norm provided the flows exist in a suitable regularity class. In this paper we show existence of regular global in time solutions, and the exponential decay of these solutions to the rest as time goes to $\infty$, when the initial data are small perturbation of the basic flow. The analysis presented here is based on estimates in Hilbert spaces.
Citation: Bum Ja Jin, Mariarosaria Padula. In a horizontal layer with free upper surface. Communications on Pure & Applied Analysis, 2002, 1 (3) : 379-415. doi: 10.3934/cpaa.2002.1.379
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