SOME ESTIMATES OF REGULAR SOLUTIONS TO THE INITIAL-BOUNDARY VALUE PROBLEM FOR THE QUASIHYDRODYNAMIC SYSTEM IN THE SHALLOW WATER APPROXIMATION

F. A. Evseev, S. G. Pyatkov

Abstract


Many important natural processes involving the motion of a fluid with a free surface in a gravitational field and characterized by a negligibly small vertical scale compared to the horizontal scale, for example, modeling tidal waves, flood flows, and tsunamis, are described by the shallow water equations. The article is devoted to studying the question of obtaining a priori estimates in Sobolev spaces for solutions to an initial-boundary value problem for a quasi-hydrodynamic system of equations in the shallow water approximation. These estimates to some extent characterize the behavior of a solution as  t to infinity . We assume that a solution to the problem exists globally in time and is regular, i.e., all generalized derivatives occurring into equations exist and summable to some power. The proofs of a priori estimates rely on the embedding theorem, interpolation properties of Sobolev spaces and, in particular, interpolation inequalities of various types.

Keywords


initial-boundary value problem; quasihydrodynamic system; shallow water approximation; existence; a priori estimate

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References


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