Estimates for the Derivatives of the Velocity and Pressure in Shallow Water Flow and Approximate Shallow Water Equations

Abstract

Estimates are derived on the time and space derivatives of the velocity and pressure of a shallow fluid flow. These estimates depend upon the velocity and pressure being bounded and the free surface having a long wavelength compared to the depth of the fluid. The technique used is to derive $L_2 $-estimates on the derivatives of the velocity and pressure and then convert these to pointwise estimates. As a consequence of these results, the horizontal velocity is shown to be independent of the depth and the pressure hydrostatic to the first approximation. Higher order estimates lead to second order approximate equations which, under additional physically motivated assumptions, correspond to the Boussinesq and Korteweg–deVries equations.