The Cauchy problem on large time for the Water Waves equations with large topography variations

Abstract

This paper shows that the long time existence of solutions to the Water Waves equations remains true with a large topography in presence of surface tension. More precisely, the dimensionless equations depend strongly on three parameters $\epsilon ,\mu ,\beta$ measuring the amplitude of the waves, the shallowness and the amplitude of the bathymetric variations respectively. In [2], the local existence of solutions to this problem is proved on a time interval of size $\frac{1}{\max (\beta ,\epsilon )}$ and uniformly with respect to μ. In presence of large bathymetric variations (typically $\beta \gg \epsilon$), the existence time is therefore considerably reduced. We remove here this restriction and prove the local existence on a time interval of size $\frac{1}{\epsilon }$ under the constraint that the surface tension parameter must be at the same order as the shallowness parameter μ. We also show that the result of [5] dealing with large bathymetric variations for the Shallow Water equations can be viewed as a particular endpoint case of our result.