A Regularized Shallow Water System

Abstract

The shallow-water system is a standard model for long waves in shallow water. The system is hyperbolic and, for a large class of initial data, solutions develop steep gradients leading to shock formation in finite time. Since such singularities violate the long-wave assumptions underlying the model, their appearance limits the regime of validity of the equations. In this work, we introduce a regularized shallow-water system in which the nonlinear terms are modified by a bounded operator. This regularization removes the standard derivative-steepening mechanism responsible for shock formation in the classical system while remaining consistent with the long-wave regime. We establish local well-posedness and small-data global well-posedness in Sobolev spaces that exclude singularity formation. In addition, numerical simulations indicate that the system admits solitary-wave solutions.

References in corpus (10)

• The Whitham Equation as a Model for Surface Water Waves
• Efficient computation of steady solitary gravity waves
• Fast accurate computation of the fully nonlinear solitary surface gravity waves
• Accurate fast computation of steady two-dimensional surface gravity waves in arbitrary depth
• A plethora of generalised solitary gravity-capillary water waves
• A comparative study of bi-directional Whitham systems
• Non-dispersive conservative regularisation of nonlinear shallow water (and isothermal Euler) equations
• Efficient computation of capillary-gravity generalized solitary waves
• Weakly singular shock profiles for a non-dispersive regularization of shallow-water equations
• Well-posedness for a Whitham-Boussinesq system with surface tension