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arXiv · 2605.30044

A Regularized Shallow Water System

Evgueni Dinvay·Henrik Kalisch
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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.

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Evgueni Dinvay, Henrik Kalisch. 2026-05-28. A Regularized Shallow Water System. https://doi.org/10.1002/pamm.70153

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