Finite-time blow-up of classical solutions to the rotating shallow water system with degenerate viscosity

Abstract

In this paper, we study the rotating shallow water system with initial data containing vacuum, the viscosity of the system is degenerate, and the Coriolis force, the capillary force and the turbulent drag force from fiction are involved. When a positive mass is surrounding by a bounded vacuum region (isolated mass group), initially, we prove that any classical solutions to the initial-boundary-value problem and periodic problem will blow up in finite time. This shows that the global weak solutions obtained in Bresch and Desjardins (Commun Math Phys 238(1–2):211–223, 2003) cannot be a classical one as long as the initial data admit an isolated mass group. It also shows that compared with the smoothing properties provided by capillarity and drag terms, the degeneracy of viscosity plays a prominent role in the global regularity problem and leads to finite-time singularity of smooth solutions.