Global existence for the Dirichlet problem for the viscous shallow water equations
The motion of a sheet of water contained in a plane-basin is mathematically described by a system of three partial differential equations in three unknown functions, that is two components of velocity \((u_1,u_2)\) and the depth of the fluid. The equations take account of the viscosity of the fluid and of the Coriolis forces due to the fact that the basin is not fixed, but itself rotating around a fixed axis, as happens for a lake on the surface of the earth. The problem has been studied by several authors, but two questions have still been unanswered, the proof that the solution decays exponentially to the equilibrium value and that the depth is always strictly positive. Through a sequence of lemmas it is possible to arrive at an energy estimate on the solution, which proves that a strong norm defined on the velocities and on the height is bounded for each time. The positivity of the height of the fluid is an immediate consequence of the fact that the velocity vector is globally of class \(C^1\).
scientific article; zbMATH DE number 930252