Bound on the advection and the height of the shallow-water problem with Dirichlet boundary conditions. (Q556921)

The authors consider solutions \((h,u)\) of \[ \begin{aligned} {\partial u\over\partial t}+ (u\cdot\nabla) u-\mu\Delta u+ a\nabla h= 0\quad &\text{in }Q= \Omega\times (0,T),\\ {\partial n\over\partial t}+ \text{div}(hu)= 0\quad &\text{in }Q= \Omega\times (0,T),\\ u= 0\quad &\text{on }\Sigma= \Gamma\times (0,T),\;\Gamma= \partial\Omega,\\ u(0,x)= u_0(x),\;h(0,x)= h_0(x)\geq 0\quad &\text{in }\Omega,\end{aligned}\tag{1} \] where \(a> 0\), \(\mu> 0\), \(T> 0\) and \(\Omega\) is a smooth open domain in \(\mathbb{R}^2\). Under some natural assumptions on the data of (1), the authors using some properties of the Hardy spaces prove a regularity result.