Dynamic boundary flux-driven shallow waters: Insights from a dissipative-dispersive system

Abstract

This paper is concerned with a shallow water system under dynamic boundary conditions:$u_t+{\left(uw\right)}_x=ϵu_{xx},x\in (0,1),t>0,w_t+{\left(u^m\right)}_x+w{w}_x=\mu w_{xx}+\delta w_{xxt},x\in (0,1),t>0,(u,w)(x,0)=(u_0,w_0)\left(x\right),x\in (0,1),u_x(0,t)=\alpha \left(t\right),u_x(1,t)=\beta \left(t\right),t>0,w(0,t)=0,w(1,t)=0,t>0.$ By constructing suitable relative entropy functionals, it is shown that under certain conditions on $\alpha \left(t\right)$ and $\beta \left(t\right)$, classical solutions with potentially large energy exist globally in time, and the solutions converge to the equilibria determined by the initial and boundary conditions. The results hold for all values of $m⩾1$ when $\mu >0$, and for $m⩾2$ when $\mu =0$. The analytic technique for studying the dispersive-regularized system ($\mu =0$) can be of independent interest and adaptable for other PDE models with similar structure.