Decay of solutions to a viscous asymptotical model for waterwaves: Kakutani–Matsuuchi model

In this article, we study a viscous asymptotical model equation for water waves $$u_t + u_x − βu_{txx} + ν(D^{1/2}u + F^{−1}(i|ξ|^{1/2}sgn(ξ)\hat{u}(ξ))) + γ uu_x = 0$$ proposed in Kakutani and Matsuuchi (1975) [6]. Theoretical questions including the existence and regularity of the solutions will be answered. Numerical simulations of its solutions will be carried out and the effects of various parameters will be investigated. We will also predict the decay rate of its solutions towards the equilibrium.

Min Chen (