Regularity and uniqueness of solutions to the Boussinesq system of equations (Q762368)

The initial value problem for the system \[ (*)\quad \rho_ t+u_ x+(u\rho)_ x=0,\quad u_ t+\rho_ x+uu_ x-u_{xxt}=0 \] on \(R\times <0,+\infty)\) is considered. It corresponds (after a transformation) to the Boussinesq model for the propagation of surface waves. The author continues the paper by \textit{M. E. Schonbek} [J. Differ. Equations 42, 325-352 (1981; Zbl 0476.35067)] where a weak solution to (*) is obtained as a limit for \(\epsilon\) \(\to 0\) of solutions \(u_{\epsilon}\) to the regularization \[ \rho_ t+u_ x+(u\rho)_ x=\epsilon \rho_{xx},\quad u_ t+\rho_ x+uu_ x-u_{xxt}=0. \] In the present paper the unicity and smoothness of this solution are proved under the assumption that the initial data are smooth.