Local well-posedness and parabolic smoothing of solutions of fully nonlinear third-order equations on the torus

Abstract

In this paper we study the initial value problem of fully nonlinear third-order equations on the torus, that is ∂tu=F∂x3u,∂x2u,∂xu,u,x,t with F a smooth function depending on the space variable x, the time variable t, the first three derivatives of u with respect to x, and u. In particular we find conditions on u(0) and F for which one can construct a local and unique solution u. In particular if F and u(0) satisfy some conditions then the equation behaves like a diffusive one and it has a parabolic smoothing property: the solution is infinitely smooth in one direction of time and the problem is ill-posed in the other direction of time. If F and u(0) satisfy some other conditions then the equation behaves like a dispersive one. We also prove continuous dependence with respect to the data. The proof relies upon energy estimates combined with a gauge transformation [6–8] and the Bona-Smith argument (Bona and Smith, 1975). © 2022 Elsevier Ltd