Local well-posedness and parabolic smoothing of solutions of fully nonlinear third-order equations on the torus. (August 2022) - Roy, Tristan
Abstract: In this paper we study the initial value problem of fully nonlinear third-order equations on the torus, that is ∂ t u = F ∂ x 3 u, ∂ x 2 u, ∂ x u, 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).