Propagation of regularity for solutions to the KP-I equation

Abstract

In this paper we prove certain regularity properties of solutions to the IVP for the KP-I equation. Specifically, we prove that if the initial data ϕ has sufficient regularity and if there exists x0∈R such that the initial data also satisfies the additional regularity property that ∂xkϕ∈L2((x0,∞)×R) for certain k, then ∂xku(⋅,t)∈L2((a,∞)×R) for all a∈R, and for all t such that 0<t≤T where T is the existence time of the solution. In particular, regularity on the initial data is transferred to the solution with infinite speed.