Convergence problem of the generalized Kadomtsev–Petviashvili II equation in anisotropic Sobolev space

Abstract

The almost everywhere pointwise and uniform convergences for the generalized KP-II equation are investigated when the initial data is in anisotropic Sobolev space \(H^{s_{1},s_{2}}({\textbf{R}}^{2})\). Firstly, we show that the solution u(x, y, t) converges pointwisely to the initial data \(f(x, y)\in H^{s_{1},s_{2}}({{\textbf{R}}}^{2}) \) for a.e. \((x, y) \in {\textbf{R}}^{2}\) when \(s_{1}\ge \frac{1}{4}\), \(s_{2}\ge \frac{1}{4}\). The proof relies upon the Strichartz estimate and high-low frequency decomposition. Secondly, We prove that \(s_{1}\ge \frac{1}{4}\), \(s_{2}\ge \frac{1}{4}\) is a necessary condition for the maximal function estimate of the generalized KP-II equation to hold. Finally, by using the Fourier restriction norm method, we establish the nonlinear smoothing estimate to show the uniform convergence of the generalized KP-II equation in \(H^{s_{1},s_{2}} ({{\textbf{R}}}^{2}) \) with \( s_{1}\ge \frac{3}{2}-\frac{\alpha }{4}+\epsilon ,\ s_{2}>\frac{1}{2}\) and \(\alpha \ge 4 \).