On the propagation of regularity of solutions of the Kadomtsev-Petviashvili equation (Q2793838)

The authors consider solutions of the initial value problem (IVP) for the Kadomtsev-Petviashvili (KPII) equation NEWLINE\[NEWLINE \partial_t u + \partial_x^3 u +\alpha \partial_x^{-1}\partial_y^2 u + u \partial_x u= 0,\quad u|_{t=0} = u_0, NEWLINE\]NEWLINE where \(\alpha=1\), \((x,y)\in\mathbb{R}^2\), \(t>0\) and \(\partial_x^{-1}\) is defined via the Fourier transform NEWLINE\[NEWLINE \widehat{\partial_x^{-1} f} (\xi,\eta) = -\frac{i}{\xi}\widehat{f}(\xi,\eta). NEWLINE\]NEWLINE According to the result of \textit{R. J. Iório jun.} and \textit{W. V. L. Nunes} [Proc. R. Soc. Edinb., Sect. A, Math. 128, No. 4, 725--743 (1998; Zbl 0911.35103)], the IVP problem for the KP equations (\(\alpha=\pm1\)) is well posed for \(u_0\in X_s(\mathbb{R}^2)\) with \(s>2\), where \(X_s(\mathbb{R}^2) = \{f\in H^s(\mathbb{R}^2): \partial_x^{-1}f \in H^s(\mathbb{R}^2)\}\). The main result of the paper under review states that for the initial data \(u_0\in X_s(\mathbb{R}^2)\), \(s>2\), whose restriction belongs to \(H^m((x_0,\infty)\times\mathbb{R})\) for some \(m\in \mathbb{Z}_{\geq 3}\) and \(x_0\in\mathbb{R}\), the restriction of the corresponding solution \(u(\cdot,t)\) belongs to \(H^m((\beta,\infty)\times\mathbb{R})\) for any \(\beta\in\mathbb{R}\) and \(t>0\).