On the Cauchy problems associated to a ZK-KP-type family equations with a transversal fractional dispersion

<p style='text-indent:20px;'>In this paper we examine the well-posedness and ill-posedeness of the Cauchy problems associated with a family of equations of ZK-KP-type</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{cases} u_{t} = u_{xxx}-\mathscr{H}D_{x}^{\alpha}u_{yy}+uu_{x}, \cr u(0) = \psi \in Z \end{cases} $\end{document} </tex-math> </disp-formula></p><p style='text-indent:20px;'>in anisotropic Sobolev spaces, where <inline-formula><tex-math id="M1">\begin{document}$ 1\le \alpha \le 1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ \mathscr{H} $\end{document}</tex-math></inline-formula> is the Hilbert transform and <inline-formula><tex-math id="M3">\begin{document}$ D_{x}^{\alpha} $\end{document}</tex-math></inline-formula> is the fractional derivative, both with respect to <inline-formula><tex-math id="M4">\begin{document}$ x $\end{document}</tex-math></inline-formula>.</p>

Sharp well-posedness of the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili equation in anisotropic Sobolev spaces

<p style='text-indent:20px;'>We consider the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili (RMKP) equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} \partial_{x}\left(u_{t}-\beta\partial_{x}^{3}u +\partial_{x}(u^{2})\right)+\partial_{y}^{2}u-\gamma u = 0 \end{align*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>in the anisotropic Sobolev spaces <inline-formula><tex-math id="M1">\begin{document}$ H^{s_{1},s_{2}}(\mathbb{R}^{2}) $\end{document}</tex-math></inline-formula>. When <inline-formula><tex-math id="M2">\begin{document}$ \beta &lt;0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ \gamma &gt;0, $\end{document}</tex-math></inline-formula> we prove that the Cauchy problem is locally well-posed in <inline-formula><tex-math id="M4">\begin{document}$ H^{s_{1}, s_{2}}(\mathbb{R}^{2}) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M5">\begin{document}$ s_{1}&gt;-\frac{1}{2} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ s_{2}\geq 0 $\end{document}</tex-math></inline-formula>. Our result considerably improves the Theorem 1.4 of R. M. Chen, Y. Liu, P. Z. Zhang(Transactions of the American Mathematical Society, 364(2012), 3395–3425.). The key idea is that we divide the frequency space into regular region and singular region. We further prove that the Cauchy problem for RMKP equation is ill-posed in <inline-formula><tex-math id="M7">\begin{document}$ H^{s_{1},0}(\mathbb{R}^{2}) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M8">\begin{document}$ s_{1}&lt;-\frac{1}{2} $\end{document}</tex-math></inline-formula> in the sense that the flow map associated to the rotation-modified Kadomtsev-Petviashvili is not <inline-formula><tex-math id="M9">\begin{document}$ C^{3} $\end{document}</tex-math></inline-formula>. When <inline-formula><tex-math id="M10">\begin{document}$ \beta &lt;0,\gamma &gt;0, $\end{document}</tex-math></inline-formula> by using the <inline-formula><tex-math id="M11">\begin{document}$ U^{p} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M12">\begin{document}$ V^{p} $\end{document}</tex-math></inline-formula> spaces, we prove that the Cauchy problem is locally well-posed in <inline-formula><tex-math id="M13">\begin{document}$ H^{-\frac{1}{2},0}(\mathbb{R}^{2}) $\end{document}</tex-math></inline-formula>.</p>

The Cauchy problem for a two-dimensional generalized Kadomtsev–Petviashvili-I equation in anisotropic Sobolev spaces

The goal of this paper is three-fold. First, we prove that the Cauchy problem for a generalized KP-I equation [Formula: see text] is locally well-posed in the anisotropic Sobolev spaces [Formula: see text] with [Formula: see text] and [Formula: see text]. Second, we prove that the Cauchy problem is globally well-posed in [Formula: see text] with [Formula: see text] if [Formula: see text]. Finally, we show that the Cauchy problem is globally well-posed in [Formula: see text] with [Formula: see text] if [Formula: see text] Our result improves the result of Saut and Tzvetkov [The Cauchy problem for the fifth order KP equations, J. Math. Pures Appl. 79 (2000) 307–338] and Li and Xiao [Well-posedness of the fifth order Kadomtsev–Petviashvili-I equation in anisotropic Sobolev spaces with nonnegative indices, J. Math. Pures Appl. 90 (2008) 338–352].

On anisotropic Sobolev spaces

We investigate two types of characterizations for anisotropic Sobolev and BV spaces. In particular, we establish anisotropic versions of the Bourgain–Brezis–Mironescu formula, including the magnetic case both for Sobolev and BV functions.