Ill-posedness results for the (generalized) Benjamin-Ono-Zakharov-Kuznetsov equation

Abstract:

Here we consider results concerning ill-posedness for the Cauchy problem associated with the Benjamin-Ono-Zakharov-Kuznetsov equation, namely, \begin{equation*} \left \{ \begin {array}{ll} u_t-\mathscr {H}u_{xx}+u_{xyy}+u^ku_x=0, \qquad (x,y)\in \mathbb {R}^2,\;\;t\in \mathbb {R}^+, \\ u(x,y,0)=\phi (x,y). \end{array} \right .\tag *{(IVP)} \end{equation*} For $k=1$, (IVP) is shown to be ill-posed in the class of anisotropic Sobolev spaces $H^{s_1,s_2}(\mathbb {R}^2), s_1,s_2\in \mathbb {R}$, while for $k\geq 2$ ill-posedness is shown to hold in $H^{s_1,s_2}(\mathbb {R}^2), 2s_1+s_2<3/2-2/k$. Furthermore, for $k=2,3$, and some particular values of $s_1,s_2$, a stronger result is also established.