×

Well-posedness for the two-dimensional modified Zakharov-Kuznetsov equation. (English) Zbl 1197.35242

Summary: We prove that the initial value problem for the two-dimensional modified Zakharov-Kuznetsov equation is locally well-posed for data in \(H^s(\mathbb R^2)\), \(s>3/4\). Even though the critical space for this equation is \(L^2(\mathbb R^2)\), we prove that well-posedness is not possible in such space. Global well-posedness and a sharp maximal function estimate are also established.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35B65 Smoothness and regularity of solutions to PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
PDFBibTeX XMLCite
Full Text: DOI