Well-posedness of Cauchy problems for Korteweg-de Vries-Benjamin-Ono equation and Hirota equation

2004 ◽  
Vol 14 (6) ◽  
pp. 472-476
Author(s):  
Zhaohui Huo ◽  
Boling Guo
Keyword(s):  
Well Posedness ◽  
Cauchy Problems ◽  
Hirota Equation ◽  
De Vries

scholarly journals Global Well-Posedness for Dissipative Korteweg-de Vries Equations

2011 ◽  
Vol 54 (1) ◽  
pp. 119-138 ◽  
Author(s):  
Stéphane Vento
Keyword(s):  
Well Posedness ◽  
De Vries

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

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jorge Morales Paredes ◽  
Félix Humberto Soriano Méndez
Keyword(s):  
Fractional Derivative ◽  
Sobolev Spaces ◽  
Hilbert Transform ◽  
Well Posedness ◽  
Cauchy Problems ◽  
Anisotropic Sobolev Spaces ◽  
The Hilbert Transform

<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>

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