On the Cauchy problem for the Kadomstev-Petviashvili equation

1993 ◽  
Vol 3 (4) ◽  
pp. 315-341 ◽  
Author(s):  
J. Bourgain
Keyword(s):  
Cauchy Problem ◽  
Petviashvili Equation ◽  
The Cauchy Problem

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

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Wei Yan ◽  
Yimin Zhang ◽  
Yongsheng Li ◽  
Jinqiao Duan
Keyword(s):  
Cauchy Problem ◽  
Sobolev Spaces ◽  
American Mathematical Society ◽  
Frequency Space ◽  
Anisotropic Sobolev Spaces ◽  
Petviashvili Equation ◽  
The Cauchy Problem ◽  
Regular Region ◽  
Flow Map ◽  
Well Posed

<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>

Cauchy Problem for Equations with Fractional Differentiation Bessel Operator in the Space of Temperate Distributions

2003 ◽  
Vol 8 (1) ◽  
pp. 61-75
Author(s):  
V. Litovchenko
Keyword(s):  
Functional Analysis ◽  
Cauchy Problem ◽  
Fundamental Solution ◽  
Initial Function ◽  
Well Posedness ◽  
Fractional Differentiation ◽  
Basic Tool ◽  
Conjugate Space ◽  
Analysis Technique ◽  
The Cauchy Problem

The well-posedness of the Cauchy problem, mentioned in title, is studied. The main result means that the solution of this problem is usual C∞ - function on the space argument, if the initial function is a real functional on the conjugate space to the space, containing the fundamental solution of the corresponding problem. The basic tool for the proof is the functional analysis technique.

Fundamental Serrin type regularity criteria for 3D MHD fluid passing through the porous medium

Filomat ◽  
2017 ◽  
Vol 31 (5) ◽  
pp. 1287-1293 ◽  
Author(s):  
Zujin Zhang ◽  
Dingxing Zhong ◽  
Shujing Gao ◽  
Shulin Qiu
Keyword(s):  
Cauchy Problem ◽  
Porous Medium ◽  
Regularity Criterion ◽  
Regularity Criteria ◽  
The Cauchy Problem ◽  
Appl Math

In this paper, we consider the Cauchy problem for the 3D MHD fluid passing through the porous medium, and provide some fundamental Serrin type regularity criteria involving the velocity or its gradient, the pressure or its gradient. This extends and improves [S. Rahman, Regularity criterion for 3D MHD fluid passing through the porous medium in terms of gradient pressure, J. Comput. Appl. Math., 270 (2014), 88-99].

Sign in / Sign up

Export Citation Format

Share Document