The Cauchy problem for Kawahara equation in Sobolev spaces with low regularity

2010 ◽  
Vol 33 (14) ◽  
pp. 1647-1660 ◽  
Author(s):  
Wei Yan ◽  
Yongsheng Li
Keyword(s):  
Cauchy Problem ◽  
Sobolev Spaces ◽  
Kawahara Equation ◽  
Low Regularity ◽  
The Cauchy Problem

On the classical solutions for a Rosenau–Korteweg-deVries–Kawahara type equation

2021 ◽  
pp. 1-23
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
Cauchy Problem ◽  
Small Amplitude ◽  
Type Equation ◽  
Discrete Systems ◽  
Finite Depth ◽  
Capillary Waves ◽  
Classical Solutions ◽  
Well Posedness ◽  
Kawahara Equation ◽  
The Cauchy Problem

The Rosenau–Korteweg-deVries–Kawahara equation describes the dynamics of dense discrete systems or small-amplitude gravity capillary waves on water of a finite depth. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem.

scholarly journals Existence and Uniqueness of Solutions for a Type of Generalized Zakharov System

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Rui Li ◽  
Xing Lin ◽  
Zongwei Ma ◽  
Jingjun Zhang
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Energy Conservation ◽  
Classical Solution ◽  
Sobolev Spaces ◽  
Existence And Uniqueness ◽  
Uniqueness Of Solutions ◽  
Zakharov System ◽  
The Cauchy Problem ◽  
Global Existence And Uniqueness

We study the Cauchy problem for a type of generalized Zakharov system. With the help of energy conservation and approximate argument, we obtain global existence and uniqueness in Sobolev spaces for this system. Particularly, this result implies the existence of classical solution for this generalized Zakharov system.

scholarly journals On regularization of the Cauchy problem for elliptic systems in weighted Sobolev spaces

2019 ◽  
Vol 27 (6) ◽  
pp. 815-834
Author(s):  
Yulia Shefer ◽  
Alexander Shlapunov
Keyword(s):  
Cauchy Problem ◽  
Sobolev Spaces ◽  
Integral Formula ◽  
Weighted Sobolev Spaces ◽  
Elliptic Systems ◽  
Fundamental Solutions ◽  
Type Integral ◽  
The Cauchy Problem ◽  
Ill Posed ◽  
Neumann Type

AbstractWe consider the ill-posed Cauchy problem in a bounded domain{\mathcal{D}}of{\mathbb{R}^{n}}for an elliptic differential operator{\mathcal{A}(x,\partial)}with data on a relatively open subsetSof the boundary{\partial\mathcal{D}}. We do it in weighted Sobolev spaces{H^{s,\gamma}(\mathcal{D})}containing the elements with prescribed smoothness{s\in\mathbb{N}}and growth near{\partial S}in{\mathcal{D}}, controlled by a real number γ. More precisely, using proper (left) fundamental solutions of{\mathcal{A}(x,\partial)}, we obtain a Green-type integral formula for functions from{H^{s,\gamma}(\mathcal{D})}. Then a Neumann-type series, constructed with the use of iterations of the (bounded) integral operators applied to the data, gives a solution to the Cauchy problem in{H^{s,\gamma}(\mathcal{D})}whenever this solution exists.

scholarly journals Low regularity well-posedness of Hartree type Dirac equations in 2, 3-dimensions

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Kiyeon Lee
Keyword(s):  
Cauchy Problem ◽  
Dirac Equation ◽  
Critical Case ◽  
Nonlinear Term ◽  
Well Posedness ◽  
Low Regularity ◽  
Bilinear Estimates ◽  
Dirac Equations ◽  
The Cauchy Problem

<p style='text-indent:20px;'>In this paper, we consider the Cauchy problem of <inline-formula><tex-math id="M1">\begin{document}$ d $\end{document}</tex-math></inline-formula>-dimension Hartree type Dirac equation with nonlinearity <inline-formula><tex-math id="M2">\begin{document}$ c|x|^{-\gamma} * \langle \psi, \beta \psi\rangle $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ c\in \mathbb R\setminus\{0\} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ 0 &lt; \gamma &lt; d $\end{document}</tex-math></inline-formula>(<inline-formula><tex-math id="M5">\begin{document}$ d = 2,3 $\end{document}</tex-math></inline-formula>). Our aim is to show the local well-posedness in <inline-formula><tex-math id="M6">\begin{document}$ H^s $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M7">\begin{document}$ s &gt; \frac{\gamma-1}2 $\end{document}</tex-math></inline-formula> with mass-supercritical cases(<inline-formula><tex-math id="M8">\begin{document}$ 1 &lt; \gamma&lt;d $\end{document}</tex-math></inline-formula>) and mass-critical case(<inline-formula><tex-math id="M9">\begin{document}$ {\gamma} = 1 $\end{document}</tex-math></inline-formula>) via bilinear estimates and angular decomposition for which we use the null structure of nonlinear term effectively. We also provide the flow of Dirac equations cannot be <inline-formula><tex-math id="M10">\begin{document}$ C^3 $\end{document}</tex-math></inline-formula> at the origin for <inline-formula><tex-math id="M11">\begin{document}$ H^s $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M12">\begin{document}$ s &lt; \frac{\gamma-1}2 $\end{document}</tex-math></inline-formula>.</p>

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