The Cauchy problem for the KdV equation with almost periodic initial data whose spectrum is nowhere dense

1994 ◽  
pp. 181-208 ◽  
Author(s):  
I. E. Egorova
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Kdv Equation ◽  
Almost Periodic ◽  
The Kdv Equation ◽  
The Cauchy Problem ◽  
Periodic Initial Data

scholarly journals Kato smoothing properties  of a class of nonlinear dispersive wave equations on a periodic domain

2021 ◽  
Author(s):  
Bingyu Zhang ◽  
Shu-Ming Sun ◽  
Xin Yang ◽  
Ning Zhong
Keyword(s):  
Cauchy Problem ◽  
Wave Equations ◽  
Kdv Equation ◽  
Dispersive Equations ◽  
Periodic Domain ◽  
Smoothing Property ◽  
The Kdv Equation ◽  
The Cauchy Problem ◽  
Real Value ◽  
Nonlinear Dispersive Wave

The solutions of the Cauchy problem of the KdV equation on a periodic domain $\T$,  \[ u_t +uu_x +u_{xxx} =0, \quad u(x,0)= \phi (x), \quad x\in \T, \ t\in \R,\]  possess neither  the sharp Kato smoothing property,  \[ \phi \in H^s (\T) \implies \partial ^{s+1}_xu \in L^{\infty}_x (\T, L^2 (0,T)),\]  nor the Kato smoothing property,  \[ \phi \in H^s (\T) \implies u\in L^2 (0,T; H^{s+1} (\T)).\]  Considered in this article is the Cauchy problem of the following dispersive equations posed on the periodic domain $\T$,  \[ u_t +uu_x +u_{xxx} - g(x) (g(x) u)_{xx} =0, \qquad u(x,0)= \phi (x), \quad x\in \T, \  t>0 \, ,\ \qquad (1) \]  where $g\in C^{\infty} (\T)$ is  a  real value function with  the support  \[ \mbox{$\omega = \{ x\in \T, \  g(x) \ne 0\}$.}\]  It is shown  that    \begin{itemize}  \item[(1)]  if $\omega\ne \emptyset$,   then the solutions of  the Cauchy problem (1) possess the Kato smoothing property;   \item[(2)] if     $g$ is a nonzero constant function,  then the solutions of  the Cauchy problem (1) possess the  sharp Kato smoothing property.   \end{itemize}

scholarly journals Positivity of solutions to the Cauchy problem for linear and semilinear biharmonic heat equations

2020 ◽  
Vol 10 (1) ◽  
pp. 353-370 ◽  
Author(s):  
Hans-Christoph Grunau ◽  
Nobuhito Miyake ◽  
Shinya Okabe
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Fourth Order ◽  
Parabolic Equations ◽  
Sufficient Conditions ◽  
Nonlinear Parabolic Equations ◽  
Heat Equations ◽  
Cauchy Problems ◽  
The Cauchy Problem ◽  
Positivity Of Solutions

Abstract This paper is concerned with the positivity of solutions to the Cauchy problem for linear and nonlinear parabolic equations with the biharmonic operator as fourth order elliptic principal part. Generally, Cauchy problems for parabolic equations of fourth order have no positivity preserving property due to the change of sign of the fundamental solution. One has eventual local positivity for positive initial data, but on short time scales, one will in general have also regions of negativity. The first goal of this paper is to find sufficient conditions on initial data which ensure the existence of solutions to the Cauchy problem for the linear biharmonic heat equation which are positive for all times and in the whole space. The second goal is to apply these results to show existence of globally positive solutions to the Cauchy problem for a semilinear biharmonic parabolic equation.

Local existence with low regularity for the 2D compressible Euler equations

2021 ◽  
Vol 18 (03) ◽  
pp. 701-728
Author(s):  
Huali Zhang
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Euler Equations ◽  
Local Existence ◽  
Compressible Euler Equations ◽  
Two Dimensional ◽  
Spatial Derivative ◽  
Compressible Euler ◽  
The Cauchy Problem ◽  
Local Solutions

We prove the local existence, uniqueness and stability of local solutions for the Cauchy problem of two-dimensional compressible Euler equations, where the initial data of velocity, density, specific vorticity [Formula: see text] and the spatial derivative of specific vorticity [Formula: see text].

Sign in / Sign up

Export Citation Format

Share Document