Local well-posedness for the quantum Zakharov system in three and higher dimensions

2021 ◽  
Vol 18 (02) ◽  
pp. 257-270
Author(s):  
Isao Kato
Keyword(s):  
Cauchy Problem ◽  
Type System ◽  
Higher Dimensions ◽  
Well Posedness ◽  
Zakharov System ◽  
Quantum Parameter ◽  
Fourier Restriction ◽  
The Cauchy Problem ◽  
The One ◽  
Quantum Zakharov System

We study the Cauchy problem associated with a quantum Zakharov-type system in three and higher spatial dimensions.Taking the quantum parameter to unit and developing Fourier restriction norm arguments, we establish local well-posedness property for wider range than the one known for the Zakharov system.

Local well-posedness for the quantum Zakharov system in one spatial dimension

2017 ◽  
Vol 14 (01) ◽  
pp. 157-192 ◽  
Author(s):  
Yung-Fu Fang ◽  
Hsi-Wei Shih ◽  
Kuan-Hsiang Wang
Keyword(s):  
Sobolev Space ◽  
Initial Data ◽  
Electric Field ◽  
Classical Result ◽  
Spatial Dimension ◽  
Well Posedness ◽  
Ion Density ◽  
Zakharov System ◽  
Quantum Parameter ◽  
Quantum Zakharov System

We consider the quantum Zakharov system in one spatial dimension and establish a local well-posedness theory when the initial data of the electric field and the deviation of the ion density lie in a Sobolev space with suitable regularity. As the quantum parameter approaches zero, we formally recover a classical result by Ginibre, Tsutsumi, and Velo. We also improve their result concerning the Zakharov system and a result by Jiang, Lin, and Shao concerning the quantum Zakharov system.

scholarly journals Well-posedness of stochastic modified Kawahara equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
P. Agarwal ◽  
Abd-Allah Hyder ◽  
M. Zakarya
Keyword(s):  
Cauchy Problem ◽  
Fixed Point ◽  
Water Wave ◽  
Well Posedness ◽  
Kawahara Equation ◽  
Fourier Restriction ◽  
The Cauchy Problem ◽  
Restriction Method ◽  
Fifth Order ◽  
Point Argument

AbstractIn this paper we consider the Cauchy problem for the stochastic modified Kawahara equation, which is a fifth-order shallow water wave equation. We prove local well-posedness for data in $H^{s}(\mathbb{R})$Hs(R), $s\geq -1/4$s≥−1/4. Moreover, we get the global existence for $L^{2}( \mathbb{R})$L2(R) solutions. Due to the non-zero singularity of the phase function, a fixed point argument and the Fourier restriction method are proposed.

scholarly journals On non-resistive limit of 1D MHD equations with no vacuum at infinity

2021 ◽  
Vol 11 (1) ◽  
pp. 702-725
Author(s):  
Zilai Li ◽  
Huaqiao Wang ◽  
Yulin Ye
Keyword(s):  
Cauchy Problem ◽  
A Priori ◽  
Strong Solutions ◽  
Mhd Equations ◽  
Well Posedness ◽  
Large Initial Data ◽  
One Dimensional ◽  
The Cauchy Problem ◽  
Global Strong Solutions ◽  
The One

Abstract In this paper, the Cauchy problem for the one-dimensional compressible isentropic magnetohydrodynamic (MHD) equations with no vacuum at infinity is considered, but the initial vacuum can be permitted inside the region. By deriving a priori ν (resistivity coefficient)-independent estimates, we establish the non-resistive limit of the global strong solutions with large initial data. Moreover, as a by-product, the global well-posedness of strong solutions for the compressible resistive MHD equations is also established.

scholarly journals Well-posedness of the Cauchy problem for the magnetic Zakharov type system

Nonlinearity ◽  
2011 ◽  
Vol 24 (8) ◽  
pp. 2191-2210 ◽  
Author(s):  
Boling Guo ◽  
Jingjun Zhang
Keyword(s):  
Cauchy Problem ◽  
Type System ◽  
Well Posedness ◽  
The Cauchy Problem

Global well-posedness of the Cauchy problem for the equations of a one-dimensional viscous heat-conducting gas with Lebesgue initial data

2004 ◽  
Vol 134 (5) ◽  
pp. 939-960 ◽  
Author(s):  
Song Jiang ◽  
Alexander Zlotnik
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Heat Conducting ◽  
Well Posedness ◽  
Global Weak Solutions ◽  
One Dimensional ◽  
Lipschitz Continuous ◽  
Viscous Heat ◽  
The Cauchy Problem ◽  
The One

We study the Cauchy problem for the one-dimensional equations of a viscous heat-conducting gas in the Lagrangian mass coordinates with the initial data in the Lebesgue spaces. We prove the existence, the uniqueness and the Lipschitz continuous dependence on the initial data of global weak solutions.

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.

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