Global existence and blow-up solutions of the radial Schrödinger maps

2020 ◽  
pp. 2050009
Author(s):  
Jian Zhai ◽  
Bo-Wen Zheng
Keyword(s):  
Cauchy Problem ◽  
Blow Up ◽  
Weighted Sobolev Space ◽  
Integrable Model ◽  
Spherically Symmetric ◽  
Well Posedness ◽  
One Dimensional ◽  
Complex Transformation ◽  
The Cauchy Problem ◽  
Symmetric Initial Data

This paper studies the Cauchy problem of radial inhomogeneous Schrödinger maps (ISM) which arises from the integrable model of the inhomogeneous spherically symmetric Heisenberg ferromagnetic spin system. Through a complex transformation the radial ISM is equivalent to an integro-differential Schrödinger equation. A new weighted Sobolev space [Formula: see text] is introduced and the well-posedness of integro-differential Schrödinger equations, including the integral radial IMS, with small spherically symmetric initial data in one-dimensional energy space [Formula: see text] is established. Furthermore, for [Formula: see text], we prove the existence of blow-up solutions for the integral radial ISM.

scholarly journals On the Blow-Up of Solutions of a Weakly Dissipative Modified Two-Component Periodic Camassa-Holm System

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Yongsheng Mi ◽  
Chunlai Mu ◽  
Weian Tao
Keyword(s):  
Cauchy Problem ◽  
Blow Up ◽  
Strong Solutions ◽  
Well Posedness ◽  
Holm Equation ◽  
Two Component ◽  
The Cauchy Problem ◽  
Blow Up Rate

We study the Cauchy problem of a weakly dissipative modified two-component periodic Camassa-Holm equation. We first establish the local well-posedness result. Then we derive the precise blow-up scenario and the blow-up rate for strong solutions to the system. Finally, we present two blow-up results for strong solutions to the system.

Well-posedness and blow-up criterion for the Chaplygin gas equations in ℝN

2019 ◽  
Vol 16 (04) ◽  
pp. 639-661 ◽  
Author(s):  
Zhen Wang ◽  
Xinglong Wu
Keyword(s):  
Cauchy Problem ◽  
Blow Up ◽  
Initial Density ◽  
Chaplygin Gas ◽  
Bmo Space ◽  
Well Posedness ◽  
Dimension Formula ◽  
The Cauchy Problem ◽  
Bounded Below ◽  
Blow Up Criterion

We establish a well-posedness theory and a blow-up criterion for the Chaplygin gas equations in [Formula: see text] for any dimension [Formula: see text]. First, given [Formula: see text], [Formula: see text], we prove the well-posedness property for solutions [Formula: see text] in the space [Formula: see text] for the Cauchy problem associated with the Chaplygin gas equations, provided the initial density [Formula: see text] is bounded below. We also prove that the solution of the Chaplygin gas equations depends continuously upon its initial data [Formula: see text] in [Formula: see text] if [Formula: see text], and we state a blow-up criterion for the solutions in the classical BMO space. Finally, using Osgood’s modulus of continuity, we establish a refined blow-up criterion of the solutions.

scholarly journals Solution concepts, well-posedness, and wave breaking for the Fornberg–Whitham equation

2020 ◽  
Author(s):  
Günther Hörmann
Keyword(s):  
Cauchy Problem ◽  
Traveling Waves ◽  
Wave Breaking ◽  
Blow Up ◽  
Strong Solutions ◽  
Well Posedness ◽  
Solution Concepts ◽  
Nonlinear Operators ◽  
The Cauchy Problem ◽  
Whitham Equation

AbstractWe discuss concepts and review results about the Cauchy problem for the Fornberg–Whitham equation, which has also been called Burgers–Poisson equation in the literature. Our focus is on a comparison of various strong and weak solution concepts as well as on blow-up of strong solutions in the form of wave breaking. Along the way we add aspects regarding semiboundedness at blow-up, from semigroups of nonlinear operators to the Cauchy problem, and about continuous traveling waves as weak solutions.

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, Blow-Up Phenomena, and Asymptotic Profile for a Weakly Dissipative Modified Two-Component Camassa-Holm Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Yongsheng Mi ◽  
Chunlai Mu
Keyword(s):  
Cauchy Problem ◽  
Blow Up ◽  
Strong Solutions ◽  
Asymptotic Behavior Of Solutions ◽  
Well Posedness ◽  
Asymptotic Profile ◽  
Holm Equation ◽  
Two Component ◽  
The Cauchy Problem ◽  
Blow Up Rate

We study the Cauchy problem of a weakly dissipative modified two-component Camassa-Holm equation. We firstly establish the local well-posedness result. Then we present a precise blow-up scenario. Moreover, we obtain several blow-up results and the blow-up rate of strong solutions. Finally, we consider the asymptotic behavior of solutions.

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.

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