scholarly journals Global Analysis for Rough Solutions to the Davey-Stewartson System

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Han Yang ◽  
Xiaoming Fan ◽  
Shihui Zhu
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Nonlinear Equation ◽  
Global Solution ◽  
Global Analysis ◽  
Space Time ◽  
Well Posedness ◽  
The Cauchy Problem ◽  
New Space ◽  
Time Bounds

The global well-posedness of rough solutions to the Cauchy problem for the Davey-Stewartson system is obtained. It reads that if the initial data is inHswiths> 2/5, then there exists a global solution in time, and theHsnorm of the solution obeys polynomial-in-time bounds. The new ingredient in this paper is an interaction Morawetz estimate, which generates a new space-timeLt,x4estimate for nonlinear equation with the relatively general defocusing power nonlinearity.

scholarly journals The Cauchy Problem for a Dissipative Periodic 2-Component Degasperis-Procesi System

2014 ◽  
Vol 2014 ◽  
pp. 1-17
Author(s):  
Sen Ming ◽  
Han Yang ◽  
Ls Yong
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Transport Equation ◽  
Wave Breaking ◽  
A Priori Estimates ◽  
A Priori ◽  
Strong Solutions ◽  
Well Posedness ◽  
Priori Estimates ◽  
The Cauchy Problem

The dissipative periodic 2-component Degasperis-Procesi system is investigated. A local well-posedness for the system in Besov space is established by using the Littlewood-Paley theory and a priori estimates for the solutions of transport equation. The wave-breaking criterions for strong solutions to the system with certain initial data are derived.

EULER SYSTEM FOR COMPRESSIBLE FLUIDS AT A JUNCTION

2008 ◽  
Vol 05 (03) ◽  
pp. 547-568 ◽  
Author(s):  
RINALDO M. COLOMBO ◽  
CRISTINA MAURI
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Euler System ◽  
Common Origin ◽  
Compressible Fluids ◽  
Well Posedness ◽  
Euler Systems ◽  
Physical Conditions ◽  
The Cauchy Problem

Consider n ducts having a common origin and filled with a fluid. Along each duct, the full Euler system describes the evolution of the fluid. At the junction, suitable physical conditions couple the n Euler systems. In this paper we prove the well posedness of the Cauchy problem for the model so obtained, provided the total varaiatio of the initial data is sufficiently small.

Dyadic bilinear estimates and applications to the well-posedness for the 2D Zakharov–Kuznetsov equation in the endpoint space 𝐻−1/4

2020 ◽  
Vol 32 (6) ◽  
pp. 1575-1598
Author(s):  
Zhaohui Huo ◽  
Yueling Jia
Keyword(s):  
Cauchy Problem ◽  
Sobolev Space ◽  
Initial Data ◽  
Well Posedness ◽  
Small Initial Data ◽  
Bilinear Estimates ◽  
The Cauchy Problem ◽  
Univariate Polynomial ◽  
Well Posed ◽  
Resonance Function

AbstractThe Cauchy problem of the 2D Zakharov–Kuznetsov equation {\partial_{t}u+\partial_{x}(\partial_{xx}+\partial_{yy})u+uu_{x}=0} is considered. It is shown that the 2D Z-K equation is locally well-posed in the endpoint Sobolev space {H^{-1/4}}, and it is globally well-posed in {H^{-1/4}} with small initial data. In this paper, we mainly establish some new dyadic bilinear estimates to obtain the results, where the main novelty is to parametrize the singularity of the resonance function in terms of a univariate polynomial.

scholarly journals Global solution to the incompressible Oldroyd-B model in hybrid Besov spaces

Filomat ◽  
2016 ◽  
Vol 30 (13) ◽  
pp. 3627-3639 ◽  
Author(s):  
Ruizhao Zia
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Global Solution ◽  
Besov Spaces ◽  
Coupling Constant ◽  
Unique Global Solution ◽  
Small Initial Data ◽  
Nonlinear Anal ◽  
The Cauchy Problem

This paper is dedicated to the Cauchy problem of the incompressible Oldroyd-B model with general coupling constant ? ?(0,1). It is shown that this set of equations admits a unique global solution in a certain hybrid Besov spaces for small initial data in ?Hs ??Bd/2 2,1 with - d/2 < s < d2-1. In particular, if d ? 3, and taking s=0, then ?H0 ? ?Bd/2 2,1 = B d/2 2,1. Since Bt2,? ? Bd/2 2,1 if t > d/2, this result extends the work by Chen and Miao [Nonlinear Anal.,68(2008), 1928-1939].

Well-posedness and limit behaviors for a stochastic higher order modified Camassa–Holm equation

2016 ◽  
Vol 16 (06) ◽  
pp. 1650019
Author(s):  
Lin Lin ◽  
Guangying Lv ◽  
Wei Yan
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Existence And Uniqueness ◽  
Local Existence ◽  
Higher Order ◽  
Well Posedness ◽  
Local Existence And Uniqueness ◽  
Holm Equation ◽  
The Cauchy Problem

This paper is devoted to the Cauchy problem for a stochastic higher order modified-Camassa–Holm equation [Formula: see text] The local existence and uniqueness with initial data [Formula: see text], [Formula: see text] and [Formula: see text], is established. The limit behaviors of the solution are examined as [Formula: see text].

scholarly journals On the Cauchy Problem for the Two-Component Novikov Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Yongsheng Mi ◽  
Chunlai Mu ◽  
Weian Tao
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Transport Equation ◽  
Besov Spaces ◽  
Well Posedness ◽  
Two Component ◽  
The Cauchy Problem ◽  
Novikov Equation

We are concerned with the Cauchy problem of two-component Novikov equation, which was proposed by Geng and Xue (2009). We establish the local well-posedness in a range of the Besov spaces by using Littlewood-Paley decomposition and transport equation theory which is motivated by that in Danchin's cerebrated paper (2001). Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time, which extend some results of Himonas (2003) to more general equations.

scholarly journals BILINEAR ESTIMATES AND APPLICATIONS TO NONLINEAR WAVE EQUATIONS

2002 ◽  
Vol 04 (02) ◽  
pp. 223-295 ◽  
Author(s):  
SERGIU KLAINERMAN ◽  
SIGMUND SELBERG
Keyword(s):  
Systematic Review ◽  
Cauchy Problem ◽  
Initial Data ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Large Data ◽  
Nonlinear Wave Equations ◽  
Well Posedness ◽  
Bilinear Estimates ◽  
The Cauchy Problem

We undertake a systematic review of results proved in [26, 27, 30-32] concerning local well-posedness of the Cauchy problem for certain systems of nonlinear wave equations, with minimal regularity assumptions on the initial data. Moreover we give a considerably simplified and unified treatment of these results and provide also complete proofs for large data. The paper is also intended as an introduction to and survey of current research in the very active area of nonlinear wave equations. The key ingredients throughout the survey are the use of the null structure of the equations we consider and, intimately tied to it, bilinear estimates.

scholarly journals Klein-Gordon Equations on Modulation Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Guoping Zhao ◽  
Jiecheng Chen ◽  
Weichao Guo
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Stability Theory ◽  
Scattering Theory ◽  
Modulation Spaces ◽  
Well Posedness ◽  
Blowup Criterion ◽  
The Cauchy Problem ◽  
Klein Gordon

We consider the Cauchy problem for a family of Klein-Gordon equations with initial data in modulation spacesMp,1a. We develop the well-posedness, blowup criterion, stability of regularity, scattering theory, and stability theory.

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