scholarly journals On the Local Well-Posedness of the Cauchy Problem for a Modified Two-Component Camassa-Holm System in Besov Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-13
Author(s):  
Jiangbo Zhou ◽  
Lu Yao ◽  
Lixin Tian ◽  
Wenbin Zhang
Keyword(s):  
Cauchy Problem ◽  
Besov Spaces ◽  
A Priori Estimates ◽  
A Priori ◽  
Cubic Nonlinearity ◽  
Priori Estimates ◽  
Linear Transport Equation ◽  
Two Component ◽  
The Cauchy Problem ◽  
Well Posed

We consider the Cauchy problem for an integrable modified two-component Camassa-Holm system with cubic nonlinearity. By using the Littlewood-Paley decomposition, nonhomogeneous Besov spaces, and a priori estimates for linear transport equation, we prove that the Cauchy problem is locally well-posed in Besov spaces Bp, rs with 1≤p, r≤+∞ and s>max{2+(1/p),5/2}.

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.

WELL-POSEDNESS AND ANALYTICITY FOR AN INTEGRABLE TWO-COMPONENT SYSTEM WITH CUBIC NONLINEARITY

2013 ◽  
Vol 10 (04) ◽  
pp. 703-723 ◽  
Author(s):  
YONGSHENG MI ◽  
CHUNLAI MU
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Besov Spaces ◽  
Well Posedness ◽  
Cubic Nonlinearity ◽  
Two Component System ◽  
Component System ◽  
Two Component ◽  
The Cauchy Problem

We study the Cauchy problem associated with a new integrable two-component system with cubic nonlinearity, which was recently proposed by Song, Qu and Qiao. We establish the local well-posedness in a range of the Besov spaces. Moreover, for analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time, which extend a result by Danchin, and Himonas et al. to more complex equations.

scholarly journals On the Cauchy problem for a class of hyperbolic operators with triple characteristics

2020 ◽  
Author(s):  
Annamaria Barbagallo ◽  
Vincenzo Esposito
Keyword(s):  
Cauchy Problem ◽  
Sobolev Spaces ◽  
Existence Result ◽  
A Priori Estimates ◽  
A Priori ◽  
Hyperbolic Operators ◽  
Priori Estimates ◽  
The Cauchy Problem

AbstractThe Cauchy problem for a class of hyperbolic operators with triple characteristics is analyzed. Some a priori estimates in Sobolev spaces with negative indexes are proved. Subsequently, an existence result for the Cauchy problem is obtained.

scholarly journals Doubly Degenerate Parabolic Equation with Time-Dependent Gradient Source and Initial Data Measures

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Lihua Deng ◽  
Xianguang Shang
Keyword(s):  
Parabolic Equation ◽  
Initial Data ◽  
Degenerate Parabolic Equation ◽  
A Priori Estimates ◽  
A Priori ◽  
Local Existence ◽  
Time Dependent ◽  
Priori Estimates ◽  
The Cauchy Problem ◽  
Degenerate Parabolic

This paper is devoted to the Cauchy problem for a class of doubly degenerate parabolic equation with time-dependent gradient source, where the initial data are Radon measures. Using the delicate a priori estimates, we first establish two local existence results. Furthermore, we show that the existence of solutions is optimal in the class considered here.

HOMOGENIZATION OF TWO HEAT CONDUCTORS WITH AN INTERFACIAL CONTACT RESISTANCE

2004 ◽  
Vol 02 (03) ◽  
pp. 247-273 ◽  
Author(s):  
PATRIZIA DONATO ◽  
SARA MONSURRÒ
Keyword(s):  
Contact Resistance ◽  
A Priori Estimates ◽  
A Priori ◽  
Limit Problem ◽  
Heat Diffusion ◽  
Stationary Heat ◽  
Priori Estimates ◽  
Two Component ◽  
Oscillating Coefficients ◽  
Interfacial Contact Resistance

In this paper we describe the asymptotic behavior of a problem depending on a small parameter ε>0 and modelling the stationary heat diffusion in a two-component conductor. The flow of heat is proportional to the jump of the temperature field, due to a contact resistance on the interface.More precisely, we give an homogenization result for the stationary heat equation with oscillating coefficients in a domain [Formula: see text] of ℝn, where [Formula: see text] is connected and [Formula: see text] is union of ε-periodic disconnected inclusions of size ε. These two sub-domains of Ω are separated by a contact surface Γε, on which we prescribe the continuity of the conormal derivatives and a jump of the solution proportional to the conormal derivative, by means of a function of order εγ.We describe the limit problem for γ>-1. The two cases -1<γ≤1 (Theorem 2.1) and γ>1 (Theorem 2.2) need to be treated separately, because of different a priori estimates.

scholarly journals The Cauchy Problem for a Weakly Dissipative 2-Component Camassa-Holm System

2014 ◽  
Vol 2014 ◽  
pp. 1-16
Author(s):  
Sen Ming ◽  
Han Yang ◽  
Yonghong Wu
Keyword(s):  
Wave Breaking ◽  
Blow Up ◽  
A Priori Estimates ◽  
A Priori ◽  
Strong Solutions ◽  
Well Posedness ◽  
Priori Estimates ◽  
The Cauchy Problem ◽  
Blow Up Rate ◽  
Global Existence Result

The weakly dissipative 2-component Camassa-Holm system is considered. A local well-posedness for the system in Besov spaces is established by using the Littlewood-Paley theory and a priori estimates for the solutions of transport equation. The wave-breaking mechanisms and the exact blow-up rate of strong solutions to the system are presented. Moreover, a global existence result for strong solutions is derived.

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 Maximum Norm Estimates of the Solution of the Navier-Stokes Equations in the Halfspace with Bounded Initial Data

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Santosh Pathak
Keyword(s):  
Initial Data ◽  
A Priori Estimates ◽  
Stokes Equations ◽  
A Priori ◽  
Maximum Norm ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Norm Estimates ◽  
Priori Estimates ◽  
The Cauchy Problem

In this paper, I consider the Cauchy problem for the incompressible Navier-Stokes equations in ℝ + n for n ≥ 3 with bounded initial data and derive a priori estimates of the maximum norm of all derivatives of the solution in terms of the maximum norm of the initial data. This paper is a continuation of my work in my previous papers, where the initial data are considered in T n and ℝ n respectively. In this paper, because of the nonempty boundary in our domain of interest, the details in obtaining the desired result are significantly different and more challenging than the work of my previous papers. This challenges arise due to the possible noncommutativity nature of the Leray projector with the derivatives in the direction of normal to the boundary of the domain of interest. Therefore, we only consider one derivative of the velocity field in that direction.

scholarly journals A priori estimates in terms of the maximum norm for the solution of the Navier-Stokes equations with periodic initial data

2019 ◽  
Vol 36 (1-2) ◽  
pp. 39-50
Author(s):  
Santosh Pathak
Keyword(s):  
Initial Data ◽  
A Priori Estimates ◽  
Stokes Equations ◽  
A Priori ◽  
Maximum Norm ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Priori Estimates ◽  
The Cauchy Problem ◽  
Periodic Initial Data

In this paper, we consider the Cauchy problem for the incompressible Navier-Stokes equations in Rn for n ≥ 3 with smooth periodic initial data and derive a priori estimtes of the maximum norm of all derivatives of the solution in terms of the maximum norm of the initial data. This paper is a special case of a paper by H-O Kreiss and J. Lorenz which also generalizes the main result of their paper to higher dimension.

scholarly journals A-priori Estimates Near the Boundary for Solutions of a class of Degenerate Elliptic Problems in Besov-type Spaces

2017 ◽  
Vol 3 (2) ◽  
pp. 149-172 ◽  
Author(s):  
Azzeddine El Baraka ◽  
Mohammed Masrour
Keyword(s):  
Besov Spaces ◽  
A Priori Estimates ◽  
A Priori ◽  
Elliptic Problems ◽  
General Frame ◽  
Priori Estimates ◽  
Special Cases ◽  
Degenerate Elliptic ◽  
Type Spaces ◽  
Besov Type Spaces

AbstractIn this paper, we give a priori estimates near the boundary for solutions of a degenerate elliptic problems in the general Besov-type spaces $B_{p,q}^{s,\tau }$, containing as special cases: Goldberg space bmo, local Morrey-Campanato spaces l2,λ and the classical Hölder and Besov spaces $B_{p,q}^s $. This work extends the results of [13, 2, 15] from Hölder and Besov spaces to the general frame of $B_{p,q}^{s,\tau }$ spaces.

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