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.

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 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.

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 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 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 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 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.

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