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].

LOCAL EXISTENCE AND UNIQUENESS THEORY FOR THE SECOND SOUND EQUATION IN ONE SPACE DIMENSION

2012 ◽  
Vol 09 (01) ◽  
pp. 177-193 ◽  
Author(s):  
KEIICHI KATO ◽  
YUUSUKE SUGIYAMA
Keyword(s):  
Cauchy Problem ◽  
Existence And Uniqueness ◽  
Space Dimension ◽  
Local Existence ◽  
Second Sound ◽  
Uniqueness Of Solutions ◽  
Local Existence And Uniqueness ◽  
The Cauchy Problem ◽  
Time Existence ◽  
Uniqueness Theory

We study the local-in-time existence and uniqueness of the Cauchy problem for the nonlinear wave equation [Formula: see text], which is called the second sound equation. Assuming that u(0, x) = φ ≥ A > 0, φ ∈ C1, and ∂xφ ∈ Hs, we establish the uniqueness of solutions without restriction on their amplitude.

scholarly journals Some study of a cauchy problem in parabolic integrodifferential equations class

2010 ◽  
Vol 7 (1) ◽  
pp. 101-105
Author(s):  
A. S. AL-FHAID
Keyword(s):  
Cauchy Problem ◽  
Existence And Uniqueness ◽  
Local Existence ◽  
Integrodifferential Equations ◽  
Local Existence And Uniqueness ◽  
The Cauchy Problem

We consider the parabolic integrodifferential equations of a form given below. We establish local existence and uniqueness and prove the convergence in L2(En) to the solution Ut of the Cauchy problem.

Local existence with low regularity for the 2D compressible Euler equations

2021 ◽  
Vol 18 (03) ◽  
pp. 701-728
Author(s):  
Huali Zhang
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Euler Equations ◽  
Local Existence ◽  
Compressible Euler Equations ◽  
Two Dimensional ◽  
Spatial Derivative ◽  
Compressible Euler ◽  
The Cauchy Problem ◽  
Local Solutions

We prove the local existence, uniqueness and stability of local solutions for the Cauchy problem of two-dimensional compressible Euler equations, where the initial data of velocity, density, specific vorticity [Formula: see text] and the spatial derivative of specific vorticity [Formula: see text].

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.

scholarly journals A LIPSCHITZ METRIC FOR THE CAMASSA–HOLM EQUATION

2020 ◽  
Vol 8 ◽  
Author(s):  
JOSÉ A. CARRILLO ◽  
KATRIN GRUNERT ◽  
HELGE HOLDEN
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Wasserstein Metric ◽  
Holm Equation ◽  
The Cauchy Problem ◽  
Lipschitz Metric

We analyze stability of conservative solutions of the Cauchy problem on the line for the Camassa–Holm (CH) equation. Generically, the solutions of the CH equation develop singularities with steep gradients while preserving continuity of the solution itself. In order to obtain uniqueness, one is required to augment the equation itself by a measure that represents the associated energy, and the breakdown of the solution is associated with a complicated interplay where the measure becomes singular. The main result in this paper is the construction of a Lipschitz metric that compares two solutions of the CH equation with the respective initial data. The Lipschitz metric is based on the use of the Wasserstein metric.

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.

HIGHER-ORDER QUASILINEAR PARABOLIC EQUATIONS WITH SINGULAR INITIAL DATA

2006 ◽  
Vol 08 (03) ◽  
pp. 331-354 ◽  
Author(s):  
V. A. GALAKTIONOV ◽  
A. E. SHISHKOV
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Quasilinear Parabolic Equation ◽  
Higher Order ◽  
Quasilinear Parabolic Equations ◽  
Semilinear Heat Equation ◽  
Basic Model ◽  
The Cauchy Problem ◽  
Singular Initial Data ◽  
Quasilinear Parabolic

As a basic model, we study the 2mth-order quasilinear parabolic equation of diffusion-absorption type [Formula: see text] where Δm,p is the 2mth-order p-Laplacian [Formula: see text]. We consider the Cauchy problem in RN × R+ with arbitrary singular initial data u0 ≠ 0 such that u0(x) = 0 for any x ≠ 0. We prove that, in the most delicate case p = q and [Formula: see text], this Cauchy problem admits the unique trivial solution u(·, t) = 0 for t > 0. For λ < λ0, such nontrivial very singular solutions are known to exist for some semilinear higher-order models. This extends the well-known result by Brezis and Friedman established in 1983 for the semilinear heat equation with p = q = m = 1.

Sign in / Sign up

Export Citation Format

Share Document