scholarly journals Well-posedness, wave breaking, Holder continuity and periodic peakons for a nonlocal sine-µ-Camassa-Holm equation

2021 ◽  
Author(s):  
Guoquan Qin ◽  
Zhenya Yan ◽  
Boling Guo
Keyword(s):  
Initial Data ◽  
Explicit Formula ◽  
Initial Value Problem ◽  
Besov Spaces ◽  
Wave Breaking ◽  
Well Posedness ◽  
Sufficient Condition ◽  
Initial Value ◽  
Solution Map ◽  
Holm Equation

In this paper, we investigate the initial value problem of a nonlocal sine-type µ-Camassa-Holm (µCH) equation, which is the µ-version of the sine-type CH equation. We first discuss its local well-posedness in the framework of Besov spaces. Then a sufficient condition on the initial data is provided to ensure the occurance of the wave-breaking phenomenon. We finally prove the H¨older continuity of the data-to-solution map, and find the explicit formula of the global weak periodic peakon solution.

WELL-POSEDNESS AND BLOWUP PHENOMENA FOR A THREE-COMPONENT CAMASSA–HOLM SYSTEM WITH PEAKONS

2012 ◽  
Vol 09 (03) ◽  
pp. 451-467 ◽  
Author(s):  
QIAOYI HU ◽  
LIYUN LIN ◽  
JI JIN
Keyword(s):  
Initial Value Problem ◽  
Strong Solutions ◽  
Well Posedness ◽  
Initial Value ◽  
Blowup Rate ◽  
Holm Equation ◽  
Two Component

First, we establish the local well-posedness of the initial value problem for a new three-component Camassa–Holm system with peakons. We then present a precise blowup scenario and several blowup results for strong solutions to that system. Finally, we determine the blowup rate of strong solutions to the system when a blowup occurs. Our results include all earlier results on the Camassa–Holm equation and on a two-component Camassa–Holm system with peakons.

scholarly journals Global well-posedness for a system of KdV-type equations with coupled quadratic nonlinearities

2014 ◽  
Vol 215 ◽  
pp. 67-149 ◽  
Author(s):  
Jerry L. bona ◽  
Jonathan Cohen ◽  
Gang Wang
Keyword(s):  
Initial Data ◽  
Sobolev Spaces ◽  
Initial Value Problem ◽  
Coupled Systems ◽  
Well Posedness ◽  
Initial Value ◽  
Quadratic Polynomials ◽  
Partial Differentiation ◽  
Quadratic Nonlinearities ◽  
Kdv Type Equations

AbstractIn this paper, coupled systemsof Korteweg-de Vries type are considered, where u = u(x, t), v = v(x, t) are real-valued functions and where x, t∈R. Here, subscripts connote partial differentiation andare quadratic polynomials in the variables u and v. Attention is given to the pure initial-value problem in which u(x, t) and v(x, t) are both specified at t = 0, namely,for x ∈ ℝ. Under suitable conditions on P and Q, global well-posedness of this problem is established for initial data in the L2-based Sobolev spaces Hs(ℝ) × Hs(ℝ) for any s > ‒3/4.

scholarly journals Global well-posedness for a system of KdV-type equations with coupled quadratic nonlinearities

2014 ◽  
Vol 215 ◽  
pp. 67-149 ◽  
Author(s):  
Jerry L. bona ◽  
Jonathan Cohen ◽  
Gang Wang
Keyword(s):  
Initial Data ◽  
Sobolev Spaces ◽  
Initial Value Problem ◽  
Coupled Systems ◽  
Well Posedness ◽  
Initial Value ◽  
Quadratic Polynomials ◽  
Partial Differentiation ◽  
Quadratic Nonlinearities ◽  
Kdv Type Equations

AbstractIn this paper, coupled systemsof Korteweg-de Vries type are considered, whereu=u(x, t),v=v(x, t) are real-valued functions and wherex, t∈R. Here, subscripts connote partial differentiation andare quadratic polynomials in the variablesuandv. Attention is given to the pure initial-value problem in whichu(x, t) andv(x, t) are both specified att= 0, namely,forx∈ ℝ. Under suitable conditions onPandQ, global well-posedness of this problem is established for initial data in theL2-based Sobolev spacesHs(ℝ) ×Hs(ℝ) for anys> ‒3/4.

scholarly journals ENERGY CRITICAL NLS IN TWO SPACE DIMENSIONS

2009 ◽  
Vol 06 (03) ◽  
pp. 549-575 ◽  
Author(s):  
J. COLLIANDER ◽  
S. IBRAHIM ◽  
M. MAJDOUB ◽  
N. MASMOUDI
Keyword(s):  
Schrödinger Equation ◽  
Initial Value Problem ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Energy Space ◽  
Well Posedness ◽  
Initial Value ◽  
Exponential Nonlinearity ◽  
Supercritical Case ◽  
Two Space Dimensions

We investigate the initial value problem for a defocusing nonlinear Schrödinger equation with exponential nonlinearity [Formula: see text] We identify subcritical, critical, and supercritical regimes in the energy space. We establish global well-posedness in the subcritical and critical regimes. Well-posedness fails to hold in the supercritical case.

Global well-posedness and self-similarity for semilinear wave equations in a time-weighted framework of Besov type

2020 ◽  
Vol 17 (01) ◽  
pp. 123-139
Author(s):  
Lucas C. F. Ferreira ◽  
Jhean E. Pérez-López
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Besov Spaces ◽  
Wave Equations ◽  
Well Posedness ◽  
Self Similarity ◽  
Semilinear Wave Equations ◽  
Semilinear Wave Equation

We show global-in-time well-posedness and self-similarity for the semilinear wave equation with nonlinearity [Formula: see text] in a time-weighted framework based on the larger family of homogeneous Besov spaces [Formula: see text] for [Formula: see text]. As a consequence, in some cases of the power [Formula: see text], we cover a initial-data class larger than in some previous results. Our approach relies on dispersive-type estimates and a suitable [Formula: see text]-product estimate in Besov spaces.

scholarly journals Some Mathematical Aspects of f(R)-Gravity with Torsion: Cauchy Problem and Junction Conditions

Universe ◽  
2019 ◽  
Vol 5 (12) ◽  
pp. 224 ◽  
Author(s):  
Stefano Vignolo
Keyword(s):  
Cauchy Problem ◽  
Initial Value Problem ◽  
Sufficient Conditions ◽  
Well Posedness ◽  
Junction Conditions ◽  
Field Equations ◽  
Initial Value ◽  
The Cauchy Problem ◽  
General Conditions

We discuss the Cauchy problem and the junction conditions within the framework of f ( R ) -gravity with torsion. We derive sufficient conditions to ensure the well-posedness of the initial value problem, as well as general conditions to join together on a given hypersurface two different solutions of the field equations. The stated results can be useful to distinguish viable from nonviable f ( R ) -models with torsion.

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