On <i>H</i> <sup>2</sup>-solutions for a Camassa-Holm type equation

Giuseppe Maria Coclite, Lorenzo di Ruvo

Type: Article

Publication Date: 2023-01-01

Citations: 5

DOI: https://doi.org/10.1515/math-2022-0577

Abstract

Abstract Camassa-Holm type equations arise as models for the unidirectional propagation of shallow water waves over a flat bottom. They also describe finite length, small amplitude radial deformation waves in cylindrical compressible hyperelastic rods. Under appropriate assumption on the initial data, on the time <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>T</m:mi> </m:math> T , and on the coefficients of such equation, we prove the well-posedness of the classical solutions for the Cauchy problem.

Locations

Open Mathematics - View - PDF

Similar Works

Action	Title	Year	Authors
+	Well-posedness and peakons for a higher-order<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml3" display="inline" overflow="scroll" altimg="si3.gif"><mml:mi>μ</mml:mi></mml:math>-Camassa–Holm equation	2018	Feng Wang Fengquan Li Zhijun Qiao
+ PDF Chat	On Well-Posedness Results for Camassa-Holm Equation on the Line: A Survey	2004	Luc Molinet
+ PDF Chat	Compactly Supported Solutions of the Camassa-Holm Equation	2005	David Henry
+	On the classical solutions for the high order Camassa-Holm type equations	2023	Giuseppe Maria Coclite Lorenzo di Ruvo
+	Global well-posedness and infinite propagation speed for the <i>N</i> − <i>abc</i> family of Camassa–Holm type equation with both dissipation and dispersion	2020	Zaiyun Zhang Zhenhai Liu Youjun Deng Chuangxia Huang Shi-you Lin Zhu Wen
+	Ill-posedness for the Cauchy problem of the Camassa-Holm equation in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msubsup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo>∞</mml:mo><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="double-struck">R</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>	2022	Yingying Guo Weikui Ye Zhaoyang Yin
+	Local Well-Posedness and Blow-Up Phenomena of Solutions of a <i>n</i>-Component <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mi>μ</mi> </math> -Camassa-Holm System	2024	亚琴高
+	Ill-posedness for the Camassa–Holm equation in $$B_{p,1}^{1}\cap C^{0,1}$$	2024	Jinlu Li Yanghai Yu Yingying Guo Weipeng Zhu
+ PDF Chat	Integrability, existence of global solutions, and wave breaking criteria for a generalization of the Camassa–Holm equation	2020	Priscila Leal da Silva Igor Leite Freire
+ PDF Chat	Camassa-Holm equation on shallow water wave equation	2017	L. Bezati Sh. Hajrulla
+	Ill-Posedness for the Cauchy Problem of the Modified Camassa-Holm Equation in $$B_{\infty ,1}^0$$	2024	He Zhen Zhaoyang Yin
+	None	2005	A. Alexandrou Himonas Gerard Misiołek
+ PDF Chat	On the Cauchy problem for a higher-order <i>μ</i>-Camassa-Holm equation	2018	Feng Wang Fengquan Li Zhijun Qiao
+	A TWO-DIMENSIONAL VERSION OF THE CAMASSA-HOLM EQUATION	2001	H.‐P. Kruse Jürgen Scheurle Wenju Du
+	The local well-posedness and existence of weak solutions for a generalized Camassa–Holm equation	2010	Shaoyong Lai Yonghong Wu
+	On uniqueness of dissipative solutions of the Camassa-Holm equation	2016	Grzegorz Jamróz
+	$H^{2}$ solutions for a Degasperis–Procesi-type equation	2024	Giuseppe Maria Coclite Lorenzo di Ruvo
+	On solutions to the Holm–Staley<i>b</i>-family of equations	2010	Yong Zhou
+	Hybrid solitary wave solutions of the Camassa–Holm equation	2022	Hugues Martial Omanda Clovis Taki Djeumen Tchaho D. Belobo Belobo
+ PDF Chat	Variational derivation of two-component Camassa–Holm shallow water system	2012	Delia Ionescu-Kruse

Works That Cite This (5)

Action	Title	Year	Authors
+ PDF Chat	What will the mathematics of tomorrow look like?	2023	S. Marano Vincenzo Vesprı
+	Energy conservation and well-posedness of the Camassa–Holm equation in Sobolev spaces	2023	Yingying Guo Weikui Ye
+ PDF Chat	Lie Symmetry Analysis, Closed-Form Solutions, and Conservation Laws for the Camassa–Holm Type Equation	2024	Jonathan Lebogang Bodibe Chaudry Masood Khalique
+	On the classical solutions for the high order Camassa-Holm type equations	2023	Giuseppe Maria Coclite Lorenzo di Ruvo
+	On the solutions for the Novikov equation	2024	Giuseppe Maria Coclite Lorenzo di Ruvo

Works Cited by This (74)

Action	Title	Year	Authors
+ PDF Chat	The Cauchy problem for an integrable shallow-water equation	2001	A. Alexandrou Himonas Gerard Misiołek
+	Symplectic structures, their Bäcklund transformations and hereditary symmetries	1981	Benno Fuchssteiner A. S. Fokas
+	Global regularity, and wave breaking phenomena in a class of nonlocal dispersive equations	2010	Hailiang Liu Zhaoyang Yin
+ PDF Chat	An explicit finite difference scheme for the Camassa-Holm equation	2008	Giuseppe Maria Coclite Kenneth H. Karlsen Nils Henrik Risebro
+	Classical solutions of the periodic Camassa—Holm equation	2002	Gerard Misiołek
+	Camassa–Holm, Korteweg–de Vries-5 and other asymptotically equivalent equations for shallow water waves	2003	Holger R. Dullin Georg A. Gottwald Darryl D. Holm
+ PDF Chat	A Singular Limit Problem for Conservation Laws Related to the Camassa–Holm Shallow Water Equation	2006	Giuseppe Maria Coclite Kenneth H. Karlsen
+ PDF Chat	GLOBAL DISSIPATIVE SOLUTIONS OF THE CAMASSA–HOLM EQUATION	2006	Alberto Bressan Adrian Constantin
+	Periodic solutions of the Degasperis–Procesi equation: Well-posedness and asymptotics	2014	Giuseppe Maria Coclite Kenneth H. Karlsen
+	ON THE UNIQUENESS AND LARGE TIME BEHAVIOR OF THE WEAK SOLUTIONS TO A SHALLOW WATER EQUATION	2002	Zhouping Xin Ping Zhang