Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the real line case

Luc Molinet, Stéphane Vento

Type: Article

Publication Date: 2011-10-05

Citations: 18

DOI: https://doi.org/10.2422/2036-2145.2011.3.02

Abstract

We complete the known results on the Cauchy problem in Sobolev spaces for the KdV-Burgers equation by proving that this equation is well-posed in H -1 (R) with a solution-map that is analytic fromwhereas it is ill-posed in H s (R), as soon as s < -1, in the sense that the flowmap u 0 → u(t) cannot be continuous from H s (R) to even D (R) at any fixed t > 0 small enough.As far as we know, this is the first result of this type for a dispersive-dissipative equation.The framework we develop here should be useful to prove similar results for other dispersive-dissipative models.

Locations

ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE - View - PDF
HAL (Le Centre pour la Communication Scientifique Directe) - View - PDF
DataCite API - View

Similar Works

Action	Title	Year	Authors
+	Sharp ill-posedness and well-posedness results for dissipative KdV equations on the real line	2019	Xavier Carvajal Pedro Gamboa Raphael Santos
+	Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: The periodic case	2012	Luc Molinet Stéphane Vento
+ PDF Chat	Sharp well-posedness and ill-posedness results for dissipative KdV equations on the real line	2021	Xavier Carvajal Pedro Gamboa Raphael Santos
+	Well-posedness of the stochastic KdV–Burgers equation	2013	Geordie Richards
+	Remark on the ill-posedness for KdV-Burgers equation in Fourier amalgam spaces	2023	Divyang G. Bhimani Saikatul Haque
+	Remark on the Ill-Posedness for KdV-Burgers Equation in Fourier Amalgam Spaces	2024	Divyang G. Bhimani Saikatul Haque
+	Well-Posedness Results and Dissipative Limit of High Dimensional KdV-Type Equations	2017	Xavier Carvajal Amin Esfahani Mahendra Panthee
+	The well-posed problem of the KdV-Burgers type equation.	2006	Ruying Xue
+	Well-posedness for a perturbation of the KdV equation	2019	Xavier Carvajal Liliana Esquivel
+ PDF Chat	Global well-posedness and inviscid limit for the Korteweg–de Vries–Burgers equation	2009	Zihua Guo Baoxiang Wang
+	Remark on well-posedness and ill-posedness for the KdV equation	2010	Takamori Kato
+ PDF Chat	On well-posedness for some Korteweg-De Vries type equations with variable coefficients	2021	Luc Molinet Raafat Talhouk Ibtissame Zaiter
+	Well-posedness of Whitham-Broer-Kaup equation with negative dispersion	2023	Nabil Bedjaoui Youcef Mammeri
+	Global well posedness and inviscid limit for the Korteweg-de Vries-Burgers equation	2008	Zihua Guo Baoxiang Wang
+	A sharp condition for the well-posedness of the linear KdV-type equation	2012	Timur Akhunov
+	A sharp condition for the well-posedness of the linear KdV-type equation	2012	Timur Akhunov
+	On well-posedness for some Korteweg–de Vries type equations with variable coefficients	2023	Luc Molinet Raafat Talhouk Ibtissame Zaiter
+	Well-posedness of Korteweg-de Vries-Burgers equation on a finite domain	2016	Jie Li Kangsheng Liu
+	Well-posedness for the KdV-type equations with low regularity data	2011	孝盛加藤
+	KdV is well-posed in $H^{-1}$	2019	Rowan Killip Monica Vişan

Works That Cite This (18)

Action	Title	Year	Authors
+ PDF Chat	On the well-posedness, ill-posedness and norm-inflation for a higher order water wave model on a periodic domain	2019	Xavier Carvajal Mahendra Panthee Ricardo A. Pastrán
+	On sharp global well-posedness and ill-posedness for a fifth-order KdV-BBM type equation	2019	Xavier Carvajal Mahendra Panthee
+ PDF Chat	Remarks on the Cauchy problem for the one-dimensional quadratic (fractional) heat equation	2015	Luc Molinet Slim Tayachi
+	Quadratic dispersive generalized Benjamin–Ono equation	2011	Yuzhao Wang
+ PDF Chat	On the well-posedness for Kadomtsev–Petviashvili–Burgers I equation	2012	Mohamad Darwich
+	Local and global well-posedness for the 2D Zakharov-Kuznetsov-Burgers equation in low regularity Sobolev space	2019	Hiroyuki Hirayama
+	Ill-posedness for the nonlinear Schrödinger equation with quadratic non-linearity in low dimensions	2014	Tsukasa Iwabuchi Takayoshi Ogawa
+	Well-posedness of the generalized Burgers equation on a finite interval	2018	Jing Li Bing‐Yu Zhang Zhixiong Zhang
+	On the ill-posedness result for the BBM equation	2010	Mahendra Panthee
+	Local and global well-posedness of the stochastic KdV-Burgers equation	2011	Geordie Richards

Works Cited by This (18)

Action	Title	Year	Authors
+ PDF Chat	Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity	2009	Nobu Kishimoto
+	On the Cauchy Problem for the Zakharov System	1997	J. Ginibre Yoshio Tsutsumi G. Velo
+	Global wellposedness of KdV in H−1(T,R)	2006	Thomas Kappeler Peter Topalov
+	Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: The periodic case	2012	Luc Molinet Stéphane Vento
+	Scattering for the quartic generalised Korteweg–de Vries equation	2006	Terence Tao
+	The Cauchy problem for the Kadomtsev-Petviashvili equation	1990	Andrei V. Faminskii
+	Global well posedness and inviscid limit for the Korteweg-de Vries-Burgers equation	2008	Zihua Guo Baoxiang Wang
+	Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation	2005	Ioan Bejenaru Terence Tao
+	On global existence and scattering for the wave maps equation	2001	Daniel Tataru
+ PDF Chat	Global well-posedness of Korteweg–de Vries equation in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mi>H</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="double-struck">R</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>	2009	Zihua Guo