Ill-posedness of the Camassa-Holm and related equations in the critical space

DOI10.1016/j.jde.2018.08.013zbMath1403.35322arXiv1805.02377OpenAlexW2885962621WikidataQ129382851 ScholiaQ129382851MaRDI QIDQ1627721

Zihua Guo, Zhaoyang Yin, Luc Molinet, Xingxing Liu

Publication date: 3 December 2018

Published in: Journal of Differential Equations (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1805.02377

zbMATH Keywords

Camassa-Holm equationill-posednessnorm inflationcritical spaceshallow water wave models

Mathematics Subject Classification ID

Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) (37K10) KdV equations (Korteweg-de Vries equations) (35Q53) Ill-posed problems for PDEs (35R25) Initial value problems for nonlinear higher-order PDEs (35G25)

Related Items (59)

Sharp ill-posedness for the generalized Camassa-Holm equation in Besov spaces ⋮ Orbital stability of peakon solutions for a generalized higher-order Camassa-Holm equation ⋮ Ill-posedness for the Cauchy problem of the Camassa-Holm equation in \(B_{\infty, 1}^1(\mathbb{R})\) ⋮ On the Cauchy problem for a Camassa-Holm type equation with cubic and quartic nonlinearities ⋮ Nonuniqueness for the \(ab\)-family of equations with peakon travelling waves on the circle ⋮ The Initial-Value Problem to the Modified Two-Component Euler--Poincaré Equations ⋮ Non-uniform dependence on initial data for the Camassa-Holm equation in the critical Besov space ⋮ Ill-posedness for the Camassa-Holm and related equations in Besov spaces ⋮ The existence and uniqueness of global admissible conservative weak solution for the periodic single-cycle pulse equation ⋮ Ill-posedness for a two-component Novikov system in Besov space ⋮ Spectral Instability of Peakons in the $b$-Family of the Camassa--Holm Equations ⋮ Energy conservation and well-posedness of the Camassa-Holm equation in Sobolev spaces ⋮ Existence and uniqueness of the globally conservative solutions for a weakly dissipative Camassa–Holm equation in time weighted H1(R) space ⋮ Non-uniform continuity on initial data for the two-component b-family system in Besov space ⋮ Norm inflation and ill-posedness for the Fornberg-Whitham equation ⋮ Ill-posedness for a generalized Camassa-Holm equation with higher-order nonlinearity in the critical Besov space ⋮ Blow-up analysis for the \(ab\)-family of equations ⋮ Well-posedness and continuity properties of the Degasperis-Procesi equation in critical Besov space ⋮ On well-posedness of nonlocal evolution equations ⋮ Curvature blow-up and peakons for the Camassa-Holm type equation ⋮ The well-posedness for the Camassa-Holm type equations in critical Besov spaces \(B_{p , 1}^{1 + \frac{ 1}{ p}}\) with \(1 \leq p < + \infty \) ⋮ On the continuity of the solution map of the Euler-Poincaré equations in Besov spaces ⋮ Global conservative solution for a dissipative Camassa-Holm type equation with cubic and quartic nonlinearities ⋮ Non-uniform dependence on initial data for the Camassa-Holm equation in Besov spaces: revisited ⋮ Optimal decay rate for higher-order derivatives of the solution to the Lagrangian-averaged Navier-Stokes-\(\alpha\) equation in \(\mathbb{R}^3\) ⋮ On the Cauchy problem for a modified Camassa-Holm equation ⋮ Blow-up phenomena and the local well-posedness and ill-posedness of the generalized Camassa-Holm equation in critical Besov spaces ⋮ Non-uniqueness for the ab-family of equations ⋮ Local well-posedness and blow-up phenomena of the generalized short pulse equation ⋮ The effect of a noise on the stochastic modified Camassa–Holm equation ⋮ Non-uniform continuity on initial data for a Camassa-Holm-type equation in Besov space ⋮ Stability of peakons for the generalized modified Camassa-Holm equation ⋮ Globally conservative solutions for the modified Camassa–Holm (MOCH) equation ⋮ The Cauchy problem of the rotation Camassa–Holm equation in equatorial water waves ⋮ Non-uniform dependence on initial data for the generalized Camassa-Holm-Novikov equation in Besov space ⋮ Blow-up phenomena and local well-posedness for a generalized Camassa-Holm equation in the critical Besov space ⋮ Global conservative solution for the periodic dispersive Hunter-Saxton equation ⋮ Blow-up phenomena, ill-posedness and peakon solutions for the periodic Euler-Poincaré equations ⋮ Local well-posedness and blow-up phenomenon for a generalization two-component Camassa-Holm system ⋮ The Cauchy problem for generalized fractional Camassa-Holm equation in Besov space ⋮ Weak well-posedness for the integrable modified Camassa-Holm equation with the cubic nonlinearity ⋮ Non-uniform continuous dependence on initial data of solutions to the Euler-Poincaré system ⋮ Global existence for the periodic dispersive Hunter-Saxton equation ⋮ Instability of \(H^1\)-stable peakons in the Camassa-Holm equation ⋮ The existence of global weak solutions for a generalized Camassa–Holm equation ⋮ Non-uniform dependence for higher dimensional Camassa-Holm equations in Besov spaces ⋮ Construction of 2-peakon solutions and nonuniqueness for a generalized mCH equation ⋮ The Cauchy problem for fractional Camassa-Holm equation in Besov space ⋮ Global solutions and blow-up phenomena for a generalized Degasperis-Procesi equation ⋮ Growth of Perturbations to the Peaked Periodic Waves in the Camassa--Holm Equation ⋮ The global Gevrey regularity of the rotation two-component Camassa-Holm system ⋮ Non-uniform dependence on initial data for the Camassa-Holm equation in Besov spaces ⋮ 2-peakon solutions and non-uniqueness of the Fokas-Olver-Rosenau-Qiao equation with higher-order nonlinearity ⋮ Wave breaking and solitary wave solutions for a generalized Novikov equation ⋮ Non-uniform continuity of the generalized Camassa-Holm equation in Besov spaces ⋮ Global existence and blow-up phenomena for a periodic modified Camassa–Holm equation (MOCH) ⋮ Global weak solutions to the Novikov equation by viscous approximation ⋮ The well-posedness, ill-posedness and non-uniform dependence on initial data for the Fornberg-Whitham equation in Besov spaces ⋮ Global existence and blow-up phenomena for the Hunter-Saxton equation on the line

Cites Work

• Unnamed Item
• Unnamed Item
• Remarks on the well-posedness of Camassa-Holm type equations in Besov spaces
• Non-uniform dependence on initial data for the CH equation on the line.
• Global conservative solutions of the Camassa-Holm equation
• Symplectic structures, their Bäcklund transformations and hereditary symmetries
• Sur quelques généralisations de l'équation de Korteweg-de Vries
• Wave breaking for nonlinear nonlocal shallow water equations
• A note on well-posedness for Camassa-Holm equation.
• Global weak solutions for a shallow water equation
• A few remarks on the Camassa-Holm equation.
• Classical solutions of the periodic Camassa-Holm equation.
• On the blow-up rate and the blow-up set of breaking waves for a shallow water equation
• Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation
• Existence of permanent and breaking waves for a shallow water equation: a geometric approach
• Strong illposedness of the incompressible Euler equation in integer \(C^m\) spaces
• Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces
• On the weak solutions to a shallow water equation
• GLOBAL DISSIPATIVE SOLUTIONS OF THE CAMASSA–HOLM EQUATION
• An integrable shallow water equation with peaked solitons
• Construction of 2-Peakon Solutions and Ill-Posedness for the Novikov Equation
• On Well-Posedness Results for Camassa-Holm Equation on the Line: A Survey
• Existence time for the Camassa-Holm equation and the critical Sobolev index
• Dispersive Perturbations of Burgers and Hyperbolic Equations I: Local Theory
• The Cauchy problem for the Novikov equation
• On the Cauchy problem for the Camassa-Holm equation

This page was built for publication: Ill-posedness of the Camassa-Holm and related equations in the critical space