Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation

1998/05/01 by Adrian Constantin, Joachim Escher · 676 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Amplitude #Applied mathematics #Class (philosophy) #Gravitational singularity #Initial value problem #Mathematical analysis #Mathematics #Navier-Stokes equation solutions #Nonlinear Waves and Solitons #Physics #Uniqueness

paper · doi:10.1002/(sici)1097-0312(199805)51:5<475::aid-cpa2>3.0.co;2-5

published in Communications on Pure and Applied Mathematics 51(5), 475-504 (Wiley)

openalex publication_date 1998/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We establish the local well-posedness of a recently derived model for small-amplitude, shallow water waves. For a large class of initial data we prove global existence of the corresponding solution. Criteria guaranteeing the development of singularities in finite time for strong solutions with smooth initial data are obtained, and an existence and uniqueness result for a class of global weak solutions is also given. © 1998 John Wiley & Sons, Inc.

Cited by

• Well-posedness and Blow-up Solutions for an Integrable Nonlinearly Dispersive Model Wave Equation
• Ill-posedness of the Camassa–Holm and related equations in the critical space
• Global existence and blow-up phenomena for the weakly dissipative Camassa–Holm equation
• The Cauchy problem for higher-order modified Camassa-Holm equations on the circle
• A stochastic variational approach to the viscous Camassa-Holm and Leray-alpha equations
• Damping-Diffusion-Noise Interactions in the Stochastic Camassa-Holm Equation
• A new result for the local well-posedness of the Camassa-Holm type equations in critial Besov spaces B^1+(1)/(p)_p,1,1≤ p<+∞
• The modified Camassa-Holm equation on the half line: a Riemann--Hilbert approach
• The periodic Camassa-Holm equation by the Riemann-Hilbert problem approach
• Non-uniform dependence for the Novikov equation in Besov spaces
• Non-uniform dependence on initial data for the Camassa-Holm equation in Besov spaces
• Non-uniform dependence for higher dimensional Camassa-Holm equations in Besov spaces
• Non-uniform dependence on initial data for the Camassa-Holm equation in the critical Besov space
• Analyticity, Gevrey regularity and unique continuation for an integrable multi-component peakon system with an arbitrary polynomial function
• The well-posedness, ill-posedness and non-uniform dependence on initial data for the Fornberg-Whitham equation in Besov spaces
• The local well-posedness, blow-up phenomena and ill-posedness of a new fifth-order Camassa-Holm type equation
• Non-uniform dependence for the two-component Camassa-Holm shallow water system
• On the blow-up structure for the generalized periodic Camassa-Holm and Degasperis-Procesi equations
• On the Blow-Up of Solutions of a Periodic Shallow Water Equation
• The Cauchy problem for the shallow water typ equations in low regularity spaces on the circle
• On the Cauchy problem for the Hunter-Saxton equation on the line
• Stability of solitary waves of a generalized two-component Camassa-Holm system
• A local-in-time theory for singular SDEs with applications to fluid models with transport noise
• The Cauchy problem for the integrable RZQ equation
• Analysis on a generalized two-component Novikov system
• Blow-up phenomena for an integrable two-component Camassa-Holm system with cubic nonlinearity and peakon solutions
• Distribution-Path Dependent Nonlinear SPDEs with Application to Stochastic Transport Type Equations
• Unique Continuation Properties for solutions to the Camassa-Holm equation and other non-local equations
• Non-uniform continuity of periodic Holm-Staley b-family of equations
• On uniqueness of dissipative solutions of the Camassa-Holm equation
• Ill-posedness for the Camassa-Holm and related equations in Besov spaces
• On the well-posedness of the Holm-Staley b-family of equations
• On the modified multi-component Camassa-Holm system in higher dimensions
• Well-posedness for the dispersive Hunter-Saxton equation
• Stability of peakons for a modified Camassa-Holm equation
• Ill-posedness for the higher dimensional Camassa-Holm equations in Besov spaces
• Global and blow-up solutions for a non-local integrable equation with applications to geometry
• Water waves and integrability
• Bi-Hamiltonian systems on the dual of the Lie algebra of vector fields of the circle and periodic shallow water equations
• On permanent and breaking waves in hyperelastic rods and rings
• Traveling surface waves of moderate amplitude in shallow water
• Non-uniform Dependence on Initial Data for the Generalized Camassa–Holm–Novikov Equation in Besov Space
• On the blow-up of solutions to the integrable modified Camassa–Holm equation
• On the Cauchy problem for a new integrable two-component system with peakon and weak kink solutions
• The Degasperis–Procesi equation as a non-metric Euler equation
• On a two-dimensional Camassa-Holm-Zakharov-Kuznetsov equation
• Geodesic flow on the diffeomorphism group of the circle
• On the blow-up rate and the blow-up set of breaking waves for a shallow water equation
• Fredholm determinants and the Camassa‐Holm hierarchy
• A blow-up result for the periodic Camassa-Holm equation
• Genesis of solitons arising from individual Flows of the Camassa‐Holm hierarchy
• On the Cauchy problem for the Camassa–Holm equation
• Traveling wave solutions of the Camassa–Holm equation
• Multi-symplectic integration of the Camassa–Holm equation
• A Note on the analytic solutions of the Camassa–Holm equation
• Finite propagation speed for the Camassa–Holm equation
• ON THE UNIQUENESS AND LARGE TIME BEHAVIOR OF THE WEAK SOLUTIONS TO A SHALLOW WATER EQUATION
• Initial Boundary Value Problems of the Camassa–Holm Equation
• Blow-up, blow-up rate and decay of the solution of the weakly dissipative Camassa-Holm equation
• Low-Regularity Solutions of the Periodic Camassa–Holm Equation
• Convergence of a Finite Difference Scheme for the Camassa–Holm Equation
• Global existence of weak solutions to the Camassa-Holm equation
• Existence time for the Camassa-Holm equation and the critical Sobolev index
• Existence of permanent and breaking waves for a shallow water equation: a geometric approach
• A Variational Approach to the Stability of Periodic Peakons
• Periodic conservative solutions of the Camassa–Holm equation
• Stability for the periodic Camassa-Holm equation

Related

• Local well-posedness and wave breaking results for periodic solutions of a shallow water equation for waves of moderate amplitude
• Global solutions to the shallow-water system
• Local well-posedness result for a class of non-local quasi-linear systems and its application to the justification of Whitham-Boussinesq systems
• On the Cauchy problem for a model equation for shallow water waves of moderate amplitude
• Global existence for quasilinear wave equations on hyperbolic space