Wave breaking for nonlinear nonlocal shallow water equations

1998/01/01 by Adrian Constantin, Joachim Escher · 75 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Nonlinear Waves and Solitons #Differential Equations and Numerical Methods

paper · pdf · doi:10.1007/bf02392586

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• The Hydrodynamical Relevance of the Camassa–Holm and Degasperis–Procesi Equations
• Global Conservative Solutions of the Camassa–Holm Equation
• 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
• Convergence of the dispersion Camassa-Holm N-Soliton
• An Optimal Transportation Metric for Solutions of the Camassa-Holm Equation
• A Rigorous Proof of Metric Blow-Up for Pseudospherical Metrics Associated with the Degasperis--Procesi 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
• Geodesic completeness of the H^3/2 metric on Diff(S¹)
• Fold catastrophe in breaking waves
• Asymptotic stability of solitary waves for the b-family of equations
• Non-uniform dependence for the Novikov equation in Besov spaces
• Stability of peakons for the generalized modified Camassa-Holm equation
• Non-uniform dependence on initial data for the Camassa-Holm equation in Besov spaces
• Existence and regularity for global solutions including breaking waves from Camassa-Holm and Novikov equations to λ-family equations
• 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
• Traffic flow models with looking ahead-behind dynamics
• Blow-up phenomenon, ill-posedness and peakon solutions for the periodic Euler-Poincaré equations
• Linear Whitham-Boussinesq modes in channels of constant cross-section
• The well-posedness, ill-posedness and non-uniform dependence on initial data for the Fornberg-Whitham equation in Besov spaces
• Existence of vortex rings in Beltrami flows
• 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 Cauchy problem for a two-component Degasperis-Procesi system
• Local well-posedness and wave breaking results for periodic solutions of a shallow water equation for waves of moderate amplitude
• Thresholds for shock formation in traffic flow models with Arrhenius look-ahead dynamics
• Discontinuous traveling waves as weak solutions to the Fornberg-Whitham\n equation
• Multipeakons of a two-component modified Camassa-Holm equation and the relation with the finite Kac-van Moerbeke lattice
• Blow-up analysis for the generalized hyperelastic rod equation
• Solution concepts, well-posedness, and wave breaking for the Fornberg-Whitham equation
• On the blow-up structure for the generalized periodic Camassa-Holm and Degasperis-Procesi equations
• Dynamical behavior near self-similar blowup waves for the generalized b-equation
• Global-in-time solvability and blow-up for a non-isospectral two-component cubic Camassa-Holm system in a critical Besov space
• On Isospectral flows related to (dual) cubic strings: Novikov, DP peakons and B, C-Toda lattices
• Blow-up phenomena for the rotation-two-component Camassa-Holm system
• On the Blow-Up of Solutions of a Periodic Shallow Water Equation
• Blow-up phenomena and global existence for a periodic two-component Hunter-Saxton system
• 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 convergent Fourier spectral Galerkin method for the fractional Camassa-Holm equation
• Inverse scattering transform for the Camassa–Holm equation
• A Local-in-Time Theory for Singular SDEs with Applications to Fluid Models with Transport Noise
• Instability of large solitary water waves
• The Cauchy problem for the integrable RZQ equation
• Analysis on a generalized two-component Novikov system
• Existence and uniqueness of global weak solutions to a generalized Camassa-Holm equation
• 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
• Algebro-Geometric Solutions of the Camassa--Holm hierarchy
• Unique Continuation Properties for solutions to the Camassa-Holm\n equation and other non-local equations
• Attractors for the viscous Camassa-Holm equation
• Non-uniform continuity of periodic Holm-Staley b-family of equations
• Well-Posedness of a kind of the free surface equation of shallow water wave
• The Camassa-Holm equation as an incompressible Euler equation: a\n geometric point of view
• Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms
• Well-posedness and persistence properties for two-component higher order Camassa-Holm systems with fractional inertia operator
• Ill-posedness for the Camassa-Holm and related equations in Besov spaces
• On the modified multi-component Camassa-Holm system in higher dimensions
• Linear guided modes and Whitham-Boussinesq model for variable topogra
• Well-posedness for the dispersive Hunter-Saxton equation
• On the wave length of smooth periodic traveling waves of the\n Camassa-Holm equation
• Global Lagrangian solutions of the Camassa-Holm equation
• A look on equations describing pseudospherical surfaces
• Wave-breaking phenomena and global existence for the generalized Fornberg-Whitham equation
• Gradient blow-up for dispersive and dissipative perturbations of the Burgers equation
• Blow-up phenomena and global existence for the periodic two-component Dullin-Gottwald-Holm system
• Blow-up analysis for a periodic two-component μ -Hunter-Saxton system. [europepmc]
• Spontaneous shock waves in pulse-stimulated flocks of Quincke rollers. [europepmc]
• Completeness and Geodesic Distance Properties for Fractional Sobolev Metrics on Spaces of Immersed Curves. [europepmc]
• One Dimensional Energy Cascades in a Fractional Quasilinear NLS. [europepmc]