On well-posedness of nonlocal evolution equations

DOI10.1007/s10013-023-00615-5zbMath1530.35280OpenAlexW4376126283MaRDI QIDQ6144956

A. Alexandrou Himonas, Fangchi Yan

Publication date: 8 January 2024

Published in: Vietnam Journal of Mathematics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s10013-023-00615-5

zbMATH Keywords

Sobolev spaceswell-posednessinitial value problemnonlocal equationsCamassa-Holm equationBourgain spacesnonuniform dependencemodified Novikov equationtrilinear estimates

Mathematics Subject Classification ID

Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) (37K10) NLS equations (nonlinear Schrödinger equations) (35Q55) Existence problems for PDEs: global existence, local existence, non-existence (35A01) Initial-boundary value problems for linear higher-order PDEs (35G16) Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness (35A02) Initial-boundary value problems for nonlinear higher-order PDEs (35G31)

Cites Work

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• Regularity properties of the cubic nonlinear Schrödinger equation on the half line
• Ill-posedness for the \(b\)-family of equations
• Continuity properties of the data-to-solution map for the periodic \(b\)-family equation.
• Non-uniform dependence on initial data for the CH equation on the line.
• On well-posedness of the Degasperis-Procesi equation
• The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations
• On the Cauchy problem for the periodic \(b\)-family of equations and of the non-uniform continuity of Degasperis-Procesi equation
• Global wellposedness of KdV in \(H^{-1}(\mathbb T,\mathbb R)\)
• Global conservative solutions of the Camassa-Holm equation
• Autonomous Ovsyannikov theorem and applications to nonlocal evolution equations and systems
• Periodic conservative solutions of the Camassa-Holm equation
• Non-uniform dependence on initial data of solutions to the Euler equations of hydrodynamics
• On a class of physically important integrable equations
• Symplectic structures, their Bäcklund transformations and hereditary symmetries
• Explicit multipeakon solutions of Novikov's cubically nonlinear integrable Camassa-Holm type equation
• A small dispersion limit to the Camassa-Holm equation: a numerical study
• Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II: The KdV-equation
• Interaction equations for short and long dispersive waves
• The geometry of peaked solitons and billiard solutions of a class of integrable PDE's
• Global existence for a new periodic integrable equation.
• Breakdown of a shallow water equation
• Multipeakons and the classical moment problem
• Ill-posedness of the Camassa-Holm and related equations in the critical space
• Non-uniqueness for the Fokas-Olver-Rosenau-Qiao equation
• The Cauchy problem for an integrable shallow-water equation
• A few remarks on the Camassa-Holm equation.
• Classical solutions of the periodic Camassa-Holm equation.
• Stability of the Camassa-Holm solitons
• Well-posedness of the Cauchy problem for a shallow water equation on the circle
• Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation
• Galilean boost and non-uniform continuity for incompressible Euler
• Properties of solutions to the Camassa-Holm equation on the line in a class containing the peakons
• Non-uniform continuity in \(H^1\) of the solution map of the CH equation
• Persistence properties and unique continuation of solutions of the Camassa-Holm equation
• Sur la géométrie différentielle des groupes de Lie de dimension infinite et ses applications à l'hydrodynamique des fluides parfaits
• Groups of diffeomorphisms and the motion of an incompressible fluid
• Traveling wave solutions of the Camassa-Holm equation
• The Cauchy problem for the Fokas-Olver-Rosenau-Qiao equation
• Vorticity and Incompressible Flow
• Multilinear weighted convolution of L 2 functions, and applications to nonlinear dispersive equations
• Norm Inflation and Ill-Posedness for the Degasperis-Procesi Equation
• Finite propagation speed for the Camassa–Holm equation
• The small dispersion limit of the Korteweg-de Vries equation. I
• GLOBAL DISSIPATIVE SOLUTIONS OF THE CAMASSA–HOLM EQUATION
• Low-Regularity Solutions of the Periodic Camassa–Holm Equation
• A new integrable equation with cuspons and W/M-shape-peaks solitons
• Well-posedness, blow-up phenomena, and global solutions for the b-equation
• Non-Uniform Dependence for the Periodic CH Equation
• Generalizations of the Camassa–Holm equation
• Commutator estimates and the euler and navier-stokes equations
• The initial-value problem for the Korteweg-de Vries equation
• The zero dispersion limit for the Korteweg-deVries KdV equation
• Multi-peakons and a theorem of Stieltjes
• A shallow water equation on the circle
• Well‐posedness and scattering results for the generalized korteweg‐de vries equation via the contraction principle
• The cauchy problem for a shallow water type equation
• Breakdown 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
• Sharp global well-posedness for KdV and modified KdV on ℝ and 𝕋
• A bilinear estimate with applications to the KdV equation
• Global Conservative Solutions of the Camassa–Holm Equation—A Lagrangian Point of View
• Existence time for the Camassa-Holm equation and the critical Sobolev index
• The Initial-Boundary Value Problem for the Korteweg–de Vries Equation
• The Cauchy problem for the Novikov equation
• Introduction to nonlinear dispersive equations
• On the Cauchy problem for the Camassa-Holm equation

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