A Refined Well-Posedness for the Fourth-Order Nonlinear Schrödinger Equation Related to the Vortex Filament

DOI10.1080/03605300701629385zbMath1133.35012OpenAlexW2090567294MaRDI QIDQ5437532

Yueling Jia, Zhaohui Huo

Publication date: 21 January 2008

Published in: Communications in Partial Differential Equations (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1080/03605300701629385

zbMATH Keywords

local well-posednessFourier restriction normfourth-order Schrödinger equation

Mathematics Subject Classification ID

Stability in context of PDEs (35B35) Ill-posed problems for PDEs (35R25) NLS equations (nonlinear Schrödinger equations) (35Q55) Viscous vortex flows (76D17)

Related Items (23)

Multi solitary waves for a fourth order nonlinear Schrödinger type equation ⋮ The probabilistic Cauchy problem for the fourth order Schrödinger equation with special derivative nonlinearities ⋮ Retracted: The probabilistic Cauchy problem for the fourth order Schrödinger equation with special derivative nonlinearities ⋮ Decay in energy space for the solution of fourth-order Hartree-Fock equations with general non-local interactions ⋮ Well‐posedness of generalized KdV and one‐dimensional fourth‐order derivative nonlinear Schrödinger equations for data with an infinite L² norm ⋮ Almost sure global well-posedness for the fourth-order nonlinear Schrödinger equation with large initial data ⋮ Refined energy inequality with application to well-posedness for the fourth order nonlinear Schrödinger type equation on torus ⋮ Dispersive limits for some perturbations of the NLS equation ⋮ Vortex filament on symmetric Lie algebras and generalized bi-Schrödinger flows ⋮ Orbital stability of a two parameter family of solitary waves for a fourth order nonlinear Schrödinger type equation ⋮ Stability of Hasimoto solitons in energy space for a fourth order nonlinear Schrödinger type equation ⋮ Well-posedness for the fourth-order nonlinear derivative Schrödinger equation in higher dimension ⋮ Random data Cauchy problem for the fourth order Schrödinger equation with the second order derivative nonlinearities ⋮ THE MASS-CRITICAL FOURTH-ORDER SCHRÖDINGER EQUATION IN HIGH DIMENSIONS ⋮ Asymptotics for the fourth-order nonlinear Schrödinger equation in the critical case ⋮ Well-posedness for the fourth-order Schrödinger equation with third order derivative nonlinearities ⋮ Well-posedness for the fifth-order shallow water equations ⋮ \(H^2\)-scattering for systems of weakly coupled Fourth-order NLS equations in low space Dimensions ⋮ Two simple criterion to obtain exact controllability and stabilization of a linear family of dispersive PDE's on a periodic domain ⋮ Well-posedness of the initial-boundary value problem for the fourth-order nonlinear Schrödinger equation ⋮ Initial value problem for the fourth order nonlinear Schrödinger type equation on torus and orbital stability of standing waves ⋮ Stability of elliptic solutions to the defocusing fourth order nonlinear Schrödinger equation ⋮ Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity

Cites Work

• Unnamed Item
• Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I: Schrödinger equations
• The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices
• The Cauchy problem for the fourth-order nonlinear Schrödinger equation related to the vortex filament
• Motion and expansion of a viscous vortex ring. Part 1. A higher-order asymptotic formula for the velocity
• Remark on well-posedness for the fourth order nonlinear Schrödinger type equation
• A bilinear estimate with applications to the KdV equation
• A soliton on a vortex filament
• A restriction theorem for the Fourier transform

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