A para-differential renormalization technique for nonlinear dispersive equations

Abstract

For α\in (1,2) we prove that the initial-value problem \partial_t u+D^α\partial_x u+\partial_x(u^2/2)=0 on \mathbb{R}_x\times\mathbb{R}_t; u(0)=ϕ, is globally well-posed in the space of real-valued L^2-functions. We use a frequency dependent renormalization method to control the strong low-high frequency interactions.

42 pages, no figures

References in corpus (3)

• Global well-posedness of the KP-I initial-value problem in the energy space
• Global well-posedness of the Benjamin-Ono equation in H^1(R)
• Local Well-posedness for dispersion generalized Benjamin-Ono equations in Sobolev spaces

Cited by in corpus (20)

• A numerical approach to Blow-up issues for dispersive perturbations of Burgers' equation
• The IVP for the dispersion generalized Benjamin-Ono equation in weighted Sobolev spaces
• Improvement of the energy method for strongly non resonant dispersive equations and applications
• Stability properties of solitary waves for fractional KdV and BBM equations
• Enhanced existence time of solutions to the fractional Korteweg-de Vries equation
• On the propagation of regularity for solutions of the Dispersion Generalized Benjamin-Ono Equation
• On the Cauchy problem for higher dimensional Benjamin-Ono and Zakharov-Kuznetsov equations
• On persistence properties in weighted spaces for solutions of the fractional Korteweg-de Vries equation
• On shorttime bilinear Strichartz estimates and applications to improve the energy method
• Local and global well-posedness of dispersion generalized Benjamin-Ono equations on the circle
• Stability of periodic waves for the fractional KdV and NLS equations
• Deep-water and shallow-water limits of the intermediate long wave equation
• Well-posedness results for the dispersion generalized Benjamin-Ono equation via the contraction principle
• One-sided Hölder regularity of global weak solutions of negative order dispersive equations
• Well-posedness for Stochastic Generalized Fractional Benjamin-Ono Equation
• On the Cauchy problem of dispersive Burgers type equations
• Norm inflation for equations of KdV type with fractional dispersion
• Dispersive perturbations of Burgers and hyperbolic equations I : local theory
• Hölder Continuity of the Data to Solution Map for HR in the Weak Topology
• Asymptotic behavior of solutions of the dispersive generalized Benjamin-Ono equation