On dispersive decay for the generalized Korteweg--de Vries equation

2025/10/02 by Kowalski, Matthew, Shan, Minjie
#35Q53 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.01728

Abstract

We prove pointwise-in-time dispersive estimates for solutions to the generalized Korteweg--de Vries (gKdV) equation. In particular, for solutions to the mass-critical model, we assume only that initial data lie in H^(1)/(4) ∩ H^-(1)/(12) and show that solutions decay in L^∞ like |t|^-(1)/(3). To accomplish this, we develop a persistence of negative regularity for solutions to gKdV and extend Lorentz--Strichartz estimates to the mixed norm case.

Citations

• Dispersive decay for the energy-critical nonlinear wave equation
• Dispersive decay for the mass-critical generalized Korteweg-de Vries equation and generalized Zakharov-Kuznetsov equations
• Dispersive decay for the mass-critical nonlinear Schrödinger equation
• Global well-posedness and scattering for the defocusing, mass - critical\n generalized KdV equation
• The Kato-Ponce Inequality
• Uniform decay estimates and the lorentz invariance of the classical wave equation

Related

• Dispersive decay for the mass-critical generalized Korteweg-de Vries equation and generalized Zakharov-Kuznetsov equations
• Propagation of regularity and decay of solutions to the k-generalized Korteweg-de Vries equation
• Dispersive decay of small data solutions for the KdV equation
• Dispersive decay bound of small data solutions to higher order scattering-supercritical KdV-type equations
• Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation