On dispersive decay for the generalized Korteweg--de Vries equation
Matthew Kowalski, Minjie Shan
TL;DR
The paper addresses pointwise dispersive decay for the generalized Korteweg--de Vries equation $u_t+u_{xxx}\pm \partial_x(u^{k+1})=0$ with $k\ge 4$, establishing $|t|^{-1/3}$ decay in $L^\infty_x$ under regime-specific initial-data hypotheses. The authors develop a framework combining persistence of negative regularity (notably for the mass-critical case $k=4$) with extended Lorentz--Strichartz estimates to backwards mixed norms, and they prove spacetime bounds that propagate regularity. They obtain dispersive decay in all regimes: for $k\ge 8$ under scaling-critical data with an $L^1$ influence, and for $4\le k<8$ under additional regularity assumptions, with optimal results in the mass-critical limit. Overall, the work connects the nonlinear gKdV dynamics to the linear Airy decay and strengthens the understanding of long-time behavior and scattering in critical and near-critical settings, aided by new harmonic analysis tools such as backwards norms and endpoint Leibniz rules.
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 $\dot{H}^{\frac{1}{4}} \cap \dot{H}^{-\frac{1}{12}}$ and show that solutions decay in $L^\infty$ like $|t|^{-\frac{1}{3}}$. To accomplish this, we develop a persistence of negative regularity for solutions to gKdV and extend Lorentz--Strichartz estimates to the mixed norm case.