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.
Categories: math.AP