$L^2({\mathbb R}) $-Unconditional well-posedness for low dispersion fractional KdV equations

TL;DR

This paper proves that low dispersion fractional KdV equations are unconditionally globally well-posed in L^2(R) for certain dispersion parameters, extending known results for classical KdV.

Contribution

It establishes unconditional well-posedness for fractional KdV equations with weaker dispersion, using refined bilinear and energy estimates.

Findings

Unconditional global well-posedness in L^2(R) for α in (55/38, 2].

Extension of well-posedness results from classical to fractional KdV equations.

Application of bilinear and Bourgain-type estimates in the proof.

Abstract

We show that the $ L^2({\mathbb R}) $-unconditional well-posedness, that is well-known for the KdV equation, is shared by KdV type equations with weaker dispersion. This is despite the difference in the nature of these equations, which are quasilinear while KdV is semilinear. More precisely we prove that the low dispersion fractional KdV equation $$ \partial_t u -D_x^\alpha \partial_x u +\partial_x(u^2)=0 $$ is unconditionally globally well-posed in $L^2({\mathbb R}) $ for $\alpha \in ]\frac{55}{38},2] $. Our method of proof combined refined bilinear estimates with the energy method enhanced with Bourgain's type estimates developed in Molinet-Vento (2015).