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

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 $${\mathrm{\partial }}_{t}u-D_x^{\alpha }{\mathrm{\partial }}_{x}u+{\mathrm{\partial }}_x(u^2)=0$$ 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).