Unconditional uniqueness for the cubic nonlinear Schrödinger equation in ${\dot{H}}^{\frac{1}{2}}({\mathbb{R}}^3)$

Abstract

While unconditional uniqueness for the cubic nonlinear Schrödinger equation (NLS) on ${\mathbb{R}}^3$ is known in the scaling-subcritical Sobolev spaces ${\dot{H}}^s({\mathbb{R}}^3)$ for $1/2<s<1$, the critical case $s=1/2$ has remained open. We resolve this problem by adapting Kato's bootstrap argument to the critical space ${\dot{B}}_{\infty ,\infty }^{-1}$. The main difficulty at this endpoint is that the natural solution class does not provide the coefficient regularity required to close the bootstrap in ${\dot{B}}_{\infty ,\infty }^{-1}$. To overcome this issue, we perform a second Duhamel iteration of Born type and estimate the resulting double Duhamel integral as a whole. This yields improved estimates that allow the bootstrap to close in ${\dot{B}}_{\infty ,\infty }^{-1}$ using only the available regularity of the coefficients.