Unconditional uniqueness for the cubic nonlinear Schr\"odinger equation in $\dot{H}^{\frac12}(\Bbb R^3)$

Abstract

While unconditional uniqueness for the cubic nonlinear Schr\"odinger equation (NLS) on $\Bbb R^3$ is known in the scaling-subcritical Sobolev spaces $\dot H^s(\Bbb R^3)$ 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 a new choice of auxiliary space, namely the critical Besov space $\dot B^{-1}_{\infty,\infty}$. The main difficulty at this endpoint is that the natural solution class does not provide the coefficient regularity required to close the bootstrap in this space. To overcome this difficulty, 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 $\dot B^{-1}_{\infty,\infty}$ using only the available regularity of the coefficients.