Uniqueness of solutions for the logarithmic Schr\"{o}dinger equation

We consider the Cauchy problem for the logarithmic Schr\"odinger equation and prove uniqueness of weak $H^s(\mathbb{R}^d)$ solutions for $s\in(0,1)$, which improves on the previous uniqueness result in $H^1(\mathbb{R}^d)$. The proof is achieved by combining a nontrivial use of integral equations, local smoothing estimates, and quantitative estimates of the sublinear effect of the nonlinearity, based on the localization argument. We also study uniqueness on the torus and uniqueness of the equation perturbed by pure power nonlinearities.