Abstract
We study an optimal mass threshold for normalizability of he Gibbs measures associated with the focusing masscritical nonlinear Schrödinger equation on the onedimensional torus. In an influential paper, Lebowitz, Rose, and Speer (1988) proposed a critical mass threshold given by the mass of the ground state on the real line. We provide a proof for the optimality of this critical mass threshold. The proof also applies to the twodimensional radial problem posed on the unit disc. In this case, we answer a question posed by Bourgain and Bulut (2014) on the optimal mass threshold.
Furthermore, in the onedimensional case, we show that the Gibbs measure is indeed normalizable at the optimal mass threshold, thus answering an open question posed by Lebowitz, Rose, and Speer (1988). This normalizability at the optimal mass threshold is rather striking in view of the minimal mass blowup solution for the focusing quintic nonlinear Schr\"odinger equation on the onedimensional torus.
Furthermore, in the onedimensional case, we show that the Gibbs measure is indeed normalizable at the optimal mass threshold, thus answering an open question posed by Lebowitz, Rose, and Speer (1988). This normalizability at the optimal mass threshold is rather striking in view of the minimal mass blowup solution for the focusing quintic nonlinear Schr\"odinger equation on the onedimensional torus.
Original language  English 

Number of pages  70 
Publication status  In preparation  2019 
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Profiles

Tadahiro Oh
 School of Mathematics  Personal Chair of Dispersive Equations
Person: Academic: Research Active