Year: 2025
Type: article
Abstract: We consider the real-valued defocusing modified Korteweg-de Vries equation (mKdV) on the circle. Based on the complete integrability of mKdV, Killip-Vişan-Zhang (2018) discovered a conserved quantity which they used to prove low regularity a priori bounds for solutions. It has been an open question if this conserved quantity can be used to define invariant measures supported at fractional Sobolev regularities. Motivated by this question, we construct probability measures supported on H s ( T ) H^s(\mathbb {T}) for 0 > s > 1 / 2 0>s>1/2 invariant under the mKdV flow. We then use the Miura transform to obtain invariant measures for the Korteweg-de Vries equation, whose supports are rougher than the white noise measure. We also obtain analogous results for the defocusing cubic nonlinear Schrödinger equation. These invariant measures cover the lowest possible regularities for which the flows of these equations are well-posed.
DOI: 10.1090/tran/9489
Authors: Andreia Chapouto, Justin Forlano
Institutions: Centre National de la Recherche Scientifique, University of California, Los Angeles, Université de Versailles Saint-Quentin-en-Yvelines, Université Paris-Saclay, Maxwell Institute for Mathematical Sciences, Laboratoire de Mathématiques de Versailles, University of Edinburgh, Monash University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G6000816776, G6505290223
OpenAlex ID: W4411085273