On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities

Tadahiro Oh; Mamoru Okamoto; Oana Pocovnicu

doi:10.3934/dcds.2019144

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2019, Volume 39, Issue 6: 3479-3520. Doi: 10.3934/dcds.2019144 open access

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On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities

1.
School of Mathematics, The University of Edinburgh and The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King's Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, United Kingdom
2.
Division of Mathematics and Physics, Faculty of Engineering, Shinshu University, 4-17-1 Wakasato, Nagano City 380-8553, Japan
3.
Department of Mathematics, Heriot-Watt University and The Maxwell Institute for the Mathematical Sciences, Edinburgh, EH14 4AS, United Kingdom

^* Corresponding author: Tadahiro Oh
^* Corresponding author: Tadahiro Oh

Received: August 31, 2018

Published: February 2019

The first author was supported by the European Research Council (grant no. 637995 "ProbDynDispEq"). The second author was supported by JSPS KAKENHI Grant number JP16K17624

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We consider the Cauchy problem for the nonlinear Schrödinger equations (NLS) with non-algebraic nonlinearities on the Euclidean space. In particular, we study the energy-critical NLS on $ \mathbb{R}^d $, $ d = 5,6 $, and energy-critical NLS without gauge invariance and prove that they are almost surely locally well-posed with respect to randomized initial data below the energy space. We also study the long time behavior of solutions to these equations: (ⅰ) we prove almost sure global well-posedness of the (standard) energy-critical NLS on $ \mathbb{R}^d $, $ d = 5, 6 $, in the defocusing case, and (ⅱ) we present a probabilistic construction of finite time blowup solutions to the energy-critical NLS without gauge invariance below the energy space.

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Mathematics Subject Classification: Primary: 35Q55.

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