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

  • * Corresponding author: Tadahiro Oh

    * Corresponding author: Tadahiro Oh 

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.

    Mathematics Subject Classification: Primary: 35Q55.


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