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

Abstract

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.

arXiv comment44 pages

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• Almost sure scattering for the radial energy critical nonlinear wave equation in three dimensions
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• Upper bound for lifespan of solutions to certain semilinear parabolic, dispersive and hyperbolic equations via a unified test function method
• Almost sure well-posedness for the cubic nonlinear Schrödinger equation in the super-critical regime on $\TTT^d$, $d\ge 3$
• Remark on upper bound for lifespan of solutions to semilinear evolution equations in a two-dimensional exterior domain
• Large global solutions for nonlinear Schrödinger equations II, mass-supercritical, energy-subcritical cases
• Probabilistic pointwise convergence problem of some dispersive equations
• Pointwise convergence problem of Ostrovsky equation with rough data and random data