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Abstract
We construct globalintime singular dynamics for the (renormalized) cubic fourth order nonlinear Schrödinger equation on the circle, having the white noise measure as an invariant measure. For this purpose, we introduce the "randomresonant / nonlinear decomposition", which allows us to single out the singular component of the solution. Unlike the classical McKean, Bourgain, Da PratoDebussche type argument, this singular component is nonlinear, consisting of arbitrarily high powers of the random initial data. We also employ a random gauge transform, leading to random Fourier restriction norm spaces. For this problem, a contraction argument does not work and we instead establish convergence of smooth approximating solutions by studying the partially iterated Duhamel formulation under the random gauge transform. We reduce the crucial nonlinear estimates to boundedness properties of certain random multilinear functionals of the white noise.
Original language  English 

Article number  e48 
Number of pages  64 
Journal  Forum of Mathematics, Sigma 
Volume  8 
DOIs  
Publication status  Published  18 Nov 2020 
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Dive into the research topics of 'Solving the 4NLS with white noise initial data'. Together they form a unique fingerprint.Projects
 1 Finished

ProbDynDispEq  Probabilistic and Dynamical Study of Nonlinear Dispersive Equations
1/03/15 → 29/02/20
Project: Research
Profiles

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