Abstract
AbstractWe construct global-in-time 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 ‘random-resonant / nonlinear decomposition’, which allows us to single out the singular component of the solution. Unlike the classical McKean, Bourgain, Da Prato-Debussche 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 the 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.
Highlights
There are many important Hamiltonian PDEs such as the Korteweg-de Vries equation (KdV) and the nonlinear Schrödinger equations (NLS), under which the 2-norm of a solution is conserved. For this type of equations, thanks to the general globalization argument introduced by Bourgain in [6, 7], if one can solve the equation locally in time with data distributed according to (1.1), one can almost surely extend the solutions for all times, and the white noise would be an invariant measure of the resulting flow
In this formulation, we successfully reduced the number of combinations; we only need to study
Non-resonant part N1 In this subsection, we evaluate the non-resonant part of the nonlinearity
Summary
There are many important Hamiltonian PDEs such as the Korteweg-de Vries equation (KdV) and the nonlinear Schrödinger equations (NLS), under which the 2-norm of a solution is conserved For this type of equations, thanks to the general globalization argument introduced by Bourgain in [6, 7], if one can solve the equation locally in time with data distributed according to (1.1), one can almost surely extend the solutions for all times, and the white noise would be an invariant measure of the resulting flow. It is worthwhile to note that in a similar discussion for the KdV equation, one can show convergence of the sequence of regularized solutions for any regularization of the white noise initial data This is because the local well-posedness analysis in [39, 50] is purely deterministic. See Remark 1.2 for a discussion in case of smoother random initial data
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