Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation

Abstract

We study the defocusing energy-critical nonlinear wave equation in four dimensions. Our main result proves the stability of the scattering mechanism under random pertubations of the initial data. The random pertubation is defined through a microlocal randomization, which is based on a unit-scale decomposition in physical and frequency space. In contrast to the previous literature, we do not require the spherical symmetry of the pertubation. The main novelty lies in a wave packet decomposition of the random linear evolution. Through this decomposition, we can adaptively estimate the interaction between the rough and regular components of the solution. Our argument relies on techniques from restriction theory, such as Bourgain's bush argument and Wolff's induction on scales.Contents MSC2010 : 35L05, 42B20, 42B37

Refined probabilistic global well-posedness for the weakly dispersive NLS

“…This also improves our previous result in Sun-Tzvetkov [28]. We rely on recent ideas of Bringmann [8] and Deng-Nahmod-Yue [17]. In particular we adapt to our situation the new resolution ansatz in [17] which captures the most singular frequency interaction parts in the X s,b type space.…”

“…Refined resolution ansatz. Refined resolution ansatz to treat the singular high-low type interaction has been recently introduced by Bringmann [8] for the wave equation and by Deng-Nahmod-Yue [17] for the 2D NLS in very different ways. The common feature in both these work is the observation that the low frequency component is independent with the high frequency linear evolution and the most singular interactions (high-low type) are removed by viewing them as part of the linear evolution for the high-frequency data and isolating them from w(t) in the previous affine ansatz u(t) = S α (t)φ ω + w(t).…”

Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity I: measures

“…The reason lies in low×low×high-interactions, which are more difficult in Schrödinger equations than in wave equations. In the last two years, we have seen new and intricate methods dealing with these interactions [10,13,14], but all of these papers heavily rely on the independence of the Fourier coefficients. In fact, overcoming this obstruction is mentioned as an open problem in [14,Section 9.1].…”

“…Remark 4. 10 We emphasize that the implicit constant does not depend on . In the application of Lemma 4.9, we will choose > 0 sufficiently small depending on δ, n, β, λ.…”

Solving the 4NLS with white noise initial data

“…In the context of stochastic PDEs, such iteration of a Duhamel formulation appears in the dispersive setting [50,31] and in the parabolic setting [34,17,43]. We also mention [8,11,12,13] on the probabilistic construction of solutions by establishing convergence of smooth solutions. In particular, the recent approach by Bourgain-Bulut [11,12] relying on the invariance of the truncated Gibbs measures even in the construction of local solutions works well for a power-type nonlinearity with positive regularity but is not suitable to our problem at hand.…”