[Paper Review] Random data wave equations

Nikolay Tzvetkov|arXiv (Cornell University)|Apr 4, 2017

Advanced Mathematical Physics Problems4 citations

TL;DR

This paper establishes probabilistic well-posedness and quasi-invariance of the 3D cubic nonlinear wave equation on the torus for initial data in supercritical Sobolev spaces below the energy threshold. By endowing low-regularity initial data with a Gaussian measure and using refined probabilistic Strichartz estimates and multi-scale Wiener chaos analysis, the authors prove almost sure global existence and uniqueness of solutions, and show the associated invariant measure is quasi-invariant under the flow, extending deterministic well-posedness results to singular regularity regimes.

ABSTRACT

Nowadays we have many methods allowing to exploit the regularising properties of the linear part of a nonlinear dispersive equation (such as the KdV equation, the nonlinear wave or the nonlinear Schroedinger equations) in order to prove well-posedness in low regularity Sobolev spaces. By well-posedness in low regularity Sobolev spaces we mean that less regularity than the one imposed by the energy methods is required (the energy methods do not exploit the dispersive properties of the linear part of the equation). In many cases these methods to prove well-posedness in low regularity Sobolev spaces lead to optimal results in terms of the regularity of the initial data. By optimal we mean that if one requires slightly less regularity then the corresponding Cauchy problem becomes ill-posed in the Hadamard sense. We call the Sobolev spaces in which these ill-posedness results hold spaces of supercritical regularity. More recently, methods to prove probabilistic well-posedness in Sobolev spaces of supercritical regularity were developed. More precisely, by probabilistic well-posedness we mean that one endows the corresponding Sobolev space of supercritical regularity with a non degenerate probability measure and then one shows that almost surely with respect to this measure one can define a (unique) global flow. However, in most of the cases when the methods to prove probabilistic well-posedness apply, there is no information about the measure transported by the flow. Very recently, a method to prove that the transported measure is absolutely continuous with respect to the initial measure was developed. In such a situation, we have a measure which is quasi-invariant under the corresponding flow. The aim of these lectures is to present all of the above described developments in the context of the nonlinear wave equation.

Motivation & Objective

• To extend well-posedness results for the 3D cubic nonlinear wave equation beyond the deterministic energy threshold using probabilistic methods.
• To establish the existence of a global flow for initial data in Sobolev spaces of supercritical regularity (below $ H^1 \times L^2 $) by assigning a non-degenerate Gaussian measure to the initial data space.
• To prove that the law of the solution flow is absolutely continuous with respect to the initial measure, establishing quasi-invariance of the Gibbs-type measure under the nonlinear wave flow.
• To develop a multi-scale probabilistic analysis combining Wiener chaos estimates and deterministic Strichartz-type bounds to control the regularity loss in the Picard iteration.

Proposed method

• Assign a non-degenerate Gaussian measure $ \widetilde{\mu}_s $ to initial data in $ H^s \times H^{s-1} $ for $ s < 1 $, enabling probabilistic well-posedness in supercritical regimes.
• Use probabilistic Strichartz estimates to control the $ L^p $-norms of nonlinear terms in the Picard iteration, with a $ \sqrt{p} $-loss in regularity from Wiener chaos estimates.
• Implement a multi-scale decomposition of frequency-localized terms in the nonlinear interaction, distinguishing cases based on dyadic frequency localization $ N_1, N_2, N_3, N_4 $ to optimize regularity gain.
• Apply bi-linear and tri-linear Wiener chaos estimates in different frequency regimes to control the $ L^p $-norms of the nonlinear terms, particularly for $ Q_1(u,v) $, with careful redistribution of derivative losses.
• Introduce renormalized energies and soft analysis techniques to handle the limit of the renormalized energy functional $ R(u) $ in the measure-theoretic framework.
• Use a differential inequality argument involving $ \dot{x}(t) \leq Cp(x(t))^{1-1/p} $ to show that sets of measure zero evolve to sets of measure zero, proving quasi-invariance in finite time and via iteration over time.

Experimental results

Research questions

• RQ1 Can the 3D cubic nonlinear wave equation be globally well-posed for initial data in Sobolev spaces of supercritical regularity below the energy threshold?
• RQ2 Is the Gibbs-type measure associated with the nonlinear wave equation quasi-invariant under the nonlinear flow, even when the deterministic flow is ill-posed?
• RQ3 What is the optimal regularity threshold for almost sure global existence of solutions when initial data are randomized via a Gaussian measure?
• RQ4 How can probabilistic Strichartz estimates and Wiener chaos analysis be combined to control the growth of nonlinear terms in low-regularity spaces?
• RQ5 What is the role of the $ \sqrt{p} $-loss in the key estimate of Theorem 4.3, and why is it critical for proving quasi-invariance?

Key findings

• The Cauchy problem for the 3D cubic wave equation is almost surely globally well-posed for initial data in $ H^s \times H^{s-1} $ with $ s < 1 $, extending beyond the deterministic threshold of $ s \geq 1 $.
• The associated Gibbs measure $ \widetilde{\mu}_s $ is quasi-invariant under the nonlinear wave flow, meaning the pushforward measure is absolutely continuous with respect to the original measure.
• The key estimate in Theorem 4.3 involves a $ \sqrt{p} $-loss in the $ L^p $-norm of nonlinear terms, which is essential for the differential inequality argument leading to quasi-invariance.
• The proof relies on a multi-scale analysis of the $ 4 $-linear expression $ Q_1(u,v) $, where different frequency regimes are treated via bi-linear or tri-linear Wiener chaos estimates to gain regularity.
• The soft analysis technique allows the limit of the renormalized energy functional $ R(u) $ to be defined in $ L^p(d\widetilde{\mu}_s) $ for all $ p < \infty $, enabling the construction of the invariant measure.
• The argument using $ \dot{x}(t) \leq Cp(x(t))^{1-1/p} $ and taking $ p \to \infty $ shows that sets of zero measure remain of zero measure, proving quasi-invariance on a time interval of order $ 1/C $, which is iterated to obtain global quasi-invariance.