[Paper Review] Stochastic nonlinear wave dynamics on compact surfaces

Tadahiro Oh, Tristan Robert+1 more|arXiv (Cornell University)|Apr 10, 2019

Advanced Mathematical Physics Problems4 citations

TL;DR

This paper establishes local and global well-posedness for stochastic and deterministic nonlinear wave equations on compact two-dimensional Riemannian manifolds, using space-dependent and space-time renormalization to handle divergences from space-time white noise and singular nonlinearities. It extends prior results on the torus to general compact manifolds by proving almost sure global existence and invariance of the Gibbs measure under the dynamics.

ABSTRACT

We study the Cauchy problem for the nonlinear wave equations (NLW) with random data and/or stochastic forcing on a two-dimensional compact Riemannian manifold without boundary. (i) We first study the defocusing stochastic damped NLW driven by additive space-time white-noise, and with initial data distributed according to the Gibbs measure. By introducing a suitable space-dependent renormalization, we prove local well-posedness of the renormalized equation. Bourgain's invariant measure argument then allows us to establish almost sure global well-posedness and invariance of the Gibbs measure for the renormalized stochastic damped NLW. (ii) Similarly, we study the random data defocusing NLW (without stochastic forcing), and establish the same results as in the previous setting. (iii) Lastly, we study the stochastic NLW without damping. By introducing a space-time dependent renormalization, we prove its local well-posedness with deterministic initial data in all subcritical spaces. These results extend the corresponding recent results on the two-dimensional torus obtained by (i) Gubinelli-Koch-Oh-Tolomeo (2018), (ii) Oh-Thomann (2017), and (iii) Gubinelli-Koch-Oh (2018), to a general class of compact manifolds. The main ingredient is the Green's function estimate for the Laplace-Beltrami operator in this setting to study regularity properties of stochastic terms appearing in each of the problems.

Motivation & Objective

• To extend recent results on the 2D torus for stochastic and deterministic nonlinear wave equations to general compact two-dimensional Riemannian manifolds without boundary.
• To establish local well-posedness of the defocusing stochastic damped nonlinear wave equation (SDNLW) with additive space-time white noise and Gibbs-distributed initial data via a space-dependent renormalization.
• To prove almost sure global well-posedness and invariance of the Gibbs measure for the renormalized SDNLW using Bourgain's invariant measure argument.
• To extend the same well-posedness and invariance results to the random data defocusing NLW without stochastic forcing.
• To establish local well-posedness for the stochastic NLW without damping using a space-time dependent renormalization, with deterministic initial data in subcritical Sobolev spaces.

Proposed method

• Introduces a space-dependent renormalization for the stochastic damped NLW to control divergences from the nonlinear term and space-time white noise.
• Employs Green's function estimates for the Laplace-Beltrami operator on compact manifolds to analyze regularity of stochastic terms.
• Applies Bourgain's invariant measure argument to prove almost sure global existence and measure invariance for the renormalized SDNLW.
• Uses functional calculus and semi-classical pseudo-differential calculus on compact manifolds to control spectral properties of the Laplace-Beltrami operator.
• Establishes probabilistic estimates for stochastic convolutions and nonlinear terms using tools from stochastic analysis and Gaussian measures.
• Applies a frequency truncation scheme and iterative convergence arguments to prove convergence of regularized solutions in critical Sobolev spaces.

Experimental results

Research questions

• RQ1 Can the well-posedness and Gibbs measure invariance results for the stochastic damped NLW on the 2D torus be extended to general compact 2D Riemannian manifolds?
• RQ2 How can space-dependent renormalization be used to handle the singularities arising from space-time white noise and nonlinearities in the NLW on compact manifolds?
• RQ3 What is the role of Green's function estimates for the Laplace-Beltrami operator in establishing regularity of stochastic terms in the wave equation?
• RQ4 Can the same global well-posedness and measure invariance results be obtained for the random data NLW without stochastic forcing on compact manifolds?
• RQ5 How does space-time dependent renormalization enable local well-posedness of the stochastic NLW without damping in subcritical Sobolev spaces?

Key findings

• The renormalized stochastic damped nonlinear wave equation on compact 2D manifolds admits local well-posedness with initial data distributed according to the Gibbs measure.
• Almost sure global well-posedness and invariance of the Gibbs measure are established for the renormalized SDNLW via Bourgain's invariant measure argument.
• The same global well-posedness and measure invariance results hold for the random data defocusing NLW without stochastic forcing on compact manifolds.
• Local well-posedness of the stochastic NLW without damping is proven with deterministic initial data in all subcritical Sobolev spaces using space-time dependent renormalization.
• The convergence of regularized solutions in $C([0,T]; ilde{ ho}^{s}( ho))$ is established via iterative estimates and Cauchy sequence arguments in the frequency-truncated setting.
• The main technical innovation lies in the Green's function estimate for the Laplace-Beltrami operator on compact manifolds, which enables control of stochastic terms and regularity in the nonlinearities.