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Abstract
We study globalintime dynamics of the stochastic nonlinear wave equations (SNLW) with an additive spacetime white noise forcing, posed on the twodimensional torus.
Our goal in this paper is twofold. (i) By introducing a hybrid argument, combining the Imethod in the stochastic setting with a Gronwalltype argument, we first prove global wellposedness of the (renormalized) cubic SNLW in the defocusing case. Our argument yields a double exponential growth bound on the Sobolev norm of a solution. (ii) We then study the stochastic damped nonlinear wave equations (SdNLW) in the defocusing case. In particular, by applying Bourgain's invariant measure argument, we prove almost sure global wellposedness of the (renormalized) defocusing SdNLW with respect to the Gibbs measure and invariance of the Gibbs measure.
Our goal in this paper is twofold. (i) By introducing a hybrid argument, combining the Imethod in the stochastic setting with a Gronwalltype argument, we first prove global wellposedness of the (renormalized) cubic SNLW in the defocusing case. Our argument yields a double exponential growth bound on the Sobolev norm of a solution. (ii) We then study the stochastic damped nonlinear wave equations (SdNLW) in the defocusing case. In particular, by applying Bourgain's invariant measure argument, we prove almost sure global wellposedness of the (renormalized) defocusing SdNLW with respect to the Gibbs measure and invariance of the Gibbs measure.
Original language  English 

Number of pages  33 
Journal  International Mathematics Research Notices 
Publication status  Accepted/In press  16 Mar 2021 
Keywords
 stochastic nonlinear wave equation
 nonlinear wave equation
 damped nonlinear wave equation
 renormalization
 white noise
 Gibbs measure
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Projects


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