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arXiv · 2312.13901

Global dynamics for the stochastic nonlinear beam equations on the four-dimensional torus

Andreia Chapouto·Guopeng Li·Ruoyuan Liu
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Abstract

We study global-in-time dynamics of the stochastic nonlinear beam equations (SNLB) with an additive space-time white noise, posed on the four-dimensional torus. The roughness of the noise leads us to introducing a time-dependent renormalization, after which we show that SNLB is pathwise locally well-posed in all subcritical and most of the critical regimes. For the (renormalized) defocusing cubic SNLB, we establish pathwise global well-posedness below the energy space, by adapting a hybrid argument of Gubinelli- Koch-Oh-Tolomeo (2022) that combines the $I$-method with a Gronwall-type argument. Lastly, we show almost sure global well-posedness and invariance of the Gibbs measure for the stochastic damped nonlinear beam equations in the defocusing case.

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Andreia Chapouto, Guopeng Li, Ruoyuan Liu. 2023-12-21. Global dynamics for the stochastic nonlinear beam equations on the four-dimensional torus. https://doi.org/10.1017/prm.2024.87

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