Global dynamics for the stochastic nonlinear beam equations on the four-dimensional torus
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...
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Veröffentlicht in: | Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 2024-11, p.1-39 |
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Sprache: | eng |
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Zusammenfassung: | 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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ISSN: | 0308-2105 1473-7124 |
DOI: | 10.1017/prm.2024.87 |