Stochastically forced Navier-Stokes equations interacting with an elastic structure
University of California at Berkeley（加州大学伯克利分校）；Technische Universität Darmstadt（达姆施塔特工业大学）；Basque Center for Applied Mathematics (BCAM)（巴斯克应用数学中心）；IKERBASQUE, Basque Foundation for Science（伊克尔巴斯基奎，巴斯克科学基金会）
We prove global-in-time strong pathwise well-posedness for a stochastic fluid-structure interaction problem coupling a two-dimensional incompressible Navier-Stokes fluid to a one-dimensional damped Kirchhoff plate. The coupling is imposed on a fixed interface through continuity of velocities and balance of normal stresses, and stochastic forcing, modeled by a cylindrical Wiener process, acts on both the fluid and structure equations. We split the problem into a linear stochastic part and a nonlinear deterministic remainder. The linear stochastic problem is treated by proving that the associated fluid-structure operator admits a bounded \(\mathcal{H}^\infty\)-calculus, yielding stochastic maximal regularity. This requires a decoupling procedure for the non-diagonal operator domain, and pressure estimates via suitable lifting constructions. The deterministic remainder is solved locally by quasilinear methods, and the resulting blow-up criterion is ruled out by higher-order a priori estimates.
This is the first global-in-time strong pathwise well-posedness result for a stochastically forced Navier-Stokes system interacting with a deformable elastic structure.
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We give an overview of the ideas central to some recent developments in the ergodic theory of the stochastically forced Navier Stokes equations and other dissipative stochastic partial differential equations. Since our desire is to make the core ideas clear, we will mostly work with a specific example: the stochastically forced Navier Stokes equations. To further clarify ideas, we will also examine in detail a toy problem. A few general theorems are given. Spatial regularity, ergodicity, exponential mixing, coupling for a SPDE, and hypoellipticity are all discussed.
Non-uniqueness in law for three-dimensional Navier-Stokes equations forced by random noise was established recently in Hofmanov$\acute{\mathrm{a}}$ et al. (2019, arXiv:1912.11841 [math.PR]). The purpose of this work is to prove non-uniqueness in law for the Boussinesq system forced by random noise. Diffusion within the equation of its temperature scalar field has a full Laplacian and the temperature scalar field can be initially smooth.
