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arXiv · 2606.28131

Stochastically forced Navier-Stokes equations interacting with an elastic structure

Felix Brandt·Matthias Hieber·Arnab Roy
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Abstract

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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Felix Brandt, Matthias Hieber, Arnab Roy. 2026-06-26. Stochastically forced Navier-Stokes equations interacting with an elastic structure. https://arxiv.org/abs/2606.28131

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