The Boussinesq system in 3-dimensional bounded rough domains: Well-posedness in critical spaces and long-time behavior

Abstract

We study the three-dimensional Boussinesq system in bounded rough domains, including bounded Lipschitz and $\mathrm{C}^{1,\alpha}$ domains, within a critical functional framework. We establish existence and uniqueness results that are global in time for small initial data and local in time for arbitrary initial data. Well-posedness in critical endpoint Besov spaces with third index equal to $\infty$ is obtained in domains with H\"older continuous boundaries, relying on $\mathrm{L}^2$-maximal regularity in time. We also prove well-posedness in critical Besov spaces with third index equal to $1$, using $\mathrm{L}^1$-maximal regularity. In this $\mathrm{L}^1$-in-time setting, the analysis applies to arbitrary bounded Lipschitz domains. In any case, we show that the fluid velocity stabilizes exponentially for large times and that the temperature converges to the initial averaged temperature of the fluid. The linear theory -- fitting the adapted product estimates and vice versa -- is properly established prior to the nonlinear analysis. With this fully prepared linear framework in hand, the nonlinear estimates that follow are then handled in the critical framework with a simplified treatment -- especially in the case where the fluid velocity and the temperature belong to slightly larger spaces than $\mathrm{L}^2(\mathrm{W}^{1,3})$ and $\mathrm{L}^2(\mathrm{L}^{3/2})$ respectively -- when compared with previously known similar results in smooth domains. This approach relies on a robust linear theory and sharp product estimates based on operator-theoretic methods and Besov space techniques. Finally, as part of the analysis, we establish several new results for the underlying linear operators, including refined characterizations for the domains of fractional powers of the Neumann Laplacian and of the Stokes operator in bounded Lipschitz domains.