Navier-Stokes Equations on Thin 3D Domains. I: Global Attractors and Global Regularity of Solutions

On singular limit equations for incompressible fluids in moving thin domains

“…We may naturally ask whether we can derive limit equations as a thin domain degenerates into a two-dimensional set and compare properties of solutions to the original three-dimensional equations and the corresponding two-dimensional limit equations. There are several works studying such problems with a three-dimensional flat thin domain [15,16,29,33] of the form…”

“…We may naturally ask whether we can derive limit equations as a thin domain degenerates into a two-dimensional set and compare properties of solutions to the original three-dimensional equations and the corresponding two-dimensional limit equations. There are several works studying such problems with a three-dimensional flat thin domain [15,16,29,33] of the formfor small ε > 0, where ω is a two-dimensional domain and g 0 and g 1 are functions on ω, and a three-dimensional thin spherical domain [34] which is a region between two concentric spheres of near radii. (We also refer to [28] for the strategy of analysis of the Euler equations in a flat and spherical thin domain and its limit equations.)…”

Dimension Reduction for Compressible Viscous Fluids

“…Although a rigorous justification of the limit passage from the 3D-fluid motion to a linear one seems of obvious practical importance, there are only a few results available in the literature, at least in the context of compressible fluids. There are numerous studies of the incompressible fluid flows on thin domains, where the limit motion becomes planar and even regular, see Iftimie, Raugel and Sell [7], Raugel and Sell [17], [15], [16], and the references therein. Obviously, the 3D to 1D limit does not make too much sense in the incompressible setting.…”

Fast Singular Oscillating Limits and Global Regularity for the 3D Primitive Equations of Geophysics

“…The technique of bootstrapping regularity of solutions of three-dimensional Navier-Stokes equations by perturbation from limit equations has been done in various contexts: thin domains [44], helical flows [39]. In these previous works, limit equations are 2-D Navier-Stokes equations for which global regularity is well known.…”