Regularity for the stationary Navier–Stokes equations over bumpy boundaries and a local wall law

Abstract

In this paper we address the large-scale regularity theory for the stationary Navier-Stokes equations in highly oscillating bumpy John domains. These domains are very rough, possibly with fractals or cusps, at the microscopic scale, but are amenable to the mathematical analysis of the Navier-Stokes equations. We prove: (i) a large-scale Calderón-Zygmund estimate, (ii) a large-scale Lipschitz estimate, (iii) large-scale higherorder regularity estimates, namely, C 1,γ and C 2,γ estimates. These nice regularity results are inherited only at mesoscopic scales, and clearly fail in general at the microscopic scales. We emphasize that the large-scale C 1,γ regularity is obtained by using first-order boundary layers constructed via a new argument. The large-scale C 2,γ regularity relies on the construction of second-order boundary layers, which allows for certain boundary data with linear growth at spatial infinity. To the best of our knowledge, our work is the first to carry out such an analysis. In the wake of many works in quantitative homogenization, our results strongly advocate in favor of considering the boundary regularity of the solutions to fluid equations as a multiscale problem, with improved regularity at or above a certain scale.

Motion of several slender rigid filaments in a Stokes flow

“…Consequently, pDpV i,α q, Dpu 5 qq 1 L 2 pFq " Y ¨∇h,Q Gpv i,α , u 5 q `GpY ¨∇h,Q v i,α , u 5 q `Gpv i,α , B t u 5 q. Thus, taking into account the regularity assumptions on the background flow u 5 , see (17), we arrive at |pDpV i,α q, Dpu 5 qq 1 L 2 pFq | ď |Y||∇ h,Q Gpv i,α , u 5 q| `C |log ε| ´1 p1 `|Y|q. Analogous considerations hold for the term ˇˇ`D pvq, Dpvq ˘1 L 2 pFq ˇˇ.…”

“…Proposition 3.2. -Given some initial disjoint positions of the centerline curves, given a smooth background flow u 5 satisfying (17), there is a unique smooth global-intime solution to (38)-( 39)-( 40) on r0, `8q.…”

“…Theorem 3.4. -We consider some initial disjoint positions of the centerline curves, a smooth background flow u 5 satisfying (17), and the solutions p h i , Q i q 1ďiďN given by Proposition 3.2. Let T in p0, `8s be the time of the first collision associated with this solution as defined in Definition 3.3.…”

“…Let us also mention that the influence of small scales in the cross sections can also be encoded by a different choice of the boundary conditions at the interface between the fluid phase and the solid phase. In this paper we concentrate on the case of the no-slip condition at the interface, but some other conditions could be considered as well, such as the Navier slip conditions, see [17] and the references therein. Hence, it would be interesting to investigate whether or not the results of Theorem 3.4 can be adapted to other boundary conditions.…”

Motion of several slender rigid filaments in a Stokes flow

“…where pu 5 , p 5 q P C `r0, `8q; 9 H 1 pR 3 qˆL 2 pR 3 qq X `W 2,8 pp0, `8q ˆR3 q ˆW 1,8 pp0, `8q ˆR3 q satisfying div u 5 " 0, (17) is the background flow, and pu p , p p q is the perturbation flow due to the filaments, whose evolution is assumed to be driven by the steady Stokes equations: ´∆u p `∇p p " 0 and div u p " 0 in Fptq, (18a)…”