On the W2,q-Regularity of Incompressible Fluids with Shear-Dependent Viscosities: The Shear-Thinning Case

Abstract

In this paper we improve the results stated in Reference [2], in this same Journal, by using -basically-the same tools. We consider a non Newtonian fluid governed by equations with p-structure and we show that second order derivatives of the velocity and first order derivatives of the pressure belong to suitable Lebesgue spaces. (2000). Primary 76A05; secondary 35B65, 35Q35. Mathematics Subject ClassificationKeywords. Shear dependent viscosity, incompressible fluid, regularity up to the boundary.

Navier–Stokes Equations with Shear Thinning Viscosity. Regularity up to the Boundary

“…In the meantime, by following the main lines established here, the results were improved in the two subsequent papers [10] and [8]. Actually, two new ideas allow interesting improvements.…”

“…Actually, two new ideas allow interesting improvements. In reference [10] the author improves the results by appealing to anisotropic embedding theorems of Sobolev type, a fruitful idea in the general context of regularity up to the boundary for p-fluid flows. Further, in reference [8], we use this last idea together with a new device that overcomes the need of results like that stated in the Lemma 3.2 below, typical in treating the shear thinning case, see [17].…”

“…Further, in reference [8], we use this last idea together with a new device that overcomes the need of results like that stated in the Lemma 3.2 below, typical in treating the shear thinning case, see [17]. This leads to an improvement of the results established in references [10] and [8]. Furthermore, in reference [9] (as above, see equation (1.21)) we obtain very accurate estimates in terms of f .…”

Existence of Strong Solutions for Incompressible Fluids with Shear Dependent Viscosities

“…The literature on this subject is very large and we focus on the papers that are mostly connected with the results we are going to prove. In particular, for the steady problem, there are several results proving existence of weak solutions [32], [26], interior regularity [1], [33] and very recently regularity up-to-the boundary for the Dirichlet problem [53], [56], [7], [8], [9], [10], [11], [12], [13], [14], [16], [21], [22]. Concerning the time-evolution Dirichlet problem in a three-dimensional domain we have recent advances on the existence of weak solutions in [58] for p > 8 5 and in [31] for p > 6 5 .…”

Shear Thinning Viscous Fluids in Cylindrical Domains. Regularity up to the Boundary

“…Moreover, we can obtain regularity results also for system (1.1) "completed" with the convective term (v ·∇)v. In fact, our regularity results still hold when adding such a term, provided that we consider a smaller range for p. We omit the details, and refer to [2], Theorem 1.5. Concerning other possible generalizations of our results, we quote here the forthcoming paper [5], where, by inserting a new device in the proof developed in reference [2] (the use of Sobolev anisotropic embedding theorems), the author obtains better integrability coefficients. Also this extension can be considered in our context, without particular difficulty.…”