Interior Differentiability of Weak Solutions to the Equations of Stationary Motion of a Class of Non-Newtonian Fluids

Abstract

In this paper, we consider weak solutions to the equations of stationary motion of a class of non-Newtonian fluids the constitutive law of which includes the "power law model" as special case. We prove the existence of second order derivatives of weak solutions to these equations. (2000). 35Q30, 76D05, 35J65. Mathematics Subject Classification

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

“…See, e.g., Berselli, Diening and Růžička [7,8] for a proof covering also the degenerate case µ = 0, hence showing stability with respect to this parameter. These same results hold, locally, for local solutions; see Naumann and Wolf [20]. Note that our "up to the boundary" results of Theorem 1.3 are weaker than the above ones.…”

Global regularity of a class of p-fluid flows in cylinders

“…As far as the regularity is concerned, the interior regularity results obtained in [15] hold here. Below we recall such results and obtain some immediate corollaries which are essential for our later study.…”

“…Further, we obtain explicit estimates on the norms involved, as they are necessary for our developments. Since these estimates are not explicitly written in reference [15] we give a short proof in Appendix A.…”

“…The idea is to overcome such difficulties as follows. We apply already known interior regularity results in Ω (see [15]), in order to achieve the global regularity of the second order derivatives of the velocity and the first order derivatives of the pressure on any cylinder Ω 0 inside Ω. Then, by applying the technique introduced in [9], we prove regularity results for solutions on an arbitrary annulus cylinder Ω \ Ω 0 inside Ω, reaching in this way the boundary of Ω.…”

The Nonlinear Stokes Problem with General Potentials Having Superquadratic Growth

“…the operator "div" has to be applied linewise. For these power law models full interior C 1,α -regularity in the 2D case has been proved by Kaplický et al [22] and Wolf [27], whereas the higher dimensional situation is studied for example in Naumann and Wolf [25]. For partial regularity results in dimensions n ≥ 3 we also refer to [10,12].…”