We consider the stationary flow of a generalized Newtonian fluid which is modelled by an anisotropic dissipative potential f . More precisely, we are looking for a solution u: Ω → R n , Ω ⊂ R n , n = 2, 3, of the following system of nonlinear partial differential equations −div T (ε(u)) + u k ∂u ∂x k + ∇π = g in Ω , div u = 0 in Ω , u = 0 on ∂Ω .
We consider equations with the simplest hysteresis operator at the right-hand side. Such equations describe the so-called processes "with memory" in which various substances interact according to the hysteresis law.We restrict our consideration on the so-called "strong solutions" belonging to the Sobolev class W 2,1 q with sufficiently large q and prove that in fact q = ∞. In other words, we establish the optimal regularity of solutions. Our arguments are based on quadratic growth estimates for solutions near the free boundary.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.