Fractional Higher Differentiability for Solutions of Stationary Stokes and Navier-Stokes Systems with Orlicz Growth

Abstract

AbstractWe consider weak solutions $(u,\pi ):{\Omega }\to \mathbb {R}^{n}\times \mathbb {R}$ ( u , π ) : Ω → ℝ n × ℝ to stationary ϕ-Navier-Stokes systems of the type $ \left \{ \begin {array}{ll} -\mathrm {div~} a(x,\mathcal {E} u)+\nabla \pi +[Du]u=f \\ \mathrm {div~} u=0 \end {array} \right . $ − div a ( x , E u ) + ∇ π + [ D u ] u = f div u = 0 in ${\Omega }\subset \mathbb {R}^{n}$ Ω ⊂ ℝ n , and to the corresponding ϕ-Stokes systems, in which the convective term [Du]u does not appear. In the above system, the function a(x,ξ) depends Hölder continuously on x and satisfies growth conditions with respect to the second variable expressed through a Young function ϕ. The notation $\mathcal {E} u$ E u is used for the symmetric part of the gradient Du. We prove results on the fractional higher differentiability of both the symmetric part of the gradient $\mathcal {E} u$ E u and of the pressure π.