Partial regularity and nonlinear potential estimates for Stokes systems with super-quadratic growth

Abstract

This paper builds a bridge between partial regularity theory and nonlinear potential theory for the following generalized stationary Stokes system with super-quadratic growth and continuous coefficients

$$\left\{ {\matrix{{ - {\rm{div}}\,{\cal A}(x,D{\boldsymbol{u}}) + \nabla \pi = {\boldsymbol{f}}} \hfill & {{\rm{in}}\,\,\Omega ,} \hfill \cr {{\rm{div}}\,{\boldsymbol{u}} = 0} \hfill & {{\rm{in}}\,\,\Omega ,} \hfill \cr } } \right.$$

where Du is the symmetric part of the gradient ∇u. We first establish an ε-regularity criterion involving both the excess functional of the symmetric gradient Du and Wolff potentials of the nonhomogeneous term f to guarantee the local vanishing mean oscillation (VMO)-regularity of Du in an open subset Ω_u of Ω with full measure. Such an ε-regularity criterion leads to a pointwise Wolff potential estimate of Du, which immediately infers that Du is partially C⁰-regular under appropriate assumptions. Finally, we give a local continuous modulus estimate of Du.