Regularity results for parabolic systems related to a class of non-Newtonian fluids.
The paper studies the regularity of a parabolic system describing electrorheological fluids \( u_t = \text{div } a(x,t,Du)=0\), where the vector field \(a\) exhibits non-standard growth. Such systes may also model other non-Newtonian fluids. A weak solution to the system is found by using additional condition on the growth of solution gradient. Partial \(C^{0,\alpha}\) regularity of spatial gradient of the weak solution is proved. The techniques developed in the paper can be extended to non-homogeneous systems provided the right-hand side satisfies appropriate growth assumptions. Local higher integrability (i.e. the fact that the weak solution and its gradient belong to certain Lebesgue and Sobolev spaces) is proved, and estimates on the parabolic Hausdorff measure of the singular set are given.
scientific article; zbMATH DE number 2057151