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Regularity results for parabolic systems related to a class of non-Newtonian fluids. (English) Zbl 1052.76004
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.

##### MSC:
 76A05 Non-Newtonian fluids 76W05 Magnetohydrodynamics and electrohydrodynamics 35Q35 PDEs in connection with fluid mechanics 35K50 Systems of parabolic equations, boundary value problems (MSC2000) 35B65 Smoothness and regularity of solutions to PDEs 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
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