# Stationary electro-rheological fluids: Low order regularity for systems with discontinuous coefficients

Bögelein V, Duzaar F, Habermann J, Scheven C (2012)

**Publication Type:** Journal article

**Publication year:** 2012

### Journal

**Publisher:** Walter de Gruyter

**Book Volume:** 5

**Pages Range:** 1-57

**Journal Issue:** 1

**DOI:** 10.1515/ACV.2011.009

### Abstract

We establish partial regularity for solutions (u, π): ℝ ^{n} ×ℝ ^{n} ⊇ Ω × Σ → ℝ ^{n} × ℝ to systems modeling electro-rheological fluids in the stationary case. As a model case our result covers the low order regularity of systems of the type (equation) where εu denotes the symmetric part of the gradient Du, π denotes the pressure, the not necessarily continuous coefficient μ is a bounded non-negative VMO-function and the variable exponent function p: Ω → (3n/n+2, ∞) fulfills the logarithmic continuity assumption, i.e., we assume that for the modulus of continuity ω p of the exponent function p there holds (equation) To be more precise, we prove Hölder continuity of the solution u outside of a negligible set. Moreover, we show that εu and the pressure belong to certain Morrey spaces on the regular set of u, i.e., the set where u is Hölder continuous. Note that under such weak assumptions partial Hölder continuity for the gradient cannot be expected. Our result is even new if the coefficient μ is continuous. © de Gruyter 2012.

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### How to cite

**APA:**

Bögelein, V., Duzaar, F., Habermann, J., & Scheven, C. (2012). Stationary electro-rheological fluids: Low order regularity for systems with discontinuous coefficients. *Advances in Calculus of Variations*, *5*(1), 1-57. https://dx.doi.org/10.1515/ACV.2011.009

**MLA:**

Bögelein, Verena, et al. "Stationary electro-rheological fluids: Low order regularity for systems with discontinuous coefficients." *Advances in Calculus of Variations* 5.1 (2012): 1-57.

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