Stationary electro-rheological fluids: Low order regularity for systems with discontinuous coefficients
Stationary electro-rheological fluids: Low order regularity for systems with discontinuous coefficients
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DOI：
10.1515/acv.2011.009
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发表时间：
2012-01
影响因子：
1.7
通讯作者：
V. Bögelein;F. Duzaar;J. Habermann;Christoph Scheven
中科院分区：
文献类型：
--
作者：
V. Bögelein;F. Duzaar;J. Habermann;Christoph Scheven
抽象的。我们为静态情况下的电流变流体建模系统的解决方案建立了部分规律性。作为模型案例，我们的结果涵盖了该类型系统的低阶正则性，其中 表示梯度的对称部分 ， 表示压力，不一定连续的系数是有界非负函数，并且可变指数函数满足对数连续性假设，即，我们假设指数函数的连续性模成立。更准确地说，我们证明 Hölder 解在可忽略的集合之外的连续性。此外，我们证明 和 压力属于正则集 上的某些莫雷空间，即 Hölder 连续的集合。请注意，在如此弱的假设下，无法预期梯度的部分霍尔德连续性。如果系数是连续的，我们的结果甚至是新的。
Abstract. We establish partial regularity for solutions 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 where denotes the symmetric part of the gradient , denotes the pressure, the not necessarily continuous coefficient is a bounded non-negative -function and the variable exponent function fulfills the logarithmic continuity assumption, i.e., we assume that for the modulus of continuity of the exponent function there holds To be more precise, we prove Hölder continuity of the solution outside of a negligible set. Moreover, we show that and the pressure belong to certain Morrey spaces on the regular set of , i.e., the set where 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 is continuous.