Abbatiello Anna, Bulíček Miroslav, Kaplický Petr
Department of Mathematics 'G. Castelnuovo', Sapienza University of Rome, Piazzale Aldo Moro 5, 00185 Rome, Italy.
Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 18675 Prague 8, Czech Republic.
Philos Trans A Math Phys Eng Sci. 2022 Nov 14;380(2236):20210351. doi: 10.1098/rsta.2021.0351. Epub 2022 Sep 26.
We consider a flow of a non-Newtonian heat conducting incompressible fluid in a bounded domain subjected to the homogeneous Dirichlet boundary condition for the velocity field and the Dirichlet boundary condition for the temperature. In three dimensions, for a power-law index greater or equal to [Formula: see text], we show the existence of a solution fulfilling the entropy equality. The entropy equality can be formally deduced from the energy equality by renormalization. However, such a procedure can be justified by the DiPerna-Lions theory only for [Formula: see text]. The main novelty is that we do not renormalize the temperature equation, but rather construct a solution which fulfils the entropy equality. This article is part of the theme issue 'Non-smooth variational problems and applications'.
我们考虑一种非牛顿导热不可压缩流体在有界区域内的流动，该区域对于速度场服从齐次狄利克雷边界条件，对于温度服从狄利克雷边界条件。在三维空间中，对于幂律指数大于或等于[公式：见原文]，我们证明了存在一个满足熵等式的解。熵等式可以通过重整化从能量等式形式上推导出来。然而，只有对于[公式：见原文]，这样的过程才能由迪佩尔纳 - 利翁斯理论来证明其合理性。主要的新颖之处在于我们不对温度方程进行重整化，而是构造一个满足熵等式的解。本文是“非光滑变分问题及应用”主题特刊的一部分。
