多尺度浓悬浮模型的适定性
Well-posedness of a multiscale model for concentrated suspensions
摘要：
在以前的工作[数学。AP/0305408]我们三个人研究了一个非线性抛物型方程中产生的介观模型的浓缩悬浮液的颗粒受到一个给定的时间依赖性剪切速率。在目前的工作中，我们扩展的模型，以允许一个更物理相关的情况下，剪切速率实际上取决于流体的宏观速度，并作为反馈的宏观速度的影响，在流体中的平均应力。考虑的几何形状是一个平面库埃特流。所研究的数学系统耦合的一维热方程和非线性Fokker-Planck型方程的非齐次，非局部和可能退化，系数。我们证明了这样一个系统的整体时间弱解的存在性和唯一性。
英文摘要：
In a previous work [math.AP/0305408] three of us have studied a nonlinear parabolic equation arising in the mesoscopic modelling of concentrated suspensions of particles that are subjected to a given time-dependent shear rate. In the present work we extend the model to allow for a more physically relevant situation when the shear rate actually depends on the macroscopic velocity of the fluid, and as a feedback the macroscopic velocity is influenced by the average stress in the fluid. The geometry considered is that of a planar Couette flow. The mathematical system under study couples the one-dimensional heat equation and a nonlinear Fokker-Planck type equation with nonhomogeneous, nonlocal and possibly degenerate, coefficients. We show the existence and the uniqueness of the global-in-time weak solution to such a system.
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