Weak Solutions of Hopf Type to 2D Maxwell Flows with Infinite Number of Relaxation Times

Abstract

A system of equations describing the motion of fluids of Maxwell type is considered:

\( \frac{\partial }{\partial t}\upsilon +\upsilon \cdot \nabla \upsilon -\underset{0}{\overset{t}{\int }}K\left(t-\tau \right) d\tau +\nabla p=f\left(x,t\right),\kern0.5em di\upsilon\;\upsilon =0. \)

Here K(t) is an exponential series \( K(t)=\sum \limits_{s=1}^{\infty }{\beta}_s{e}^{-{\alpha}_st} \). The existence of a weak solution for the initial boundary value problem

\( {\left.\begin{array}{ccc}\upsilon \left(x,0\right)={\upsilon}_0(x),& {\left.\upsilon \cdot n\right|}_{\partial \varOmega }=0,& rot\end{array}\;\upsilon \right|}_{\partial \varOmega }=0 \)

is proved.