Nonlocal problems for the equations of Kelvin–Voight fluids and their $\varepsilon$-approximations

In this paper, we study some nonlocal problems for the Kelvin–Voight equations (1) and the penalized Kelvin–Voight equations (2): the first and second initial boundary-value problems and the first and second time periodic boundary problems. We prove that these problems have global smooth solutions of the class $W^1_\infty(\mathbb R^+;W_2^{2+k}(\Omega))$, $k=1,2,…$; $\Omega\subset\mathbb R^3$. Bibliography: 25 titles.