Existence and uniqueness of classical solutions of the equations of motion modified second-grade fluids and instability of the rest state (Q1576993)

The aim of this paper is to study the stability of the rest state for the considered fluids. The authors prove the instability of this rest state, in the Lyapunov sense, provided that the negative second normal stress modulus \(\alpha_1\), is large enough. Further, it is proved the existence and the uniqueness of classical solutions of the main problem \((12)-(14)\) using some tools from a paper of \textit{G. P. Galdi}, \textit{M. Grobbelaar-van Dalsen} and \textit{N. Sauer} [Arch. Ration. Mech. Anal. 124, 221-237 (1993; Zbl 0804.76003)] provided that \(\alpha_1>0.\) The authors show that if the initial data of the ``linear problem'', \(w_0\), and the involved domain are sufficiently smooth, there exists a classical solution for at least a limited time interval. Moreover, if \(w_0\) is restricted in size and a coefficient \(\gamma\) is large enough, then this solution exists for all times. Finally, the authors are interested in the case when \(\alpha_1<0\) and theu prove the local existence of the classical solution.