On the possibility of global solutions for variant models of viscous flow on unbounded domains

Abstract

In previous papers [1, 2], the authors established global nonexistence backward in time for solutions on bounded regions of R 3 of the three alternate models for viscous fluid flow suggested by Ladyzhenskaya [3] and for the same problem forward in time for a fluid of third grade when the normal stress coefficient α 1 is negative. Exponential growth in time for a fluid of second and of third grade under the same conditions on α 1 was previously shown by Dunn and Fosdick [4] and by Fosdick and Rajagopal [5]. In this paper we derive a similar result for the solution of the first model equation of Ladyzhenskaya in all of R n, n = 2, 3. For the other two models it is shown that if a solution exists in R n ( n = 2, 3) from all previous time then the limit of the L 2 integral of the solution as t → − ∞ can be bounded below in terms of the data at t = 0. In Section 5 a related nonstandard exterior initial-boundary value problem in R 2 is shown to have no global solutions. Finally, in Section 6 we show that the L 2 integral of the velocity gradients for the second or third grade model is bounded below by a growing exponential function of time when the region is either all of R 3 or an exterior region in R 3.