Possibility of Global Solutions for Variant Models of Viscous Flow on Unbounded Domains.

摘要

The authors have established global nonexistence backward in time for solutions on bounded regions of R-cubed of the three alternate models for viscous fluid flow suggested by Ladyzhenskaya and the same problem forward in time for a fluid of third grade when the normal stress coefficient alpha is negative. Exponential growth in time for a fluid of second and third grade under the same conditions on alpha, was previously shown by Dunn and Fosdick and by Fosdick and Rajagopal. In this paer we derive a similar result for the solution of the first model equation of Ladyzhenskaya in all of R to the Nth power, n = 2,3. For the other two models it is shown that if a solution exists in R to the nth power (n=2,3) from all previous time then the limit of the L2 integral of the solution as t approaches - infinity can be bounded below in terms of the data at t = O. In Section 5 a related nonstandard exterior initial=boundary value problem in R-sq is shown to have no global solutions. Finally, in Section 6 we show that the L2 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-cubed or an exterior region in R-cubed. Keywords: Reprints.