Re. Grundy et R. Mclaughlin, GLOBAL BLOW-UP OF SEPARABLE SOLUTIONS OF THE VORTICITY EQUATION, IMA journal of applied mathematics, 59(3), 1997, pp. 287-307
In this paper we construct solutions to the equation () delta(3)u/del
ta t delta y(2) = epsilon delta(4)u/delta y(4) + delta(3)u/delta y(3)
- delta u delta(2)u/delta y delta y(2), epsilon > 0 on a finite interv
al in y which blow-up globally in finite time. This equation arises in
a number of physical situations and can be derived from the vorticity
equation by looking for stagnation-point type separable solutions for
the two-dimensional streamfunction of the form xu(y, t). In the parti
cular application which has prompted the investigation reported in thi
s paper, () is solved subject to boundary conditions involving delta(
2)u/delta y(2). For this type of boundary condition the phenomenon of
blow-up was first observed numerically by solving the initial-boundary
-value problem for (). These computations reveal that, depending on t
he parameter combinations chosen, the solution to the initial-value pr
oblem may either blow-up globally in finite time or approach a steady
state as t --> infinity. Using the computations as a guide we construc
t the analytic behaviour of the solution close to the blow-up time usi
ng the methods of formal asymptotics.