In the small diffusion limit ε→0, metastable
dynamics is studied for the generalized Burgers problemformula hereHere u=u(x, t) and f(u)
is smooth, convex, and satisfies f(0)=f′(0)=0.
The choice
f(u)=u2/2 has been
shown previously to arise in connection with the physical problem of
upward flame-front propagation in a vertical channel in a particular parameter
regime. In this
context, the shape y=y(x, t)
of the flame-front interface satisfies u=−yx.
For this problem, it is shown that the principal eigenvalue associated
with the
linearization around an equilibrium solution corresponding
to a parabolic-shaped flame-front interface is exponentially small.
This exponentially small eigenvalue then leads to a metastable behaviour
for the time-
dependent problem. This behaviour is studied quantitatively by deriving
an asymptotic
ordinary differential equation characterizing the slow motion of the tip
location of a
parabolic-shaped interface. Similar metastability results are
obtained for more general f(u). These
asymptotic results are shown to compare very favourably with full numerical
computations.
Flame Front Propagation in an Optical GDI Engine under Stoichiometric and Lean Burn Conditions
