S. Engelberg, An analytical proof of the linear stability of the viscous shock profile of the burgers equation with fourth-order viscosity, SIAM J MATH, 30(4), 1999, pp. 927-936
In this paper we establish the exponential decay of solutions of the equati
on
u(t) + phi(x)u(x) = -partial derivative(x)(4)u
in an exponentially weighted norm. Here phi(x) is the viscous shock profile
corresponding to the Burgers equation with fourth-order viscosity:
u(t) + uu(x) = -partial derivative(x)(4)u.
Because of the fact that the profile is not monotone, showing the stability
is nontrivial. We extend the techniques of Koppel and Howard (Adv. Math. 1
8 (1975), pp. 306-358), techniques that they employ to prove the existence
of the viscous shock profile, and we use the techniques to prove the stabil
ity of the viscous shock profile. We have previously shown that the viscous
shock profile is a stable solution in an exponentially weighted norm by ma
king use of numerical results. The main advantage of our current method is
that it is analytical. One sees more clearly what properties of the viscous
shock profile cause it to be a stable solution of the PDE.