D. Pelinovsky

Enstrophy growth in the viscous Burgers equation

Dynamics of PDEs 9, 305-340 (2012)

We study bounds on the enstrophy growth for solutions of the viscous Burgers equation on the unit circle. Using the variational formulation of Lu and Doering, we prove that the maximizer of the enstrophy’s rate of change is sharp in the limit of large enstrophy up to a numerical constant but does not saturate the PoincarŽe inequality for mean-zero 1-periodic functions. Using the dynamical system methods, we give an asymptotic representation of the maximizer in the limit of large enstrophy as a viscous shock on the background of a linear rarefactive wave. This asymptotic construction is used to prove that a larger growth of enstrophy can be achieved when the initial data to the viscous Burgers equation saturates the Poincare inequality up to a numerical constant.

An exact solution of the Burgers equation is constructed to describe formation of a metastable viscous shock on the background of a linear rarefactive wave. When we consider the Burgers equation on an infinite line subject to the nonzero (shock-type) boundary conditions, we prove that the maximum enstrophy achieved in the time evolution is scaled as E3/2, where E is the large initial enstrophy, whereas the time needed for reaching the maximal enstrophy is scaled as E-1/2 log(E). Similar but slower rates hold on the unit circle.

Viscous Burgers equation, enstrophy, shocks, Cole-Hopf transformation, heat equation.