  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 Poincar\'e 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 Poincar\'e inequality up to a numerical
constant.
  An exact self-similar 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
$\mathcal{E}^{3/2}$, where $\mathcal{E}$ is the large initial enstrophy,
whereas the time needed for reaching the maximal enstrophy is scaled as
$\mathcal{E}^{-1/2} \log(\mathcal{E})$. Similar but slower rates are proved on
the unit circle.
