  In this work, we are interested in the small time global null controllability
for the viscous Burgers' equation y_t - y_xx + y y_x = u(t) on the line segment
[0,1]. The second-hand side is a scalar control playing a role similar to that
of a pressure. We set y(t,1) = 0 and restrict ourselves to using only two
controls (namely the interior one u(t) and the boundary one y(t,0)). In this
setting, we show that small time global null controllability still holds by
taking advantage of both hyperbolic and parabolic behaviors of our system. We
use the Cole-Hopf transform and Fourier series to derive precise estimates for
the creation and the dissipation of a boundary layer.
