  In this work, we are interested in the small time local null controllability
for the viscous Burgers' equation $y_t - y_{xx} + y y_x = u(t)$ on the line
segment $[0,1]$, with null boundary conditions. The second-hand side is a
scalar control playing a role similar to that of a pressure. In this setting,
the classical Lie bracket necessary condition $[f_1,[f_1,f_0]]$ introduced by
Sussmann fails to conclude. However, using a quadratic expansion of our system,
we exhibit a second order obstruction to small time local null controllability.
This obstruction holds although the information propagation speed is infinite
for the Burgers equation. Our obstruction involves the weak $H^{-5/4}$ norm of
the control $u$. The proof requires the careful derivation of an integral
kernel operator and the estimation of residues by means of weakly singular
integral operator estimates.
