We consider complex-valued solutions of the n-dimensional Burgers' system, $n \gt 1$. We show that there exists an open set in the space of $n^2 + 5n - 2/2$-parameter families of initial conditions such that for each family from this set there are values of parameters for which the solution develops blow up in finite time.

We survey recent advances in the analysis of the large data global (and asymptotic) behaviour of nonlinear dispersive equations such as the nonlinear wave (NLW), nonlinear Schrödinger (NLS), wave maps (WM), Schrödinger maps (SM), generalised Korteweg-de Vries (gKdV), Maxwell-Klein-Gordon (MKG), and Yang-Mills (YM) equations. The classification of the nonlinearity as subcritical (weaker than the linear dispersion at high frequencies), critical (comparable to the linear dispersion at all frequencies), or supercritical (stronger than the linear dispersion at high frequencies) is fundamental to this analysis, and much of the recent progress has pivoted on the case when there is a critical conservation law. We discuss how one synthesises a satisfactory critical (scale-invariant) global theory, starting the basic building blocks of perturbative analysis, conservation laws, and monotonicity formulae, but also incorporating more advanced (and recent) tools such as gauge transforms, concentration-compactness, and induction on energy.