Exploding solutions of the complex two-dimensional Burgers equations: Computer simulations

We study by computer simulations the complex solutions of the two-dimensional Burgers equations in the whole plane in absence of external forces. For such model the existence of singularities, corresponding to a divergence of the total energy at a finite time, is proved by Li and Sinai “Singularities of complex-valued solutions of the two-dimensional Burgers system,” J. Math. Phys. 51, 015205 (2010)10.1063/1.3276099 for a large class of initial data. The simulations show that the blow-up takes place in a very short time, of the order of 10−5 time units. Moreover near the blow-up time the support of the solution in Fourier space moves out to infinity along a straight line. In x-space the solutions are concentrated in a finite region, with large space derivatives, as one would expect for physical phenomena such as tornadoes.