Abstract : Many basic PDE of physical interest, such as the three-dimensional Navier-Stokes equations, are "supercritical" in that the known conserved or bounded quantities for these equations allow the nonlinear components of the PDE to dominate the linear ones at fine scales. Because of this, almost none of the known methods for establishing global regularity for such equations can work, and global regularity for Navier-Stokes in particular is a notorious open problem. We present here some ways to show that if one allows some modifications to these supercritical PDE, one can in fact construct solutions that blow up in finite time (while still obeying conservation laws such as conservation of energy). This does not directly impact the global regularity question for the unmodified equations, but it does rule out some potential approaches to establish such regularity. |