Prof J D Gibbon, Mathematics Department
Recent preprints and papers
One of the most interesting & unexplained phenomenon in 3D incompressible Navier-Stokes isotropic turbulence is the problem of regularity of solutions for arbitrarily long times. As one of the major unsolved problems of modern applied mathematics it is listed as on of the 7 AMS-Clay million-dollar Prize problems. This issue is intimately related to the problem of dissipation-range intermittency in solutions of the Navier-Stokes equations at high Reynolds numbers. Vorticity has the tendency to accumulate on "thin sets" in the form of quasi one-dimensional tubes & quasi two-dimensional sheets; these structures dominate the local vortical topology. This phenomenon is closely related to the regularity issue because the direction of vorticity is one of the factors that determines the size of the vortex stretching term. It is our inability to bound this term in a satisfactory fashion that leads to our failure to prove that the 3D Navier-Stokes equations are regular (see Doering & Gibbon Applied Analysis of the Navier-Stokes equations CUP '95). Some of those in the list of newer papers above together with some older ones below address these issues.
Many active quantities cluster into concentrated sets, like bubbles in a Swiss cheese. Vorticity in a fluid is just one example; the number of examples in physics and biology is manifold. The paper below, co-authored with Edriss Titi of Israel's Weizmann Institute, shows that competition between members of a hierarchy of length scales in complex multi-scale systems gives generically induces clustering that dominates the intermittent structure. The clusters have halo-like surfaces that have scaling exponents lower than that of their kernels, which can be as high as the domain dimension.
Work on Quaternions and particle dynamics in the Euler fluid equations below hints that the three-dimensional Euler equations has a geometric structure. Some of those in the list of newer papers above together with some older ones below address these issues.
Another interesting problem is that of the behaviour of solutions of the 3D incompressible Euler equations (and related equations) where there is evidence to suggest that finite energy solutions may become singular in a finite time (at least with well-prepared initial data), but no proof is known and little is known about the process. The following papers discuss the infinite energy blow-up process. The original paper in this series is the last.