The three-dimensional Euler fluid equations: where do we stand?

John David Gibbon, Department of Mathematics, Imperial College, London

Despite their deceptive simplicity, the three-dimensional Euler fluid equations remain one of the most puzzling sets of equations in mathematical physics. While it has long been suspected that they develop a singularity in finite time, only limited rigorous results exist (such as the BKM theorem) and numerical evidence is contradictory. I will survey some of the results in this area but confine myself to remarks on the incompressible case. I will then move on to considering the motion of fluid particles governed by the Euler equations as an example of particle dynamics in a Lagrangian flow. Hamilton's quaternions are a unifying theme in this area as they are now widely used in the aero/astro and computer animation industries to understand the motion of rapidly moving objects undergoing three-axis rotations. Thus, other problems closely related to the Euler fluid equations, such as ideal MHD, barotropic compressible Euler and mixing can all be considered together.

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