A note on deformations of 2D fluid motions using 3D Born-Infeld equations

Classical fluid motions can be described either by time dependent diffeomorphisms (Lagrangian description) or by the corresponding generating vector fields (Eulerian description). We are interested in interpolating two such fluid motions. For both analytic and numerical purposes, we would like the corresponding interpolating vector fields to evolve according to some hyperbolic evolution PDEs. In the 2D case, it turns out that a convenient set of such PDEs can be deduced from the Born-Infeld (BI) theory of Electromagnetism. The 3D BI equations were originally designed as a nonlinear correction to the linear Maxwell equations allowing finite electric fields for point charges. They depend on a parameter A. As lambda goes to infinity, the classical Maxwell equations are recovered. It turns out that in the opposite case, as lambda goes to zero, the BI equations provide a solution to our problem. These equations, as lambda = 0, also describe classical strings (extremal surfaces) evolving in the 4D Minkowski space-time. In their static version, they can be interpreted in the framework of optimal transportation theory.