Information geometric regularization of the barotropic Euler equation

A key numerical difficulty in compressible fluid dynamics is the formation of shock waves in supersonic flows. Shock waves feature jump discontinuities in the velocity and density of the fluid and thus preclude the existence of classical solutions to the compressible Euler equations. Weak solutions are commonly defined by viscous regularization, but even small amounts of viscosity can substantially change the long-term behavior of the solution through excessive dissipation, motivating the search for inviscid regularizations.

We revisit this classical problem by interpreting the Euler equations as flows on diffeomorphism manifolds based on the pioneering work of Vladimir Arnold. From this perspective, the formation of shock waves results from the lack of geodesic completeness of the diffeomorphism manifold. We restore the geodesic completeness of the diffeomorphism manifold by embedding it into a suitable information geometry. By deriving the Euler equations on this modified geometry, we obtain a regularization that avoids the formation of singularities without adding viscosity.