On Euler's equation and 'EPDiff'

Peter W. Michor, David Mumford
2013 Journal of Geometric Mechanics (JGM)  
We study a family of approximations to Euler's equation depending on two parameters ε, η ≥ 0. When ε = η = 0 we have Euler's equation and when both are positive we have instances of the class of integro-differential equations called EPDiff in imaging science. These are all geodesic equations on either the full diffeomorphism group Diff H ∞ (R n ) or, if ε = 0, its volume preserving subgroup. They are defined by the right invariant metric induced by the norm on vector fields given by . All
more » ... iven by . All geodesic equations are locally well-posed, and the Lε,η-equation admits solutions for all time if η > 0 and p ≥ (n + 3)/2. We tie together solutions of all these equations by estimates which, however, are only local in time. This approach leads to a new notion of momentum which is transported by the flow and serves as a generalization of vorticity. We also discuss how delta distribution momenta lead to"vortexsolitons", also called "landmarks" in imaging science, and to new numeric approximations to fluids. 2010 Mathematics Subject Classification. Primary 35Q31, 58B20, 58D05.
doi:10.3934/jgm.2013.5.319 fatcat:frliqsmjpvcvrhe6a7fcggk4ya