The Euler–Korteweg system consists of a quasi-linear, dispersive perturbation of the Euler equations. The Cauchy problem has been studied in any dimension d ≥ 1 by Benzoni, Danchin, and Descombes, who obtained local well-posedness results when the velocity is in Hs for s > d/2 + 1. They noticed that one may expect to find some smoothing effect due to the dispersive effects, but there was no pro...

We study the smoothing effect of averaging over spheres for solutions of kinetic transport equations in hyperbolic Sobolev spaces.

In optical flow estimation, an additional constraint to the constant brightness assumption is required to uniquely determine both components of the flow. Typically, these constraints impose a smoothness requirement on the flow estimate. Since the smoothness constraint may be inconsistent with the brightness constraint, a smoothing parameter is introduced to control the tradeoff between satisfyi...