$c$-cyclical monotonicity is the most important optimality condition for an optimal transport plan. While the proof of necessity is relatively easy, the proof of sufficiency is often more difficult or even elusive. We present here a new approach, and we show how known results are derived in this new framework and how this approach allows to prove sufficiency in situations previously not treatable.
We propose a new fractional Laplacian for bounded domains, expressed as a conservation law and thus particularly suited to finite-volume schemes. Our approach permits the direct prescription of no-flux boundary conditions. We first show the well-posedness theory for the fractional heat equation. We also develop a numerical scheme, which correctly captures the action of the fractional Laplacian and its anomalous diffusion effect. We benchmark numerical solutions for the Lévy–Fokker–Planck equation against known analytical solutions. We conclude by numerically exploring properties of these equations with respect to their stationary states and long-time asymptotics.
We consider heat or mass transport from a circular cylinder under a uniform crossflow at small Reynolds numbers, 
$mathrm{Re}ll 1$. This problem has been thwarted in the past by limitations inherent in the classical analyses of the singular flow problem, which have used asymptotic expansions in inverse powers of 
$log mathrm{Re}$. We here make use of the hybrid approximation of Kropinski, Ward & Keller [(1995) SIAM J. Appl. Math. 55, 1484], based upon a robust asymptotic expansion in powers of 
$mathrm{Re}$. In that approximation, the “inner” streamfunction is provided by the product of a pre-factor 
$S$, a slowly varying function of 
$mathrm{Re}$, with a 
$mathrm{Re}$-independent “canonical” solution of a simple mathematical form. The pre-factor, in turn, is determined as an implicit function of 
$log mathrm{Re}$ via asymptotic matching with a numerical solution of the nonlinear single-scaled “outer” problem, where the cylinder appears as a point singularity. We exploit the hybrid approximation to analyse
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