As V. I. Arnold observed in the 1960s, the Euler equations of incompressible fluid flow correspond formally to geodesic equations in a group of volume-preserving diffeomorphisms. Working in an Eulerian framework, we study incompressible flows of shapes as critical paths for action (kinetic energy) along transport paths constrained to be shape densities (characteristic functions). The formal geodesic equations for this problem are Euler equations for incompressible, inviscid potential flow of fluid with zero pressure and surface tension on the free boundary. The problem of minimizing this action exhibits an instability associated with microdroplet formation, with the following outcomes: Any two shapes of equal volume can be approximately connected by an Euler spray---a countable superposition of ellipsoidal geodesics. The infimum of the action is the Wasserstein distance squared, and is almost never attained except in dimension 1. Every Wasserstein geodesic between bounded densities of compact support provides a solution of the (compressible) pressureless Euler system that is a weak limit of (incompressible) Euler sprays. Each such Wasserstein geodesic is also the unique minimizer of a relaxed least-action principle for a two-fluid mixture theory corresponding to incompressible fluid mixed with vacuum.

We study convolution groups generated by completely monotone sequences and completely monotone functions. Using a convolution group, we define a fractional calculus for a certain class of distributions. When acting on causal functions, this definition agrees with the traditional Riemann-Liouville definition for t>0 but includes some singularities at t=0 so that the group property holds. Using this group, we are able to extend the definition of Caputo derivatives of order in (0,1) to a certain class of locally integrable functions without using the first derivative. The group property allows us to de-convolve the fractional differential equations to integral equations with completely monotone kernels, which then enables us to prove the general Gronwall inequality (or comparison principle) with the most general conditions. This then opens the door of a priori energy estimates of fractional PDEs. Some other fundamental results for fractional ODEs are also established within this frame under very weak conditions. Besides, we also obtain some interesting results about completely monotone sequences.

Abstract We study a class of fourth order nonlinear parabolic equations which include the thin-film equation and the quantum drift-diffusion model as special cases. We investigate these equations by first developing functional inequalities of the type $$ \int_\Omega u^{2\gamma-\alpha-\beta}\Delta\ua\Delta\ub dx \geq c\int_\Omega|\Delta \ur |^2dx, $$ which seem to be of interest on their own right.

A longstanding question in the dual Brunn–Minkowski theory is “What are the dual analogues of Federer’s curvature measures for convex bodies?” The answer to this is provided. This leads naturally to dual versions of Minkowski-type problems: What are necessary and sufficient conditions for a Borel measure to be a dual curvature measure of a convex body? Sufficient conditions, involving measure concentration, are established for the existence of solutions to these problems.

In this paper we are concerned with the existence of a weak solution to the initial boundary value problem for the equation ∂ t u=Δ(Δu) −3 . This problem arises in the mathematical modeling of the evolution of a crystal surface. Existence of a weak solution u with Δu≥0 is obtained via a suitable substitution. Our investigations reveal the close connection between this problem and the equation ∂ t ρ+ρ 2 Δ 2 ρ 3 =0 , another crystal surface model first proposed by H. Al Hajj Shehadeh, R. V. Kohn and J. Weare in [1].