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.

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A two species interacting system motivated by the density functional theory for triblock copolymers contains long range interaction that affects the two species differently. In a two species periodic assembly of discs, the two species appear alternately on a lattice. The minimal two species periodic assembly is the one with the least energy per lattice cell area. There is a parameter $b$ in $[0,1]$ and the type of the lattice associated with the minimal assembly varies depending on $b$. There are several threshold defined by a number $B=0.1867...$ If $b \in [0, B)$, the minimal assembly is associated with a rectangular lattice whose ratio of the longer side and the shorter side is in $[\sqrt{3}, 1)$; if $b \in [B, 1-B]$, the minimal assembly is associated with a square lattice; if $b \in (1-B, 1]$, the minimal assembly is associated with a rhombic lattice with an acute angle in $[\frac{\pi}{3}, \frac{\pi}{2})$. Only when $b=1$, this rhombic lattice is the hexagonal lattice. None of the other values of $b$ yields the hexagonal lattice, a sharp contrast to the situation for one species interacting systems, where the hexagonal lattice is ubiquitously observed.

This paper investigates infinite-time spreading and finite-time blow-up for the Keller-Segel system. For 0 < m ≤ 2 − 2 / d, the L p space for both dynamic and steady solutions are detected with ${p:=\frac{d(2-m)}{2} }$ . Firstly, the global existence of the weak solution is proved for small initial data in L p . Moreover, when m > 1 − 2 / d, the weak solution preserves mass and satisfies the hyper-contractive estimates in L q for any p < q < ∞. Furthermore, for slow diffusion 1 < m ≤ 2 − 2/d, this weak solution is also a weak entropy solution which blows up at finite time provided by the initial negative free energy. For m > 2 − 2/d, the hyper-contractive estimates are also obtained. Finally, we focus on the L p norm of the steady solutions, it is shown that the energy critical exponent m = 2d/(d + 2) is the critical exponent separating finite L p norm and infinite L p norm for the steady state solutions.

In this paper, we study entire solutions of the Allen-Chan equation in one-dimensional Euclidean space. This equation is a scalar reaction-diffusion equation with a bistable nonlinearity. It is well-known that this equation admits three different types of traveling fronts connecting two of its three constant states. Under certain conditions on the wave speeds, the existence of entire solutions with merging these three traveling fronts is shown by constructing a suitable pair of super-sub-solutions.