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Despite longstanding excitement and progress toward understanding liquid–liquid phase separation in natural and artificial membranes, fundamental questions have persisted about which molecules are required for this phenomenon. Except in extraordinary circumstances, the smallest number of components that has produced large-scale, liquid–liquid phase separation in bilayers has stubbornly remained at three: a sterol, a phospholipid with ordered chains, and a phospholipid with disordered chains. This requirement of three components is puzzling because only two components are required for liquid–liquid phase separation in lipid monolayers, which resemble half of a bilayer. Inspired by reports that sterols interact closely with lipids with ordered chains, we tested whether phase separation would occur in bilayers in which a sterol and lipid were replaced by a single, joined sterol–lipid. By evaluating a panel of sterol–lipids, some of which are present in bacteria, we found a minimal bilayer of only two components (PChemsPC and diPhyPC) that robustly demixes into micron-scale, liquid phases. It suggests an additional role for sterol–lipids in nature, and it reveals a membrane in which tie-lines (and, therefore, the lipid composition of each phase) are straightforward to determine and will be consistent across multiple laboratories. 

We study the Muskat problem for one fluid in an arbitrary dimension, bounded below by a flat bed and above by a free boundary given as a graph. In addition to a fixed uniform gravitational field, the fluid is acted upon by a generic force field in the bulk and an external pressure on the free boundary, both of which are posited to be in traveling wave form. We prove that, for sufficiently small force and pressure data in Sobolev spaces, there exists a locally unique traveling wave solution in Sobolev-type spaces. The free boundary of the traveling wave solutions is either periodic or asymptotically flat at spatial infinity. Moreover, we prove that small periodic traveling wave solutions induced by external pressure only are asymptotically stable. These results provide the first class of nontrivial stable solutions for the problem. 

Abstract The free boundary problem for a two‐dimensional fluid permeating a porous medium is studied. This is known as the one‐phase Muskat problem and is mathematically equivalent to the vertical Hele‐Shaw problem driven by gravity force. We prove that if the initial free boundary is the graph of a periodic Lipschitz function, then there exists a global‐in‐time Lipschitz solution in the strong sense and it is the unique viscosity solution. The proof requires quantitative estimates for layer potentials and pointwise elliptic regularity in Lipschitz domains. This is the first construction of unique global strong solutions for the Muskat problem with initial data of arbitrary size. 

We study two-dimensional Rayleigh–Bénard convection with Navier-slip, fixed temperature boundary conditions and establish bounds on the Nusselt number. As the slip-length varies with Rayleigh number R a , this estimate interpolates between the Whitehead–Doering bound by R a 5 12 for free-slip conditions (Whitehead & Doering. 2011 Ultimate state of two-dimensional Rayleigh–Bénard convection between free-slip fixed-temperature boundaries. Phys. Rev. Lett. 106 , 244501) and the classical Doering–Constantin R a 1 2 bound (Doering & Constantin. 1996 Variational bounds on energy dissipation in incompressible flows. III. Convection. Phys. Rev. E 53 , 5957–5981). This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 1)’.