Bulletin of Mathematical Sciences

In this short survey, we describe some recent developments on the modeling of propagation by reaction-differential equations with free boundaries, which involve local as well as nonlocal diffusion. After the pioneering works of Fisher, Kolmogorov–Petrovski–Piskunov (KPP) and Skellam, the use of reaction–diffusion equations to model propagation and spreading speed has been widely accepted, with remarkable progresses achieved in several directions, notably on propagation in heterogeneous media, models for interacting species including epidemic spreading, and propagation in shifting environment caused by climate change, to mention but a few. Such models involving a free boundary to represent the spreading front have been studied only recently, but fast progress has been made. Here, we will concentrate on these free boundary models, starting with those where spatial dispersal is represented by local diffusion. These include the Fisher–KPP model with free boundary and related problems, where both the one space dimension and high space dimension cases will be examined; they also include some two species population models with free boundaries, where we will show how the long-time dynamics of some competition models can be fully determined. We then consider the nonlocal Fisher–KPP model with free boundary, where the diffusion operator $\mathrm{\Delta }u$ is replaced by a nonlocal one involving a kernel function. We will show how a new phenomenon, known as accelerated spreading, can happen to such a model. After that, we will look at some epidemic models with nonlocal diffusion and free boundaries, and show how the long-time dynamics can be rather fully described. Some remarks and comments are made at the end of each section, where related problems and open questions will be briefly discussed.

We are concerned with free boundary problems arising from the analysis of multidimensional transonic shock waves for the Euler equations in compressible fluid dynamics. In this expository paper, we survey some recent developments in the analysis of multidimensional transonic shock waves and corresponding free boundary problems for the compressible Euler equations and related nonlinear partial differential equations (PDEs) of mixed type. The nonlinear PDEs under our analysis include the steady Euler equations for potential flow, the steady full Euler equations, the unsteady Euler equations for potential flow, and related nonlinear PDEs of mixed elliptic–hyperbolic type. The transonic shock problems include the problem of steady transonic flow past solid wedges, the von Neumann problem for shock reflection–diffraction, and the Prandtl–Meyer problem for unsteady supersonic flow onto solid wedges. We first show how these longstanding multidimensional transonic shock problems can be formulated as free boundary problems for the compressible Euler equations and related nonlinear PDEs of mixed type. Then we present an effective nonlinear method and related ideas and techniques to solve these free boundary problems. The method, ideas, and techniques should be useful to analyze other longstanding and newly emerging free boundary problems for nonlinear PDEs.

Let $𝔛$ be a class of finite groups closed under taking subgroups, homomorphic images and extensions. It is known that if $A$ is a normal subgroup of a finite group $G$ then the image of an $𝔛$-maximal subgroup $H$ of $G$ in $G/A$ is not, in general, $𝔛$-maximal in $G/A$. We say that the reduction $𝔛$-theorem holds for a finite group $A$ if, for every finite group $G$ that is an extension of $A$ (i.e. contains $A$ as a normal subgroup), the number of conjugacy classes of $𝔛$-maximal subgroups in $G$ and $G/A$ is the same. The reduction $𝔛$-theorem for $A$ implies that $HA/A$ is $𝔛$-maximal in $G/A$ for every extension $G$ of $A$ and every $𝔛$-maximal subgroup $H$ of $G$. In this paper, we prove that the reduction $𝔛$-theorem holds for $A$ if and only if all $𝔛$-maximal subgroups of $A$ are conjugate in $A$ and classify the finite groups with this property in terms of composition factors.

It is shown that the Baer–Kaplansky Theorem can be extended to all abelian groups provided that the rings of endomorphisms of groups are replaced by trusses of endomorphisms of corresponding heaps. That is, every abelian group is determined up to isomorphism by its endomorphism truss and every isomorphism between two endomorphism trusses associated to some abelian groups $G$ and $H$ is induced by an isomorphism between $G$ and $H$ and an element from $H$. This correspondence is then extended to all modules over a ring by considering heaps of modules. It is proved that the truss of endomorphisms of a heap associated to a module $M$ determines $M$ as a module over its endomorphism ring.

This paper is concerned with the zero-viscosity limit of the three-dimensional (3D) incompressible stationary magnetohydrodynamics (MHD) equations in the 3D unbounded domain $\mathrm{\Omega }:={ℝ}^{+}\times {ℝ}^2$. The main result of this paper establishes that the solution of 3D incompressible stationary MHD equations converges to the solution of the 3D incompressible stationary Euler equations as the viscosity coefficient goes to zero.