The article presents a survey of mathematical problems, techniques and challenges arising in thermoacoustic tomography and its sibling photoacoustic tomography.

It is generally believed that in order to solve initial value problems using Lie symmetry methods, the initial condition needs to be left invariant by the infinitesimal symmetry generator that admits the invariant solution. This is not so. In this paper we incorporate the imposed initial value as a side condition to find ‘infinitesimals’ from which solutions satisfying the initial value can be recovered, along with the corresponding symmetry generator.

Dense, dry granular avalanches are very efficient at sorting the larger particles towards the free surface of the flow, and finer grains towards the base, through the combined processes of kinetic sieving and squeeze expulsion. This generates an inversely graded particle-size distribution, which is fundamental to a variety of pattern formation mechanisms, as well as subtle size-mobility feedback effects, leading to the formation of coarse-grained lateral levees that create channels in geophysical flows, enhancing their run-out. In this paper we investigate some of the properties of a recent model [Gray, J. M. N. T. & Thornton, A. R. (2005) A theory for particle size segregation in shallow granular free-surface flows. Proc. R. Soc. 461, 1447–1473]; [Thornton, A. R., Gray, J. M. N. T. & Hogg, A. J. (2006) A three-phase mixture theory for particle size segregation in shallow granular free-surface flows. J. Fluid. Mech. 550, 1–25] for the segregation of particles of two sizes but the same density in a shear flow typical of shallow avalanches. The model is a scalar conservation law in space and time, for the volume fraction of smaller particles, with non-constant coefficients depending on depth within the avalanche. It is proved that for steady flow from an inlet, complete segregation occurs beyond a certain finite distance down the slope, no matter what the mixture at the inlet. In time-dependent flow, dynamic shock waves can develop; they are interfaces separating different mixes of particles. Shock waves are shown to be stable if and only if there is a greater concentration of large particles above the interface than below. Constructions with shocks and rarefaction waves are demonstrated on a pair of physically relevant initial boundary value problems, in which a region of all small particles is penetrated from the inlet by either a uniform mixture of particles or by a layer of small particles over a layer of large particles. In both cases, and under a linear shear flow, solutions are constructed for all time and shown to have similar structure for all choices of parameters.

We study the coupled surface and grain boundary motion in a bi-crystal in the context of the ‘quarter loop’ geometry. Two types of normal curve velocities are involved in this model: motion by mean curvature and motion by surface diffusion. Three curves meet at points where junction conditions are given. A formulation that describes the coupled normal motion of the curves and preserves arc length parametrisation up to scaling is proposed. The formulation is shown to be well-posed in a simple, linear setting. Equations and junction conditions are approximated by finite difference methods. Numerical convergence to exact travelling wave solutions is shown. The method is applied to other problems of physical interest.

We review simple models of oil reservoirs and suggest some ideas for theoretical and numerical study of this important inverse problem. These models are formed by a system of an elliptic and a parabolic (or first-order hyperbolic) quasilinear partial differential equations. There are and probably there will be serious theoretical and computational difficulties mainly due to the degeneracy of the system. The practical value of the problem justifies efforts to improve the methods for its solution. We formulate ‘history matching’ as a problem of identification of two coefficients of this system. We consider global and local versions of this inverse problem and propose some approaches, including the use of the inverse conductivity problem and the structure of fundamental solutions. The global approach looks for properties of the ground in the whole domain, while the local one is aimed at recovery of these properties near wells. We discuss the use of the model proposed by Muskat which is a difficult free boundary problem. The inverse Muskat problem combines features of inverse elliptic and hyperbolic problems. We analyse its linearisation about a simple solution and show uniqueness and exponential instability for the linearisation.

In this paper we consider a 2π-periodic and two-dimensional Hele-Shaw flow describing the motion of a viscous, incompressible fluid. The free surface is moving under the influence of surface tension and gravity. The motion of the fluid is modelled using a modified version of Darcy's law for Stokesian fluids. The bottom of the cell is assumed to be impermeable. We prove the existence of a unique classical solution for a domain which is a small perturbation of a cylinder. Moreover, we identify the equilibria of the flow and study their stability.

Linear stability theory is developed for an activator–inhibitor model where fractional derivative operators of generally different exponents act both on diffusion and reaction terms. It is shown that in the short wave limit the growth rate is a power law of the wave number with decoupled time scales for distinct anomaly exponents of the different species. With equal anomaly exponents an exact formula for the anomalous critical value of reactants diffusion coefficients' ratio is obtained.

A steady-state bubble solution to the constrained mass conserving Allen–Cahn equation in a two-dimensional domain is constructed in the limit of small diffusivity. The solution is asymptotically constant inside a circle of radius rb centred at some unknown location x0 and has a sharp interface at the bubble radius that allows for a transition to a different asymptotically constant state outside the bubble. In a study by M. J. Ward (Metastable bubble solutions for the Allen–Cahn equation with mass conservation. SIAM J. Appl. Math. 56, 1996, 247–1279), the bubble centre was determined by a limiting solvability condition. The solution found by Ward suggests the existence of a corner type boundary layer where the normal derivative of the bubble solution readjusts to satisfy the no-flux condition at the boundary of the domain. This work is concerned with the details of the readjustment. A variational approach similar to the one of W. L. Kath, C. Knessl and B. J. Matkowsky (A variational approach to nonlinear singularly perturbed boundary-value problems. Stud. Appl. Math. 77, 1987, 61–88) shows the formation of a corner layer (for the derivative of the solution) which influences as a high-order correction the available determination of the bubble centre. This corner layer describes to leading order the readjustment of the level lines of the bubble to lines parallel to the boundary of the container; moreover, it provides to leading order a smooth solution across the corner layer.

In this paper, we study basic properties of the diffusive wave approximation of the shallow water equations (DSW). This equation is a doubly non-linear diffusion equation arising in shallow water flow models. It has been used as a model to simulate water flow driven mainly by gravitational forces and dominated by shear stress, that is, under uniform and fully developed turbulent flow conditions. The aim of this work is to present a survey of relevant results coming from the studies of doubly non-linear diffusion equations that can be applied to the DSW equation when topographic effects are ignored. In fact, we present proofs of the most relevant results existing in the literature using constructive techniques that directly lead to the implementation of numerical algorithms to obtain approximate solutions.