We present a mathematical model describing the distribution of monomer and micellar surfactant in a steady straining flow beneath a fixed free surface. The model includes adsorption of monomer surfactant at the surface and a single-step reaction whereby $n$ monomer molecules combine to form each micelle. The equations are analysed asymptotically and numerically and the results are compared with experiments. Previous studies of such systems have often assumed equilibrium between the monomer and micellar phases, i.e. that the reaction rate is effectively infinite. Our analysis shows that such an approach inevitably fails under certain physical conditions and also cannot accurately match some experimental results. Our theory provides an improved fit with experiments and allows the reaction rates to be estimated.

We prove the uniqueness for the inverse problem of determining a coefficient $q(x)$ in $\partial _t^2 u(x,t) = \uDelta u(x,t) - q(x)u(x,t)$ for $x \in R^n$ and $t > 0$, from observations of $u\vert_{\Gamma\times(0,T)}$ and the normal derivative $\frac{\partial u}{\partial \nu}\vert_{\Gamma\times(0,T)}$ where $\Gamma$ is an arbitrary $C^{\infty}$-hypersurface. Our main result asserts the uniqueness of $q$ over $R^n$ provided that $T > 0$ is sufficiently large and $q$ is analytic near $\Gamma$ and outside a ball. The proof depends on Fritz John's global Holmgren theorem and the uniqueness by a Carleman estimate.

Movement of biological organisms is frequently initiated in response to a diffusible or otherwise transported signal, and in its simplest form this movement can be described by a diffusion equation with an advection term. In systems in which the signal is localized in space the question arises as to whether aggregation of a population of indirectly-interacting organisms, or localization of a single organism, is possible under suitable hypotheses on the transition rules and the production of a control species that modulates the transition rates. It has been shown [25] that continuum approximations of reinforced random walks show aggregation and even blowup, but the connections between solutions of the continuum equations and of the master equation for the corresponding lattice walk were not studied. Using variational techniques and the existence of a Lyapunov functional, we study these connections here for certain simplified versions of the model studied earlier. This is done by relating knowledge about the shape of the minimizers of a variational problem to the asymptotic spatial structure of the solution.

The steady-state viscous quantum hydrodynamic model in one space dimension is studied. The model consists of the continuity equations for the particle and current densities, coupled to the Poisson equation for the electrostatic potential. The equations are derived from a Wigner–Fokker–Planck model and they contain a third-order quantum correction term and second-order viscous terms. The existence of classical solutions is proved for “weakly supersonic” quantum flows. This means that a smallness condition on the particle velocity is still needed but the bound is allowed to be larger than for classical subsonic flows. Furthermore, the uniqueness of solutions and various asymptotic limits (semiclassical and inviscid limits) are investigated. The proofs are based on a reformulation of the problem as a fourth-order elliptic equation by using an exponential variable transformation.

We prove short-time well-posedness of a Hele–Shaw system with two fluids and no surface tension (this is also known as the Muskat problem). We restrict our attention here to the stable case of the problem. That is, in order for the motion to be well-posed, the initial data must satisfy a sign condition which is a generalization of a condition of Saffman and Taylor. This sign condition essentially means that the more viscous fluid must displace the less viscous fluid. The proof uses the formulation introduced in the numerical work of Hou, Lowengrub, and Shelley, and relies on energy methods.