Turing conditions for pattern forming systems on evolving manifolds

The study of pattern-forming instabilities in reaction-diffusion systems on growing or otherwise time-dependent domains arises in a variety of settings, including applications in developmental biology, spatial ecology, and experimental chemistry. Analyzing such instabilities is complicated, as there is a strong dependence of any spatially homogeneous base states on time, and the resulting structure of the linearized perturbations used to determine the onset of instability is inherently non-autonomous. We obtain general conditions for the onset and structure of diffusion driven instabilities in reaction-diffusion systems on domains which evolve in time, in terms of the time-evolution of the Laplace-Beltrami spectrum for the domain and functions which specify the domain evolution. Our results give sufficient conditions for diffusive instabilities phrased in terms of differential inequalities which are both versatile and straightforward to implement, despite the generality of the studied problem. These conditions generalize a large number of results known in the literature, such as the algebraic inequalities commonly used as a sufficient criterion for the Turing instability on static domains, and approximate asymptotic results valid for specific types of growth, or specific domains. We demonstrate our general Turing conditions on a variety of domains with different evolution laws, and in particular show how insight can be gained even when the domain changes rapidly in time, or when the homogeneous state is oscillatory, such as in the case of Turing-Hopf instabilities. Extensions to higher-order spatial systems are also included as a way of demonstrating the generality of the approach

Topological Sound and Flocking on Curved Surfaces

Active systems on curved geometries are ubiquitous in the living world. In the presence of curvature orientationally ordered polar flocks are forced to be inhomogeneous, often requiring the presence of topological defects even in the steady state due to the constraints imposed by the topology of the underlying surface. In the presence of spontaneous flow the system additionally supports long-wavelength propagating sound modes which get gapped by the curvature of the underlying substrate. We analytically compute the steady state profile of an active polar flock on a two-sphere and a catenoid, and show that curvature and active flow together result in symmetry protected topological modes that get localized to special geodesics on the surface (the equator or the neck respectively). These modes are the analogue of edge states in electronic quantum Hall systems and provide unidirectional channels for information transport in the flock, robust against disorder and backscattering.Comment: 15 pages, 6 figure

On Existence of L1L^1-solutions for Coupled Boltzmann Transport Equation and Radiation Therapy Treatment Optimization

The paper considers a linear system of Boltzmann transport equations modelling the evolution of three species of particles, photons, electrons and positrons. The system is coupled because of the collision term (an integral operator). The model is intended especially for dose calculation (forward problem) in radiation therapy. It, however, does not apply to all relevant interactions in its present form. We show under physically relevant assumptions that the system has a unique solution in appropriate ($L^1$-based) spaces and that the solution is non-negative when the data (internal source and inflow boundary source) is non-negative. In order to be self-contained as much as is practically possible, many (basic) results and proofs have been reproduced in the paper. Existence, uniqueness and non-negativity of solutions for the related time-dependent coupled system are also proven. Moreover, we deal with inverse radiation treatment planning problem (inverse problem) as an optimal control problem both for external and internal therapy (in general $L^p$-spaces). Especially, in the case $p=2$ variational equations for an optimal control related to an appropriate differentiable convex object function are verified. Its solution can be used as an initial point for an actual (global) optimization.Comment: Corrected typos. Added a new section 3. Revised the argument of Example 7.