Mathematical Analysis of a System for Biological Network Formation

Motivated by recent physics papers describing rules for natural network formation, we study an elliptic-parabolic system of partial differential equations proposed by Hu and Cai. The model describes the pressure field thanks to Darcy's type equation and the dynamics of the conductance network under pressure force effects with a diffusion rate $D$ representing randomness in the material structure. We prove the existence of global weak solutions and of local mild solutions and study their long term behaviour. It turns out that, by energy dissipation, steady states play a central role to understand the pattern capacity of the system. We show that for a large diffusion coefficient $D$, the zero steady state is stable. Patterns occur for small values of $D$ because the zero steady state is Turing unstable in this range; for $D=0$ we can exhibit a large class of dynamically stable (in the linearized sense) steady states

Well posedness and Maximum Entropy Approximation for the Dynamics of Quantitative Traits

We study the Fokker-Planck equation derived in the large system limit of the Markovian process describing the dynamics of quantitative traits. The Fokker-Planck equation is posed on a bounded domain and its transport and diffusion coefficients vanish on the domain's boundary. We first argue that, despite this degeneracy, the standard no-flux boundary condition is valid. We derive the weak formulation of the problem and prove the existence and uniqueness of its solutions by constructing the corresponding contraction semigroup on a suitable function space. Then, we prove that for the parameter regime with high enough mutation rate the problem exhibits a positive spectral gap, which implies exponential convergence to equilibrium. Next, we provide a simple derivation of the so-called Dynamic Maximum Entropy (DynMaxEnt) method for approximation of moments of the Fokker-Planck solution, which can be interpreted as a nonlinear Galerkin approximation. The limited applicability of the DynMaxEnt method inspires us to introduce its modified version that is valid for the whole range of admissible parameters. Finally, we present several numerical experiments to demonstrate the performance of both the original and modified DynMaxEnt methods. We observe that in the parameter regimes where both methods are valid, the modified one exhibits slightly better approximation properties compared to the original one.Comment: 28 pages, 4 tables, 5 figure

Decay to equilibrium for energy-reaction-diffusion systems

We derive thermodynamically consistent models of reaction-diffusion equations coupled to a heat equation. While the total energy is conserved, the total entropy serves as a driving functional such that the full coupled system is a gradient flow. The novelty of the approach is the Onsager structure, which is the dual form of a gradient system, and the formulation in terms of the densities and the internal energy. In these variables it is possible to assume that the entropy density is strictly concave such that there is a unique maximizer (thermodynamical equilibrium) given linear constraints on the total energy and suitable density constraints. We consider two particular systems of this type, namely, a diffusion-reaction bipolar energy transport system, and a drift-diffusion-reaction energy transport system with confining potential. We prove corresponding entropy-entropy production inequalities with explicitely calculable constants and establish the convergence to thermodynamical equilibrium, at first in entropy and further in $L^1$ using Cziszar-Kullback-Pinsker type inequalities.Comment: 40 page

Notes on a PDE System for Biological Network Formation

We present new analytical and numerical results for the elliptic-parabolic system of partial differential equations proposed by Hu and Cai, which models the formation of biological transport networks. The model describes the pressure field using a Darcy's type equation and the dynamics of the conductance network under pressure force effects. Randomness in the material structure is represented by a linear diffusion term and conductance relaxation by an algebraic decay term. The analytical part extends the results of Haskovec, Markowich and Perthame regarding the existence of weak and mild solutions to the whole range of meaningful relaxation exponents. Moreover, we prove finite time extinction or break-down of solutions in the spatially onedimensional setting for certain ranges of the relaxation exponent. We also construct stationary solutions for the case of vanishing diffusion and critical value of the relaxation exponent, using a variational formulation and a penalty method. The analytical part is complemented by extensive numerical simulations. We propose a discretization based on mixed finite elements and study the qualitative properties of network structures for various parameters values. Furthermore, we indicate numerically that some analytical results proved for the spatially one-dimensional setting are likely to be valid also in several space dimensions.Comment: 33 pages, 12 figure

On the XFEL Schroedinger Equation: Highly Oscillatory Magnetic Potentials and Time Averaging

We analyse a nonlinear Schr\"odinger equation for the time-evolution of the wave function of an electron beam, interacting selfconsistently through a Hartree-Fock nonlinearity and through the repulsive Coulomb interaction of an atomic nucleus. The electrons are supposed to move under the action of a time dependent, rapidly periodically oscillating electromagnetic potential. This can be considered a simplified effective single particle model for an X-ray Free Electron Laser (XFEL). We prove the existence and uniqueness for the Cauchy problem and the convergence of wave-functions to corresponding solutions of a Schr\"odinger equation with a time-averaged Coulomb potential in the high frequency limit for the oscillations of the electromagnetic potential