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We study an elliptic-parabolic system of partial differential equations describing formation of biological network structures. The model takes into consideration the evolution of the permeability tensor under the influence of a diffusion term, representing randomness in the material structure, a decay term describing metabolic cost and a pressure force. A Darcy's law type equation describes the pressure field. In the spatially two-dimensional setting, we present a constructive, formal derivation of the PDE system from the discrete network formation model in the refinement limit of a sequence of unstructured triangulations. Moreover, we show that the PDE system is a formal $L^2$-gradient flow of an energy functional with biological interpretation, and study its convexity properties. For the case when the energy functional is convex, we construct unique global weak solutions of the PDE system. Finally, we construct steady state solutions in one- and multi-dimensional settings and discuss their stability properties., 17 pages, 1 figure

We provide an overview of the results on Hughes' model for pedestrian movements available in the literature. After the first successful approaches to solving a regularised version of the model, researchers focused on the structure of the Riemann problem, which led to local-in-time existence results for Riemann-type data and paved the way for a WFT (Wave-Front Tracking) approach to the solution semigroup. In parallel, a DPA (Deterministic Particles Approximation) approach was developed in the spirit of follow-the-leader approximation results for scalar conservation laws. Beyond having proved to be powerful analytical tools, the WFT and the DPA approaches also led to interesting numerical results. However, only existence theorems on very specific classes of initial data (essentially ruling out non-classical shocks) have been available until very recently. A proper existence result using a DPA approach was proven not long ago in the case of a linear coupling with the density in the eikonal equation. Shortly after, a similar result was proven via a fixed point approach. We provide a detailed statement of the aforementioned results and sketch the main proofs. We also provide a brief overview of results that are related to Hughes' model, such as the derivation of a dynamic version of the model via a mean-field game strategy, an alternative optimal control approach, and a localized version of the model. We also present the main numerical results within the WFT and DPA frameworks.

We study self-regulating processes modeling biological transportation networks. Firstly, we write the formal $L^2$-gradient flow for the symmetric tensor valued diffusivity $D$ of a broad class of entropy dissipations associated with a purely diffusive model. The introduction of a prescribed electric potential leads to the Fokker-Planck equation, for whose entropy dissipations we also investigate the formal $L^2$-gradient flow. We derive an integral formula for the second variation of the dissipation functional, proving convexity (in dependence of diffusivity tensor) for a quadratic entropy density modeling Joule heating. Finally, we couple in the Poisson equation for the electric potential obtaining the Poisson-Nernst-Planck system. The formal gradient flow of the associated entropy loss functional is derived, giving an evolution equation for $D$ coupled with two auxiliary elliptic PDEs., 16 pages

This paper presents a partial differential equation framework for deep residual neural networks and for the associated learning problem. This is done by carrying out the continuum limits of neural networks with respect to width and depth. We study the wellposedness, the large time solution behavior, and the characterization of the steady states of the forward problem. Several useful time-uniform estimates and stability/instability conditions are presented. We state and prove optimality conditions for the inverse deep learning problem, using standard variational calculus, the Hamilton-Jacobi-Bellmann equation and the Pontryagin maximum principle. This serves to establish a mathematical foundation for investigating the algorithmic and theoretical connections between neural networks, PDE theory, variational analysis, optimal control, and deep learning., Comment: 27. arXiv admin note: text overlap with arXiv:1807.01083 by other authors

We demonstrate the validity of Murray's law, which represents a scaling relation for branch conductivities in a transportation network, for discrete and continuum models of biological networks. We first consider discrete networks with general metabolic coefficient and multiple branching nodes and derive a generalization of the classical 3/4-law. Next we prove an analogue of the discrete Murray's law for the continuum system obtained in the continuum limit of the discrete model on a rectangular mesh. Finally, we consider a continuum model derived from phenomenological considerations and show the validity of the Murray's law for its linearly stable steady states.

We investigate the non-uniqueness of weak solutions of the Quantum-Hydrodynamic system. This form of ill-posedness is related to the change of the number of connected components of the support of the position density (called nodal domains) of the weak solution throughout its time evolution. We start by considering a scenario consisting of initial and final time, showing that if there is a decrease in the number of connected components, then we have non-uniqueness. This result relies on the Brouwer invariance of domain theorem. Then we consider the case in which the results involve a time interval and a full trajectory (position-current densities). We introduce the concept of trajectory-uniqueness and its characterization., 10 pages

In this paper, we classify the fundamental solutions for a class of Schrodinger operators., Comment: 25

This paper provides an elementary proof of the classical limit of the Schr��dinger equation with WKB type initial data and over arbitrary long finite time intervals. We use only the stationary phase method and the Laptev-Sigal simple and elegant construction of a parametrix for Schr��dinger type equations [A. Laptev, I. Sigal, Review of Math. Phys. 12 (2000), 749-766]. We also explain in detail how the phase shifts across caustics obtained when using the Laptev-Sigal parametrix are related to the Maslov index., 21 pages

In this paper, we prove the existence of minimizers of a class of multi-constrained variational problems. We consider systems involving a nonlinearity that does not satisfy compactness, monotonicity, neither symmetry properties. Our approach hinges on the concentration-compactness approach. In the second part, we will treat orthogonal constrained problems for another class of integrands using density matrices method.

We investigate the behavior of solutions of the complex Gross-Pitaevskii equation, a model that describes the dynamics of pumped decaying Bose-Einstein condensates. The stationary radially symmetric solutions of the equation are studied and their linear stability with respect to two-dimensional perturbations is analyzed. Using numerical continuation, we calculate not only the ground state of the system, but also a number of excited states. Accurate numerical integration is employed to study the general nonlinear evolution of the system from the unstable stationary solutions to the formation of stable vortex patterns., Comment: 26 pages, 23 figures

We present and discuss a mathematical model for the operation of bilayer organic photovoltaic devices. Our model couples drift-diffusion-recombination equations for the charge carriers (specifically, electrons and holes) with a reaction-diffusion equation for the excitons/ polaron pairs and Poisson's equation for the self-consistent electrostatic potential. The material difference (i.e. the HOMO/LUMO gap) of the two organic substrates forming the bilayer device are included as a work-function potential. Firstly, we perform an asymptotic analysis of the scaled one-dimensional stationary state system i) with focus on the dynamics on the interface and ii) with the goal of simplifying the bulk dynamics away for the interface. Secondly, we present a twodimensional Hybrid Discontinuous Galerkin Finite Element numerical scheme which is very well suited to resolve i) the material changes ii) the resulting strong variation over the interface and iii) the necessary upwinding in the discretization of drift-diffusion equations. Finally, we compare the numerical results with the approximating asymptotics., Comment: 33 page, 11, Figures

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., Comment: 17 pages, submitted

We discuss local and global existence and uniqueness for the price formation free boundary model with homogeneous Neumann boundary conditions introduced by Lasry & Lions in 2007. The results are based on a transformation of the problem to the heat equation with nonstandard boundary conditions. The free boundary becomes the zero level set of the solution of the heat equation. The transformation allows us to construct an explicit solution and discuss the behavior of the free boundary. Global existence can be verified under certain conditions on the free boundary and examples of non-existence are given.

We discuss global existence and asymptotic behaviour of a price formation free boundary model introduced by Lasry & Lions in 2007. Our results are based on a construction which transforms the problem into the heat equation with specially prepared initial datum. The key point is that the free boundary present in the original problem becomes the zero level set of this solution. Using the properties of the heat operator we can show global existence, regularity and asymptotic results of the free boundary.

The aim of this paper is to study a free boundary problem for a uniformly elliptic fully non-linear operator. Under certain assumptions we show that free and fixed boundaries meet tangentially at contact points., Comment: 11 pges

We consider the Cauchy-problem for a class of scalar linear dispersive equations with rapidly oscillating initial data. The problem of high-frequency asymptotics of such models is reviewed,in particular we highlight the difficulties in crossing caustics when using (time-dependent) WKB-methods. Using Wigner measures we present an alternative approach to such asymptotic problems. We first discuss the connection of the naive WKB solutions to transport equations of Liouville type (with mono-kinetic solutions) in the prebreaking regime. Further we show that the Wigner measure approach can be used to analyze high-frequency limits in the post-breaking regime, in comparison with the traditional Fourier integral operator method. Finally we present some illustrating examples., 38 pages

We consider the Schr��dinger-Poisson system in the attractive (plasma physics) Coulomb case. Given a steady state from a certain class we prove its nonlinear stability, using an appropriately defined energy-Casimir functional as Lyapunov function. To obtain such steady states we start with a given Casimir functional and construct a new functional which is in some sense dual to the corresponding energy-Casimir functional. This dual functional has a unique maximizer which is a steady state of the Schr��dinger-Poisson system and lies in the stability class. The steady states are parametrized by the equation of state, giving the occupation probabilities of the quantum states as a strictly decreasing function of their energy levels., 16 pages