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Abstract
We propose a mesoscopic modeling framework for optimal transportation networks with biological applications.
The network is described in terms of a joint probability measure on the phase space of tensor-valued conductivity and position in physical space.
The energy expenditure of the network is given by a functional consisting of a pumping (kinetic) and metabolic power-law term, constrained by a Poisson equation accounting for local mass conservation.
We establish convexity and lower semicontinuity of the functional on appropriate sets.
We then derive its gradient flow with respect to the 2-Wasserstein topology on the space of probability measures, which leads to a transport equation, coupled to the Poisson equation.
To lessen the mathematical complexity of the problem, we derive a reduced Wasserstein gradient flow, taken with respect to the tensor-valued conductivity variable only. We then construct equilibrium measures of the resulting PDE system.
Finally, we derive the gradient flow of the constrained energy functional with respect to the Fisher-Rao (or Hellinger-Kakutani) metric, which gives a reaction-type PDE. We calculate its equilibrium states, represented by measures concentrated on a hypersurface in the phase space.
Then, we study a macroscopic generalization of the mesoscopic model. This one is a 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.
In particular, we focus on the 1D setting for Dirichlet and Neumann boundary conditions. We prove an existence and uniqueness result under the assumption of positivity of the diffusivity $D$. We explore systematically various scenarios and gain insights into the behavior of $D$ and its impact on the studied system. This involves analyzing the system with a signed measure distribution of sources and sinks. Finally, we perform several numerical tests in which the solution $D$ touches zero, confirming the previous hints of local existence in particular cases.
Finally, we proved a multi-dimensional local well-posedness result for the problem in Hölder spaces employing Schauder and semigroup theory. Then, after a suitable parameter reduction through scaling, we computed the numerical solution for the proposed system in 2D using a recently developed ghost nodal finite element method. An interesting aspect emerges when the solution is very articulated and the branches occupy a wide region of the domain.
Brief Biography
Simone Portaro is a Ph.D. Student in the Computer, Electrical, Mathematical Sciences and Engineering (CEMSE) department at the Applied Mathematics and Computational Sciences (AMCS) program under the supervision of Professor Peter Markowich at King Abdullah University of Science and Technology. Simone received his B.S. degree in Industrial Engineering from the University of Catania, Italy in 2017. In 2020, he received his M.S. degree in Mathematical Engineering from Politecnico di Torino, Italy.
