Title: Towards a new mathematical model of the visual cycle
preprint available at hal.archives-ouvertes.fr/hal-03517553/
Abstract: The visual cycle allows photoreceptors in the retina to convert light into electrical signals and subsequently to return to the dark state. George Wald obtained the Nobel Prize in 1967 for his pioneering studies on this process and it has been an active field of research in Ophthalmology and Biochemistry ever since. I will discuss the key aspects of the visual cycle in photoreceptor cells and present a new mathematical model involving ODEs and PDEs for such process in rod cells. The goal is to give a quantitative description of the kinetics of the main photo-sensitive molecules after exposure to light. I will present analytical and numerical results and explain how the model can be extended in order to account for the accumulation of toxic byproducts (e.g. A2E and lipofuscin) in the eye in connection with degenerative retinal diseases.
Title: Existence and regularity for a system of porous medium equations with small cross-diffusion and nonlocal drifts
Authors:L. Alasio, M. Bruna, S. Fagioli, S. Schulz
Journal: Nonlinear Analysis, Volume 223, 2022,
preprint available at arxiv.org/abs/2105.14037
Abstract: We prove the existence and Sobolev regularity of solutions of a nonlinear system of degenerate-parabolic PDEs with self- and cross-diffusion, transport/confinement and nonlocal interaction terms. The macroscopic system of PDEs models the evolution of an arbitrary number of species with quadratic porous-medium interactions in a bounded domain Ω in any spatial dimension and originates from a many-particle system. The cross interactions between different species are scaled by a parameter δ<1, with the δ=0 case corresponding to no interactions across species. A smallness condition on δ ensures existence of solutions up to an arbitrary time T>0 in a subspace of L2(0,T;H1(Ω)). This is shown via a Schauder fixed point argument for a regularised system followed by a vanishing diffusivity approach. The proof uses the lower semicontinuity of the Fisher information in combination with the div–curl Lemma. An ad hoc weak–strong uniqueness result ensures equivalence between weak formulations of the regularised problem; this is proved by studying a related dual problem. We provide numerical evidence showing blow-up of the Sobolev norm for δ→1.
Title: Trend to Equilibrium for Systems with Small Cross-Diffusion
Authors:L. Alasio, H. Ranetbauer, M. Schmidtchen, M.-T. Wolfram
Journal: ESAIM: M2AN, Volume 54, Number 5, 2020,
preprint available at arxiv.org/abs/1906.08060
Abstract: This paper presents new analytical results for a class of nonlinear parabolic systems of partial different equations with small cross-diffusion which describe the macroscopic dynamics of a variety of large systems of interacting particles. Under suitable assumptions, we prove existence of classical solutions and we show exponential convergence in time to the stationary state. Furthermore, we consider the special case of one mobile and one immobile species, for which the system reduces to a nonlinear equation of Fokker-Planck type. In this framework, we improve the convergence result obtained for the general system and we derive sharper bounds for the solutions in two spatial dimensions. We conclude by illustrating the behaviour of solutions with numerical experiments in one and two spatial dimensions.
Title: Global existence for a class of viscous systems of conservation laws
Authors: L. Alasio, S. Marchesani
Journal: NoDEA 26:32, 2019,
preprint available at arxiv.org/abs/1902.02714
Abstract: We prove existence and boundedness of classical solutions for a family of viscous conservation laws in one space dimension for arbitrarily large time. The result relies on H. Amann’s criterion for global existence of solutions and on suitable uniform-in-time estimates for the solution. We also apply Jüngel’s boundedness-by-entropy principle in order to obtain global existence for systems with possibly degenerate diffusion terms. This work is motivated by the study of a physical model for the space-time evolution of the strain and velocity of an anharmonic spring of finite length.
Title: The role of strong confining potentials in a nonlinear Fokker-Planck equation
Authors: L. Alasio, M. Bruna, J. A. Carrillo
Journal: Nonlinear Analysis, Vol. 193, special issue Nonlocal and Fractional Phenomena, 2020
preprint available at arxiv.org/abs/1807.11055
Abstract: We show that solutions of nonlinear nonlocal Fokker–Planck equations in a bounded domain with no-flux boundary conditions can be approximated by Cauchy problems with increasingly strong confining potentials defined in the whole space. Two different approaches are analyzed, making crucial use of uniform estimates for L2 energy functionals and free energy (or entropy) functionals respectively. In both cases, we prove that the weak formulation of the problem in a bounded domain can be obtained as the weak formulation of a limit problem in the whole space involving a suitably chosen sequence of large confining potentials. The free energy approach extends to the case degenerate diffusion.
Title: Stability estimates for a family of cross-diffusion systems
Authors: L. Alasio, M. Bruna, Y. Capdeboscq
Journal: ESAIM: M2AN, Volume 52, Number 3, 2018,
preprint available at arxiv.org/abs/1801.06470
Abstract: We discuss the analysis and stability of a family of cross-diffusion boundary value problems with nonlinear diffusion and drift terms. We assume that these systems are close, in a suitable sense, to a set of decoupled and linear problems. We focus on stability estimates, that is, continuous dependence of solutions with respect to the nonlinearities in the diffusion and in the drift terms. We establish well-posedness and stability estimates in an appropriate Banach space. Under additional assumptions we show that these estimates are time independent. These results apply to several problems from mathematical biology; they allow comparisons between the solutions of different models a priori. For specific cell motility models from the literature, we illustrate the limit of the stability estimates we have derived numerically, and we document the behaviour of the solutions for extremal values of the parameters.