Prof. Adrian Muntean Department of Mathematics and Computer Science, Karlstad University, Karlstad, Sweden.

Diffusion in the presence of microstructures: Does

vesicle micro-dynamics enhance the signalling among

plants macro-transport?

The interplay between microscopic (lattice-based, SDEs, PDEs, etc.) models and macroscopic evolution equations leads to interesting questions in the classical theory of diffusion. Some of them are still in search for rigorous

answers. Upscaling of microscopic models, via a variety of techniques like renormalization, hydrodynamic limits, or homogenization, is usually a preferred methodological path. However, some problem settings are not the result of an averaging procedure and require the explicit handling of models posed on separate space scales. In this seminar, we study a transport problem for signalling among plants in the context of measure-valued equations. We report on preliminary results concerning the modelling and mathematical analysis of a reaction-diffusion scenario involving the macroscopic diffusion of signalling molecules enhanced by the presence of a finite number of microscopic vesicles - pockets with own dynamics able to capture and release signals as a relay system. The coupling between the macroscopic and microscopic spatial scales relies on the use of a

two-scale transmission condition and benefits of the posing of the problem in terms of measures. Mild solutions to our problem will turn to exist and will also be positive weak solutions. A couple of open questions at the modeling, mathematical analysis, and simulation levels will be pointed out. This is a joint work with Sander Hille (Leiden, NL) and Omar Richardson (Karlstad,

Sweden), supported financially by the KvA’s G. S. Magnussonsfond.

References:

Evers, Joep; Hille, Sander; Muntean, Adrian; Measure-valued mass evolution problems with flux boundary conditions and solution-dependent velocities, SIAM Journal of Mathematical Analysis, 48 (2016), no. 3, 1929-1953.

Cirillo, Emilio N. M.; de Bonis, Ida; Muntean, Adrian; Richardson, Omar; Upscaling the interplay between diffusion and polynomial drifts through a composite thin strip with periodic microstructure. Meccanica 55 (2020), no. 11, 2159-2178.

Lind, Martin; Muntean, Adrian; Richardson, Omar; A semidiscrete Galerkin scheme for a coupled two-scale elliptic-parabolic system: well-posedness and convergence approximation rates. BIT 60 (2020), no. 4, 999-1031.

Prof. Alexander Nazarov

Department of Steklov Institute, St. Petersburg State University, St. Petersburg, Russia.

Alexandrov maximum principle and applications. Besides their intrinsic mathematical interest and beauty, maximum principles and specifi- cally the Alexandrov-Bakelman-Pucci maximum principle, can be applied to problems with non-smooth coefficients or data, and therefore in the general setting of inhomogeneous problems. Suitable versions of such tools do hold for anomalous diffusion processes, modeled by the Fractional Laplacian, in its several different definitions. These models have recently drawn an increasing interest, owing to their relevance to physics, finance and probability.

References:

Musina, Roberta; Nazarov, Alexander I. Strong maximum principles for frac- tional Laplacians. Proc. Roy. Soc. Edinburgh Sect. A 149 (2019), no. 5, 1223-1240.

Apushkinskaya, Darya E.; Nazarov, Alexander I.; Palagachev, Dian K.; Soft- ova, Lubomira G.; Venttsel Boundary Value Problems with Discontinuous Data. SIAM J. Math. Anal. 53 (2021), no. 1, 221-252.