• Thematic program: Kinetic Transport Theory: Analysis and Applications (2011)

• Thematic program: Kinetic Transport Theory: Analysis and Applications (2011)

I will talk about how to incorporate the nonlinear terms into non-Markovian fractional partial differential equations corresponding to subdiffusive transport. I will discuss applications of these equations in biology: chemotaxis, subdiffusion in dendrites, etc. I will show that the standard subdiffusive fractional equations with constant anomalous exponent are not structurally stable with respect to the non-homogeneous variations of exponent.

• Thematic program: Kinetic Transport Theory: Analysis and Applications (2011)
• Event: Workshop "Fractional Diffusion and Application" (2012)

Scalar one-dimensional conservation laws with a nonlocal diffusion term corresponding to a Riesz-Feller differential operator are considered. Solvability results for the Cauchy problem with essentially bounded initial datum are adapted from the case of a fractional derivative with homogeneous symbol. The main interest of this work is the investigation of smooth shock profiles. In case of a genuinely nonlinear smooth flux function we prove the existence of such traveling waves, which are monotone and satisfy the standard entropy condition. Moreover, the dynamic nonlinear stability of the traveling waves under small perturbations is proven, similarly to the case of the standard diffusive regularization, by constructing a Lyapunov functional. We will provide an example of a single layer shallow water flow, where the pressure is governed by a nonlinear conservation law with the aforementioned nonlocal diffusion term and additional dispersion term and report on the recent progress in the analysis of smooth shock profiles.

• Thematic program: Kinetic Transport Theory: Analysis and Applications (2011)
• Event: Workshop "Fractional Diffusion and Application" (2012)

The binding of individual components to form aggregate structures is a ubiquitous phenomenon in physics, chemistry and material science. Nucleation events may be heterogeneous, where particles are attracted to an initial exogenous site or homogeneous where identical particles spontaneously cluster upon contact. Particle nucleation and cluster growth have been extensively studied in the past decades, often assuming infinitely large numbers of building blocks and unbounded cluster sizes. These assumptions led to the use of mass-action, mean field descriptions such as the well known Becker Doering equations. In cellular biology, however, nucleation events often take place in confined spaces, with a finite number of components, so that discrete and stochastic effects must be taken into account. In this talk we examine finite sized homogeneous nucleation by considering a fully stochastic master equation, solved via Monte-Carlo simulations and via analytical insight. We find striking differences between the mean cluster sizes obtained from our discrete, stochastic treatment and those predicted by mean field treatments. We also consider heterogeneous nucleation stochastic treatments, first passage time results and possible applications to prion unfolding and clustering dynamics.

• Thematic program: Kinetic Transport Theory: Analysis and Applications (2011)