Speaker
Description
This talk is based on an ongoing project with Z. Brzezniak, where we study a stochastic diffusion-transport problem on a finite star graph with a nonlinear dynamic condition at the junction and a reflecting obstacle acting on the junction variable. A natural physical interpretation is the evolution of a concentration in a network of thin channels connected to a small central reservoir. Along the channels, the concentration diffuses and is transported by a nonlinear flux. At the junction, the reservoir exchanges mass with all channels, is subject to random fluctuations, and is constrained to remain nonnegative. Problems with dynamic conditions on selected vertices are naturally formulated on a product space containing both the edge variables and the nodal variables, with the coupling imposed through trace conditions. The main difficulty of our model is introduced by the reflection acting on the dynamic variable at the junction. This variable is subject to both a stochastic forcing and a reflecting constraint, but its evolution is not autonomous: it is coupled to the diffusion--transport dynamics on the edges through the boundary fluxes at the common vertex. Thus, although the constraint itself is deceptively simple, since it only requires the scalar junction variable to remain nonnegative, the resulting obstacle problem couples a finite-dimensional reflected component with an infinite-dimensional evolution equation on the network. In particular, it cannot be reduced to the standard one-dimensional problem of reflection on the half-line $[0,\infty)$.