arXiv subjects

Sander C. Hille

Publications and source records attributed to Sander C. Hille.

22 records · Page 2Linked to original sources

arXiv2016

Measure-valued mass evolution problems with flux boundary conditions and solution-dependent velocities

In this paper we prove well-posedness for a measure-valued continuity equation with solution-dependent velocity and flux boundary conditions, posed on a bounded one-dimensional domain. We generalize the results of [Evers, Hille and Muntean. Journal of Differential Equations, 259:1068-1097, 2015] to settings where the dynamics are driven by interactions. In a forward-Euler-like approach, we construct a time-discretized version of the original problem and employ the results of [Evers, Hille and Muntean. Journal of Differential Equations, 259:1068-1097, 2015] as a building block within each subinterval. A limit solution is obtained as the mesh size of the time discretization goes to zero. Moreover, the limit is independent of the specific way of partitioning the time interval $[0,T]$. This paper is partially based on results presented in Chapter 5 of [Evers. PhD thesis, Eindhoven University of Technology, 2015], while a number of issues that were still open there, are now resolved.

math.AP↗

arXiv2015

Law of the Iterated Logarithm for some Markov operators

The Law of the Iterated Logarithm for some Markov operators, which converge exponentially to the invariant measure, is established. The operators correspond to iterated function systems which, for example, may be used to generalize the cell cycle model examined by A. Lasota and M.C. Mackey, J. Math. Biol. (1999).

math.PR↗

arXiv2014

Mild solutions to a measure-valued mass evolution problem with flux boundary conditions

We investigate the well-posedness and approximation of mild solutions to a class of linear transport equations on the unit interval $[0,1]$ endowed with a linear discontinuous production term, formulated in the space $\mathcal{M}([0,1])$ of finite Borel measures. Our working technique includes a detailed boundary layer analysis in terms of a semigroup representation of solutions in spaces of measures able to cope with the passage to the singular limit where thickness of the layer vanishes. We obtain not only a suitable concept of solutions to the chosen measure-valued evolution problem, but also derive convergence rates for the approximation procedure and get insight in the structure of flux boundary conditions for the limit problem.

math.FA↗

arXiv2014

Modelling with measures: Approximation of a mass-emitting object by a point source

We consider a linear diffusion equation on $Ω:=\mathbb{R}^2\setminus\bar{Ω_\mathcal{O}}$, where $Ω_\mathcal{O}$ is a bounded domain. The time-dependent flux on the boundary $Γ:=\partialΩ_\mathcal{O}$ is prescribed. The aim of the paper is to approximate the dynamics by the solution of the diffusion equation on the whole of $\mathbb{R}^2$ with a measure-valued point source in the origin and provide estimates for the quality of approximation. For all time $t$, we derive an $L^2([0,t];L^2(Γ))$-bound on the difference in flux on the boundary. Moreover, we derive for all $t>0$ an $L^2(Ω)$-bound and an $L^2([0,t];H^1(Ω))$-bound for the difference of the solutions to the two models.

math.AP↗