Institute of Applied Analysis
University Ulm
Helmholtzstraße 18
89081 Ulm
Room E.10
Tel.: +49 731 50-23560
andre.schlichting@uni-ulm.de
News and Updates
- Preprint: Derivation of the fourth-order DLSS equation with nonlinear mobility via chemical reactions
Together with Alexander Mielke and Artur Stephan, we provide a derivation of the fourth-order DLSS equation based on an interpretation as a chemical reaction network. We consider the rate equation on the discretized circle for a process in which pairs of particles occupying the same site simultaneously jump to the two neighboring sites; the reverse process involves pairs of particles at adjacent sites simultaneously jumping back to the site located between them. Depending on the rates, in the vanishing-mesh-size limit we obtain either the classical DLSS equation or a variant with nonlinear mobility of power type. Via EDP convergence, we identify the limiting gradient structure to be driven by entropy with respect to a generalization of diffusive transport with nonlinear mobility. Interestingly, the DLSS equation with power-type mobility shares qualitative similarities with the fast diffusion and porous medium equation, since we find traveling wave solutions with algebraic tails or compactly supported polynomials, respectively.
For details see arXiv:2510.07149
- Preprint: Existence and Non-existence for Continuous Generalized Exchange-Driven Growth model
Together with Chun Yin “Solomon Lam, after the derivation of the continuous generalized exchange-driven growth model in arXiv:2503.21572, we continue the study of the existence and uniqueness of solutions for kernels with superlinear growth at infinity and singularity at the origin. Moreover, we show the non-existence of solutions for kernels with sufficiently rapid growth. The latter result is shown via the finite-time gelation and instantaneous gelation in the sense of moment blow-up. For details see arXiv:2509.05262.
- Extended updated preprint: Variational convergence for an irreversible exchange-driven stochastic particle system
Together with Jasper Hoeksema and Chin Yin Lam, we considerably extended and updated the previous version of the arXiv preprint 2401.06696.
We show the variational convergence of an irreversible Markov jump process describing a finite stochastic particle system to the solution of a countable infinite system of deterministic time-inhomogeneous quadratic differential equations known as the exchange-driven growth model, which has two conserved quantities. As a bounded perturbation of the reversible kernel, the variational formulation is a generalization of the gradient flow formulation of the reversible process and can be interpreted as the large deviation functional of the Markov jump process. As a consequence of the variational convergence result, we show the propagation of chaos of the Markov processes to the limiting equation and the -convergence of the energy functional. The latter convergence is consistent with related results for reversible coagulation-fragmentation equations and reveals the connection of stochastic processes to the long-time condensation phenomena in the limit equation. - Preprint: Convergence of a Stochastic Particle System to the Continuous Generalized Exchange-Driven Growth Model
Together with Chun Yin Lam, we study the continuous generalized exchange-driven growth model (CGEDG) is a system of integro-differential equations describing the evolution of cluster mass under mass exchange. The rate of exchange depends on the masses of the clusters involved and the mass being exchanged. This can be viewed as both a continuous generalization of the exchange-driven growth model and a coagulation-fragmentation equation that generalizes the continuous Smoluchowski equation.
Starting from a Markov jump process that describes a finite stochastic interacting particle system with exchange dynamics, we prove the weak law of large numbers for this process for sublinearly growing kernels in the mean-field limit. We establish the tightness of the stochastic process on a measure-valued Skorokhod space induced by the 1-Wasserstein metric, from which we deduce the existence of solutions to the (CGEDG) system. The solution is shown to have a Lebesgue density under suitable assumptions on the initial data. Moreover, within the class of solutions with density, we establish the uniqueness under slightly more restrictive conditions on the kernel.
arXiv:2503.21572 - Preprint: Diffusive transport on the real line: semi-contractive gradient flows and their discretization
Together with Daniel Matthes and Eva-Maria Rott, we introduce the diffusive transport distance, a novel pseudo-metric between probability measures on the real line. It generalizes Martingale optimal transport, and forms a hierarchy with the Hellinger and the Wasserstein metrics. We observe that certain classes of parabolic PDEs, among them the porous medium equation of exponent two, are formally semi-contractive metric gradient flows in the new distance. This observation is made rigorous for a suitable spatial discretization of the considered PDEs: these are semi-contractive gradient flows with respect to an adapted diffusive transport distance for measures on the point lattice. The main result is that the modulus of convexity is uniform with respect to the lattice spacing. Particularly for the quadratic porous medium equation, this is in contrast to what has been observed for discretizations of the Wasserstein gradient flow structure.
arXiv:2501.14527
