arXiv subjects

Noah Geltner

Publications and source records attributed to Noah Geltner.

4 recordsLinked to original sources

arXiv2026

Improved weak-strong uniqueness for general cross-diffusion systems

The weak-strong uniqueness of bounded solutions is established for a broad class of cross-diffusion systems, with or without volume-filling effects, and with or without full coercivity. Compared to existing results, the regularity and positivity assumptions on the strong solution are significantly relaxed. The analysis also extends to non-coercive systems, with the volume-filling constraint compensating for the lack of coercivity. The proof is based on two key ideas: The positivity condition is eliminated through the construction of a glued entropy, while the regularity requirement can be relaxed by using the gradient estimate and the Gagliardo-Nirenberg inequality.

math.AP↗

arXiv2026

Analysis of a multispecies cross-diffusion Keller-Segel system with volume filling

A chemotaxis-driven multiphase multispecies diffusion system, arising in the formation of vascular-like structures, is analyzed. The model couples porous-medium-type cross-diffusion equations for the volume fractions of the cellular components with multispecies Keller-Segel equations governing the chemoattractant concentrations, posed in a bounded domain with no-flux boundary conditions. The system is derived within a multiphase framework based on mass and force balance laws, together with a characterization of the mixture pressure gradient, which follows from the volume-filling constraint. The existence of a global weak solution, the weak-strong uniqueness property, the exponential decay to the constant steady state, and the vanishing diffusion limit are established. The existence proof extends the boundedness-by-entropy method to solution codomains that are bounded in some directions only, while the other results are based on various entropy estimates and uniform dissipation bounds.

math.AP↗

arXiv2026

Well-posedness and asymptotic limits for a degenerate Keller-Segel system with volume filling

A class of parabolic-parabolic Keller-Segel systems with degenerate diffusion and volume filling is studied in a bounded domain subject to no-flux boundary conditions. The equations are derived from a multiphase fluid model. The interplay between nonlinear diffusion and density saturation leads to a rich variety of behaviors across different parameter regimes. We establish the existence of global weak solutions, a weak-strong uniqueness result, the exponential convergence to the homogeneous steady state, pattern formation in one spatial dimension, as well as the parabolic-elliptic and vanishing diffusion limits. The analysis relies on a priori estimates derived from suitable entropy functionals. Pattern formation is demonstrated by reducing the system to a first-order equation and conducting a detailed analysis of the resulting nonlinearity. Numerical simulations from a one-dimensional finite-volume scheme illustrate the asymptotic regimes.

math.AP↗

arXiv2025

Chemotaxis guidance of random walkers modeling self-wiring of neural networks

A stochastic walker model is proposed to describe the chemotactic guidance of growth cones, i.e. the tips of developing neurites. The model accounts for the influence of both attractive and repulsive chemical cues, which are emitted by the growth cones and the somas. The system couples stochastic differential equations governing the motion of the growth cones with reaction-diffusion equations that describe the dynamics of the chemical concentrations. The existence of a unique solution to this coupled system is proved. Numerical experiments are performed to investigate the sensitivity of the model to key biological parameters. The impact of the nonlocal regularization of point sources in the reaction-diffusion equations is analyzed in a simplified deterministic setting.

math.AP↗