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Luca Melzi

Publications and source records attributed to Luca Melzi.

3 recordsLinked to original sources
arXiv2026

Arnold stability and rigidity in Zeitlin's model of hydrodynamics

Zeitlin's model is a discretisation of the 2-D Euler equations that preserves the underlying geometric structure. This feature makes it suitable for studying the qualitative behaviour of the dynamics. Here, we utilise Arnold's geometric approach to prove Lyapunov stability of steady states in Zeitlin's model. Furthermore, we show that such Arnold stable stationary solutions are subject to a rigidity condition that enforces a specific form of the matrix describing the state. Our argument relies on matrix theory and is therefore detached, and conceptually different, from the nonlinear stability analysis as developed for the 2-D Euler equations. Nevertheless, our results concur with those known for the 2-D Euler equations, which hints at links between matrix theory and nonlinear PDE techniques. Furthermore, our results show that the Zeitlin's model, as a numerical discretisation, is reliable for studying stationary solutions.

math.AP↗
arXiv2025

On a non-local phase-field model for tumour growth with single-well Lennard-Jones potential

In the present work, we develop a comprehensive and rigorous analytical framework for a non-local phase-field model that describes tumour growth dynamics. The model is derived by coupling a non-local Cahn-Hilliard equation with a parabolic reaction-diffusion equation, which accounts for both phase segregation and nutrient diffusion. Previous studies have only considered symmetric potentials for similar models. However, in the biological context of cell-to-cell adhesion, single-well potentials, like the so-called Lennard-Jones potential, seem physically more appropriate. The Cahn-Hilliard equation with this kind of potential has already been analysed. Here, we take a step forward and consider a more refined model. First, we analyse the model with a viscous relaxation term in the chemical potential and subject to suitable initial and boundary conditions. We prove the existence of solutions, a separation property for the phase variable, and a continuous dependence estimate with respect to the initial data. Finally, via an asymptotic analysis, we recover the existence of a weak solution to the initial and boundary value problem without viscosity, provided that the chemotactic sensitivity is small enough.

math.AP↗