arXiv subjects

Andrea Di Primio

Publications and source records attributed to Andrea Di Primio.

9 recordsLinked to original sources

arXiv2026

Stochastic phase separation driven by transport noise

Phase separation phenomena occurring in complex fluids are known to be sensitive to hydrodynamic effects: when a binary fluid mixture undergoes spinodal decomposition, the emerging pattern of domains is no longer dictated by interfacial dynamics alone, but is reshaped by advection. In several applications of interest, advection effects may even cause the onset of turbulence, further affecting the segregation process on a wide range of spatial scales. Motivated by this, we propose a Cahn-Hilliard model driven by transport noise able to capture the intrinsic stochasticity of turbulence. The model is analyzed in its thermodynamically-relevant framework, namely employing a singular Flory-Huggins potential and a possibly degenerate mobility, and the noise is considered both in Ito and Stratonovich form. In this work, as a first-step investigation, we establish well-posedness of the system in two and three spatial dimensions and its thermodynamical consistency.

math.AP↗

arXiv2026

The stochastic nonlocal Cahn-Hilliard equation with regular potential and multiplicative noise

In this work, we deal with the stochastic counterpart of the nonlocal Cahn-Hilliard equation with regular potential in a smooth bounded one-, two- or three-dimensional domain. The problem is endowed with homogeneous Neumann boundary conditions and random initial data. Furthermore, the system is driven by cylindrical noise of multiplicative type. For the resulting system, we are able to show the existence of probabilistically-weak (or martingale) solutions in two and three dimensions, that are unique and probabilistically-strong under suitable assumptions on the stochastic diffusion. Moreover, we investigate the nonlocal-to-local asymptotics toward solutions of the local stochastic Cahn-Hilliard equations, establishing, under regularity conditions, a precise rate of convergence as well.

math.AP↗

arXiv2026

An Allen-Cahn equation with jump-diffusion noise for biological damage and repair processes

This paper analyzes a stochastic Allen--Cahn equation for the dynamics of biomolecular damage and repair. The system is driven by two distinct noise processes: a multiplicative cylindrical Wiener process, modeling continuous background stochastic fluctuations, and a jump-type noise, modeling the abrupt, localized damage induced by external shocks. The drift of the equation is singular and covers the typical logarithmic Flory-Huggins potential required in phase-separation dynamics. We prove well-posedness of the model in a strong probabilistic sense, and analyze its long-time behavior in terms of existence and uniqueness of invariant measures, ergodicity, and mixing properties. Eventually, we present an Euler--Maruyama scheme to simulate the model and illustrate how it captures fundamental biological phenomena, such as damage clustering, stress-induced topology perturbations, and damage dynamics.

math.PR↗

arXiv2025

Existence, uniqueness and asymptotic stability of invariant measures for the stochastic Allen-Cahn-Navier-Stokes system with singular potential

We study the long-time behaviour of a stochastic Allen-Cahn-Navier-Stokes system modelling the dynamics of binary mixtures of immiscible fluids. The model features two stochastic forcings, one on the velocity in the Navier-Stokes equation and one on the phase variable in the Allen-Cahn equation, and includes the thermodynamically-relevant Flory-Huggins logarithhmic potential. We first show existence of ergodic invariant measures and characterise their support by exploiting ad-hoc regularity estimates and suitable Feller-type and Markov properties. Secondly, we prove that if the noise acting in the Navier-Stokes equation is non-degenerate along a sufficiently large number of low modes, and the Allen-Cahn equation is highly dissipative, then the stochastic flow admits a unique invariant measure and is asymptotically stable with respect to a suitable Wasserstein metric.

math.PR↗

arXiv2024

On a diffuse interface model for diblock copolymers interacting with an electric field

We consider a diffuse interface model describing a ternary system constituted by a conductive diblock copolymer and a homopolymer acting as solvent. The resulting dynamics is modeled by two Cahn--Hilliard--Oono equations for the copolymer blocks, accounting for long-range interactions; a classical Cahn--Hiliard equation for the homopolymer and the Maxwell equation for the electric displacement field. A multiphase singular potential is employed in order to ensure physical consistency. First, we show existence of global weak solutions in two and three dimensions. Uniqueness of weak solutions is established in the constant mobility case, and a conditional result is given in the general case. Instantaneous regularization and long-time behavior are also investigated, the latter in the case of affine-linear electric permittivity, showing in particular that solutions converge to a single stationary state.

math.AP↗

arXiv2023

Stochastic Cahn-Hilliard and conserved Allen-Cahn equations with logarithmic potential and conservative noise

We investigate the Cahn-Hilliard and the conserved Allen-Cahn equations with logarithmic type potential and conservative noise in a periodic domain. These features ensure that the order parameter takes its values in the physical range and, albeit the stochastic nature of the problems, that the total mass is conserved almost surely in time. For the Cahn-Hilliard equation, existence and uniqueness of probabilistically-strong solutions is shown up to the three-dimensional case. For the conserved Allen-Cahn equation, under a restriction on the noise magnitude, existence of martingale solutions is proved even in dimension three, while existence and uniqueness of probabilistically-strong solutions holds in dimension one. The analysis is carried out by studying the Cahn-Hilliard/conserved Allen-Cahn equations jointly, that is a linear combination of both the equations, which has an independent interest.

math.AP↗

arXiv2022

A stochastic Allen-Cahn-Navier-Stokes system with singular potential

We investigate a stochastic version of the Allen-Cahn-Navier-Stokes system in a smooth two- or three-dimensional domain with random initial data. The system consists of a Navier-Stokes equation coupled with a convective Allen-Cahn equation, with two independent sources of randomness given by general multiplicative-type Wiener noises. In particular, the Allen-Cahn equation is characterized by a singular potential of logarithmic type as prescribed by the classical thermodynamical derivation of the model. The problem is endowed with a no-slip boundary condition for the (volume averaged) velocity field, as well as a homogeneous Neumann condition for the order parameter. We first prove the existence of analytically weak martingale solutions in two and three spatial dimensions. Then, in two dimensions, we also estabilish pathwise uniqueness and the existence of a unique probabilistically-strong solution. Eventually, by exploiting a suitable generalisation of the classical De Rham theorem to stochastic processes, existence and uniqueness of a pressure is also shown.

math.AP↗

arXiv2022

On a system of coupled Cahn-Hilliard equations

We consider a system which consists of a Cahn-Hilliard equation coupled with a Cahn-Hilliard-Oono equation in a bounded domain of $\mathbb{R}^d$, $d = 2, 3$. This system accounts for macrophase and microphase separation in a polymer mixture through two order parameters $u$ and $v$. The free energy of this system is a bivariate interaction potential which contains the mixing entropy of the two order parameters and suitable coupling terms. The equations are endowed with initial conditions and homogeneous Neumann boundary conditions both for $u,v$ and for the corresponding chemical potentials. We first prove that the resulting problem is well posed in a weak sense. Then, in the conserved case, we establish that the weak solution regularizes instantaneously. Furthermore, in two spatial dimensions, we show the strict separation property for $u$ and $v$, namely, they both stay uniformly away from the pure phases $\pm 1$ in finite time. Finally, we investigate the long-time behavior of a finite energy solution showing, in particular, that it converges to a single stationary state.

math.AP↗

arXiv2022

Well-posedness for a Navier-Stokes-Cahn-Hilliard System for Incompressible Two-phase Flows with Surfactant

We investigate a diffuse-interface model that describes the dynamics of incompressible two-phase viscous flows with surfactant. The resulting system of partial differential equations consists of a sixth-order Cahn-Hilliard equation for the difference of local concentrations of the binary fluid mixture coupled with a fourth-order Cahn-Hilliard equation for the local concentration of the surfactant. The former has a smooth potential, while the latter has a singular potential. Both equations are coupled with a Navier-Stokes system for the (volume averaged) fluid velocity. The evolution system is endowed with suitable initial conditions, a no-slip boundary condition for the velocity field and homogeneous Neumann boundary conditions for the phase functions as well as for the chemical potentials. We first prove the existence of a global weak solution, which turns out to be unique in two dimensions. Stronger regularity assumptions on the initial data allow us to prove the existence of a unique global (resp. local) strong solution in two (resp. three) dimensions. In the two dimensional case, we can derive a continuous dependence estimate with respect to the norms controlled by the total energy. Then we establish instantaneous regularization properties of global weak solutions for $t>0$. In particular, we show that the surfactant concentration stays uniformly away from the pure states $0$ and $1$ after some positive time.

math.AP↗