arXiv subjects

Luca Scarpa

Publications and source records attributed to Luca Scarpa.

At least 19 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

Strong Feller property, irreducibility, and uniqueness of the invariant measure for stochastic PDEs with degenerate multiplicative noise

We establish strong Feller property and irreducibility for the transition semigroup associated to a class of nonlinear stochastic partial differential equations with multiplicative degenerate noise. As a by-product, we prove uniqueness of the invariant measure under no strong-dissipativity assumptions. The drift of the equation diverges exactly where the noise coefficient vanishes, resulting in a competition between the dissipative effects and the degeneracy of the noise. We propose a method to measure the accumulation of the solution towards the potential barriers, allowing to give rigorous meaning to the inverse of the degenerate noise coefficient. From the mathematical perspective, this is one of the first contributions in the literature establishing strong Feller properties and irreducibility in the multiplicative degenerate case, and opens up novel investigation paths in the direction of regularisation effects and ergodicity in the degenerate-noise framework. From the application perspective, the models cover interesting scenarios in physics, in the context of evolution of relative concentrations of mixtures, under the influence of thermodynamically-relevant potentials of Flory-Huggins type.

math.PR↗

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

Pathwise uniqueness by noise for singular stochastic PDEs

Pathwise uniqueness for stochastic PDEs with drift in differential form is a main open problem in the recent literature on regularisation by noise. This paper establishes a self-contained theory in the framework of stochastic evolution equations on separable Hilbert spaces and provides a first result to address such an issue. The singularity of the drift allows to achieve novel uniqueness results for several classes of examples, ranging from fluid-dynamics to phase-separation models.

math.PR↗

arXiv2025

Local asymptotics for the nonlocal Swift-Hohenberg equation

The nonlocal-to-local asymptotics investigation for evolutionary problems is a central topic both in the theory of PDEs and in functional analysis. More recently, it became the main core of the mathematical analysis of phase-separation models. In this paper we focus on the Swift-Hohenberg equations which are key benchmark models in pattern formation problems and amplitude equations. We prove well-posedness of the nonlocal Swift-Hohenberg equation, and study the nonlocal-to-local asymptotics with one and two nonlocal contributions under homogeneous Neumann boundary conditions using suitable energy estimates on the nonlocal problems.

math.AP↗

arXiv2025

Weak stability by noise for approximations of doubly nonlinear evolution equations

Doubly nonlinear stochastic evolution equations are considered. Upon assuming the additive noise to be rough enough, we prove the existence of probabilistically weak solutions of Friedrichs type and study their uniqueness in law. This entails stability for approximations of stochastic doubly nonlinear equations in a weak probabilistic sense. Such effect is a genuinely stochastic, as doubly nonlinear equations are not even expected to exhibit uniqueness in the deterministic case.

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

Cahn-Hilliard equations with singular potential, reaction term and pure phase initial datum

We consider local and nonlocal Cahn-Hilliard equations with constant mobility and singular potentials including, e.g., the Flory-Huggins potential, subject to no-flux (or periodic) boundary conditions. The main goal is to show that the presence of a suitable class of reaction terms allows to establish the existence of a weak solution to the corresponding initial and boundary value problem even though the initial condition is a pure state. In other words, the separation process takes place even in presence of a pure phase, provided that it is triggered by a convenient reaction term. This fact was already observed by the authors in a previous contribution devoted to a specific biological model. In this context, we generalize the previously-mentioned concept by examining the essential assumptions required for the reaction term to apply the new strategy. Also, we explore the scenario involving the nonlocal Cahn-Hilliard equation and provide illustrative examples that contextualize within our abstract framework.

math.AP↗

arXiv2024

Nonlocal to local convergence of phase field systems with inertial term

This paper deals with a nonlocal model for a hyperbolic phase field system coupling the standard energy balance equation for temperature with a dynamic for the phase variable: the latter includes an inertial term and a nonlocal convolution-type operator where the family of kernels depends on a small parameter. We rigorously study the asymptotic convergence of the system as the approximating parameter tends to zero and we obtain at the limit the local system with the elliptic laplacian operator acting on the phase variable. Our analysis is based on some asymptotic properties on nonlocal-to-local convergence that have been recently and successfully applied to families of Cahn--Hilliard models.

math.AP↗

arXiv2023

Analysis and computations of a stochastic Cahn-Hilliard model for tumor growth with chemotaxis and variable mobility

In this work, we present and analyze a system of PDEs, which models tumor growth by considering chemotaxis, active transport, and random effects. The stochasticity of the system is modelled by random initial data and Wiener noises that appear in the tumor and nutrient equations. The volume fraction of the tumor is governed by a stochastic phase-field equation of Cahn-Hilliard type, and the mass density of the nutrients is modelled by a stochastic reaction-diffusion equation. We allow a variable mobility function and non-increasing growth functions, such as logistic and Gompertzian growth. Via approximation and stochastic compactness arguments, we prove the existence of a probabilistic weak solution and, in the case of constant mobilities, the well-posedness of the model in the strong probabilistic sense. Lastly, we propose a numerical approximation based on the Galerkin finite element method in space and the semi-implicit Euler-Maruyama scheme in time. We illustrate the effects of the stochastic forcing in the tumor growth in several numerical simulations.

math.AP↗

arXiv2023

Local asymptotics and optimal control for a viscous Cahn-Hilliard-Reaction-Diffusion model for tumor growth

In this paper we study nonlocal-to-local asymptotics for a tumor-growth model coupling a viscous Cahn-Hilliard equation describing the tumor proportion with a reaction-diffusion equation for the nutrient phase parameter. First, we prove that solutions to the nonlocal Cahn-Hilliard system converge, as the nonlocality parameter tends to zero, to solutions to its local counterpart. Second, we provide first-order optimality conditions for an optimal control problem on the local model, accounting also for chemotaxis, and both for regular or singular potentials, without any additional regularity assumptions on the solution operator. The proof is based on an approximation of the local control problem by means of suitable nonlocal ones, and on proving nonlocal-to-local convergence both for the corresponding dual systems and for the associated first-order optimality conditions.

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↗

arXiv2023

Weak uniqueness by noise for singular stochastic PDEs

We prove weak uniqueness of mild solutions for general classes of SPDEs on a Hilbert space. The main novelty is that the drift is only defined on a Sobolev-type subspace and no H\"older-continuity assumptions are required. This framework turns out to be effective to achieve novel uniqueness results for several specific examples. Such wide range of applications is obtained by exploiting either coloured or rougher-than-cylindrical noises.

math.PR↗

arXiv2023

Strict separation and numerical approximation for a non-local Cahn-Hilliard equation with single-well potential

In this paper we study a non-local Cahn-Hilliard equation with singular single-well potential and degenerate mobility. This results as a particular case of a more general model derived for a binary, saturated, closed and incompressible mixture, composed by a tumor phase and a healthy phase, evolving in a bounded domain. The general system couples a Darcy-type evolution for the average velocity field with a convective reaction-diffusion type evolution for the nutrient concentration and a non-local convective Cahn-Hilliard equation for the tumor phase. The main mathematical difficulties are related to the proof of the separation property for the tumor phase in the Cahn-Hilliard equation: up to our knowledge, such problem is indeed open in the literature. For this reason, in the present contribution we restrict the analytical study to the Cahn-Hilliard equation only. For the non-local Cahn- Hilliard equation with singular single-well potential and degenerate mobility, we study the existence and uniqueness of weak solutions for spatial dimensions $d\leq 3$. After showing existence, we prove the strict separation property in three spatial dimensions, implying the same property also for lower spatial dimensions, which opens the way to the proof of uniqueness of solutions. Finally, we propose a well posed and gradient stable continuous finite element approximation of the model for $d\leq 3$, which preserves the physical properties of the continuos solution and which is computationally efficient, and we show simulation results in two spatial dimensions which prove the consistency of the proposed scheme and which describe the phase ordering dynamics associated to the system.

math.AP↗

arXiv2023

A note on regularity and separation for the stochastic Allen-Cahn equation with logarithmic potential

We prove refined space-time regularity for the classical stochastic Allen-Cahn equation with logarithmic potential. This allows to establish a random separation property, i.e. that the trajectories of the solution are strictly separated from the potential barriers. The present contribution extends the results obtained in Bertacco et al. (2022), where separation was proved only for $p$-Laplace operators with $p$ greater than the space-dimension.

math.PR↗

arXiv2022

A Cahn-Hilliard system with forward-backward dynamic boundary condition and non-smooth potentials

A system with equation and dynamic boundary condition of Cahn-Hilliard type is considered. This system comes from a derivation performed in Liu-Wu (Arch. Ration. Mech. Anal. 233 (2019), 167--247) via an energetic variational approach. Actually, the related problem can be seen as a transmission problem for the phase variable in the bulk and the corresponding variable on the boundary. The asymptotic behavior as the coefficient of the surface diffusion acting on the boundary phase variable goes to 0 is investigated. By this analysis we obtain a forward-backward dynamic boundary condition at the limit. We can deal with a general class of potentials having a double-well structure, including the non-smooth double-obstacle potential. We illustrate that the limit problem is well-posed by also proving a continuous dependence estimate. Moreover, in the case when the two graphs, in the bulk and on the boundary, exhibit the same growth, we show that the solution of the limit problem is more regular and we prove an error estimate for a suitable order of the diffusion parameter.

math.AP↗

arXiv2022

Degenerate Kolmogorov equations and ergodicity for the stochastic Allen-Cahn equation with logarithmic potential

Well-posedness \`a la Friedrichs is proved for a class of degenerate Kolmogorov equations associated to stochastic Allen-Cahn equations with logarithmic potential. The thermodynamical consistency of the model requires the potential to be singular and the multiplicative noise coefficient to vanish at the respective potential barriers, making thus the corresponding Kolmogorov equation not uniformly elliptic in space. First, existence and uniqueness of invariant measures and ergodicity are discussed. Then, classical solutions to some regularised Kolmogorov equations are explicitly constructed. Eventually, a sharp analysis of the blow-up rates of the regularised solutions and a passage to the limit with a specific scaling yield existence \`a la Friedrichs for the original Kolmogorov equation.

math.PR↗

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↗