arXiv subjects

Andrea Poiatti

Publications and source records attributed to Andrea Poiatti.

At least 19 recordsLinked to original sources

arXiv2026

De Giorgi varifold solutions to Mean Curvature Flow: a minimizing movements approach

We propose an alternative existence proof of global weak solutions to mean curvature flow and volume preserving mean curvature flow. We prove for the first time for a minimizing movements scheme the unconditional convergence towards a varifold solution, here a De Giorgi solution. The argument is purely variational and does not rely on comparison principles. The key novelty is an alternative proxy for the completely degenerate $L^2$ distance that is more robust than the one of Almgren-Taylor-Wang and Luckhaus-Sturzenhecker.

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arXiv2026

Weak and strong solutions for a class of quasilinear Allen--Cahn systems

We consider a quasilinear Allen--Cahn system which arises when the gradient energy term in the Ginzburg--Landau energy also contains zero order terms. Such systems offer significant advantages in applications, since surface tensions and mobilities can be easily calibrated. The analysis for these systems is highly challenging, partly due to the fact that the gradient term in the energy is non-convex and since gradient terms appear quadratically in the weak formulation. This explains why an existence theory has been lacking for nearly thirty years. In this paper, we give the first existence and uniqueness results for such systems. Firstly, we prove existence and uniqueness of local-in-time strong solutions using the theory of maximal regularity. Here, non-standard techniques have to be applied due to the fact that linear constraints on the solution are involved and due to nonlinear boundary conditions. Secondly, using a minimizing movement approach we show the existence of global-in-time weak solutions. Here, the main difficulty arises from the fact that the underlying energy is not $λ$-convex. We overcome this issue by proving higher integrability of the gradient of the solution, first showing that solutions are bounded and then using an approach by Giaquinta and Modica. This finally allows us to pass to the limit in the time-discrete approximation. Using the de Giorgi interpolation technique, we are also able to show a sharp energy decay property despite the lack of convexity of the energy.

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arXiv2026

Well-posedness and longtime behavior of the conserved Navier--Stokes--Allen--Cahn equations with unmatched viscosities and singular potential

We consider an incompressible Navier--Stokes system nonlinearly coupled with a conserved Allen--Cahn equation with a singular potential (e.g., of Flory--Huggins type). This model describes a mass-conserving two-phase flow with constant density and non-constant viscosity. First, in three spatial dimensions, we prove the existence and uniqueness of local-in-time strong solutions to the associated initial--boundary value problem, subject to no-slip boundary conditions for the velocity field and homogeneous boundary conditions for the phase field. Next, by means of a relative energy approach, we establish a conditional weak--strong uniqueness principle in three dimensions, as well as unconditional uniqueness of weak solutions in two dimensions. Finally, building on recent seminal results by the first and third authors, we prove for the first time that, in both two and three dimensions and for general singular potentials, every global-in-time weak solution asymptotically separates from the pure phases and converges to a unique equilibrium. This result is obtained under minimal assumptions on the viscosity coefficient. Moreover, under additional regularity assumptions on the viscosity, we combine the asymptotic strict separation property with the conditional weak--strong uniqueness principle to show that weak solutions undergo asymptotic regularization. As a consequence, convergence to equilibrium also holds in higher-order norms.

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arXiv2026

Singularities in phase separation models: a spectral element approach for the nonlocal Cahn-Hilliard equation

The nonlocal Cahn-Hilliard equation provides a natural extension of the classical model for phase separation by incorporating long-range interactions through a singular convolution kernel. While this formulation admits a rich existence and regularity theory, its numerical approximation remains challenging: discretisation of the nonlocal term leads to dense operators, and the singularity of the kernel requires special treatment in collocation-based schemes. In this work, we develop an efficient and error-controlled numerical framework for the nonlocal Cahn-Hilliard system with constant mobility, logarithmic potential, Newtonian interaction kernel, and no-flux boundary conditions. Our approach is based on a pseudospectral multishape method that accurately approximates the action of singular convolution operators. We present high-resolution numerical solutions for this nonlocal system of equations that can be achieved with limited computational resources.

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arXiv2026

Asymptotic stabilization of weak solutions to phase-field equations with non-degenerate mobility and singular potential

A common paradigm in phase-field models with singular potentials is that global-in-time weak solutions converge to a single equilibrium only after undergoing asymptotic regularization. However, in arXiv:2510.17296 we introduced a novel method to establish the convergence to a single equilibrium for solutions to Cahn--Hilliard equations, and some related coupled systems, with non-degenerate mobility and singular potentials, under very general assumptions: we only require the existence of a global weak solution satisfying an energy inequality and then we make use of a Lojasiewicz--Simon inequality. Here we take a non-trivial step further. We relax the assumptions needed to prove the precompactness of trajectories, which is an essential ingredient of the complete proof. Thanks to this generalization, we can handle all the main phase-field models, with fully general singular potentials, in a three-dimensional domain, whose asymptotic behavior has so far remained an open problem. Namely, we apply the new method to the Cahn--Hilliard equation with nonlinear diffusion, the conserved Allen--Cahn equation, and the nonlocal Cahn--Hilliard equation. In the case of second-order equations, De Giorgi's iteration argument is crucial to show that weak solutions stay asymptotically uniformly away from pure phases (strict separation property), which is the key ingredient to apply the Lojasiewicz--Simon inequality. We expect this new technique to have a wider range of applications to coupled problems, including hydrodynamic models like the conserved Allen--Cahn--Navier--Stokes system or the nonlocal Abels--Garcke--Grün system with non-degenerate mobility. Further applications to multi-component models are also possible.

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arXiv2026

The Cahn--Hilliard--Darcy--Forchheimer system with surfactant: Existence and long-time behavior of global weak solutions

We consider a diffuse-interface model for two-phase incompressible viscous flows with a soluble surfactant in a bounded porous medium. This hydrodynamic system consists of a Darcy--Forchheimer equation for the seepage velocity $\boldsymbol{u}$ coupled with two Cahn--Hilliard equations involving Flory--Huggins type singular potentials, one for the phase-field variable $ϕ$, the difference in volume fractions of the two fluids, and the other for the surfactant concentration $ψ$. We study the initial boundary value problem in two or three dimensions, with impermeability boundary conditions for $\boldsymbol{u}$ and homogeneous Neumann boundary conditions for $(ϕ, ψ)$ and their associated chemical potentials. First, we establish the existence of global weak solutions via an implicit-explicit time-discretization scheme based on the energy dissipation law. Furthermore, applying the seminal results of the first and third authors (arXiv:2510.17296), we prove that every weak solution satisfying an energy inequality converges to a single equilibrium as time tends to infinity. In sharp contrast with the available literature on similar models, in this case weak solutions are enough to guarantee the uniqueness of asymptotic limits, without the necessity of any further eventual regularization.

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arXiv2026

Long-time dynamics of a bulk-surface convective Cahn--Hilliard system: Pullback attractors and convergence to equilibrium

We study the long-time dynamics of a bulk-surface convective Cahn--Hilliard system describing phase separation processes with bulk-surface interaction. The presence of convection terms leads to a non-autonomous dynamical system and prevents the associated free energy from being a Lyapunov functional, which makes the analysis of the asymptotic behavior considerably more challenging. First, we establish an instantaneous regularization property for weak solutions. Next, interpreting the evolution as a continuous two-parameter process, we prove the existence of a minimal pullback attractor. Finally, under suitable decay assumptions on the velocity fields, we show that every solution converges as $t\to\infty$ to a single steady state. The proof of this convergence relies on the Łojasiewicz--Simon inequality combined with customized decay estimates that compensate for the lack of a monotone energy functional.

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arXiv2026

Varifold solutions to Volume-Preserving Mean Curvature Flow: existence and weak-strong uniqueness

In this contribution we introduce a novel weak solution concept for two-phase volume-preserving mean curvature flow, having both properties of unconditional global-in-time existence and weak-strong uniqueness. These solutions extend the ones proposed by Hensel-Laux [J. Differential Geom. 130, 209-268 (2025)] for the standard mean curvature flow, and consist in evolving varifolds coupled with the phase volumes by a transport equation. First, we show that, in the same setting as in Takasao [Arch. Ration. Mech. Anal. 247, 52 (2023)], any sharp interface limit of solutions to a slightly modified nonlocal Allen-Cahn equation is a varifold solution according to our new definition. Secondly, we crucially introduce a new notion of volume-preserving gradient-flow calibrations, allowing the extended velocity vector field to point in the normal direction on the interface. We show that any sufficiently regular strong solution is calibrated in this sense. Finally, we prove that any classical solution to volume-preserving mean curvature flow, which is then automatically a calibrated flow, is unique in the class of our new varifold solutions.

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arXiv2026

Convergence to equilibrium of weak solutions to the Cahn--Hilliard equation with non-degenerate mobility and singular potential

We consider the classical initial and boundary value problem for the Cahn--Hilliard equation with non-degenerate mobility and singular (e.g., logarithmic) potential. We prove that any weak solution converges to a single equilibrium using only minimal assumptions, that is, the existence of a global weak solution which satisfies an energy inequality. This result appears to be new in the literature and also holds in the three-dimensional case, which was an open problem due to the lack of regularity results, especially when the mobility is just a continuous function. We then prove the same result for a Cahn--Hilliard-Navier--Stokes type system with unmatched densities and viscosities proposed by Abels, Garcke, and Grün (Math. Models Methods Appl. Sci. 22, 2012), always assuming a non-degenerate mobility. We expect that this novel method can be used to analyze the same issue for other models where the regularization properties of the solutions are unknown or unlikely.

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arXiv2025

Diffuse Interface Model for Two-Phase Flows on Evolving Surfaces with Different Densities: Local Well-Posedness

A Cahn-Hilliard-Navier-Stokes system for two-phase flow on an evolving surface with non-matched densities is derived using methods from rational thermodynamics. For a Cahn-Hilliard energy with a singular (logarithmic) potential short time well-posedness of strong solutions together with a separation property is shown, under the assumption of a priori prescribed surface evolution. The problem is reformulated with the help of a pullback to the initial surface. Then a suitable linearization and a contraction mapping argument for the pullback system are used. In order to deal with the linearized system, it is necessary to show maximal $L^2$-regularity for the surface Stokes operator in the case of variable viscosity and to obtain maximal $L^p$-regularity for the linearized Cahn-Hilliard system.

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arXiv2025

Well-posedness and sharp interface limit of a non-isothermal Navier--Stokes/Allen--Cahn model

We propose a thermodynamically consistent phase-field model for the flow of a mixture of two different viscous incompressible fluids of equal density in a bounded domain. We prove the well-posedness of local-in-time strong solutions by means of maximal regularity and contraction mapping arguments. We introduce a suitable entropic weak formulation of the problem, replacing the heat equation by the total energy inequality and an entropy production inequality, and we rigorously prove global-in-time existence of such weak solutions, developing a novel approximation scheme. We also show that an entropic weak solution to this non-isothermal phase-field model converges to a distributional (or $BV$) solution to a non-isothermal Navier--Stokes/mean curvature flow, under an energy convergence assumption.

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arXiv2025

Weak solutions and sharp interface limit of the anisotropic Cahn-Hilliard equation with disparate mobility and inhomogeneous potential

We study the existence of weak solutions and the corresponding sharp interface limit of an anisotropic Cahn-Hilliard equation with disparate mobility, i.e., the mobility is degenerate in one of the two pure phases, making the diffusion in that phase vanish. The double-well potential is polynomial and is weighted by a spatially inhomogeneous coefficient. In the limit when the parameter of the interface width tends to zero, and under an energy convergence assumption, we prove that the weak solutions converge to BV solutions of a weighted anisotropic Hele-Shaw flow. We also add some numerical simulations to analyze the effects of anisotropy on the Cahn-Hilliard equation.

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arXiv2025

Multi-component phase separation and small deformations of a spherical biomembrane

We focus on the derivation and analysis of a model for multi-component phase separation occurring on biological membranes, inspired by observations of lipid raft formation. The model integrates local membrane composition with local membrane curvature, describing the membrane's geometry through a perturbation method represented as a graph over an undeformed Helfrich minimising surface, such as a sphere. The resulting energy consists of a small deformation functional coupled to a Cahn-Hilliard functional. By applying Onsager's variational principle, we obtain a multi-component Cahn-Hilliard equation for the vector $\boldsymbolφ$ of protein concentrations coupled to an evolution equation for the small deformation $u$ along the normal direction to the reference membrane. Then, in the case of a constant mobility matrix, we consider the Cauchy problem and we prove that it is (globally) well posed in a weak setting. We also demonstrate that any weak solution regularises in finite time and satisfies the so-called "strict separation property". This property allows usto show that any weak solution converges to a single stationary state by a suitable version of the Lojasiewicz-Simon inequality.

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arXiv2025

Weak Solutions to a Sharp Interface Model for a Two-Phase Flow of Incompressible Viscous Fluids with Different Densities

In this paper we consider the flow of two incompressible, viscous and immiscible fluids in a bounded domain, with different densities and viscosities. This model consists of a coupled system of Navier-Stokes and Mullins-Sekerka type parts, and can be obtained from the sharp interface limit of the diffuse interface model proposed by the first author, Garcke, and Grün (Math. Models Methods Appl. Sci. 22, 2012). We introduce a new notion of weak solutions and prove its global in time existence, together with a consistency result of smooth weak solutions with the classical Navier-Stokes-Mullins-Sekerka system. Our new notion of solution allows to include the case of different densities of the two fluids, a sharp energy dissipation principle à la De Giorgi, together with a weak formulation of the constant contact angle condition at the boundary, which were left open in the previous notion of solution proposed by the first author and Röger (Ann. Inst. H. Poincaré Anal. Non Linéaire 26, 2009).

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arXiv2025

Non-isothermal non-Newtonian fluids: the stationary case

The stationary Navier-Stokes equations for a non-Newtonian incompressible fluid are coupled with the stationary heat equation and subject to Dirichlet type boundary conditions. The viscosity is supposed to depend on the temperature and the stress depends on the strain through a suit-able power law depending on $p \in (1,2)$ (shear thinning case). For this problem we establish the existence of a weak solution as well as we prove some regularity results both for the Navier-Stokes and the Stokes cases. Then, the latter case with the Carreau power law is approximated through a FEM scheme and some error estimates are obtained. Such estimates are then validated through some two-dimensional numerical experiments.

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arXiv2024

Long-Time behavior of the tangential surface Navier-Stokes equation

We investigate the initial-value problem for the incompressible tangential Navier-Stokes equation with variable viscosity on a given two-dimensional surface without boundary. Existence of global weak and strong solutions under inhomogeneous forcing is proved by a fixed-point and continuation argument. Continuous dependence on data, backward uniqueness, and instantaneous regularization are also discussed. Depending on the effect of the inhomogeneous forcing on the dissipative and the nondissipative components of the system, we investigate the long-time behavior of solutions. We prove the existence and properties of the $σ$-global attractor, in the case of bounded trajectories, and of the so-called unbounded attractor, for unbounded trajectories.

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arXiv2024

Diffuse Interface Model for Two-Phase Flows on Evolving Surfaces with Different Densities: Global Well-Posedness

We show existence and uniqueness of strong solutions to a Navier-Stokes/Cahn-Hilliard type system on a given two-dimensional evolving surface in the case of different densities and a singular (logarithmic) potential. The system describes a diffuse interface model for a two-phase flow of viscous incompressible fluids on an evolving surface. We also establish the validity of the instantaneous strict separation property from the pure phases. To show these results we use our previous achievements on local well-posedness together with suitable novel regularity results for the convective Cahn-Hilliard equation. The latter allows to obtain higher-order energy estimates to extend the local solution globally in time. To this aim the time evolution of energy type quantities has to be calculated and estimated carefully.

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arXiv2024

Nonlocal-to-local convergence rates for strong solutions to a Navier-Stokes-Cahn-Hilliard system with singular potential

The main goal of this paper is to establish the nonlocal-to-local convergence of strong solutions to a Navier--Stokes--Cahn--Hilliard model with singular potential describing immiscible, viscous two-phase flows with matched densities, which is referred to as the Model H. This means that we show that the strong solutions to the nonlocal Model H converge to the strong solution to the local Model H as the weight function in the nonlocal interaction kernel approaches the delta distribution. Compared to previous results in the literature, our main novelty is to further establish corresponding convergence rates. Before investigating the nonlocal-to-local convergence, we first need to ensure the strong well-posedness of the nonlocal Model H. In two dimensions, this result can already be found in the literature, whereas in three dimensions, it will be shown in the present paper. Moreover, in both two and three dimensions, we establish suitable uniform bounds on the strong solutions of the nonlocal Model H, which are essential to prove the nonlocal-to-local convergence results.

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