arXiv subjects

Antonio Segatti

Publications and source records attributed to Antonio Segatti.

At least 19 recordsLinked to original sources

arXiv2026

On $\alpha$-entropy solutions of a nonlocal thin film equation: existence and finite speed of propagation

We consider an initial-boundary value problem for a class of nonlocal thin film equations governed by the spectral fractional Laplacian with homogeneous Neumann boundary conditions. We were the first to establish an $\alpha$-entropy estimate for nonlocal thin film equations, which yields essential a priori bounds for the regularity and long-time behavior of weak solutions. By developing a localized version of this estimate, we prove finite speed of propagation, showing that the support of nonnegative solutions remains compact for positive times. Furthermore, we find a sufficient condition for a waiting time phenomenon, whereby the solution remains identically zero in a region for a nontrivial time interval. These results highlight new features in the interaction between nonlocal effects and classical thin film dynamics.

math.AP↗

arXiv2026

Heat flow of harmonic maps into CAT($0$)-spaces

We introduce a new approach to prove the global existence and uniqueness of suitable weak solutions of the heat flow of harmonic mappings into CAT(0) metric spaces. Our method allows also to prove Lipschitz continuity in spatial variables for such solutions into any CAT$(0)$-space, answering a long-standing open problem in the field. Our approach is based on an elliptic regularization of the gradient flow of the Dirichlet energy and even in the case of smooth Riemannian targets provides a novel viewpoint, together with a new Dynamical Variational Principle and a new proof of the celebrated Eells-Sampson theorem. The spatial Lipschitz regularity for such weak solutions is achieved by fully exploiting the variational structure of the problem at the regularized level and introducing a parabolic frequency function of Almgren-Poon type. Our contribution is the first instance of the use of monotonicity methods for parabolic deformations of maps into singular targets.

math.AP↗

arXiv2025

Global attractor for a Cahn-Hilliard-chemotaxis model with logistic degradation

We consider a mathematical model coupling the Cahn-Hilliard system for phase separation with an additional equation describing the diffusion process of a chemical quantity whose concentration influences the physical process. The main application of the model refers to tumor progression, where the phase variable denotes the local proportion of active cancer cells and the chemical concentration may refer to a nutrient transported by the blood flow or to a drug administered to the patient. The resulting system is characterized by cross-diffusion effects similar to those appearing in the Keller-Segel model for chemotaxis; in particular, the nutrient tends to be attracted towards the regions where more active tumor cells are present (and consume it in a quickier way). Complementing various recent results on related models, we investigate here the long-time behavior of solutions under the perspective of infinite-dimensional dynamical systems. To this aim, we first identify a regularity setting in which the system is well posed and generates a closed semigroup according to the terminology introduced by Pata and Zelik. Then, partly based on the approach introduced by Rocca and the first author for the Cahn-Hilliard system with singular potential, we prove that the semigroup is strongly dissipative and asymptotically compact so guaranteeing the existence of the global attractor in a suitable phase space. Finally, we discuss the sign properties of the nutrient and prove that, under additional assumptions on the initial data, its concentration is uniformly larger than some strictly positive constant at least on finite time intervals.

math.AP↗

arXiv2024

Existence and finite speed of propagation of solutions for a multi-dimensional fractional thin-film equation

In this paper, we discuss existence and finite speed of propagation for the solutions to an initial-boundary value problem for a family of fractional thin-film equations in a bounded domain in $\mathbb{R}^d$. The nonlocal operator we consider is the spectral fractional Laplacian with Neumann boundary conditions. In the case of a ``strong slippage'' regime with ``complete wetting'' interfacial conditions, we prove local entropy estimates that entail finite speed of propagation of the support and a lower bound for the waiting time phenomenon.

math.AP↗

arXiv2023

Dynamics of Ginzburg-Landau vortices for vector fields on surfaces

In this paper we consider the gradient flow of the following Ginzburg-Landau type energy \[ F_\varepsilon(u) := \frac{1}{2}\int_{M}\vert D u\vert_g^2 +\frac{1}{2\varepsilon^2}\left(\vert u\vert_g^2-1\right)^2\mathrm{vol}_g. \] This energy is defined on tangent vector fields on a $2$-dimensional closed and oriented Riemannian manifold $M$ (here $D$ stands for the covariant derivative) and depends on a small parameter $\varepsilon>0$. If the energy satisfies proper bounds, when $\varepsilon\to 0$ the second term forces the vector fields to have unit length. However, due to the incompatibility for vector fields on $M$ between the Sobolev regularity and the unit norm constraint, critical points of $F_\varepsilon$ tend to generate a finite number of singular points (called vortices) having non-zero index (when the Euler characteristic is non-zero). These types of problems have been extensively analyzed in a recent paper by R. Ignat and R. Jerrard. As in Euclidean case, the position of the vortices is ruled by the so-called renormalized energy. In this paper we are interested in the dynamics of vortices. We rigorously prove that the vortices move according to the gradient flow of the renormalized energy, which is the limit behavior when $\varepsilon\to 0$ of the gradient flow of the Ginzburg-Landau energy.

math.AP↗

arXiv2023

Dimensional Reduction and emergence of defects in the Oseen-Frank model for nematic liquid crystals

In this paper we discuss the behavior of the Oseen-Frank model for nematic liquid crystals in the limit of vanishing thickness. More precisely, in a thin slab~$Ω\times (0,h)$ with~$Ω\subset \mathbb{R}^2$ and $h>0$ we consider the one-constant approximation of the Oseen-Frank model for nematic liquid crystals. We impose Dirichlet boundary conditions on the lateral boundary and weak anchoring conditions on the top and bottom faces of the cylinder~$Ω\times (0,h)$. The Dirichlet datum has the form $(g,0)$, where $g\colon\partialΩ\to \mathbb{S}^1$ has non-zero winding number. Under appropriate conditions on the scaling, in the limit as~$h\to 0$ we obtain a behavior that is similar to the one observed in the asymptotic analysis of the two-dimensional Ginzburg-Landau functional. More precisely, we rigorously prove the emergence of a finite number of defect points in $Ω$ having topological charges that sum to the degree of the boundary datum. Moreover, the position of these points is governed by a Renormalized Energy, as in the seminal results of Bethuel, Brezis and Hélein.

math.AP↗

arXiv2022

Partial regularity of the heat flow of half-harmonic maps and applications to harmonic maps with free boundary

We introduce a heat flow associated to half-harmonic maps, which have been introduced by Da Lio and Rivière. Those maps exhibit integrability by compensation in one space dimension and are related to harmonic maps with free boundary. We consider a new flow associated to these harmonic maps with free boundary which is actually motivated by a rather unusual heat flow for half-harmonic maps. We construct then weak solutions and prove their partial regularity in space and time via a Ginzburg-Landau approximation. The present paper complements the study initiated by Struwe and Chen-Lin.

math.AP↗

arXiv2021

Motion of vortices for the extrinsic Ginzburg-Landau flow for vector fields on surfaces

We consider the gradient flow of a Ginzburg-Landau functional of the type \[ F_\varepsilon^{\mathrm{extr}}(u):=\frac{1}{2}\int_M \left|D u\right|_g^2 + \left|\mathscr{S} u\right|^2_g +\frac{1}{2\varepsilon^2}\left(\left|u\right|^2_g-1\right)^2\mathrm{vol}_g \] which is defined for tangent vector fields (here $D$ stands for the covariant derivative) on a closed surface $M\subseteq\mathbb{R}^3$ and includes extrinsic effects via the shape operator $\mathscr{S}$ induced by the Euclidean embedding of~$M$. The functional depends on the small parameter $\varepsilon>0$. When $\varepsilon$ is small it is clear from the structure of the Ginzburg-Landau functional that $\left|u\right|_g$ ''prefers'' to be close to $1$. However, due to the incompatibility for vector fields on $M$ between the Sobolev regularity and the unit norm constraint, when $\varepsilon$ is close to $0$, it is expected that a finite number of singular points (called vortices) having non-zero index emerges (when the Euler characteristic is non-zero). This intuitive picture has been made precise in the recent work by R. Ignat \& R. Jerrard [7]. In this paper we are interested the dynamics of vortices generated by $F_\varepsilon^{\mathrm{extr}}$. To this end we study the behavior when $\varepsilon\to 0$ of the solutions of the (properly rescaled) gradient flow of $F_\varepsilon^{\mathrm{extr}}$. In the limit $\varepsilon\to 0$ we obtain the effective dynamics of the vortices. The dynamics, as expected, is influenced by both the intrinsic and extrinsic properties of the surface $M\subseteq\mathbb{R}^3$.

math.AP↗

arXiv2020

On a fractional thin film equation

This paper deals with a nonlinear degenerate parabolic equation of order $α$ between 2 and 4 which is a kind of fractional version of the Thin Film Equation. Actually, this one corresponds to the limit value $α=4$ while the Porous Medium Equation is the limit $α=2$. We prove existence of a nonnegative weak solution for a general class of initial data, and establish its main properties. We also construct the special solutions in self-similar form which turn out to be explicit and compactly supported. As in the porous medium case, they are supposed to give the long time behaviour or the wide class of solutions. This last result is proved to be true under some assumptions. Lastly, we consider nonlocal equations with the same nonlinear structure but with order from 4 to 6. For these equations we construct self-similar solutions that are positive and compactly supported, thus contributing to the higher order theory.

math.AP↗

arXiv2018

Weighted Energy-Dissipation principle for gradient flows in metric spaces

This paper develops the so-called Weighted Energy-Dissipation (WED) variational approach for the analysis of gradient flows in metric spaces. This focuses on the minimization of the parameter-dependent global-in-time functional of trajectories \[ \mathcal{I}_\varepsilon[u] = \int_0^{\infty} e^{-t/\varepsilon}\left( \frac12 |u'|^2(t) + \frac1{\varepsilon}ϕ(u(t)) \right) \dd t, \] featuring the weighted sum of energetic and dissipative terms. As the parameter $\varepsilon$ is sent to~$0$, the minimizers $u_\varepsilon$ of such functionals converge, up to subsequences, to curves of maximal slope driven by the functional $ϕ$. This delivers a new and general variational approximation procedure, hence a new existence proof, for metric gradient flows. In addition, it provides a novel perspective towards relaxation.

math.AP↗

arXiv2018

Convergence of solutions for the fractional Cahn-Hilliard system

This paper deals with the Cauchy-Dirichlet problem for the fractional Cahn-Hilliard equation. The main results consist of global (in time) existence of weak solutions, characterization of parabolic smoothing effects (implying under proper condition eventual boundedness of trajectories), and convergence of each solution to a (single) equilibrium. In particular, to prove the convergence result, a variant of the so-called Łojasiewicz-Simon inequality is provided for the fractional Dirichlet Laplacian and (possibly) non-analytic (but $C^1$) nonlinearities.

math.AP↗

arXiv2017

Convergent tangent plane integrators for the simulation of chiral magnetic skyrmion dynamics

We consider the numerical approximation of the Landau-Lifshitz-Gilbert equation, which describes the dynamics of the magnetization in ferromagnetic materials. In addition to the classical micromagnetic contributions, the energy comprises the Dzyaloshinskii-Moriya interaction, which is the most important ingredient for the enucleation and the stabilization of chiral magnetic skyrmions. We propose and analyze three tangent plane integrators, for which we prove (unconditional) convergence of the finite element solutions towards a weak solution of the problem. The analysis is constructive and also establishes existence of weak solutions. Numerical experiments demonstrate the applicability of the methods for the simulation of practically relevant problem sizes.

math.NA↗

arXiv2017

A gradient flow approach to the porous medium equation with fractional pressure

We consider a family of fractional porous media equations, recently studied by Caffarelli and Vázquez. We show the construction of a weak solution as Wasserstein gradient flow of a square fractional Sobolev norm. Energy dissipation inequality, regularizing effect and decay estimates for the $L^p$ norms are established. Moreover, we show that a classical porous medium equation can be obtained as a limit case.

math.AP↗

arXiv2017

Defects in Nematic Shells: a $Γ$-convergence discrete-to-continuum approach

In this paper we rigorously investigate the emergence of defects on Nematic Shells with genus different from one. This phenomenon is related to a non trivial interplay between the topology of the shell and the alignment of the director field. To this end, we consider a discrete $XY$ system on the shell $M$, described by a tangent vector field with unit norm sitting at the vertices of a triangulation of the shell. Defects emerge when we let the mesh size of the triangulation go to zero, namely in the discrete-to-continuum limit. In this paper we investigate the discrete-to-continuum limit in terms of $Γ$-convergence in two different asymptotic regimes. The first scaling promotes the appearance of a finite number of defects whose charges are in accordance with the topology of shell $M$, via the Poincaré-Hopf Theorem. The second scaling produces the so called Renormalized Energy that governs the equilibrium of the configurations with defects.

math.AP↗

arXiv2016

Analysis of a variational model for nematic shells

We analyze an elastic surface energy which was recently introduced by G. Napoli and L.Vergori to model thin films of nematic liquid crystals. We show how a novel approach that takes into account also the extrinsic properties of the surfaces coated by the liquid crystal leads to considerable differences with respect to the classical intrinsic energy. Our results concern three connected aspects: i) using methods of the calculus of variations, we establish a relation between the existence of minimizers and the topology of the surface; ii) we prove, by a Ginzburg-Landau approximation, the well-posedness of the gradient flow of the energy; iii) in the case of a parametrized axisymmetric torus we obtain a stronger characterization of global and local minimizers, which we supplement with numerical experiments.

math-ph↗

arXiv2016

Quantitative estimates on localized finite differences for the fractional Poisson problem, and applications to regularity and spectral stability

We establish new quantitative estimates for localized finite differences of solutions to the Poisson problem for the fractional Laplace operator with homogeneous Dirichlet conditions of solid type settled in bounded domains satisfying the Lipschitz cone regularity condition. We then apply these estimates to obtain (i)~regularity results for solutions of fractional Poisson problems in Besov spaces; (ii)~quantitative stability estimates for solutions of fractional Poisson problems with respect to domain perturbations; (iii)~quantitative stability estimates for eigenvalues and eigenfunctions of fractional Laplace operators with respect to domain perturbations.

math.AP↗

arXiv2015

On a singular heat equation with dynamic boundary conditions

In this paper we analyze a nonlinear parabolic equation characterized by a singular diffusion term describing very fast diffusion effects. The equation is settled in a smooth bounded three-dimensional domain and complemented with a general boundary condition of dynamic type. This type of condition prescribes some kind of mass conservation; hence extinction effects are not expected for solutions that emanate from strictly positive initial data. Our main results regard existence of weak solutions, instantaneous regularization properties, long-time behavior, and, under special conditions, uniqueness.

math.AP↗

arXiv2015

Morse's index formula in VMO for compact manifolds with boundary

In this paper, we study Vanishing Mean Oscillation vector fields on a compact manifold with boundary. Inspired by the work of Brezis and Niremberg, we construct a topological invariant - the index - for such fields, and establish the analogue of Morse's formula. As a consequence, we characterize the set of boundary data which can be extended to nowhere vanishing VMO vector fields. Finally, we show briefly how these ideas can be applied to (unoriented) line fields with VMO regularity, thus providing a reasonable framework for modelling a surface coated with a thin film of nematic liquid crystals.

math.FA↗