arXiv subjects

Antonio Segatti

Publications and source records attributed to Antonio Segatti.

31 records · Page 2Linked to original sources

arXiv2015

Non-existence and instantaneous extinction of solutions for singular nonlinear fractional diffusion equations

We show non-existence of solutions of the Cauchy problem in $\mathbb{R}^N$ for the nonlinear parabolic equation involving fractional diffusion $\partial_t u + (-Δ)^s ϕ(u)= 0,$ with $0 0$, or $ϕ(u) = \log u$, and we take nonnegative $L^1$ initial data, there is no (nonnegative) solution of the problem in any dimension $N\ge 2$. We find the range of non-existence when $N=1$ in terms of $s$ and $n$. The range of exponents that we find for non-existence both for parabolic and elliptic equations are optimal. Non-existence is then proved for more general nonlinearities $ϕ$, and it is also extended to the related elliptic problem of nonlinear nonlocal type: $u + (-Δ)^s ϕ(u) = f$ with the same type of nonlinearity $ϕ$.

math.AP↗

arXiv2015

Fractional Cahn-Hilliard, Allen-Cahn and porous medium equations

We introduce a fractional variant of the Cahn-Hilliard equation settled in a bounded domain $Ω$ of $R^N$ and complemented with homogeneous Dirichlet boundary conditions of solid type (i.e., imposed in the entire complement of $Ω$). After setting a proper functional framework, we prove existence and uniqueness of weak solutions to the related initial-boundary value problem. Then, we investigate some significant singular limits obtained as the order of either of the fractional Laplacians appearing in the equation is let tend to 0. In particular, we can rigorously prove that the fractional Allen-Cahn, fractional porous medium, and fractional fast-diffusion equations can be obtained in the limit. Finally, in the last part of the paper, we discuss existence and qualitative properties of stationary solutions of our problem and of its singular limits.

math.AP↗

arXiv2014

Equilibrium configurations of nematic liquid crystals on a torus

The topology and the geometry of a surface play a fundamental role in determining the equilibrium configurations of thin films of liquid crystals. We propose here a theoretical analysis of a recently introduced surface Frank energy, in the case of two-dimensional nematic liquid crystals coating a toroidal particle. Our aim is to show how a different modeling of the effect of extrinsic curvature acts as a selection principle among equilibria of the classical energy, and how new configurations emerge. In particular, our analysis predicts the existence of new stable equilibria with complex windings.

math-ph↗

arXiv2013

Modeling and analysis of a phase field system for damage and phase separation processes in solids

In this work, we analytically investigate a multi-component system for describing phase separation and damage processes in solids. The model consists of a parabolic diffusion equation of fourth order for the concentration coupled with an elliptic system with material dependent coefficients for the strain tensor and a doubly nonlinear differential inclusion for the damage function. The main aim of this paper is to show existence of weak solutions for the introduced model, where, in contrast to existing damage models in the literature, different elastic properties of damaged and undamaged material are regarded. To prove existence of weak solutions for the introduced model, we start with an approximation system. Then, by passing to the limit, existence results of weak solutions for the proposed model are obtained via suitable variational techniques.

math.AP↗

arXiv2013

The Penrose-Fife phase-field model with dynamic boundary conditions

In this paper we derive, starting from the basic principles of Thermodynamics, an extended version of the nonconserved Penrose-Fife phase transition model, in which dynamic boundary conditions are considered in order to take into account interactions with walls. Moreover, we study the well-posedness and the asymptotic behavior of the Cauchy problem for the PDE system associated to the model, allowing the phase configuration of the material to be described by a singular function.

math.AP↗

arXiv2011

Asymptotic uniform boundedness of energy solutions to the Penrose-Fife model

We study a Penrose-Fife phase transition model coupled with homogeneous Neumann boundary conditions. Improving previous results, we show that the initial value problem for this model admits a unique solution under weak conditions on the initial data. Moreover, we prove asymptotic regularization properties of weak solutions.

math.AP↗

arXiv2010

Finite dimensional reduction and convergence to equilibrium for incompressible Smectic-A liquid crystal flows

We consider a hydrodynamic system that models the Smectic-A liquid crystal flow. The model consists of the Navier-Stokes equation for the fluid velocity coupled with a fourth-order equation for the layer variable $\vp$, endowed with periodic boundary conditions. We analyze the long-time behavior of the solutions within the theory of infinite-dimensional dissipative dynamical systems. We first prove that in 2D, the problem possesses a global attractor $\mathcal{A}$ in certain phase space. Then we establish the existence of an exponential attractor $\mathcal{M}$ which entails that the global attractor $\mathcal{A}$ has finite fractal dimension. Moreover, we show that each trajectory converges to a single equilibrium by means of a suitable Lojasiewicz--Simon inequality. Corresponding results in 3D are also discussed.

math.AP↗

arXiv2010

Constant $σ_{k}$-curvature metrics with Delaunay type ends

In this paper we produce families of complete non compact Riemannian metrics with positive constant $σ_k$-curvature by performing the connected sum of a finite number of given $n$-dimensional Delaunay type solutions, provided $2 \leq 2k < n$. The problem is equivalent to solve a second order fully nonlinear elliptic equation.

math.AP↗

arXiv2009

Global attractors for gradient flows in metric spaces

We develop the long-time analysis for gradient flow equations in metric spaces. In particular, we consider two notions of solutions for metric gradient flows, namely energy and generalized solutions. While the former concept coincides with the notion of curves of maximal slope, we introduce the latter to include limits of time-incremental approximations constructed via the Minimizing Movements approach. For both notions of solutions we prove the existence of the global attractor. Since the evolutionary problems we consider may lack uniqueness, we rely on the theory of generalized semiflows introduced by J.M. Ball. The notions of generalized and energy solutions are quite flexible and can be used to address gradient flows in a variety of contexts, ranging from Banach spaces to Wasserstein spaces of probability measures. We present applications of our abstract results by proving the existence of the global attractor for the energy solutions both of abstract doubly nonlinear evolution equations in reflexive Banach spaces, and of a class of evolution equations in Wasserstein spaces, as well as for the generalized solutions of some phase-change evolutions driven by mean curvature.

math.AP↗

arXiv2009

Finite-dimensional global and exponential attractors for the reaction-diffusion problem with an obstacle potential

A reaction-diffusion problem with an obstacle potential is considered in a bounded domain of $\R^N$. Under the assumption that the obstacle $\K$ is a closed convex and bounded subset of $\mathbb{R}^n$ with smooth boundary or it is a closed $n$-dimensional simplex, we prove that the long-time behavior of the solution semigroup associated with this problem can be described in terms of an exponential attractor. In particular, the latter means that the fractal dimension of the associated global attractor is also finite.

math.AP↗

arXiv2007

Attractors for gradient flows of non convex functionals and applications

This paper addresses the long-time behavior of gradient flows of non convex functionals in Hilbert spaces. Exploiting the notion of generalized semiflows by J. M. Ball, we provide some sufficient conditions for the existence of a global attractor. The abstract results are applied to various classes of non convex evolution problems. In particular, we discuss the long-time behavior of solutions of quasi-stationary phase field models and prove the existence of a global attractor.

math.AP↗

arXiv2006

Attractors for the semiflow associated with a class of doubly nonlinear parabolic equations

A doubly nonlinear parabolic equation of the form $α(u_t)-Δu+W'(u)= f$, complemented with initial and either Dirichlet or Neumann homogeneous boundary conditions, is addressed. The two nonlinearities are given by the maximal monotone function $α$ and by the derivative $W'$ of a smooth but possibly nonconvex potential $W$; $f$ is a known source. After defining a proper notion of solution and recalling a related existence result, we show that from any initial datum emanates at least one solution which gains further regularity for $t>0$. Such "regularizing solutions" constitute a semiflow $S$ for which uniqueness is satisfied for strictly positive times and we can study long time behavior properties. In particular, we can prove existence of both global and exponential attractors and investigate the structure of $ω$-limits of single trajectories.

math.AP↗

arXiv2006

Well-posedness and long-time behavior for a class of doubly nonlinear equations

This paper addresses a doubly nonlinear parabolic inclusion of the form $A(u_t)+B(u)\ni f$. Existence of a solution is proved under suitable monotonicity, coercivity, and structure assumptions on the operators $A$ and $B$, which in particular are both supposed to be subdifferentials of functionals on $L^2(Ω)$. Moreover, under additional hypotheses on $B$, uniqueness of the solution is proved. Finally, a characterization of $ω$-limit sets of solutions is given and we investigate the convergence of trajectories to limit points.

math.AP↗