Related papers: Fractional multi-phase transitions and nonlocal mi…

The nonequilibrium phase transition in a system of diffusing, coagulating particles in the presence of a steady input and evaporation of particles is studied. The system undergoes a transition from a phase in which the average number of…

In this set of notes, we present some recent developments on the fractional Allen-Cahn equation $$ (-\Delta)^s u = u-u^3,$$ with special attention to $\Gamma$-convergence results, energy and density estimates, convergence of level sets,…

We construct the exact partition function of the Potts model on a complete graph subject to external fields with linear and nematic type couplings. The partition function is obtained as a solution to a linear diffusion equation and the free…

This paper focuses on the study of semilinear fractional diffusion-wave equations in the context of critical nonlinearities. Firstly, we address the issue of local well-posedness for the problem, examine spatial regularity, and the…

A non-isothermal phase field model that captures both displacive and diffusive phase transformations in a unified framework is presented. The model is developed in a formal thermodynamic setting, which provides guidance on admissible…

In the study of phase transitions a very few models are accessible to exact solution. In the most cases analytical simplifications have to be done or some numerical technique has to be used to get insight about their critical properties.…

First-order nonequilibrium phase transitions observed in active matter, fluid dynamics, biology, climate science, and other systems with irreversible dynamics are challenging to analyze because they cannot be inferred from a simple free…

In this paper, the mathematical properties and numerical discretizations of multiphase models that simulate the phase separation of an $N$-component mixture are studied. For the general choice of phase variables, the unisolvent property of…

Comparing with the classical local gradient flow and phase field models, the nonlocal models such as nonlocal Cahn-Hilliard equations equipped with nonlocal diffusion operator can describe more practical phenomena for modeling phase…

In this work we analyze a class of nonlinear fractional elliptic systems involving Hardy--type potentials and coupled by critical Hardy-Sobolev--type nonlinearities in $\mathbb{R}^N$. Due to the lack of compactness at the critical exponent…

This work is devoted to the study of the existence of at least one weak solution to nonlocal equations involving a general integro-differential operator of fractional type. As a special case, we derive an existence theorem for the…

In this work we study a general shape optimization problem where the state equation is given in terms of a nonlocal operator. Examples of the problems considered are monotone combinations of fractional eigenvalues. Moreover, we also analyze…

This paper studies the numerical analysis of a parameter identification problem governed by elliptic equations with power-type nonlinearity. We propose a numerical reconstruction via a suitable least-squares minimization problem based on…

This paper concerns with a class of elliptic equations on fractal domains depending on a real parameter. Our approach is based on variational methods. More precisely, the existence of at least two non-trivial weak (strong) solutions for the…

We review some fractional free boundary problems that were recently considered for modeling anomalous phase-transitions. All problems are of Stefan type and involve fractional derivatives in time according to Caputo's definition. We survey…

In statistical physics, phase transitions are arguably among the most extensively studied phenomena. In the computational approach to this field, the development of algorithms capable of estimating entropy across the entire energy spectrum…

This paper addresses the approximation of fractional harmonic maps. Besides a unit-length constraint, one has to tackle the difficulty of nonlocality. We establish weak compactness results for critical points of the fractional Dirichlet…

We discuss computational and qualitative aspects of the fractional Plateau and the prescribed fractional mean curvature problems on bounded domains subject to exterior data being a subgraph. We recast these problems in terms of energy…

In this paper we study some boundary value problems for a fractional analogue of second order elliptic equation with an involution perturbation in a rectangular domain. Theorems on existence and uniqueness of a solution of the considered…

In this paper we consider a nonlinear Fokker-Planck equation with asymptotically small parameters. It describes the diffusion of finite-size particles in the presence of a fixed distribution of obstacles in the limit of low-volume fraction.…