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We study the Fokker-Planck equation derived in the large system limit of the Markovian process describing the dynamics of quantitative traits. The Fokker-Planck equation is posed on a bounded domain and its transport and diffusion…
A Collision-Avoiding flocking particle system proposed in [8] is studied in this paper. The global wellposedness of its corresponding Vlasov-type kinetic equation is proved. As a corollary of the global stability result, the mean field…
We are interested in studying the stationary solutions and phase transitions of aggregation equations with degenerate diffusion of porous medium-type, with exponent $1 < m < \infty$. We first prove the existence of possibly infinitely many…
We consider a stochastically perturbed reaction diffusion equation in a bounded interval, with boundary conditions imposing the two stable phases at the endpoints. We investigate the asymptotic behavior of the front separating the two…
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…
A modified Poisson-Nernst-Planck system in a bounded domain with mixed Dirichlet-Neumann boundary conditions is analyzed. It describes the concentrations of ions immersed in a polar solvent and the correlated electric potential due to the…
Phase diagrams of [001] and [110] field-cooled (FC) (1-x)Pb(Mg1/3Nb2/3O3)-xPbTiO3 or PMN-xPT crystals have been constructed, based on high-resolution x-ray diffraction data. Comparisons reveal several interesting findings. First, a region…
A new solution to the mono-dimensional diffusion equation for time-variable first kind boundary condition is presented where the time-variable function at the surface is derived proposing a surface saturation model. This solution may be…
We present a derivation of the sharp-interface limit of a generic fluctuating phase-field model for solidification. As a main result, we obtain a sharp-interface projection which presents noise terms in both the diffusion equation and in…
We study solution techniques for an evolution equation involving second order derivative in time and the spectral fractional powers, of order $s \in (0,1)$, of symmetric, coercive, linear, elliptic, second-order operators in bounded domains…
This paper develops three linear and energy-stable schemes for a modified phase field crystal model with a strong nonlinear vacancy potential (VMPFC model). This sixth-order phase-field model enables realistic crystal growth simulation.…
We consider the mean field Fokker-Planck equation subject to nonlinear no-flux boundary conditions, which necessarily arise when subjecting a system of Brownian particles interacting via a pair potential in a bounded domain. With the…
We consider a lattice regularization for an ill-posed diffusion equation with trilinear constitutive law and study the dynamics of phase interfaces in the parabolic scaling limit. Our main result guarantees for a certain class of…
We investigate statistics of occupation times for an over-damped Brownian particle in an external force field. A backward Fokker-Planck equation introduced by Majumdar and Comtet describing the distribution of occupation times is solved.…
This paper presents a mathematical foundation for physical models in nonlinear optics through the lens of evolutionary equations. It focuses on two key concepts: well-posedness and exponential stability of Maxwell equations, with models…
The phase-field crystal model is by now widely used in order to predict crystal nucleation and growth. For colloidal solidification with completely overdamped individual particle motion, we show that the phase-field crystal dynamics can be…
We study an abstract family of advection-diffusion equations within the framework of the fractional Laplacian. The system involves two independent diffusion parameters: one introduced via a damping operator acting on the scalar unknown and…
This paper studies the large time behavior of aggregation-diffusion equations. For one spatial dimension with certain assumptions on the interaction potential, the diffusion index $m$, and the initial data, we prove the convergence to the…
This paper introduces a time-domain combined field integral equation for electromagnetic scattering by a perfect electric conductor. The new equation is obtained by leveraging the quasi-Helmholtz projectors, which separate both the unknown…
The long wavelength limit of a recent microscopic phase field crystal (PFC) theory of a binary alloy mix- ture is used to derive an analytical approximation for the segregation coefficient as a function of the interface velocity, and relate…