Papers by Danielle Hilhorst
Tunisian journal of mathematics, Dec 31, 2022
We consider an Allen-Cahn equation with nonlinear diffusion, motivated by the study of the scalin... more We consider an Allen-Cahn equation with nonlinear diffusion, motivated by the study of the scaling limit of certain interacting particle systems. We investigate its singular limit and show the generation and propagation of an interface in the limit. The evolution of this limit interface is governed by mean curvature flow with a novel, homogenized speed in terms of a surface tension-mobility parameter emerging from the nonlinearity in our equation.
HAL (Le Centre pour la Communication Scientifique Directe), Aug 10, 2017
Discrete and Continuous Dynamical Systems-series B, 2023
HAL (Le Centre pour la Communication Scientifique Directe), Jun 26, 2018
In this paper, we prove a well posedness result for an initial boundary value problem for a stoch... more In this paper, we prove a well posedness result for an initial boundary value problem for a stochastic nonlocal reaction-diffusion equation with nonlinear diffusion together with a nul-flux boundary condition in an open bounded domain of R n with a smooth boundary. We suppose that the additive noise is induced by a Q-Brownian motion. f (a + b)a ≤ −C 1 a 2p + f 2 (b), |f 2 (b)| ≤ C 2 (b 2p + 1), for all a, b ∈ R (F 2) There exist positive constants C 3 andC 3 (M) such that |f (s)| ≤ C 3 |s − M | 2p−1 +C 3 (M) 1 (F 3) There exists a positive constant C 4 such that f (s) ≤ C 4. We will check in the Appendix that the function f (s) = 2p−1 r=0 b r s r with b 2p−1 < 0, p ≥ 2 satisfies the properties (F 1) − (F 3). • We assume that A = ∇ v Ψ(v) : R n → R n for some strictly convex function Ψ ∈ C 1,1 (i.e. Ψ(v) ∈ C 1 (R n) and ∇Ψ(v) is Lipschitz-continuous) satisfying A(0) = ∇Ψ(0) = 0, Ψ(0) = 0 D 2 Ψ L ∞ (R n ;R n×n) ≤ c 1 , (1.1) for some constant c 1 > 0. We remark that (1.1) implies that |A(a) − A(b)| ≤ C|a − b| (1.2) for all a, b ∈ R n , where C is a positive constant, and that the strict convexity of Ψ implies that A is strictly monotone, namely there exists a positive constant C 0 such that (A(a) − A(b))(a − b) ≥ C 0 |a − b| 2 , (1.3) for all a, b ∈ R n. We remark that if A is the identity matrix, the nonlinear diffusion operator − div(A(∇u)) reduces to the linear operator −∆u. • The function W = W (x, t) is a Q-Brownian motion. More precisely, let Q be a nonnegative definite symmetric operator on L 2 (D), {e l } l≥1 be an orthonormal basis in L 2 (D) diagonalizing Q, and {λ l } l≥1 be the corresponding eigenvalues, so that Qe l = λ l e l for all l ≥ 1. Since Q is of trace-class, it follows that Tr Q = ∞ l=1 Qe l , e l L 2 (D) = ∞ l=1 λ l ≤ Λ 0 , (1.4) for some positive constant Λ 0. We suppose furthermore that e l ∈ H 1 (D) ∩ L ∞ (D) for l = 1, 2... and that there exist positive constants Λ 1 and Λ 2 such that ∞ l=1 λ l e l 2 L ∞ (D) ≤ Λ 1 , (1.5)
Communications in Mathematical Physics, Jun 28, 2022
We derive a continuum mean-curvature flow as a certain hydrodynamic scaling limit of a class of G... more We derive a continuum mean-curvature flow as a certain hydrodynamic scaling limit of a class of Glauber+Zero-range particle systems. The Zero-range part moves particles while preserving particle numbers, and the Glauber part governs the creation and annihilation of particles and is set to favor two levels of particle density. When the two parts are simultaneously seen in certain different timescales , the Zero-range part being diffusively scaled while the Glauber part is speeded up at a lesser rate, a mean-curvature interface flow emerges, with a homogenized 'surface tension-mobility' parameter reflecting microscopic rates, between the two levels of particle density. We use relative entropy methods, along with a suitable 'Boltzmann-Gibbs' principle, to show that the random microscopic system may be approximated by a 'discretized' Allen-Cahn PDE with nonlinear diffusion. In turn, we show the behavior, especially generation and propagation of interface properties, of this 'discretized' PDE.
HAL (Le Centre pour la Communication Scientifique Directe), Jul 13, 2023
A one-dimensional model for simulating localized corrosion is developed, incorporating time-depen... more A one-dimensional model for simulating localized corrosion is developed, incorporating time-dependent diffusion-migration transport. The model is applied to iron with the Butler-Volmer formula as the dissolution law, and both crevice and pit configurations are simulated. A finite-difference ALE scheme is used for the numerical computation of the solutions of this free boundary problem. The results show that the dissolution rate increases with chloride concentration and metal potential, and migration plays, most of the time, a significant role in species transport for both crevice and pit configurations. The evolution of the repassivation potential with pit depth is computed and is in good agreement with experimental results. The time-dependent model under consideration provides more accurate quantitative results for concentration profile and crevice depth than stationary models.
Interdisciplinary Information Sciences
European Journal of Applied Mathematics
Steel corrosion plays a central role in different technological fields. In this article, we consi... more Steel corrosion plays a central role in different technological fields. In this article, we consider a simple case of a corrosion phenomenon which describes a pure iron dissolution in sodium chloride. This article is devoted to prove rigorously that under rather general hypotheses on the initial data, the solution of this iron dissolution model converges to a self-similar profile as $t\rightarrow +\infty$ . We will do so for an equivalent formulation as presented in the book of Avner Friedman about parabolic equations (Friedman (1964) Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, NJ.). In order to prove the convergence result, we apply a comparison principle together with suitable upper and lower solutions.
SIAM Journal on Mathematical Analysis, 2020
In this paper, we study the generation of interface for the solution of the mass conserved Allen-... more In this paper, we study the generation of interface for the solution of the mass conserved Allen--Cahn equation involving a nonlocal integral term. We show that, for a rather general class of initi...
We consider a nonlocal reaction-diffusion equation with mass conservation, which was originally p... more We consider a nonlocal reaction-diffusion equation with mass conservation, which was originally proposed by Rubinstein and Sternberg as a model for phase separation in a binary mixture. We study the large time behavior of the solution and show that it converges to a stationary solution as t tends to infinity. We also evaluate the rate of convergence. In some special case, we show that the limit solution is constant.
Discrete & Continuous Dynamical Systems - A, 2019
We establish a general theory on the existence of fixed points and the convergence of orbits in o... more We establish a general theory on the existence of fixed points and the convergence of orbits in order-preserving semi-dynamical systems having a certain mass conservation property (or, equivalently, a first integral). The base space is an ordered metric space and we do not assume differentiability of the system nor do we even require linear structure in the base space. Our first main result states that any orbit either converges to a fixed point or escapes to infinity (convergence theorem). This will be shown without assuming the existence of a fixed point. Our second main result states that the existence of one fixed point implies the existence of a continuum of fixed points that are totally ordered (structure theorem). This latter result, when applied to a linear problem for which 0 is always a fixed point, automatically implies the existence of positive fixed points. Our result extends the earlier related works by Arino (1991), Mierczyński (1987) and Banaji-Angeli (2010) considerably with exceedingly simpler proofs. We apply our results to a number of problems including molecular motor models with time-periodic or autonomous coefficients, certain classes of reaction-diffusion systems and delay-differential equations.
We present some error estimates for a finite volume scheme applied to a nonlinear advection-react... more We present some error estimates for a finite volume scheme applied to a nonlinear advection-reaction-diffusion equation, where the velocity is deduced from Darcy’s law together with a uniformly parabolic equation for the liquid pressure. This system describes the transport of a contaminant in groundwater flow.
We prove a contraction in L 1 property for the solutions of a nonlinear reaction-diffusion system... more We prove a contraction in L 1 property for the solutions of a nonlinear reaction-diffusion system whose special cases include intercellular transport as well as reversible chemical reactions. Assuming the existence of stationary solutions we show that the solutions stabilize as t tends to infinity. Moreover, in the special case of linear reaction terms, we prove the existence and the uniqueness (up to a multiplicative constant) of the stationary solution.
Nonlinear Differential Equations and Applications NoDEA, 2016
A new approach is used to describe the large time behavior of the nonlocal differential equation ... more A new approach is used to describe the large time behavior of the nonlocal differential equation initially studied in [3]. Our approach is based upon the existence of infinitely many Lyapunov functionals and allows us to extend the analysis performed in [3].
European Journal of Applied Mathematics, 2015
We consider a cell growth model involving a nonlinear system of partial differential equations wh... more We consider a cell growth model involving a nonlinear system of partial differential equations which describes the growth of two types of cell populations with contact inhibition. Numerical experiments show that there is a parameter regime where, for a large class of initial data, the large time behaviour of the solutions is described by a segregated travelling wave solution with positive wave speed c. Here, the word segregated expresses the fact that the different types of cells are spatially segregated, and that the single densities are discontinuous at the moving interface which separates the two populations. In this paper, we show that, for each wave speed c > c, there exists an overlapping travelling wave solution, whose profile is continuous and no longer segregated. We also show that, for a large class of initial functions, the overlapping travelling wave solutions cannot represent the large time profile of the solutions of the system of partial differential equations. The...
Cornell University - arXiv, Dec 24, 2021
We consider an Allen-Cahn equation with nonlinear diffusion, motivated by the study of the scalin... more We consider an Allen-Cahn equation with nonlinear diffusion, motivated by the study of the scaling limit of certain interacting particle systems. We investigate its singular limit and show the generation and propagation of an interface in the limit. The evolution of this limit interface is governed by mean curvature flow with a novel, homogenized speed in terms of a surface tension-mobility parameter emerging from the nonlinearity in our equation.
We consider a variant of a nonlinear parabolic system, proposed by Gilad, von Hardenberg, Provenz... more We consider a variant of a nonlinear parabolic system, proposed by Gilad, von Hardenberg, Provenzale, Shachak and Meron, in desertification studies, in which there is a strong absorption. The system models the mutual interaction between the biomass, the soil-water content w and the surfacewater height which is diffused by means of the degenerate operator ∆hm with m ≥ 2. The main novelty in this article is that the absorption is given in terms of an exponent α ∈ (0, 1), in contrast to the case α = 1 considered in the previous literature. Thanks to this, some new qualitative behavior of the dynamics of the solutions can be justified. After proving the existence of non-negative solutions for the system with Dirichlet and Neumann boundary conditions, we demonstrate the possible extinction in finite time and the finite speed of propagation for the surfacewater height component h(t, x). Also, we prove, for the associate stationary problem, that if the precipitation datum p(x) grows near t...
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Papers by Danielle Hilhorst