arXiv subjects
Publications and source records attributed to Mark A. Peletier.
This is a former PhD student's take on his teacher's scientific philosophy. I describe a set of 'principles' that I believe are conducive to good applied mathematics, and that I have learnt myself from observing Hans van Duijn in action.
In recent work [1] we uncovered intriguing connections between Otto's characterisation of diffusion as entropic gradient flow [16] on one hand and large-deviation principles describing the microscopic picture (Brownian motion) on the other. In this paper, we sketch this connection, show how it generalises to a wider class of systems, and comment on consequences and implications. Specifically, we connect macroscopic gradient flows with large deviation principles, and point out the potential of a bigger picture emerging: we indicate that in some non- equilibrium situations, entropies and thermodynamic free energies can be derived via large deviation principles. The approach advocated here is different from the established hydrodynamic limit passage but extends a link that is well known in the equilibrium situation.
We study a singular-limit problem arising in the modelling of chemical reactions. At finite ε > 0, the system is described by a Fokker-Planck convection-diffusion equation with a double-well convection potential. This potential is scaled by 1/ε, and in the limit ε -> 0, the solution concentrates onto the two wells, resulting into a limiting system that is a pair of ordinary differential equations for the density at the two wells. This convergence has been proved in Peletier, Savaré, and Veneroni, SIAM Journal on Mathematical Analysis, 42(4):1805-1825, 2010, using the linear structure of the equation. In this paper we re-prove the result by using solely the Wasserstein gradient-flow structure of the system. In particular we make no use of the linearity, nor of the fact that it is a second-order system. The first key step in this approach is a reformulation of the equation as the minimization of an action functional that captures the property of being a curve of maximal slope in an integrated form. The second important step is a rescaling of space. Using only the Wasserstein gradient-flow structure, we prove that the sequence of rescaled solutions is pre-compact in an appropriate topology. We then prove a Gamma-convergence result for the functional in this topology, and we identify the limiting functional and the differential equation that it represents. A consequence of these results is that solutions of the ε-problem converge to a solution of the limiting problem.
We present the second of two articles on the small volume fraction limit of a nonlocal Cahn-Hilliard functional introduced to model microphase separation of diblock copolymers. After having established the results for the sharp-interface version of the functional (arXiv:0907.2224), we consider here the full diffuse-interface functional and address the limit in which epsilon and the volume fraction tend to zero but the number of minority phases (called particles) remains O(1). Using the language of Gamma-convergence, we focus on two levels of this convergence, and derive first- and second-order effective energies, whose energy landscapes are simpler and more transparent. These limiting energies are only finite on weighted sums of delta functions, corresponding to the concentration of mass into `point particles'. At the highest level, the effective energy is entirely local and contains information about the size of each particle but no information about their spatial distribution. At the next level we encounter a Coulomb-like interaction between the particles, which is responsible for the pattern formation. We present the results in three dimensions and comment on their two-dimensional analogues.
We develop a gradient-flow framework based on the Wasserstein metric for a parabolic moving-boundary problem that models crystal dissolution and precipitation. In doing so we derive a new weak formulation for this moving-boundary problem and we show that this formulation is well-posed. In addition, we develop a new uniqueness technique based on the framework of gradient flows with respect to the Wasserstein metric. With this uniqueness technique, the Wasserstein framework becomes a complete well-posedness setting for this parabolic moving-boundary problem.
We shall prove new contraction properties of general transportation costs along nonnegative measure-valued solutions to Fokker-Planck equations in $R^d$, when the drift is a monotone (or $λ$-monotone) operator. A new duality approach to contraction estimates has been developed: it relies on the Kantorovich dual formulation of optimal transportation problems and on a variable-doubling technique. The latter is used to derive a new comparison property of solutions of the backward Kolmogorov (or dual) equation. The advantage of this technique is twofold: it directly applies to distributional solutions without requiring stronger regularity and it extends the Wasserstein theory of Fokker-Planck equations with gradient drift terms started by Jordan-Kinderlehrer-Otto (1998) to more general costs and monotone drifts, without requiring the drift to be a gradient and without assuming any growth conditions.
We study the limit of high activation energy of a special Fokker-Planck equation, known as Kramers-Smoluchowski (K-S) equation. This equation governs the time evolution of the probability density of a particle performing a Brownian motion under the influence of a chemical potential H/e. We choose H having two wells corresponding to two chemical states A and B. We prove that after a suitable rescaling the solution to (K-S) converges, in the limit of high activation energy (e -> 0), to the solution of a simple system modeling the diffusion of A and B, and the reaction A <-> B. The aim of this paper is to give a rigorous proof of Kramer's formal derivation and to embed chemical reactions and diffusion processes in a common variational framework which allows to derive the former as a singular limit of the latter, thus establishing a connection between two worlds often regarded as separate. The singular limit is analysed by means of Gamma-convergence in the space of finite Borel measures endowed with the weak-* topology.
We study the stability of layered structures in a variational model for diblock copolymer-homopolymer blends. The main step consists of calculating the first and second derivative of a sharp-interface Ohta-Kawasaki energy for straight mono- and bilayers. By developing the interface perturbations in a Fourier series we fully characterise the stability of the structures in terms of the energy parameters. In the course of our computations we also give the Green's function for the Laplacian on a periodic strip and explain the heuristic method by which we found it.
We study the H^{-1}-norm of the function 1 on tubular neighbourhoods of curves in R^2. We take the limit of small thickness epsilon, and we prove two different asymptotic results. The first is an asymptotic development for a fixed curve in the limit epsilon to 0, containing contributions from the length of the curve (at order epsilon^3), the ends (epsilon^4), and the curvature (epsilon^5). The second result is a Gamma-convergence result, in which the central curve may vary along the sequence epsilon to 0. We prove that a rescaled version of the H^{-1}-norm, which focuses on the epsilon^5 curvature term, Gamma-converges to the L^2-norm of curvature. In addition, sequences along which the rescaled norm is bounded are compact in the W^{1,2} -topology. Our main tools are the maximum principle for elliptic equations and the use of appropriate trial functions in the variational characterisation of the H^{-1}-norm. For the Gamma-convergence result we use the theory of systems of curves without transverse crossings to handle potential intersections in the limit.
We study a new formulation for the eikonal equation |grad u| =1 on a bounded subset of R^2. Considering a field P of orthogonal projections onto 1-dimensional subspaces, with divergence bounded in L^2, we prove existence and uniqueness for solutions of the equation P div P=0. We give a geometric description, comparable with the classical case, and we prove that such solutions exist only if the domain is a tubular neighbourhood of a regular closed curve. This formulation provides a useful approach to the analysis of stripe patterns. It is specifically suited to systems where the physical properties of the pattern are invariant under rotation over 180 degrees, such as systems of block copolymers or liquid crystals.
We consider a pattern-forming system in two space dimensions defined by an energy G_e. The functional G_e models strong phase separation in AB diblock copolymer melts, and patterns are represented by {0,1}-valued functions; the values 0 and 1 correspond to the A and B phases. The parameter e is the ratio between the intrinsic, material length scale and the scale of the domain. We show that in the limit (as e goes to 0) any sequence u_e of patterns with uniformly bounded energy G_e(u_e) becomes stripe-like: the pattern becomes locally one-dimensional and resembles a periodic stripe pattern of periodicity O(e). In the limit the stripes become uniform in width and increasingly straight. Our results are formulated as a convergence theorem, which states that the functional G_e Gamma-converges to a limit functional G_0. This limit functional is defined on fields of rank-one projections, which represent the local direction of the stripe pattern. The functional G_0 is only finite if the projection field solves a version of the Eikonal equation, and in that case it is the L^2-norm of the divergence of the projection field, or equivalently the L^2-norm of the curvature of the field. At the level of patterns the converging objects are the jump measures |grad(u_e)| combined with the projection fields corresponding to the tangents to the jump set. The central inequality from Peletier & Roeger, (Archive for Rational Mechanics and Analysis, to appear), provides the initial estimate and leads to weak measure-function-pair convergence. We obtain strong convergence by exploiting the non-intersection property of the jump set.
We study a new formulation for the eikonal equation |grad u| =1 on a bounded subset of R^2. Instead of a vector field grad u, we consider a field P of orthogonal projections on 1-dimensional subspaces, with div P in L^2. We prove existence and uniqueness for solutions of the equation P div P=0. We give a geometric description, comparable with the classical case, and we prove that such solutions exist only if the domain is a tubular neighbourhood of a regular closed curve. The idea of the proof is to apply a generalized method of characteristics introduced in Jabin, Otto, Perthame, "Line-energy Ginzburg-Landau models: zero-energy states", Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1 (2002), to a suitable vector field m satisfying P = m \otimes m. This formulation provides a useful approach to the analysis of stripe patterns. It is specifically suited to systems where the physical properties of the pattern are invariant under rotation over 180 degrees, such as systems of block copolymers or liquid crystals.
We study a variational model for a diblock-copolymer/homopolymer blend. The energy functional is a sharp-interface limit of a generalisation of the Ohta-Kawasaki energy. In one dimension, on the real line and on the torus, we prove existence of minimisers of this functional and we describe in complete detail the structure and energy of stationary points. Furthermore we characterise the conditions under which the minimisers may be non-unique. In higher dimensions we construct lower and upper bounds on the energy of minimisers, and explicitly compute the energy of spherically symmetric configurations.
We derive a new criterion for a real-valued function $u$ to be in the Sobolev space $W^{1,2}(\R^n)$. This criterion consists of comparing the value of a functional $\int f(u)$ with the values of the same functional applied to convolutions of $u$ with a Dirac sequence. The difference of these values converges to zero as the convolutions approach $u$, and we prove that the rate of convergence to zero is connected to regularity: $u\in W^{1,2}$ if and only if the convergence is sufficiently fast. We finally apply our criterium to a minimization problem with constraints, where regularity of minimizers cannot be deduced from the Euler-Lagrange equation.
We review and compare different computational variational methods applied to a system of fourth order equations that arises as a model of cylinder buckling. We describe both the discretization and implementation, in particular how to deal with a 1 dimensional null space. We show that we can construct many different solutions from a complex energy surface. We examine numerically convergence in the spatial discretization and in the domain size. Finally we give a physical interpretation of some of the solutions found.
Partial localization is the phenomenon of self-aggregation of mass into high-density structures that are thin in one direction and extended in the others. We give a detailed study of an energy functional that arises in a simplified model for lipid bilayer membranes. We demonstrate that this functional, defined on a class of two-dimensional spatial mass densities, exhibits partial localization and displays the `solid-like' behavior of cell membranes. Specifically, we show that density fields of moderate energy are partially localized, i.e. resemble thin structures. Deviation from a specific uniform thickness, creation of `ends', and the bending of such structures all carry an energy penalty, of different orders in terms of the thickness of the structure. These findings are made precise in a Gamma-convergence result. We prove that a rescaled version of the energy functional converges in the zero-thickness limit to a functional that is defined on a class of planar curves. Finiteness of the limit enforces both optimal thickness and non-fracture; if these conditions are met, then the limit value is given by the classical Elastica (bending) energy of the curve.
We revisit the classical problem of the buckling of a long thin axially compressed cylindrical shell. By examining the energy landscape of the perfect cylinder we deduce an estimate of the sensitivity of the shell to imperfections. Key to obtaining this is the existence of a mountain pass point for the system. We prove the existence on bounded domains of such solutions for all most all loads and then numerically compute example mountain pass solutions. Numerically the mountain pass solution with lowest energy has the form of a single dimple. We interpret these results and validate the lower bound against some experimental results available in the literature.
In this paper we consider a system of equations that describes a class of mass-conserving aggregation phenomena, including gravitational collapse and bacterial chemotaxis. In spatial dimensions strictly larger than two, and under the assumptions of radial symmetry, it is known that this system has at least two stable mechanisms of singularity formation (see e.g. M.P. Brenner et al.1999, Nonlinearity, 12, 1071-1098); one type is self-similar, and may be viewed as a trade-off between diffusion and attraction, while in the other type the attraction prevails over the diffusion and a non-self-similar shock wave results. Our main result identifies a class of initial data for which the blow-up behaviour is of the former, self-similar type. The blow-up profile is characterized as belonging to a subset of stationary solutions of the associated ordinary differential equation.