Algebraic Rate of Decay for the Excess Free Energy and Stability of Fronts for a Nonlocal Phase Kinetics Equation with a Conservation Law. I

This is the first of two papers devoted to the study of a nonlocal evolution equation that describes the evolution of the local magnetization in a continuum limit of an Ising spin system with Kawasaki dynamics and Kac potentials. We consider subcritical temperatures, for which there are two local equilibria, and begin the proof of a local nonlinear stability result for the minimum free energy profiles for the magnetization at the interface between regions of these two different local equilibria; i.e., the fronts. We shall show in the second paper that an initial perturbation v ₀ of a front that is sufficiently small in L ² norm, and sufficiently localized that x ² v ₀(x)² dx<, yields a solution that relaxes to another front, selected by a conservation law, in the L ¹ norm at an algebraic rate that we explicitly estimate. There we also obtain rates for the relaxation in the L ² norm and the rate of decrease of the excess free energy. Here we prove a number of estimates essential for this result. Moreover, the estimates proved here suffice to establish the main result in an important special case.on leave from     

Approximate Solutions of the Cahn-Hilliard Equation via Corrections to the Mullins-Sekerka Motion

We develop an alternative method to matched asymptotic expansions for the construction of approximate solutions of the Cahn-Hilliard equation suitable for the study of its sharp interface limit. The method is based on the Hilbert expansion used in kinetic theory. Besides its relative simplicity, it leads to calculable higher order corrections to the interface motion. An erratum to this article is available at .     

Kinetics of a Model Weakly Ionized Plasma in the Presence of Multiple Equilibria

We study, globally in time, the velocity distribution f(v,t) of a spatially homogeneous system that models a system of electrons in a weakly ionized plasma, subjected to a constant external electric field E. The density f satisfies a Boltzmann-type kinetic equation containing a fully nonlinear electron‐electron collision term as well as linear terms representing collisions with reservoir particles having a specified Maxwellian distribution. We show that when the constant in front of the nonlinear collision kernel, thought of as a scaling parameter, is sufficiently strong, then the L ¹ distance between f and a certain time-dependent Maxwellian stays small uniformly in t. Moreover, the mean and variance of this time‐dependent Maxwellian satisfy a coupled set of nonlinear ordinary differential equations that constitute the “hydrodynamical” equations for this kinetic system. This remains true even when these ordinary differential equations have non‐unique equilibria, thus proving the existence of multiple stable stationary solutions for the full kinetic model. Our approach relies on scale‐independent estimates for the kinetic equation, and entropy production estimates. The novel aspects of this approach may be useful in other problems concerning the relation between the kinetic and hydrodynamic scales globally in time. (Accepted September 3, 1996)     

Optimal hypercontractivity for fermi fields and related non-commutative integration inequalities

Optimal hypercontractivity bounds for the fermion oscillator semigroup are obtained. These are the fermion analogs of the optimal hypercontractivity bounds for the boson oscillator semigroup obtained by Nelson. In the process, several results of independent interest in the theory of non-commutative integration are established.Work supported by U.S. National Science Foundation grant no. PHY90-19433-A01.     

Mesoscopic Analysis of Droplets in Lattice Systems with Long-Range Kac Potentials

We investigate the geometry of typical equilibrium configurations for a lattice gas in a finite macroscopic domain with attractive, long range Kac potentials. We focus on the case when the system is below the critical temperature and has a fixed number of occupied sites. We connect the properties of typical configurations to the analysis of the constrained minimizers of a mesoscopic non-local free energy functional, which we prove to be the large deviation functional for a density profile in the canonical Gibbs measure with prescribed global density. In the case in which the global density of occupied sites lies between the two equilibrium densities that one would have without a constraint on the particle number, a “droplet” of the high (low) density phase may or may not form in a background of the low (high) density phase. We determine the critical density for droplet formation, and the nature of the droplet, as a function of the temperature and the size of the system, by combining the present large deviation principle with the analysis of the mesoscopic functional given in Nonlinearity 22, 2919–2952 (2009).     

Propagation of Smoothness and the Rate of Exponential Convergence to Equilibrium for a Spatially Homogeneous Maxwellian Gas

We prove an inequality for the gain term in the Boltzmann equation for Maxwellian molecules that implies a uniform bound on Sobolev norms of the solution, provided the initial data has a finite norm in the corresponding Sobolev space. We then prove a sharp bound on the rate of exponential convergence to equilibrium in a weak norm. These results are then combined, using interpolation inequalities, to obtain the optimal rate of exponential convergence in the strong L¹ norm, as well as various Sobolev norms. These results are the first showing that the spectral gap in the linearized collision operator actually does govern the rate of approach to equilibrium for the full non-linear Boltzmann equation, even for initial data that is far from equilibrium.     

Subadditivity of The Entropy and its Relation to Brascamp–Lieb Type Inequalities

We prove a general duality result showing that a Brascamp–Lieb type inequality is equivalent to an inequality expressing subadditivity of the entropy, with a complete correspondence of best constants and cases of equality. This opens a new approach to the proof of Brascamp–Lieb type inequalities, via subadditivity of the entropy. We illustrate the utility of this approach by proving a general inequality expressing the subadditivity property of the entropy on ${\mathbb {R}^n}We prove a general duality result showing that a Brascamp–Lieb type inequality is equivalent to an inequality expressing subadditivity of the entropy, with a complete correspondence of best constants and cases of equality. This opens a new approach to the proof of Brascamp–Lieb type inequalities, via subadditivity of the entropy. We illustrate the utility of this approach by proving a general inequality expressing the subadditivity property of the entropy on \mathbb Rⁿ{\mathbb {R}^n}, and fully determining the cases of equality. As a consequence of the duality mentioned above, we obtain a simple new proof of the classical Brascamp–Lieb inequality, and also a fully explicit determination of all of the cases of equality. We also deduce several other consequences of the general subadditivity inequality, including a generalization of Hadamard’s inequality for determinants. Finally, we also prove a second duality theorem relating superadditivity of the Fisher information and a sharp convolution type inequality for the fundamental eigenvalues of Schr?dinger operators. Though we focus mainly on the case of random variables in \mathbb Rⁿ{\mathbb {R}^n} in this paper, we discuss extensions to other settings as well.     

Conservative diffusions

In Nelson's stochastic mechanics, quantum phenomena are described in terms of diffusions instead of wave functions. These diffusions are formally given by stochastic differential equations with extremely singular coefficients. Using PDE methods, we prove the existence of solutions. This result provides a rigorous basis for stochastic mechanics.     

Sharp uniform convexity and smoothness inequalities for trace norms

Summary We prove several sharp inequalities specifying the uniform convexity and uniform smoothness properties of the Schatten trace idealsC _p , which are the analogs of the Lebesgue spacesL _p in non-commutative integration. The inequalities are all precise analogs of results which had been known inL _p , but were only known inC _p for special values ofp. In the course of our treatment of uniform convexity and smoothness inequalities forC _p we obtain new and simple proofs of the known inequalities forL _p .Oblatum 7-VII-1993Work partially supported by US National Science Foundation grant DMS 88-07243Work partially supported by US National Science Foundation grant DMS 92-07703Work partially supported by US National Science Foundation grant PHY90-19433 A02     

Competing symmetries,the logarithmic HLS inequality and Onofri's inequality ons n

The sharp version of the logarithmic Hardy-Littlewood-Sobolev inequality including the cases of equality is established. We then show that this implies Beckner's generalization of Onofri's inequality to arbitrary dimensions and determines the cases of equality.The work of the second author is partially supported by the U.S. National Science Foundation under Grant DMS-90-05729.