Global existence and decay estimates of the Boltzmann equation with frictional force: The general results

In this paper, we give the existence theory and the optimal time convergence rates of the solutions to the Boltzmann equation with frictional force near a global Maxwellian. We generalize our previous results on the same problem for hard sphere model into both hard potential and soft potential case. The main method used in this paper is the classic energy method combined with some new time–velocity weight functions to control the large velocity growth in the nonlinear term for the case of interactions with hard potentials and to deal with the singularity of the cross-section at zero relative velocity for the soft potential case.     

On the relation between rates of relaxation and convergence of wild sums for solutions of the Kac equation

In the case of Maxwellian molecules, the Wild summation formula gives an expression for the solution of the spatially homogeneous Boltzmann equation in terms of its initial data F as a sum . Here, is an average over n-fold iterated Wild convolutions of F. If M denotes the Maxwellian equilibrium corresponding to F, then it is of interest to determine bounds on the rate at which tends to zero. In the case of the Kac model, we prove that for every ε>0, if F has moments of every order and finite Fisher information, there is a constant C so that for all n, where Λ is the least negative eigenvalue for the linearized collision operator. We show that Λ is the best possible exponent by relating this estimate to a sharp estimate for the rate of relaxation of f(·,t) to M. A key role in the analysis is played by a decomposition of into a smooth part and a small part. This depends in an essential way on a probabilistic construction of McKean. It allows us to circumvent difficulties stemming from the fact that the evolution does not improve the qualitative regularity of the initial data.     

Spectral gap estimates for interacting particle systems via a Bochner-type identity

We develop a general technique, based on a Bochner-type identity, to estimate spectral gaps of a class of Markov operator. We apply this technique to various interacting particle systems. In particular, we give a simple and short proof of the diffusive scaling of the spectral gap of the Kawasaki model at high temperature. Similar results are derived for Kawasaki-type dynamics in the lattice without exclusion, and in the continuum. New estimates for Glauber-type dynamics are also obtained.     

Global a priori estimates for the inhomogeneous Landau equation with moderately soft potentials

We establish a priori upper bounds for solutions to the spatially inhomogeneous Landau equation in the case of moderately soft potentials, with arbitrary initial data, under the assumption that mass, energy and entropy densities stay under control. Our pointwise estimates decay polynomially in the velocity variable. We also show that if the initial data satisfies a Gaussian upper bound, this bound is propagated for all positive times.     

Propagation of and Maxwellian weighted bounds for derivatives of solutions to the homogeneous elastic Boltzmann equation

We consider the n-dimensional space homogeneous Boltzmann equation for elastic collisions for variable hard potentials with Grad (angular) cutoff. We prove sharp moment inequalities, the propagation of L¹-Maxwellian weighted estimates, and consequently, the propagation L^∞-Maxwellian weighted estimates to all derivatives of the initial value problem associated to the afore mentioned equation. More specifically, we extend to all derivatives of the initial value problem associated to this class of Boltzmann equations corresponding sharp moment (Povzner) inequalities and time propagation of L¹-Maxwellian weighted estimates as originally developed Bobylev [A.V. Bobylev, Moment inequalities for the Boltzmann equation and applications to spatially homogeneous problems, J. Statist. Phys. 88 (1997) 1183–1214] in the case of hard spheres in 3 dimensions. To achieve this goal we implement the program presented in Bobylev–Gamba–Panferov [A.V. Bobylev, I.M. Gamba, V. Panferov, Moment inequalities and high-energy tails for Boltzmann equation with inelastic interactions, J. Statist. Phys. 116 (5–6) (2004) 1651–1682], which includes a full analysis of the moments by means of sharp moment inequalities and the control of L¹-exponential bounds, in the case of stationary states for different inelastic Boltzmann related problems with ‘heating’ sources where high energy tail decay rates depend on the inelasticity coefficient and the type of ‘heating’ source. More recently, this work was extended to variable hard potentials with angular cutoff by Gamba–Panferov–Villani [I.M. Gamba, V. Panferov, C. Villani, Upper Maxwellian bounds for the spatially homogeneous Boltzmann equation, ARMA (2008), in press] in the elastic case collision case where the L¹-Maxwellian weighted norm was shown to propagate if initial states have such property. In addition, we also extend to all derivatives the propagation of L^∞-Maxwellian weighted estimates, proven in [I.M. Gamba, V. Panferov, C. Villani, Upper Maxwellian bounds for the spatially homogeneous Boltzmann equation, ARMA (2008), in press], to solutions of the initial value problem to the Boltzmann equations for elastic collisions for variable hard potentials with Grad (angular) cutoff.     

Spectral analysis of diffusions with jump boundary

In this paper we consider one-dimensional diffusions with constant coefficients in a finite interval with jump boundary and a certain deterministic jump distribution. We use coupling methods in order to identify the spectral gap in the case of a large drift and prove that there is a threshold drift above which the bottom of the spectrum no longer depends on the drift. As a corollary to our result we are able to answer two questions concerning elliptic eigenvalue problems with non-local boundary conditions formulated previously by Iddo Ben-Ari and Ross Pinsky.     

Spectral analysis of a family of second-order elliptic operators with nonlocal boundary condition indexed by a probability measure

Let D⊂Rd be a bounded domain and let

     

Gaussian bounds for higher-order elliptic differential operators with Kato type potentials

Let P(D)$P(D)$ be a nonnegative homogeneous elliptic operator of order 2m   with real constant coefficients on Rⁿ${\mathbb{R}}^n$ and V   be a suitable real measurable function. In this paper, we are mainly devoted to establish the Gaussian upper bound for Schrödinger type semigroup e^−tH$e^{-tH}$ generated by H=P(D)+V$H=P(D)+V$ with Kato type perturbing potential V  , which naturally generalizes the classical result for Schrödinger semigroup e^−t(Δ+V)$e^{-t(\mathrm{\Delta }+V)}$ as V∈_K2(Rⁿ)$V\in K_2({\mathbb{R}}^n)$, the famous Kato potential class. Our proof significantly depends on the analyticity of the free semigroup e^−tP(D)$e^{-tP(D)}$ on L¹(Rⁿ)$L^1({\mathbb{R}}^n)$. As a consequence of the Gaussian upper bound, the L^p$L^p$-spectral independence of H is concluded.     

A Finite-difference Solution of the Ginzburg–Landau Equation

Two finite-difference methods, which differ only in the way that they approximate the derivative boundary conditions, are developed for solving a particular form of the complex Ginzburg–Landau equation of superconductivity. The non-linear term in this equation is linearized in a way familiar to readers of Professor Mickens' work, and the numerical solution is obtained at each time step by solving a linear algebraic system. Consistency and stability are discussed and some numerical results are reported.     

Gaussian estimates for elliptic operators with unbounded drift

We consider a strictly elliptic operator

     

Acoustic limit for the Boltzmann equation in the whole space

This paper is devoted to the following rescaled Boltzmann equation in the acoustic time scaling in the whole space

(0.1)     

Hard and soft constraints for reasoning about qualitative conditional preferences

Many real life optimization problems are defined in terms of both hard and soft constraints, and qualitative conditional preferences. However, there is as yet no single framework for combined reasoning about these three kinds of information. In this paper we study how to exploit classical and soft constraint solvers for handling qualitative preference statements such as those captured by the CP-nets model. In particular, we show how hard constraints are sufficient to model the optimal outcomes of a possibly cyclic CP-net, and how soft constraints can faithfully approximate the semantics of acyclic conditional preference statements whilst improving the computational efficiency of reasoning about these statements. This material is based in part upon works supported by the Science Foundation Ireland under Grant No. 00/PI.1/C075     

Resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds III: Global-in-time Strichartz estimates without lossRésolvante et mesure spectrale sur des variétés non-captives asymptotiquement hyperboliques III : Estimations de Strichartz sans perte et globales en temps

In the present paper, we investigate global-in-time Strichartz estimates without loss on non-trapping asymptotically hyperbolic manifolds. Due to the hyperbolic nature of such manifolds, the set of admissible pairs for Strichartz estimates is much larger than usual. These results generalize the works on hyperbolic space due to Anker–Pierfelice and Ionescu–Staffilani. However, our approach is to employ the spectral measure estimates, obtained in the author's joint work with Hassell, to establish the dispersive estimates for truncated/microlocalized Schrödinger propagators as well as the corresponding energy estimates. Compared with hyperbolic space, the crucial point here is to cope with the conjugate points on the manifold. Additionally, these Strichartz estimates are applied to the $L^2$ well-posedness and $L^2$ scattering for nonlinear Schrödinger equations with power-like nonlinearity and small Cauchy data.     

Non-selfadjoint matrix Sturm-Liouville operators with spectral singularities

In this paper we investigate discrete spectrum of the non-selfadjoint matrix Sturm-Liouville operator L generated in L²(R₊,S) by the differential expression

     

Spectral radii of refinement and subdivision operators

The spectral radii of refinement and subdivision operators considered on the space can be estimated by using norms of their symbols. In several cases, including those arising in wavelet analysis, the exact value of the spectral radius is found. For example, if is the unit circle and if the symbol of a refinement operator satisfies the conditions , and then the spectral radius of this operator is equal to

     

Spectral gap for the Kac model with hard sphere collisions

We prove the analog of the Kac conjecture for hard sphere collisions, giving a computable, close estimate on the spectral gap that is independent of the number of particles. Previous work has focused on the case in which the collision rates are independent of the particle velocities, the case of so-called Maxwellian molecules. The new methods introduced here allow us to deal with collision rates that are not bounded from below. We also obtain information on the structure of the gap eigenfunction.