On the Cauchy Problem for the Inelastic Boltzmann Equation with External Force

In this paper, the Cauchy problem for the inelastic Boltzmann equation with external force is considered for near vacuum data. Under the assumptions on the bicharacteristic generated by external force which can be arbitrarily large, we prove the global existence of mild solution for initial data small enough with respect to the sup norm with exponential weight by using the contraction mapping theorem. Furthermore, we prove the uniform L ¹ stability of the mild solution following from the exponential decay estimate and the Gronwall’s inequality for the case of soft potentials.     

A Revision on Classical Solutions to the Cauchy Boltzmann Problem for Soft Potentials

This short note complements the recent paper of the authors (Alonso, Gamba in J. Stat. Phys. 137(5–6):1147–1165, 2009). We revisit the results on propagation of regularity and stability using L ^p estimates for the gain and loss collision operators which had the exponent range misstated for the loss operator. We show here the correct range of exponents. We require a Lebesgue’s exponent α>1 in the angular part of the collision kernel in order to obtain finiteness in some constants involved in the regularity and stability estimates. As a consequence the L ^p regularity associated to the Cauchy problem of the space inhomogeneous Boltzmann equation holds for a finite range of p≥1 explicitly determined.     

The BGK Model with External Confining Potential: Existence,Long-Time Behaviour and Time-Periodic Maxwellian Equilibria

We study global existence and long time behaviour for the inhomogeneous nonlinear BGK model for the Boltzmann equation with an external confining potential. For an initial datum f ₀≥0 with bounded mass, entropy and total energy we prove existence and strong convergence in L ¹ to a Maxwellian equilibrium state, by compactness arguments and multipliers techniques. Of particular interest is the case with an isotropic harmonic potential, in which Boltzmann himself found infinitely many time-periodic Maxwellian steady states. This behaviour is shared with the Boltzmann equation and other kinetic models. For all these systems we study the multistability of the time-periodic Maxwellians and provide necessary conditions on f ₀ to identify the equilibrium state, both in L ¹ and in Lyapunov sense. Under further assumptions on f, these conditions become also sufficient for the identification of the equilibrium in L ¹.     

A Quantum Boltzmann Equation for Haldane Statistics and Hard Forces; the Space-Homogeneous Initial Value Problem

The paper considers equations of Boltzmann type for Haldane exclusion statistics. Existence and some basic properties of the solutions are studied for the space homogeneous initial value problem with hard forces and angular cut-off. The approach uses strong L ¹ compactness. Some of the technical estimates are based on L ^∞ decay properties, and the control of the filling factor on range estimates for the solutions.     

Inviscid Limits¶of the Complex Ginzburg–Landau Equation

In the inviscid limit the generalized complex Ginzburg–Landau equation reduces to the nonlinear Schr?dinger equation. This limit is proved rigorously with H ¹ data in the whole space for the Cauchy problem and in the torus with periodic boundary conditions. The results are valid for nonlinearities with an arbitrary growth exponent in the defocusing case and with a subcritical or critical growth exponent at the level of L ² in the focusing case, in any spatial dimension. Furthermore, optimal convergence rates are proved. The proofs are based on estimates of the Schr?dinger energy functional and on Gagliardo–Nirenberg inequalities. Received: 2 April 1999 / Accepted: 29 March 2000     

Global Existence of Classical Solutions to the Fokker-Planck-BGK Equation

A kinetic model of the Fokker-Planck-Boltzmann equation is introduced by replacing the original Boltzmann collision operator with the Bhatnagar-Gross-Krook collision model (BGK collision model). This model equation, which we call the Fokker-Planck-BGK equation, has many physical features that the Fokker-Planck-Boltzmann equation possesses. We first establish an L ^∞ existence result for this equation, by which we construct the approximate solutions. Then, by means of the regularizing effects of the linear Fokker-Planck operator and L ^p estimates of local Maxwellians, we obtain some uniform estimates of the approximate solutions. Finally, combining those estimates and regularizing effects, we prove by a compactness argument that the equation has a global classical solution under rather general initial conditions. Supported by the Scientific Research Foundation of Huazhong University of Science and Technology (HUST-SRF).     

Asymptotic Stability of the Relativistic Boltzmann Equation for the Soft Potentials

In this paper it is shown that unique solutions to the relativistic Boltzmann equation exist for all time and decay with any polynomial rate towards their steady state relativistic Maxwellian provided that the initial data starts out sufficiently close in L^￥_l{L^\infty_\ell}. If the initial data are continuous then so is the corresponding solution. We work in the case of a spatially periodic box. Conditions on the collision kernel are generic in the sense of Dudyński and Ekiel-Jeżewska (Commun Math Phys 115(4):607–629, 1985); this resolves the open question of global existence for the soft potentials.     

Global Existence and Full Regularity of the Boltzmann Equation Without Angular Cutoff

We prove the global existence and uniqueness of classical solutions around an equilibrium to the Boltzmann equation without angular cutoff in some Sobolev spaces. In addition, the solutions thus obtained are shown to be non-negative and C ^∞ in all variables for any positive time. In this paper, we study the Maxwellian molecule type collision operator with mild singularity. One of the key observations is the introduction of a new important norm related to the singular behavior of the cross section in the collision operator. This norm captures the essential properties of the singularity and yields precisely the dissipation of the linearized collision operator through the celebrated H-theorem.     

Convolution Inequalities for the Boltzmann Collision Operator

We study integrability properties of a general version of the Boltzmann collision operator for hard and soft potentials in n-dimensions. A reformulation of the collisional integrals allows us to write the weak form of the collision operator as a weighted convolution, where the weight is given by an operator invariant under rotations. Using a symmetrization technique in L ^p we prove a Young’s inequality for hard potentials, which is sharp for Maxwell molecules in the L ² case. Further, we find a new Hardy-Littlewood-Sobolev type of inequality for Boltzmann collision integrals with soft potentials. The same method extends to radially symmetric, non-increasing potentials that lie in some L^s_weak{L^{s}_{weak}} or L ^s . The method we use resembles a Brascamp, Lieb and Luttinger approach for multilinear weighted convolution inequalities and follows a weak formulation setting. Consequently, it is closely connected to the classical analysis of Young and Hardy-Littlewood-Sobolev inequalities. In all cases, the inequality constants are explicitly given by formulas depending on integrability conditions of the angular cross section (in the spirit of Grad cut-off). As an additional application of the technique we also obtain estimates with exponential weights for hard potentials in both conservative and dissipative interactions.     

On the linearized relativistic Boltzmann equation

The linearized relativistic Boltzmann equation inL ²(r,p) is investigated. The detailed analysis of the collision operatorL is carried out for a wide class of scattering cross sections.L is proved to have a form of the multiplication operatorv(p) plus the compact inL ²(p) perturbationK. The collisional frequencyv(p) is analysed to discriminate between relativistic soft and hard interactions. Finally, the existence and uniqueness of the solution to the linearized relativistic Boltzmann equation is proved.     

Global existence inL 1 for the generalized Enskog equation

Various existence theorems are given for the generalized Enskog equation inR ³ and in a bounded spatial domain with periodic boundary conditions. A very general form of the geometric factorY is allowed, including an explicit space, velocity, and time dependence. The method is based on the existence of a Liapunov functional, an analog of theH-function in the Boltzmann equation, and utilizes a weak compactness argument inL ¹.     

A priori estimates for the Hill and Dirac operators

The Hill operator Ty = −y″ + q′(t)y is considered in L ²(ℝ), where q ∈ L ²(0, 1) is a periodic real potential. The spectrum of T is absolutely continuous and consists of bands separated by gaps. We obtain a priori estimates of gap lengths, effective masses, and action variables for the KDV equation. In the proof of these results, the analysis of a conformal mapping corresponding to quasimomentum of the Hill operator is used. Similar estimates for the Dirac operator are obtained.     

Distributional and Classical Solutions to the Cauchy Boltzmann Problem for Soft Potentials with Integrable Angular Cross Section

This paper focuses on the study of existence and uniqueness of distributional and classical solutions to the Cauchy Boltzmann problem for the soft potential case assuming S ^n?1 integrability of the angular part of the collision kernel (Grad cut-off assumption). For this purpose we revisit the Kaniel–Shinbrot iteration technique to present an elementary proof of existence and uniqueness results that includes the large data near local Maxwellian regime with possibly infinite initial mass. We study the propagation of regularity using a recent estimate for the positive collision operator given in (Alonso et al. in Convolution inequalities for the Boltzmann collision operator. arXiv:0902.0507 [math.AP]) , by E. Carneiro and the authors, that allows us to show such propagation without additional conditions on the collision kernel. Finally, an L ^p -stability result (with 1≤p≤∞) is presented assuming the aforementioned condition.     

Direct neutron decay of the isoscalar giant dipole resonance

The direct and statistical neutron decay of the isoscalar giant dipole resonance has been studied in ⁹⁰Zr, ¹¹⁶Sn, and ²⁰⁸Pb using the (α, α’ n) reaction at a bombarding energy of 200 MeV. The spectra of fast decay neutrons populating valence hole states of the Z, N − 1 nuclei were analyzed, and estimates for the branching ratios were determined. The observation of the nucleon-direct-decay channels helped to select giant-resonance strengths and suppress the underlying background and continuum, which led to an indication of the existence of a new mode with L = 2 character, presumably the overtone of the isoscalar giant quadrupole resonance. The text was submitted by the authors in English.     

Fundamental Solutions for the Klein-Gordon Equation in de Sitter Spacetime

In this article we construct the fundamental solutions for the Klein-Gordon equation in de Sitter spacetime. We use these fundamental solutions to represent solutions of the Cauchy problem and to prove L ^p  − L ^q estimates for the solutions of the equation with and without a source term.     

Equilibrium Solution to the Inelastic Boltzmann Equation Driven by a Particle Bath

We show the existence of smooth stationary solutions for the inelastic Boltzmann equation under the thermalization induced by a host medium with a fixed distribution. This is achieved by controlling the L ^p -norms, the moments and the regularity of the solutions to the Cauchy problem together with arguments related to a dynamical proof for the existence of stationary states.     

On Lieb-Thirring Inequalities for Schrödinger Operators with Virtual Level

We consider the operator H=−Δ−V in L₂(ℝ^d), d≥3. For the moments of its negative eigenvalues we prove the estimate Similar estimates hold for the one-dimensional operator with a Dirichlet condition at the origin and for the two-dimensional Aharonov-Bohm operator.     

Existence of the scattering matrix for the linearized Boltzmann equation

Following Hejtmanek, we consider neutrons in infinite space obeying a linearized Boltzmann equation describing their interaction with matter in some compact setD. We prove existence of theS-matrix and subcriticality of the dynamics in the (weak-coupling) case where the mean free path is larger than the diameter ofD uniform in the velocity. We prove existence of theS-matrix also for the case whereD is convex and filled with uniformly absorbent material. In an appendix, we present an explicit example where the dynamics is not invertible onL ₊ ¹ , the cone of positive elements inL ¹.A. Sloan fellow; research partially supported by the U.S. NSF under Grant GP 39048     

Scalar Curvature Deformation and a Gluing Construction for the Einstein Constraint Equations

On a compact manifold, the scalar curvature map at generic metrics is a local surjection [F-M]. We show that this result may be localized to compact subdomains in an arbitrary Riemannian manifold. The method is extended to establish the existence of asymptotically flat, scalar-flat metrics on ℝ ⁿ (n≥ 3) which are spherically symmetric, hence Schwarzschild, at infinity, i.e. outside a compact set. Such metrics provide Cauchy data for the Einstein vacuum equations which evolve into nontrivial vacuum spacetimes which are identically Schwarzschild near spatial infinity. Received: 8 November 1999 / Accepted: 27 March 2000