Global existence inL 1 for the modified nonlinear Enskog equation in ℝ3

A global existence theorem with large initial data inL ¹ is given for the modified Enskog equation in ³. The method, which is based on the existence of a Liapunov functional (analog of theH-Boltzmann theorem), utilizes a weak compactness argument inL ¹ in a similar way to the DiPerna-Lions proof for the Boltzmann equation. The existence theorem is obtained under certain condition on the behavior of the geometric factorY. The condition onY amounts to the fact that theL ¹ norm of the collision term grows linearly when the local density tends to infinity.     

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.     

Global existence proof for relativistic Boltzmann equation

The existence and causality of solutions to the relativistic Boltzmann equation inL ¹ and inL _loc ¹ are proved. The solutions are shown to satisfy physically naturala priori bounds, time-independent inL ¹. The results rely upon new techniques developed for the nonrelativistic Boltzmann equation by DiPerna and Lions.     

Small data existence for the Enskog equation inL 1

An existence theorem for the Enskog equation with small initial data is proved in anL ¹ setting. This type of result is not available for the Boltzmann equation.     

The Broadwell model for initial values inL + 1 (ℝ)

The Cauchy problem for the Broadwell model is shown to have a global mild solution for initial data inL ₊ ¹ () with smallL ¹-norm, and a local solution for arbitrary initial data inL ₊ ¹ (). For data which are small inL ¹(), the asymptotic behaviour of the solutions ast is determined. Moreover, it is shown that a global solution exists for all initial values inL ₊ ¹ () with finite entropy if theH-Theorem holds.     

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.     

Existence and uniqueness for nonlinear boundary value problems in kinetic theory

A boundary value problem for the stationary nonlinear Boltzmann equation in a slab has been examined in a weightedL space. It has been proved that the problem possesses a unique solution for boundary data small enough. The proof is based on the implicit function theorem. It has also been shown that for the linearized problem the Fredholm alternative applies.     

Coincidence theorem for the direct correlation function of hard-particle fluids

The Mayerf-function for purely hard particles of arbitrary shape satisfiesf ²(1, 2)=–f(1, 2). This relation can be introduced into the graphical expansion of the direct correlation functionc(1, 2) to obtain a graphical expression for the case of exact coincidence, in position and orientation, of two identical hard cores. The resulting expression forc(1, 1)+1 contains only graphsG fromc(1), the sum of irreducible graphs with one labeled point. Relative to its coefficient inc(1),G occurs inc(1, 1) with an additional factorR _c which is 1 for the leading graph in the expansion and of the form 2–2L(G) for all other graphs. HereL(G)=0, 1, 2,..., is a nonnegative integer. Topological analysis is used to derive an expression forL(G) in terms of the connectivity properties ofG.     

Connections withL P bounds on curvature

We show by means of the implicit function theorem that Coulomb gauges exist for fields over a ball inR ⁿ when the integralL ^n/2 field norm is sufficiently small. We then are able to prove a weak compactness theorem for fields on compact manifolds withL ^p integral norms bounded,p>n/2.     

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     

Anomalous internal conversion ofE1 transitions in outer atomic shells

The experimentalN andO shell conversion coefficients of highly hinderedE1 transitions were found to be anomalous. The discrepancies as well as those for theL andM shells were removed by using the following values of the nuclear current parameter: 9·2±0·1 for the 6·21 keV transition in¹⁸¹Ta, 8·5±0·5 or –7·5±0·5 for the 84·20 keV transition in²³¹Pa and 1·9±0·2 for the 26·38 keV one in²³⁷Np.     

Schrödinger operators withL loc p -potentials

We discuss the question of when the closure of the Schrödinger operator, –+V, acting inL ^p(R ^l,d ^lx), generates a strongly continuous contraction semigroup. We prove a series of theorems proving the stability for –:L ^pL ^p of the property of having am-accretive closure under perturbations by functions inL _loc ^q (1<pq). The connection with form sums and the Trotter product formula are considered. These results generalize earlier results of Kato, Kalf-Walter, Semenov and Beliy-Semenov in that we allow more general local singularities, including arbitrary singularities at one point, and arbitrary growth at infinity. We exploit bilinear form methods, Kato's inequality and certain properties of infinitesimal generators of contractions.     

A self-consistent kinetic modeling of a 1-D, bounded, plasma in equilibrium

A self-consistent kinetic treatment is presented here, where the Boltzmann equation is solved for a particle conserving Krook collision operator. The resulting equations have been implemented numerically. The treatment solves for the entire quasineutral column, making no assumptions about λ_mfp/L, where λ_mfp is the ion-neutral collision mean free path and L the size of the device. Coulomb collisions are neglected in favour of collisions with neutrals, and the particle source is modeled as a uniform Maxwellian. Electrons are treated as an inertialess but collisional fluid. The ion distribution function for the trapped and the transiting orbits is obtained. Interesting findings include the anomalous heating of ions as they approach the presheath, the development of strongly non-maxwellian features near the last λ_mfp, and strong modifications of the sheath criterion.     

Conservation of Energy,Entropy Identity,and Local Stability for the Spatially Homogeneous Boltzmann Equation

For nonsoft potential collision kernels with angular cutoff, we prove that under the initial condition f ₀(v)(1+|v|²+|logf ₀(v)|)L ¹(R ³), the classical formal entropy identity holds for all nonnegative solutions of the spatially homogeneous Boltzmann equation in the class L ([0, ); L ¹ ₂(R ³))C ¹([0, ); L ¹(R ³)) [where L ¹ _s (R ³)={ff(v)(1+|v|²) ^s/2L ¹(R ³)}], and in this class, the nonincrease of energy always implies the conservation of energy and therefore the solutions obtained all conserve energy. Moreover, for hard potentials and the hard-sphere model, a local stability result for conservative solutions (i.e., satisfying the conservation of mass, momentum, and energy) is obtained. As an application of the local stability, a sufficient and necessary condition on the initial data f ₀ such that the conservative solutions f belong to L ¹ _loc([0, ); L ¹ ₂₊ (R ³)) is also given.     

Realization of a unique time evolution unitary operator in Klein Gordon theory

The scattering theory for the Klein Gordon equation, with time-dependent potential and in a non-static space-time, is considered. Using the Klein Gordon equation formulated in the Hubert spaceL ²(R ³) and the Einstein’s relativistic equation in the spaceL ²(R ³, dx) and establishing the equivalence of the vacuum states of their linearized forms in the Hubert spaceL ²(R ³) with the help of unique symmetric symplectic operator, the time evolution unitary operatorU(t) has been fixed for the Klein Gordon equation, incorporating either the positive or negative frequencies, in the infinite dimensional Hubert spaceL ²(R ³).     

Muonium formation in xenon and argon up to 60 atmospheres

Results of muon polarization studies in xenon and argon up to 60 atm are reported. In argon for pressures up to 10 atm, the muon polarization is best explained by an epithermalcharge exchange model. Above this pressure, the decrease inP _D and increase inP _L are ascribed to charge neutralization and spin exchange reactions, respectively, in the radiolysis track. Measurements with Xe/He mixtures with a xenon pressure of 1 atm indicate that the lost polarization in the pure xenon at this pressure is due to inefficient moderation of the muon. As the pressure in pure xenon is increased above 10 atm, we find thatP _L remains roughly constant andP _D begins to increase. The lost fraction may be due to the formation of a XeMu Van der Waals type complex, whileP _D is ascribed to XeMu⁺ formation. This suggests that spur processes appear to be less important in xenon than in argon.     

Dynamic models of pest propagation and pest control

This paper proposes a pest propagation model to investigate the evolution behaviours of pest aggregates. A pest aggregate grows by self-monomer birth, and it may fragment into two smaller ones. The kinetic evolution behaviours of pest aggregates are investigated by the rate equation approach based on the mean-field theory. For a system with a self-birth rate kernel I(k)=Ik and a fragmentation rate kernel L(i,j)=L, we find that the total number M₀^A(t) and the total mass of the pest aggregates M₁^A(t) both increase exponentially with time if L ≠ 0. Furthermore, we introduce two catalysis-driven monomer death mechanisms for the former pest propagation model to study the evolution behaviours of pest aggregates under pesticide and natural enemy controlled pest propagation. In the pesticide controlled model with a catalyzed monomer death rate kernel J₁(k)=J₁k, it is found that only when I<J₁B₀ (B₀ is the concentration of catalyst aggregates) can the pests be killed off. Otherwise, the pest aggregates can survive. In the model of pest control with a natural enemy, a pest aggregate loses one of its individuals and the number of natural enemies increases by one. For this system, we find that no matter how many natural enemies there are at the beginning, pests will be eliminated by them eventually.     

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.     

Global existence of solutions for a model Boltzmann equation

A model recently introduced by Ianiro and Lebowitz is shown to have a global solution for initial data having a finiteH-functional and belonging toL ¹ (L _x ). Methods previously introduced by Tartar to deal with discrete velocity models are used.     

On Diffusive Equilibria in Generalized Kinetic Theory

We investigate the solvability of equations Q(f,f)+ ² f=0 in term of nonnegative integrable densities fL ¹ ₊(R ³). Here, Q(f, f) is a generalized collision operator. If Q is the Boltzmann operator, the only solution is 0. In contrast, we show that if Q is the pseudo-Maxwellian collision operator for granular flow, then there are non -trivial weak solutions of ().