A kinetic model for polytropic gases with internal energy

We consider a mixture of N ideal, polytropic gases. Each species is described by a distribution function f_i(t, x, v, I) ≥ 0, 1 ≤ i ≤ N, defined on , and its evolution is governed by a Boltzmann-type equation. In order to recover the energy law of polytropic gases, the authors of [4] proposed a kinetic model in the framework of a weighted L¹ space. Another approach has been developed in [3] in the context of polyatomic gases. Following this previous lead, our model provides a L² framework in both variables v and I, to eventually perform a mathematical study of the diffusion asymptotics, as it was done in [2] for a model without energy exchange. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A quantum model with one fermionic degree of freedom

A quantum model with one fermionic degree of freedom is discussed in detail. The operator action of the model has local operator gauge symmetry. A group of constrains on operator gauge potentialB ₀ and gauge transformation operatorU from some physical requirement are obtained. The Euler-Lagrange equation of motion of fermionic operator φ is just the usual equation of motion of fermion type. And the Euler-Lagrange equation of motion of operator gauge potentialB ₀ is just a constraint, which is just. the canonical quantization condition of fermion.     

Hypocoercivity for kinetic equations with linear relaxation terms

This Note is devoted to a simple method for proving the hypocoercivity associated to a kinetic equation involving a linear time relaxation operator. It is based on the construction of an adapted Lyapunov functional satisfying a Gronwall-type inequality. The method clearly distinguishes the coercivity at microscopic level, which directly arises from the properties of the relaxation operator, and a spectral gap inequality at the macroscopic level for the spatial density, which is connected to the diffusion limit. It improves on previously known results. Our approach is illustrated by the linear BGK model and a relaxation operator which corresponds at macroscopic level to the linearized fast diffusion. To cite this article: J. Dolbeault et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Ill-posedness and local ill-posedness concepts in hilbert spaces *

In this paper, we study ill-posedness concepts of nonlinear and linear operator equations in a Hilbert space setting. Such ill-posedness information may help to select appropriate optimization approaches for the stable approximate solution of inverse problems, which are formulated by the operator equations. We define local ill-posedness of a nonlinear operator equation F(x) = y ₀ in a solution point x ₀:and consider the interplay between the nonlinear problem and its linearization using the Fréchet derivative F′(x ₀). To find a corresponding ill-posedness concept for the linearized equation we define intrinsic ill-posedness for linear operator equations A x = y and compare this approach with the ill-posedness definitions due to Hadamard and Nashed     

Existence of Positive Solutions for Operator Equations and Applications to Semipositone Problems

In this paper we study the existence of positive solutions of the following operator equation in a Banach space E: where G(x, λ) = λKFx+e₀, K: E↦ E is a linear completely continuous operator, F: P↦ E is a nonlinear continuous , bounded operator, e₀∈ E, λ is a parameter and P is a cone of Banach space E. Since F is not assumed to be positive and e₀ may be a negative element, the operator equation is a so-called semipositone problem. We prove that under certain super-linear conditions on the operator F the operator equation has at least one positive solution for λ > 0 sufficiently small, and that under certain sub-linear conditions on the operator F the operator equation has at least one positive solution for λ > 0 sufficiently large. In addition, we briefly outline an application of our results which simplify previous theorems in the literature.     

Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations

We study the asymptotic behavior of linear evolution equations of the type t_∂g=Dg+Lg−λg, where L is the fragmentation operator, D is a differential operator, and λ is the largest eigenvalue of the operator Dg+Lg. In the case Dg=−x_∂g, this equation is a rescaling of the growth-fragmentation equation, a model for cellular growth; in the case Dg=−x_∂(xg), it is known that λ=1 and the equation is the self-similar fragmentation equation, closely related to the self-similar behavior of solutions of the fragmentation equation t_∂f=Lf.By means of entropy–entropy dissipation inequalities, we give general conditions for g to converge exponentially fast to the steady state G of the linear evolution equation, suitably normalized. In other words, the linear operator has a spectral gap in the natural L² space associated to the steady state. We extend this spectral gap to larger spaces using a recent technique based on a decomposition of the operator in a dissipative part and a regularizing part.     

Nonlocal fractional elliptic equations and applications

The L_p-coercive properties of a nonlocal fractional elliptic equation is studied. Particularly, it is proved that the fractional elliptic operator generated by this equation is sectorial in L_p space and also is a generator of an analytic semigroup. Moreover, by using the L_p-separability properties of the given elliptic operator the maximal regularity of the corresponding nonlocal fractional parabolic equation is established.     

A classification of well-posed kinetic layer problems

In the first part of this paper, we study the half space boundary value problem for the Boltzmann equation with an incoming distribution, obtained when considering the boundary layer arising in the kinetic theory of gases as the mean free path tends to zero. We linearize it about a drifting Maxwellian and prove that, as conjectured by Cercignani [4], the problem is well-posed when the drift velocity u exceeds the sound speed c, but that one (respectively four, five) additional conditions must be imposed when 0 < u < c (respectively ?c < u < 0 and u < ?c). In the second part, we show that the well-posedness and the asymptotic behavior results for kinetic layers equations with prescribed incoming flux can be extended to more general and realistic boundary conditions.     

The mean field kinetic equation for interacting particle systems with non-Lipschitz force

In this paper, we prove the global existence of the weak solution to the mean field kinetic equation derived from the N-particle Newtonian system. For L¹∩L^∞ initial data, the solvability of the mean field kinetic equation can be obtained by using uniform estimates and compactness arguments while the difficulties arising from the nonlocal nonlinear interaction are tackled appropriately using the Aubin-Lions compact embedding theorem.     

The Schr?dinger equation on cylinders and the n-torus

In this paper, we study the solutions to the Schr?dinger equation on some conformally flat cylinders and on the n-torus. First, we apply an appropriate regularization procedure. Using the Clifford algebra calculus with an appropriate Witt basis, the solutions can be expressed as multiperiodic eigensolutions to the regularized parabolic-type Dirac operator. We study their fundamental properties, give representation formulas of all these solutions in terms of multiperiodic generalizations of the elliptic functions in the context of the regularized parabolic-type Dirac operator. Furthermore, we also develop some integral representation formulas. In particular, we set up a Green type integral formula for the solutions to the homogeneous regularized Schr?dinger equation on cylinders and n-tori. Then, we treat the inhomogeneous Schr?dinger equation with prescribed boundary conditions in Lipschitz domains on these manifolds. We present an L _p -decomposition where one of the components is the kernel of the first-order differential operator that factorizes the cylindrical (resp. toroidal) Schr?dinger operator. Finally, we study the behavior of our results in the limit case where the regularization parameter tends to zero.     

A simple and efficient kinetic spanner

We present a new and simple (1+ε)-spanner of size O(n/ε²) for a set of n points in the plane, which can be maintained efficiently as the points move. Assuming the trajectories of the points can be described by polynomials whose degrees are at most s, the number of topological changes to the spanner is O((n/ε²)λ_s+2(n)), and at each event the spanner can be updated in O(1) time.     

Higher moments for kinetic equations: The Vlasov–Poisson and Fokker–Planck cases

Let us consider a solution f(x,v,t)?L¹(?^2N × [0,T]) of the kinetic equation where |v|^α+1 f_o,|v|^α ?L¹ (?2N × [0, T]) for some α< 0. We prove that f has a higher moment than what is expected. Namely, for any bounded set K_x, we have We use this result to improve the regularity of the local density ρ(x,t) = ∫?dν for the Vlasov–Poisson equation, which corresponds to g = E?, where E is the force field created by the repartition ? itself. We also apply this to the Bhatnagar-Gross-;Krook model with an external force, and we prove that the solution of the Fokker-Pianck equation with a source term in L² belongs to L²([0, T]; H^1/2(?)).     

Global hypoelliptic estimates for fractional order kinetic equation

In this paper we study a class of fractional order kinetic equation, which is a linear model of spatially inhomogeneous Boltzmann equation without angular cutoff. Using the multiplier method introduced by F. Hérau and K. Pravda‐Starov (J. Math. Pures et Appl., 2011), we establish the optimal global hypoelliptic estimates with weights for the linear model operator.     

Trace theorems and kinetic equations for non divergence free external forces

For a general class of time dependent linear Boltzmann type equations with (i) an external, non divergence free force terma a ?u/?ξ (ii) a collision term which can be written as the difference of a gain term involving a general nonnegative "collision frequencyn h(x,ξ,t) and a loss term involving an arbitrary bounded linear operator J, and (iii) a general boundary operator K which is a (strict) contraction, the method of characteristics and perturbation techniques are used to obtain the well-posed- ness of the initial-boundary value problem, provided the divergence b of a is bounded above. The functional setting is L^p, 1o-semigroup on L_p(Σdμ). The results are proven by generalizing a recently established theory of time dependent kinetic equations where the external force is divergence free with respect to velocity. Solutions on spaces of measures are discussed briefly.     

The Neumann problem for some degenerate elliptic equations

In the paper we study the equation L _u = f, where L is a degenerate elliptic operator, with Neumann boundary condition in a bounded open set μ. We prove existence and uniqueness of solutions in the space H(μ) for the Neumann problem.     

On a Class of Operator Equations

We assume that E₁ and E₂ are Banach spaces, a:E₁ E₂ is a continuous linear surjective operator, f:E₁ E₂ is a nonlinear completely continuous operator. In this paper, we study existence problems for the equation a(x)=f(x) and estimate the topological dimension dim of the set of solutions.     

The Point Spectrum and Spectral Geometry for Riemannian Manifolds with Cusps

The bifurcation of a solution of the equation f(x, λ) = 0 at the point (x₀, λ₀) is investigated. In the case that B: = –f_x(x₀, λ₀) is a FREDHOLM operator by the method of LJAPUNOV/SCHMIDT the original equation is equivalent to a system consisting of a locally uniquely solvable equation and an equation in a finit dimensional subspace, the so-called bifurcation equation. For analytical/recursion formulas are deduced to determine the locally unique solution. In the case of FREDHOLM operators B with index zero practicable criteria are given for the applicability of a theorem of IZE being a generalization of a well known theorem of KRASNOSELSKIJ.     

A kinetic method for strictly nonlinear conservation laws

Ideas from kinetic theory are used to construct a new solution method for nonlinear conservation laws of the formu ₁+f(u)_x=0. We choose a class of distribution functionsG=G(t, x, ), which are compactly supported with respect to the artificial velocity. This can be done in an optimal way, i.e. so that the-integral of the solution of the linear kinetic equationG _t+G_x=0 solves the nonlinear conservation law exactly.Introducing a time step and variousx-discretisations one easily obtains a variety of numerical schemes. Among them are interesting new methods as well as known upstream schemes, which get a new interpretation and the possibility to incorporate boundary value problems this way.