In this paper we study the pointwise and uniform convergence, and convergence in the pseudo metric of L (R)(0, 1), for 0 < r < 1, of orthogonal series with respect to multiplicative systems with coefficients of generalized bounded variation. We also study the L-1-convergence with some Tauberian conditions.
Let (Phi,Psi) be a complementary pair of N-functions and (Theta, Upsilon) be a complementary pair of M-functions. Let L-Phi and L-Theta the be Orlicz and Orlicz sequence spaces on a measure space. We define an Orlicz space on L-Phi such that it exhibits L-Theta behavior at infinity, and we refer to it as amalgams of L-Phi and L-Theta, denoted by (L-Phi, L-Theta). We establish that (L-Phi, L-Theta) is a Banach space, and under certain conditions, we demonstrate that its dual space is (L Psi, L Upsilon). For a locally compact group G with the Haar measure mu, we define (L-Phi (G), L-Theta (G)), and we prove that if G is abelian, Phi is an element of triangle(2), and (L-Phi (G), L-Theta (G)) forms a Banach algebra with the convolution product, then G must be compact.
We prove that the function x -> zeta(cosh(x)) is strictly log-convex on (0, infinity). An application of this result leads to zeta ( root (x + 1)(y + 1) +root (x 1)(y 1)/2 ) < root zeta(x)zeta(y) (x, y > 1, x not equal y) which refines the well-known functional inequality zeta (x +y/2) zeta< root zeta(x)zeta(y) (x,y > 1, x not equal y).
The vector field method has been a central tool in the analysis of geometric wave equations since its earlier development by Klainerman in the 1980s. We give here an overview of vector field methods in the context of kinetic equations, with applications to classical systems, such as the Vlasov-Poisson system. These notes emanate from the lectures given by the author during the 11th Summer School on Methods and Models of Kinetic Theory, Pesaro, June 2022.
In this paper, we establish an inequality for the doubly warped product submanifolds of Riemannian manifold with mixed generalized quasi constant curvature. We also investigate its equality case. Finally, we construct a non trivial example of mixed generalized quasi Einstein manifold for its existence.
The derivation of effective macroscopic theories approximating microscopic systems of interacting particles is a major question in non-equilibrium statistical mechanics. In these notes we present an approximation of systems made by many fermions interacting via inverse power law potentials in the mean-field and semiclassical regimes, reviewing the material presented at the 11th summer school "Methods and Models of Kinetic Theory" held in Pesaro in June 2022. More precisely, we focus on weakly interacting fermions whose collective effect can be approximated by an averaged potential in convolution form, and review recent mean-field techniques based on second quantization approaches. As a first step we obtain a reduced description given by the time-dependent Hartree-Fock equation. As a second step we look at longer time scales where a semiclassical description starts to be relevant and approximate the many-body dynamics with the Vlasov equation, which describes the evolution of the effective probability density of particles on the one particle phase space.
We present here two methods for improving the accuracy of the solution reconstruction in particle methods. First, we recall the mathematical theory of the transport equation, which is used in the following, and the principle of the smooth particle method. We then present the LTP method, where the shape functions of the particles are transformed according to a local expansion in order to follow the local variation of the flow. This method allows to obtain a strong convergence for the reconstructed solution without extensive overlapping of the particles. We then present a second method, the FBL method, which uses the backward Lagrangian representation of the solution and also the local expansions of the flow around the particles. The convergence analysis shows a theoretical gain of at least one order of convergence compared to the LTP method.
In this paper, by giving rate of convergence under certain conditions we approximate the function by partial integrals and Norlund means of its Fourier integral. Also, in particular as corollaries, we get the rate of convergence of Harmonic means and of (C, 1) means of the Fourier integral.
We are concerned with the impulsive singular nonlinear q-Sturm-Liouville equation (with impulsive conditions at d and boundary conditions at +/-infinity) [- 1/qD(q-1)D(q) + v(eta)] y(eta) = Upsilon (eta, y),eta is an element of J, where J := J(1) boolean OR J(2), J(1) := (-infinity, d), J(2) := (d, infinity), d > 0 and y = y (eta) is a sought solution. Under various assumptions on Upsilon we prove the existence an uniqueness of solutions for this problem.
We discuss a functional kinetic theory approach to perform ensemble simulations on quantum computers. It is argued that the approach requires several hundreds of logical noiseless qubits, hence commanding major technological breakthroughs in noise correction and mitigation practices.
In this note we show that on introducing an appropriate point transformation, any solitary wave solution of a 1+1 partial differential equation (PDE) in a single unknown is equivalent to the sech(2) soliton of the Korteweg-de Vries (KdV) equation.
The 1-D steady detonation problem is studied for a mixture of four gases, undergoing a reversible bimolecular reaction. The multi- velocity and multi-temperature governing system provides an Euler level description, which is derived as a principal subsystem of the hyperbolic balance laws obtained from the 13-moment Grad approximation of the reactive Boltzmann equations. Suitable jump conditions for the onset of detonation processes are analysed and the occurrence of compatible stable sub-shocks along the travelling wave solution in the transient is detected. The results confirm the clear drawbacks in dealing with multi-component flows and with implicit constraints induced by chemical equilibrium conditions. The paper aims to highlight the key points to be sorted out to ensure the well-posedness of the detonation problem and to manage sub-shock occurrence.
This paper presents a review of the results which illustrate the interplay of macroscopic (continuum) approach and kinetic theory of gases in solving the closure problem. Continuum approach to the closure problem is based upon entropy principle, and it is limited since phenomenological coefficients cannot be explicitly determined. On the other hand, kinetic theory provides closed systems of equations as approximate solutions to the Boltzmann equation, but it is mainly limited to rarefied gases. Combined closure procedure, reviewed in this paper, proposes a systematic matching procedure in which advantages of both approaches are taken into account. It is illustrated by the classical examples of Navier-Stokes-Fourier system and 13 moments model, but also with novel applications to the multi-temperature mixture of Euler fluids.
Quantum corrections to the semiclassical drift-diffusion equation are obtained for electrons in graphene with a regularized energy-band. The derivation starts from the single-particle, single-band Wigner equation and exploits the quantum maximum entropy principle together with the classical Chapman-Enskog method. The functional calculus in phase-phase space is then used to expand the model to second order in the scaled Planck's constant. The model is shown to be singular in the limit where the regularization parameter goes to zero.
The paper is devoted to study of asymptotic properties (for large values of energy) of radially symmetric solutions to the spatially homogeneous Landau equation for Coulomb forces. The main result of the paper is the proof of propagation in time of the exponential moment of the third order and some explicit time-dependent estimates of this moment. Roughly speaking, this means that the high energy tails of the form exp[-b(t)vk] with some k >= 3 are typical for solutions of the Landau equation with initial data having compact support. A comparison with related results for similar kinetic equations is briefly discussed.
We consider a multicellular system with spatial structure described by the kinetic theory for active particles such that the microscopic state includes dependence on position and velocity, besides the biological activity of the considered populations. The changes in velocity are described by appropriate integral turning operators that include some effects like a velocity-jump process and a volume-filling effect for one population, and a random motion of particles leading to diffusion for another population. The model describes the migration of T-cells driven by cytokines and the possible lesion of particular tissues or organs resulting from inflammation in the response of the immune system within an autoimmune disease. We then derive the hydrodynamic limit of the kinetic system in a diffusive regime and obtain a diffusion-chemotaxis macroscopic model. The stability analysis of the macroscopic system without diffusion is developed, the Turing instability of the complete system and appearance of spatial patterns are investigated. Some numerical simulations are also performed in view of illustrating our theoretical results.
The purpose of this article is to review recent progress on the regularity theory for kinetic models of Fokker-Planck type. Such equations are known to be hypoelliptic, and our aim is to explain how De Giorgi-Nash-Moser iterations can be used on such problems. Most of this note is based on our recent joint work with C. Imbert, C. Mouhot and A. Vasseur [Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), Vol. XIX (2019), 253-295].
Models of the boundary conditions for the Boltzmann equation were constructed systematically in a recent paper by the present authors (K. Aoki et al., in: Phys. Rev. E 106:035306, 2022) using an iteration scheme for the half-space problem of a linear kinetic equation describing the behavior of gas and physisorbed molecules in a thin layer adjacent to a solid surface (physisorbate layer). In the present paper, special attention is focused on the model based on the second iteration that was only touched on in the above reference. The model is presented in an explicit form, and its properties are investigated numerically. In particular, it is shown by the comparison with the numerical solution of the half-space problem that the model is a significant improvement compared with the model based on the first iteration and is accurate and useful.
This paper is inspired by that of Leonetti, Russo and Somaglia [Dense lineability and spaceability in certain subsets of pound, Bull. London Math. Soc. 55 (2023), 2283-2303] and the lineability problems raised therein. It concerns the properties of pound subsets defined by cluster points of sequences. Using the fact that the set of cluster points of a sequence x depends only on its equivalence class in l(infinity)/c(0) and that the quotient space l(infinity)/c(0) is isometrically isomorphic to C(beta N \ N), we are able to translate lineability problems from l(infinity) to C(beta N \ N). We prove that for a compact space K with properties similar to those of beta N \ N, the sets of continuous functions f in C(K) with |rng(f ) = omega and those f with |rng(f ) = c contain, up to zero function, an isometric copy of c(0)(kappa) for uncountable cardinal kappa. Specializing those results to some closed subspaces K of beta N \ N we are able to generalize known results to their ideal versions.
We present a complete analysis of the dispersion relations for longitudinal and transverse waves in a rarefied polyatomic gas based on Rational Extended Thermodynamics (RET), which describes the evolution of a non-polytropic gas in nonequilibrium. Observability of the second mode of the longitudinal wave and the transverse wave is discussed although these waves are usually not payed much attention. The cases of CO$_2$ gas and para-H$_2$ gas are specifically analyzed as typical examples.
