Convergence to equilibrium for linear spatially homogeneous Boltzmann equation with hard and soft potentials: A semigroup approach in L1‐spaces

We investigate the large time behavior of solutions to the spatially homogeneous linear Boltzmann equation from a semigroup viewpoint. Our analysis is performed in some (weighted) L¹‐spaces. We deal with both the cases of hard and soft potentials (with angular cut‐off). For hard potentials, we provide a new proof of the fact that, in weighted L¹‐spaces with exponential or algebraic weights, the solutions converge exponentially fast towards equilibrium. Our approach uses weak‐compactness arguments combined with recent results of the second author on positive semigroups in L¹‐spaces. For soft potentials, in L¹‐spaces, we exploit the convergence to ergodic projection for perturbed substochastic semigroup to show that, for very general initial datum, solutions to the linear Boltzmann equation converges to equilibrium in large time. Moreover, for a large class of initial data, we also prove that the convergence rate is at least algebraic. Notice that, for soft potentials, no exponential rate of convergence is expected because of the absence of spectral gap.     

Global existence and nonexistence of solutions for a system of nonlinear damped wave equations

We consider the Cauchy problem for the system of semilinear damped wave equations with small initial data:

We show that a critical exponent which classifies the global existence and the finite time blow up of solutions indeed coincides with the one to a corresponding semilinear heat systems with small data. The proof of the global existence is based on the L^p–L^q estimates of fundamental solutions for linear damped wave equations [K. Nishihara, L^p–L^q estimates of solutions to the damped wave equation in 3-dimensional space and their application, Math. Z. 244 (2003) 631–649; K. Marcati, P. Nishihara, The L^p–L^q estimates of solutions to one-dimensional damped wave equations and their application to compressible flow through porous media, J. Differential Equations 191 (2003) 445–469; T. Hosono, T. Ogawa, Large time behavior and L^p–L^q estimate of 2-dimensional nonlinear damped wave equations, J. Differential Equations 203 (2004) 82–118; T. Narazaki, L^p–L^q estimates for damped wave equations and their applications to semilinear problem, J. Math. Soc. Japan 56 (2004) 585–626]. And the blow-up is shown by the Fujita–Kaplan–Zhang method [Q. Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case, C. R. Acad. Sci. Paris 333 (2001) 109–114; F. Sun, M. Wang, Existence and nonexistence of global solutions for a nonlinear hyperbolic system with damping, Nonlinear Anal. 66 (12) (2007) 2889–2910; T. Ogawa, H. Takeda, Non-existence of weak solutions to nonlinear damped wave equations in exterior domains, Nonlinear Anal. 70 (10) (2009) 3696–3701].     

Hardy and Rellich-Type Inequalities for Metrics Defined by Vector Fields

Let {X _i }, i=1,...,m be a system of locally Lipschitz vector fields on DR ⁿ , such that the corresponding intrinsic metric is well-defined and continuous w.r.t. the Euclidean topology. Suppose that the Lebesgue measure is doubling w.r.t. the intrinsic balls, that a scaled L¹ Poincaré inequality holds for the vector fields at hand (thus including the case of Hörmander vector fields) and that the local homogeneous dimension near a point x ₀ is sufficiently large. Then weighted Sobolev–Poincaré inequalities with weights given by power of (,x ₀) hold; as particular cases, they yield non-local analogues of both Hardy and Sobolev–Okikiolu inequalities. A general argument which shows how to deduce Rellich-type inequalities from Hardy inequalities is then given: this yields new Rellich inequalities on manifolds and even in the uniformly elliptic case. Finally, applications of Sobolev–Okikiolu inequalities to heat kernel estimates for degenerate subelliptic operators and to criteria for the absence of bound states for Schrödinger operators H=–L+V are given.     

Strong convergence towards homogeneous cooling states for dissipative Maxwell models

We show the propagation of regularity, uniformly in time, for the scaled solutions of the inelastic Maxwell model for small inelasticity. This result together with the weak convergence towards the homogeneous cooling state present in the literature implies the strong convergence in Sobolev norms and in the L¹ norm towards it depending on the regularity of the initial data. The strategy of the proof is based on a precise control of the growth of the Fisher information for the inelastic Boltzmann equation. Moreover, as an application we obtain a bound in the L¹ distance between the homogeneous cooling state and the corresponding Maxwellian distribution vanishing as the inelasticity goes to zero.     

On the Strong L1 Trend to Equilibrium for the Boltzmann Equation

This paper considers the Boltzmann equation for hard and soft (k > 2) forces having an angular cutoff. The main result is strong L¹ convergence to global Maxwellians when time tends to infinity..     

Weighted Hardy and Opial-type inequalities

An integral condition on weights u and v is given which is equivalent to the boundedness of the Hardy operator between the weighted Lebesgue spaces L_u^p and L_v^q with 0 < q < 1 < p < ∞. The Hardy inequalities are applied to give easily verified weight conditions which imply inequalities of Opial type.     

The Navier–Stokes limit of the Boltzmann equation for bounded collision kernels

The present work establishes a Navier–Stokes limit for the Boltzmann equation considered over the infinite spatial domain R ³. Appropriately scaled families of DiPerna-Lions renormalized solutions are shown to have fluctuations whose limit points (in the w-L ¹ topology) are governed by Leray solutions of the limiting Navier–Stokes equations. This completes the arguments in Bardos-Golse-Levermore [Commun. Pure Appl. Math. 46(5), 667–753 (1993)] for the steady case, and in Lions-Masmoudi [Arch. Ration. Mech. Anal. 158(3), 173–193 (2001)] for the time-dependent case.Mathematics Subject Classification (2000) 35Q35, 35Q30, 82C40     

Some inequalities for algebraic polynomials with the Laguerre weight

There is a series of publications which have considered inequalities of Markov–Bernstein–Nikolskii type for algebraic polynomials with the Jacobi weight (see [N.K. Bari, A generalization of the Bernstein and Markov inequalities, Izv. Akad. Nauk SSSR Math. Ser. 18 (2) (1954) 159–176; B.D. Bojanov, An extension of the Markov inequality, J. Approx. Theory 35 (1982) 181–190; P. Borwein, T. Erdélyi, Polynomials and Polynomial Inequalities, Springer, New York, 1995; I.K. Daugavet, S.Z. Rafalson, Some inequalities of Markov–Nikolskii type for algebraic polynomials, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 1 (1972) 15–25; A. Guessab, G.V. Milovanovic, Weighted L²-analogues of Bernstein's inequality and classical orthogonal polynomials, J. Math. Anal. Appl. 182 (1994) 244–249; I.I. Ibragimov, Some inequalities for algebraic polynomials, in: V.I. Smirnov (Ed.), Fizmatgiz, 1961, Research on Modern Problems of Constructive Functions Theory; G.K. Lebed, Inequalities for polynomials and their derivatives, Dokl. Akad. Nauk SSSR 117 (4) (1957) 570–572; G.I. Natanson, To one theorem of Lozinski, Dokl. Akad. Nauk SSSR 117 (1) (1957) 32–35; M.K. Potapov, Some inequalities for polynomials and their derivatives, Vestnik Moskov. Univ. Ser. Mat. Mekh. 2 (1960); E. Schmidt, Über die nebst ihren Ableitungen orthogonalen Polynomsysteme und das zugehörige Extremum, Math. Ann. 119 (1944) 165–209; P. Turán, Remark on a theorem of Erhard Schmidt, Mathematica 2 (25) (1960) 373–378]). In this paper we find an inequality of the same type for algebraic polynomials on (0,∞) with the Laguerre weight function e^-xx^α (α>-1).     

Weighted sharp boundedness for multilinear commutators

We obtain sharp estimates for some multilinear commutators related to certain sublinear integral operators. These operators include the Littlewood-Paley operator and Marcinkiewicz operator. As an application, we obtain weighted L ^p (p > 1) inequalities and an L log L-type estimate for multilinear commutators. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 10, pp. 1419–1431, October, 2007.     

Smoothness of the Solution of the Spatially Homogeneous Boltzmann Equation without Cutoff

Abstract

For regularized hard potentials cross sections, the solution of the spatially homogeneous Boltzmann equation without angular cutoff lies in Schwartz's space 𝒮(? ^N ) for all (strictly positive) time. The proof is presented in full detail for the two-dimensional case, and for a moderate singularity of the cross section. Then we present those parts of the proof for the general case, where the dimension, or the strength of the singularity play an essential role.     

The stationary Boltzmann equation for a two‐component gas in the slab

The stationary Boltzmann equation for hard forces in the context of a two‐component gas is considered in the slab. An L¹ existence theorem is proved when one component satisfies a given indata profile and the other component satisfies diffuse reflection at the boundaries. Weak L¹ compactness is extracted from the control of the entropy production term. Trace at the boundaries are also controlled. Copyright © 2007 John Wiley & Sons, Ltd.     

Asymptotics of the solutions to theN-particle Kolmogorov-Feller equations and the asymptotics of the solution to the Boltzmann equation in the region of large deviations

We construct a representation in which the asymptotics of the solution to the Kolmogorov-Feller equation in the Fock space Γ(L ₁(ℝ ⁿ )) is of a form similar to the WKB asymptotic expansion; namely, the Boltzmann equation inL ₁(ℝ ⁿ ) plays the role of the Hamilton equations, the linearized Boltzmann equation extended to Γ(L ₁(ℝ ⁿ )) plays the role of the transport equation, and the Hamilton-Jacobi equation follows from the conservation of the total probability for the solutions of the Boltzmann equation. We also construct the asymptotics of the solution to the Boltzmann equation with small transfer of momentum; this asymptotics is given by the tunnel canonical operator corresponding to the self-consistent characteristic equation. Translated fromMatematicheskie Zametki, Vol. 58, No. 5, pp. 694–709, November, 1995. The author is deeply grateful to Prof. A. M. Chebotarev, whose assistance has made the writing of this paper possible. This work was financially supported by the International Science Foundation under grants Nos. MFO000 and MFO300.     

A robust semi-explicit difference scheme for the Kuramoto–Tsuzuki equation

In this paper, we propose a robust semi-explicit difference scheme for solving the Kuramoto–Tsuzuki equation with homogeneous boundary conditions. Because the prior estimate in L_∞-norm of the numerical solutions is very hard to obtain directly, the proofs of convergence and stability are difficult for the difference scheme. In this paper, we first prove the second-order convergence in L₂-norm of the difference scheme by an induction argument, then obtain the estimate in L_∞-norm of the numerical solutions. Furthermore, based on the estimate in L_∞-norm, we prove that the scheme is also convergent with second order in L_∞-norm. Numerical examples verify the correction of the theoretical analysis.     

A Degenerate Fourth-Order Parabolic Equation Modeling Bose-Einstein Condensation Part II: Finite-Time Blow-Up

A degenerate fourth-order parabolic equation modeling condensation phenomena related to Bose-Einstein particles is analyzed. The model can be motivated from the spatially homogeneous and isotropic Boltzmann-Nordheim equation by a formal Taylor expansion of the collision integral. It maintains some of the main features of the kinetic model, namely mass and energy conservation and condensation at zero energy. The existence of local-in-time weak solutions satisfying a certain entropy inequality is proven. The main result asserts that if a weighted L ¹ norm of the initial data is sufficiently large and the initial data satisfies some integrability conditions, the solution blows up with respect to the L ^∞ norm in finite time. Furthermore, the set of all such blow-up enforcing initial functions is shown to be dense in the set of all admissible initial data. The proofs are based on approximation arguments and interpolation inequalities in weighted Sobolev spaces. By exploiting the entropy inequality, a nonlinear integral inequality is proved which implies the finite-time blow-up property.     

Global existence and uniqueness to the Cauchy problem of the BGK equation with infinite energy

The Bhatnagar–Gross–Krook model of the Boltzmann equation is of great importance in the kinetic theory of rarefied gases. Various existence and uniqueness results have been built under the boundedness of energy. In this paper, we will establish several global existence results to the Bhatnagar–Gross–Krook equation with infinite energy. It heavily relies on a new moments lemma and a new existence and uniqueness theorem of weighted velocity‐spatial L^∞ solutions. Copyright © 2015 John Wiley & Sons, Ltd.     

Commutators of singular integrals on spaces of homogeneous type

In this work we prove some sharp weighted inequalities on spaces of homogeneous type for the higher order commutators of singular integrals introduced by R. Coifman, R. Rochberg and G. Weiss in Factorization theorems for Hardy spaces in several variables, Ann. Math. 103 (1976), 611–635. As a corollary, we obtain that these operators are bounded on L ^p (w) when w belongs to the Muckenhoupt’s class A _p , p > 1. In addition, as an important tool in order to get our main result, we prove a weighted Fefferman-Stein type inequality on spaces of homogeneous type, which we have not found previously in the literature.     

Global Well-Posedness for Higher-Order Schrödinger Equations in Weighted L2 Spaces

We obtain the global well-posedness for Schrödinger equations of higher orders in weighted L² spaces. This is based on weighted L² Strichartz estimates for the corresponding propagator with higher-order dispersion. Our method is also applied to the Airy equation which is the linear component of Korteweg-de Vries type equations.     

A large data existence result for the stationary Boltzmann equation in a cylindrical geometry

AnL ¹-existence theorem is proved for the nonlinear stationary Boltzmann equation with hard forces and no small velocity truncation—only the Grad angular cut-off-in a setting between two coaxial rotating cylinders when the indata are given on the cylinders.     

On equivalence of moduli of smoothness of polynomials in ,

It is well known that for functions , 1p∞. For general functions fL_p, it does not hold for 0<p<1, and its inverse is not true for any p in general. It has been shown in the literature, however, that for certain classes of functions the inverse is true, and the terms in the inequalities are all equivalent. Recently, Zhou and Zhou proved the equivalence for polynomials with p=∞. Using a technique by Ditzian, Hristov and Ivanov, we give a simpler proof to their result and extend it to the L_p space for 0<p∞. We then show its analogues for the Ditzian–Totik modulus of smoothness and the weighted Ditzian–Totik modulus of smoothness for polynomials with .     

Weighted Positivity of Second Order Elliptic Systems

Integral inequalities that concern the weighted positivity of a differential operator have important applications in qualitative theory of elliptic boundary value problems. Despite the power of these inequalities, however, it is far from clear which operators have this property. In this paper, we study weighted integral inequalities for general second order elliptic systems in ℝ ⁿ (n ≥ 3) and prove that, with a weight, smooth and positive homogeneous of order 2–n, the system is weighted positive only if the weight is the fundamental matrix of the system, possibly multiplied by a semi-positive definite constant matrix.