Energy method for Boltzmann equation

Liu, Tai-Ping; Yang, Tong; Yu, Shih‐Hsien

doi:10.1016/j.physd.2003.07.011

Cited by 213 publications

(201 citation statements)

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“…After the completion of the results in this paper we learned that Yang and Yu [20] have removed those restrictions and further shown a time decay rate to Maxwellian of O(t −1/2 ). The method in [19,20] makes use of techniques from Liu and S.-H. Yu [14] and Liu, Yang and S.-H. Yu [15] as well as the study in conservation laws. But there seems to be serious obstacles to generalizing the method of [19,20] to the full system (1) due to difficulties involved in controlling the electromagnetic field, in particular B(t, x).…”

Section: The Vlasov-maxwell-boltzmann Systemmentioning

confidence: 99%

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The Vlasov–Maxwell–Boltzmann System in the Whole Space

Strain

2006

Commun. Math. Phys.

101

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Abstract. The Vlasov-Maxwell-Boltzmann system is a fundamental model to describe the dynamics of dilute charged particles, where particles interact via collisions and through their self-consistent electromagnetic field. We prove the existence of global in time classical solutions to the Cauchy problem near Maxwellians. The Vlasov-Maxwell-Boltzmann SystemThe Vlasov-Maxwell-Boltzmann system is a very fundamental model to describe the dynamics of dilute charged particles (e.g. electrons and ions):(1)The initial conditions areHere F ± (t, x, v) ≥ 0 are number density functions for ions (+) and electrons (-) respectively at time t ≥ 0, positionThe constants e ± and m ± are the magnitude of their charges and masses, and c is the speed of light.The self-consistent electromagnetic field [E(t, x), B(t, x)] in (1) is coupled with F (t, x, v) through the celebrated Maxwell system:Let g A (v), g B (v) be two number density functions for two types of particles A and B with masses m i and diameters σ i (i ∈ {A, B}). In this article we consider the Boltzmann collision operator with hard-sphere interactions [3]:Here ω ∈ S 2 and the post-collisional velocities areThen the pre-collisional velocities are v, u and vice versa.

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Section: The Vlasov-maxwell-boltzmann Systemmentioning

confidence: 99%

“…And Lg = 0 if and only if g = Pg, where P was defined in (15). Furthermore there is a δ > 0 such that…”

Section: Local Solutionsmentioning

confidence: 99%

The Vlasov–Maxwell–Boltzmann System in the Whole Space

Strain

2006

Commun. Math. Phys.

101

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“…All these works have been carried out on bounded domains, essentially the torus apart from [12] where various types of boundary conditions are considered. For completeness we also refer to a recent preprint dealing with the problem in whole space [22] and the article [27] for a related non-linear energy method.…”

Section: Introductionmentioning

confidence: 99%

Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus

Mouhot¹,

Neumann²

2006

Nonlinearity

164

275

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Abstract. For a general class of linear collisional kinetic models in the torus, including in particular the linearized Boltzmann equation for hard spheres, the linearized Landau equation with hard and moderately soft potentials and the semi-classical linearized fermionic and bosonic relaxation models, we prove explicit coercivity estimates on the associated integro-differential operator for some modified Sobolev norms. We deduce existence of classical solutions near equilibrium for the full non-linear models associated, with explicit regularity bounds, and we obtain explicit estimates on the rate of exponential convergence towards equilibrium in this perturbative setting. The proof are based on a linear energy method which combines the coercivity property of the collision operator in the velocity space with transport effects, in order to deduce coercivity estimates in the whole phase space.

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“…Although there is extensive mathematical literature for the Boltzmann theory (see [1,4,8,9,15,21,31,27,28,35] and their references), much less is known for soft potentials γ < 0. Global smooth small-amplitude solutions of the Vlasov-Poisson-Boltzmann system near vacuum were constructed with −3 ≤ γ < −2 in [11].…”

mentioning

confidence: 99%

“…Our construction of the global solution, almost positive definite for the solution to (1.3), is based on a recent nonlinear energy method developed in [14,16,21] with a new a priori estimate for the linearized collision operator L itself, not its time integration [13,15,12].…”

mentioning

confidence: 99%

On the Cauchy problem of the Boltzmann and Landau equations with soft potentials

Hsiao

2007

Quart. Appl. Math.

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Energy method for Boltzmann equation

Cited by 213 publications

References 10 publications

The Vlasov–Maxwell–Boltzmann System in the Whole Space

The Vlasov–Maxwell–Boltzmann System in the Whole Space

Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus

On the Cauchy problem of the Boltzmann and Landau equations with soft potentials

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