Optimal Exponential Decay for the Linearized Ellipsoidal BGK Model in Weighted Sobolev Spaces
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Optimal Exponential Decay for the Linearized Ellipsoidal BGK Model in Weighted Sobolev Spaces Fucai Li1 · Baoyan Sun2 Received: 31 January 2020 / Accepted: 15 June 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This paper deals with the asymptotic behavior of solution to the linearized ellipsoidal BGK model in torus. We prove that the solution converges exponentially to the equilibrium in the weighted Sobolev spaces with polynomial weight. Our exponential decay rate e−λt is optimal in the sense that λ > 0 equals to the spectral gap of the linearized operator in the standard Hilbert space. Our strategy is taking advantage of the quantitative spectral gap estimates in a smaller reference Hilbert space, the factorization method, and the enlargement of the functional space for the associated semigroup. Keywords Ellipsoidal BGK model · Polynomial weight · Spectral gap · Exponential rate Mathematics Subject Classification 82C40 · 47H20 · 35B40
1 Introduction and Main Results In this paper we consider the following ellipsoidal BGK model [20]: ∂t f + v · ∇ x f = A ν M ν ( f ) − f ,
(1.1)
f (0, x, v) = f in (x, v) ≥ 0.
(1.2)
In (1.1), the unknown function f = f (t, x, v) ≥ 0 is the distribution function for the gas particles with the time t ≥ 0, the position x ∈ T3 and the velocity v ∈ R3 . Aν denotes the collision frequency:
Communicated by Isabelle Gallagher.
B
Baoyan Sun [email protected] Fucai Li [email protected]
1
Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
2
School of Mathematics and Information Sciences, Yantai University, Yantai 264005, People’s Republic of China
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F. Li and B. Sun
Aν =
1 ρT , ν ∈ − ,1 . 1−ν 2
The nonisotropic Gaussian Mν is defined as follows: Mν ( f ) = √
(v−u)T Tν−1 (v−u) ρ 2 e− , det(2π Tν )
where ρ denotes the macroscopic density, u the bulk velocity, the superscript T denotes the transposition, and the temperature tensor Tν is defined as the linear combination of the temperature T and the stress tensor : Tν = (1 − ν)T Id + ν
⎛
⎞ ν12 ν13 (1 − ν)T + ν11 ⎠, ν21 (1 − ν)T + ν22 ν23 =⎝ ν31 ν32 (1 − ν)T + ν33
(1.3)
where Id is the 3 × 3 identity matrix. Here ρ, u, T , and are defined by
f (v) dv, ρ(t, x) = R3
1 v f (v) dv, u(t, x) = ρ R3
1 T (t, x) = f (v)|v − u|2 dv, 3ρ R3 and (t, x) =
1 ρ
R3
f (v)(v − u) ⊗ (v − u) dv,
respectively. As shown in [3], the restriction on the range of ν in (1.3) is imposed to guarantee that the temperature tensor Tν remains positive definite. We note that the temperature T is recovered as the trace of Tν : T =
1 1 1 (11 + 22 + 33 ) = tr = trTν . 3 3 3
Thanks to the property
T3 ×R3
Mν ( f ) − f {1, v, |v|2 } d x dv = 0,
we know that the ellipsoidal BGK model (1.1) satisfies the conservation of mass, momentum, and energy:
d f (t, x, v){1, v, |v|2 } d x dv = 0. dt T3 ×R3 Moreover, the entropy dissipation property, i.e. the celebrated H-theorem, also holds [3]:
d f ln f d x dv ≤ 0. dt T3 ×R3
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Optimal Exponentia
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