On the Cutoff Approximation for the Boltzmann Equation with Long-Range Interaction
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On the Cutoff Approximation for the Boltzmann Equation with Long-Range Interaction Ling-Bing He1
· Jin-Cheng Jiang2 · Yu-Long Zhou3
Received: 7 July 2020 / Accepted: 19 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The Boltzmann collision operator for long-range interactions is usually employed in its “weak form” in the literature. However the weak form utilizes the symmetry property of the spherical integral and thus should be understood more or less in the principle value sense especially for strong angular singularity. To study the integral in the Lebesgue sense, it is natural to define the collision operator via the cutoff approximation. In this way, we give a rigorous proof to the local well-posedness of the Boltzmann equation with the long-range interactions. The result has the following main features and innovations: (1). The initial data is not necessarily a small perturbation around equilibrium but satisfies compatible conditions. (2). A quasilinear method instead of the standard linearization method is used to prove existence and non-negativity of the solution in a suitably designed energy space depending heavily on the initial data. In such space, we derive the first uniqueness result for the equation in particular for hard potential case. Keywords Cutoff approximation · Long-range interaction · Cauchy problem Mathematics Subject Classification 35Q20 · 35A01 · 35A02 · 35A09
Communicated by Eric A. Carlen.
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Ling-Bing He [email protected] Jin-Cheng Jiang [email protected] Yu-Long Zhou [email protected]
1
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People’s Republic of China
2
Department of Mathematics, National Tsing Hua University, Hsinchu 30013, Taiwan, ROC
3
School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
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L.-B. He et al.
1 Introduction The Boltzmann collision operator Q is a bilinear operator which acts only on the velocity variable, B(v − v∗ , σ )(g∗ f − g∗ f )dσ dv∗ . (1.1) Q(g, f )(v) := R3
S2
Here we use the standard shorthands f = f (v), g∗ = g(v∗ ), f = f (v ), g∗ = g(v∗ ) where (v, v∗ ) and (v , v∗ ) are the velocities of particles before and after the collision. And v and v∗ are given by v =
|v − v∗ | v + v∗ + σ, 2 2
v∗ =
|v − v∗ | v + v∗ − σ, 2 2
σ ∈ S2 ,
(1.2)
which comes from the physical laws of elastic collision. In (1.1), the collision kernel B verifies that B(v − v∗ , σ ) = B(|v − v∗ |, cos θ ) where v−v∗ cos θ := |v−v · σ . Mathematically, one assumes ∗| B(v − v∗ , σ ) = |v − v∗ |γ b(cos θ ),
(1.3)
where b(cos θ ) sin θ ∼ θ −1−2s , 0 < s < 1, −3 < γ ≤ 2. Note that the parameter range 0 < s < 1, −3 < γ ≤ 2 covers the inverse power law model where the intermolecular force is given by U (r ) = r − p , p > 2. The angular function b is not integrable near the zero, π b(cos θ ) sin θ dθ = ∞. 0
This means that Q cannot be decomposed into the gain term and the loss term, a fact which invokes Grad’s
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