The Boltzmann equation with large-amplitude initial data and specular reflection boundary condition

Duan, Renjun; Ko, Gyounghun; Lee, Dong-Hyun

doi:10.48550/arxiv.2011.01503

Cited by 5 publications

(6 citation statements)

References 17 publications

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“…By defining mild solution, low regularity L ∞ solutions have been studied after [12]. There are lots of references [2,4,5,6,8,18,19,22,23], where the low regularity solutions were studied for the cut-off type Boltzmann equation. We also refer to recent results [1,7,13,21], etc., which deal with low regularity solution of the kinetic equation whose collision operator have regularizing effect such as non-cutoff Boltzmann or Landau equation.…”

Section: Introductionmentioning

confidence: 99%

On $C^2$ solution of the free-transport equation in a disk

Ko¹,

Lee²

2021

Preprint

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The free transport operator of probability density function f (t, x, v) is one the most fundamental operator which is widely used in many areas of PDE theory including kinetic theory, in particular. When it comes to general boundary problems in kinetic theory, however, it is well-known that high order regularity is very hard to obtain in general. In this paper, we study the free transport equation in a disk with the specular reflection boundary condition. We obtain initial-boundary compatibility conditions for C 1 t,x,v and C 2 t,x,v regularity of the solution. We also provide regularity estimates.of Theorem 1.2 4. Initial-boundary compatibility condition for C 2 t,x,v 4.1. Condition for ∇ xv 4.2. Condition for ∇ vv 4.3. Condition for ∇ xx 4.4. Condition for ∇ vx 4.5. Compatibility conditions for transpose : ∇ T xv = ∇ vx and ∇ T xx = ∇ xx 4.6. Conditions including ∂ t 4.7. Proof of Theorem 1.3 5. Regularity estimate of f 5.1. First order estimates of characteristics 5.2. Second order estimates of characteristics 5.3. Proof of Theorem 1.6 References

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Section: Introductionmentioning

confidence: 99%

On $C^2$ solution of the free-transport equation in a disk

Ko¹,

Lee²

2021

Preprint

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“…x,v data. And recently, combining with idead of [25], the large amplitude problem with specular reflection in C 3 convex domain with small relative entropy (1.7) has been resolved in [14].…”

Section: Introductionmentioning

confidence: 99%

“…(Lemma 2.4.) Unlike to hard potential case ( [13], [14]), however, Γ − can be treated in another way. Nonlinear Γ − contains local term f (v), so we see factor v γ .…”

Section: Introductionmentioning

confidence: 99%

The large amplitude solution of the Boltzmann equation with soft potential

Ko,

Lee,

Park

2021

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In this paper, we deal with the (angular cut-off) Boltzmann equation with soft potential (−3 < γ < 0). In particular, we construct a unique global solution in L ∞x,v which converges to global equilibrium asymptotically provided that initial data has a large amplitude but with sufficienlty small relative entropy. Because frequency multiplier is not uniformly positive anymore, unlike hard potential case, time-involved velocity weight will be used to derive sub-exponential decay of the solution. Motivated by recent development of L 2 -L ∞ approach also, we introduce some modified estimates of quadratic nonlinear terms. Linearized collision kernel will be treated in a subtle manner to control singularity of soft potential kernel.

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“…In the end, we also mention extensive studies of the conditional regularity of solutions to the spatially inhomogeneous Boltzmann equation for general initial data in [39][40][41][42]56] by Silvestre together with his collaborators. It would be interesting to develop a self-contained theory of both existence and regularity without any extra condition on solutions to the spatially inhomogeneous non-cutoff Boltzmann equation with initial data allowing to have possibly large oscillations in space variable, for instance, see [22,23] in the cutoff case where the L ∞ norm can be arbitrarily large but the relative entropy is small enough. 1.4.…”

Section: Introductionmentioning

confidence: 99%

Gevrey regularity of mild solutions to the non-cutoff Boltzmann equation

Duan¹,

Li²,

Liu³

2021

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In the paper, for the Cauchy problem on the non-cutoff Boltzmann equation in torus, we establish the global-in-time Gevrey smoothness in velocity and space variables for a class of low-regularity mild solutions near Maxwellians with the Gevrey index depending only on the angular singularity. This together with [24] provides a self-contained well-posedness theory for both existence and regularity of global solutions for initial data of low regularity in the framework of perturbations. For the proof we treat in a subtle way the commutator between the regularization operators and the Boltzmann collision operator involving rough coefficients, and this enables us to combine the classical Hörmander's hypoelliptic techniques together with the global symbolic calculus established for the linearized Boltzmann operator so as to improve the regularity of solutions at positive time.

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The Boltzmann equation with large-amplitude initial data and specular reflection boundary condition

Cited by 5 publications

References 17 publications

On $C^2$ solution of the free-transport equation in a disk

On $C^2$ solution of the free-transport equation in a disk

The large amplitude solution of the Boltzmann equation with soft potential

Gevrey regularity of mild solutions to the non-cutoff Boltzmann equation

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