Global Well-Posedness in Spatially Critical Besov Space for the Boltzmann Equation

Duan, Renjun; Liu, Shuangqian; Xu, Jiang

doi:10.1007/s00205-015-0940-4

Cited by 22 publications

(30 citation statements)

References 34 publications

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“…Let 1 ≤ ̺, q, r ≤ ∞ and s > 0. Then we have∇ x · L ̺ T (Ḃ s p,r ) ∼ · L ̺ T ) .Finally, it follows from[12] thatLemma 5.11. Let s ∈ R and 1 ≤ ̺ 1 , ̺ 2 , p, r ≤ ∞.…”

mentioning

confidence: 76%

“…In [18], the authors conjectured that it remained still open to determine whether the regularity indices 1 + 1 2s is sharp or not. On the other hand, Duan-Liu-Xu [12] and Morimoto-Sakamoto [22] studied the Cauchy problem for the Boltzmann equation with the initial datum belonging to critical Besov space. Motivated by those works, we intend to study the inhomogeneous non-cutoff Kac equation in critical Besov space and then improve the Gelfand-Shilov regularizing properties and Gevrey regularizing properties.…”

Section: Introductionmentioning

confidence: 99%

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The Cauchy problem for the inhomogeneous non-cutoff Kac equation in critical Besov space

Cao

et al. 2020

Journal of Differential Equations

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In this work, we investigate the Cauchy problem for the spatially inhomogeneous non-cutoff Kac equation. If the initial datum belongs to the spatially critical Besov space, we can prove the well-posedness of weak solution under a perturbation framework. Furthermore, it is shown that the solution enjoys Gelfand-Shilov regularizing properties with respect to the velocity variable and Gevrey regularizing properties with respect to the position variable. In comparison with the recent result in [18], the Gelfand-Shilov regularity index is improved to be optimal. To the best of our knowledge, our work is the first one that exhibits smoothing effect for the kinetic equation in Besov spaces.

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mentioning

confidence: 76%

Section: Introductionmentioning

confidence: 99%

The Cauchy problem for the inhomogeneous non-cutoff Kac equation in critical Besov space

Cao

et al. 2020

Journal of Differential Equations

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“…Second, as pointed out in [8], the most interesting value of the index s under consideration is d/2. One may not expect to take this value if one seeks a solution in the usual Sobolev space H d/2 , since H d/2 is not embedded into L ∞ while the Besov space B d/2 2,1 is.…”

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confidence: 91%

“…Indeed, the perturbation theory of the Boltzmann equation near global Maxwellians has been well established so far. Interested readers may refer to [8] as well as [10] for an almost complete list of references on the subject. In what follows we would only mention some research works most related to our current study in this paper.…”

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confidence: 99%

Solution to the Boltzmann equation in velocity-weighted Chemin-Lerner type spaces

Duan¹,

Sakamoto²

2018

Kinetic &Amp; Related Models

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In this paper we study the Boltzmann equation near global Maxwellians in the d-dimensional whole space. A unique global-in-time mild solution to the Cauchy problem of the equation is established in a Chemin-Lerner type space with respect to the phase variable (x, v). Both hard and soft potentials with angular cutoff are considered. The new function space for global well-posedness is introduced to essentially treat the case of soft potentials, and the key point is that the velocity variable is taken in the weighted supremum norm, and the space variable is in the s-order Besov space with s ≥ d/2 including the spatially critical regularity. The proof is based on the time-decay properties of solutions to the linearized equation together with the bootstrap argument. Particularly, the linear analysis in case of hard potentials is due to the semigroup theory, where the extra time-decay plays a role in coping with initial data in L 2 with respect to the space variable. In case of soft potentials, for the time-decay of linear equations we borrow the results based on the pure energy method and further extend them to those in L ∞ framework through the technique of L 2 -L ∞ interplay. In contrast to hard potentials, L 1 integrability in x of initial data is necessary for soft potentials in order to obtain global solutions to the nonlinear Cauchy problem.

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“…In this paper, we consider (1.1) without angular cutoff by using the triple norm, introduced by Alexandre, Morimoto, Ukai, Xu, and Yang (AMUXY in what follows) [4,7]. The triple norm was originally adopted to discuss the equation in L ∞ t (0, ∞; H k l (R 6 x,v )) for suitable l, k ∈ N. We combine both ideas [13] and [7](see also [5,6,8]) to discuss the Cauchy problem without angular cutoff. For this purpose, we later introduce a Chemin-Lerner type triple norm, which enables us to take full advantage of the two ideas.…”

Section: Introductionmentioning

confidence: 99%

Global solutions in the critical Besov space for the non-cutoff Boltzmann equation

Morimoto

Sakamoto

2016

Journal of Differential Equations

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The Boltzmann equation is studied without the cutoff assumption. Under a perturbative setting, a unique global solution of the Cauchy problem of the equation is established in a critical Chemin-Lerner space. In order to analyse the collisional term of the equation, a Chemin-Lerner norm is combined with a non-isotropic norm with respect to a velocity variable, which yields an apriori estimate for an energy estimate. Together with local existence following from commutator estimates and the Hahn-Banach extension theorem, the desired solution is obtained. Also, the non-negativity of the solution is rigorously shown.

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Global Well-Posedness in Spatially Critical Besov Space for the Boltzmann Equation

Cited by 22 publications

References 34 publications

The Cauchy problem for the inhomogeneous non-cutoff Kac equation in critical Besov space

The Cauchy problem for the inhomogeneous non-cutoff Kac equation in critical Besov space

Solution to the Boltzmann equation in velocity-weighted Chemin-Lerner type spaces

Global solutions in the critical Besov space for the non-cutoff Boltzmann equation

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