Estimates for the kinetic transport equation in hyperbolic Sobolev spaces
Estimates for the kinetic transport equation in hyperbolic Sobolev spaces
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双曲 Sobolev 空间中的动力学输运方程的估计
DOI：
10.1016/j.matpur.2018.03.007
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发表时间：
2018
影响因子：
2.3
通讯作者：
Lee Sanghyuk
中科院分区：
文献类型：
--
作者：
Bennett Jonathan;Bez Neal;Gutierrez Susana;Lee Sanghyuk
我们在双曲Soblev空间的框架下建立了动力学输运方程解的速度平均算子ρ的光滑化估计。如果速度域是单位球或单位球，则对于任意指数q和r，我们找到了指数β+和β−的特征，但可能的端点情况除外，其中D+β+D−β−ρ是从空间速度L x，v2到时空L t q L x r的有界的。这里，D+和D−分别是经典和双曲导数算子。事实上，我们将提供一个统一这些速度域的论点，并且证明了在任何一种情况下速度平均估计都等价于作用在L 2上的锥乘子算子的混合范数界。我们从几个方面进一步发展了我们的想法，包括对位于某些贝索夫空间中的初始数据的估计，证明的一个关键工具是Bourain和Demeter最近建立的尖锐的ℓp解耦定理。我们还表明，如果我们限制对空间变量径向对称的初始数据的关注，则允许的平滑程度显着增加。
We establish smoothing estimates in the framework of hyperbolic Sobolev spaces for the velocity averaging operator ρ of the solution of the kinetic transport equation. If the velocity domain is either the unit sphere or the unit ball, then, for any exponents q and r, we find a characterisation of the exponents β+ and β−, except possibly for an endpoint case, for which D+ β+ D− β− ρ is bounded from space–velocity L x, v 2 to space–time L t q L x r. Here, D+ and D− are the classical and hyperbolic derivative operators, respectively. In fact, we shall provide an argument which unifies these velocity domains and the velocity averaging estimates in either case are shown to be equivalent to mixed-norm bounds on the cone multiplier operator acting on L 2. We develop our ideas further in several ways, including estimates for initial data lying in certain Besov spaces, for which a key tool in the proof is the sharp ℓ p decoupling theorem recently established by Bourgain and Demeter. We also show that the level of permissible smoothness increases significantly if we restrict attention to initial data which are radially symmetric in the spatial variable.