Vlasov方程的多相公式及其应用
Multiphasic formulation of Vlasov equations and applications
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中文总结 AI 辅助
本文研究Vlasov方程的多相公式，将其化为耦合无压力Euler系统，在低正则性下建立局部适定性统一理论，并应用于Vlasov-Poisson与Vlasov-Navier-Stokes系统，证明极限与稳定性结果。
中文摘要 AI 辅助
本文是对Vlasov方程多相公式的数学研究，该公式将这一无碰撞动力学方程重新表述为耦合的无压力Euler方程组。该框架允许考虑仅在速度变量上取测度值的解，因此适用于处理自然出现粗糙速度分布（如Dirac质量）的物理问题。我们特别研究非线性Vlasov方程的情形，其中力场比解的某些速度矩多一个导数的正则性。在此关键假设下，我们从头为多相公式发展了一个有限正则性（典型地在Sobolev空间中）的局部适定性统一理论，从而为相关Vlasov方程的Cauchy问题在低正则性下提供了相应结果。我们将这一抽象理论彻底应用于两类方程，即Vlasov-Poisson型系统和Vlasov-Navier-Stokes型系统。除了证明单动能极限的合理性外，该理论还为每类方程带来了具体应用，允许可能粗糙的速度分布。对于Vlasov-Poisson型系统，我们证明了从Hartree方程出发的半经典极限，并描述了均匀平衡态的非线性不稳定性。对于Vlasov-Navier-Stokes型系统，我们建立了单动能剖面附近的非线性渐近稳定性。我们证明了允许空间与速度正则性分离的新结果，并在此过程中提供了已知结果的新证明，并将其推广到粗糙解的情形。最后，利用该框架的灵活性，我们将其扩展到更复杂的系统，如离子Vlasov-Poisson方程，或与可压缩Navier-Stokes系统耦合的Vlasov方程。
英文摘要
This work is a mathematical study of the multiphasic formulation of the Vlasov equation, which consists in recasting this collisionless kinetic equation as a system of coupled pressureless Euler equations. This framework allows to consider solutions that are only measure-valued in the velocity variable and is thus relevant to tackle physical problems where rough velocity distributions, such as Dirac masses, naturally arise. We specifically address the case of nonlinear Vlasov equations where the force field is one derivative more regular than some moments in velocity of the solution. Under this key assumption, a unified theory of local well-posedness at finite regularity (typically in Sobolev spaces) is developed from scratch for the multiphasic formulation, yielding corresponding results for the Cauchy problem of the associated Vlasov equation at low regularity. We thoroughly apply this abstract theory to two classes of equations, namely Vlasov-Poisson type systems, and Vlasov-Navier-Stokes type systems. In addition to the justification of the monokinetic limit, it also leads to specific applications for each class of equations, allowing possibly rough velocity distributions. For Vlasov-Poisson type systems, we justify the semiclassical limit from Hartree equations, and we describe the nonlinear instability of homogeneous equilibria. For Vlasov-Navier-Stokes type systems, we establish nonlinear asymptotic stability near monokinetic profiles. New results allowing a separation between the regularity in space and in velocity are proven, and along the way, we also provide new proofs of known results and generalize them to the case of rough solutions. Finally, the flexibility of the framework is exploited to obtain extensions to more sophisticated systems such as the Vlasov-Poisson equation for ions, or the Vlasov equation coupled with the compressible Navier-Stokes system.
发表机构
- CNRS, Institut Camille Jordan, Université Claude Bernard Lyon 1（法国国家科学研究中心，卡米尔·让研究所，里昂第一大学）
- CEREMADE, CNRS, Université Paris-Dauphine, PSL Research University（数学与决策分析研究中心，法国国家科学研究中心，巴黎第九大学，PSL研究大学）
- CNRS, Laboratoire de Mathématiques Jean Leray, Nantes Université（法国国家科学研究中心，让·勒雷数学实验室，南特大学）
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