磁化等离子体混合Vlasov-MHD模型整体弱解的存在性
Existence of Global Weak Solutions to a Hybrid Vlasov-MHD Model for Magnetized Plasmas
摘要：
证明了三维空间中不可压缩混合Vlasov-磁流体动力学模型大数据有限能量弱解的整体时间存在性。该模型耦合磁化等离子体的三个基本成分：一个传输方程的密度函数，模型的能量稀薄粒子的一个物种;不可压缩的Navier-Stokes系统的散装流体;和一个抛物线发展方程，涉及磁扩散，磁场。给出了模型的物理推导。它还表明，弱解，其存在的成立，具有非递增的总能量，它满足一些物理相关的属性，包括守恒的总动量，守恒的总质量，和非负的密度函数的高能粒子。证明是基于一个一级近似计划，这是精心设计的，以避免增加的总能量的序列的近似解决方案，结合一个弱紧的论点序列的近似解决方案。数学模型分析中的关键技术挑战是弗拉索夫型粒子方程的非耗散性质和多线性耦合项中的弱极限。
英文摘要：
We prove the global-in-time existence of large-data finite-energy weak solutions to an incompressible hybrid Vlasov-magnetohydrodynamic model in three space dimensions. The model couples three essential ingredients of magnetized plasmas: a transport equation for the probability density function, which models energetic rarefied particles of one species; the incompressible Navier--Stokes system for the bulk fluid; and a parabolic evolution equation, involving magnetic diffusivity, for the magnetic field. The physical derivation of our model is given. It is also shown that the weak solution, whose existence is established, has nonincreasing total energy, and that it satisfies a number of physically relevant properties, including conservation of the total momentum, conservation of the total mass, and nonnegativity of the probability density function for the energetic particles. The proof is based on a one-level approximation scheme, which is carefully devised to avoid increase of the total energy for the sequence of approximating solutions, in conjunction with a weak compactness argument for the sequence of approximating solutions. The key technical challenges in the analysis of the mathematical model are the nondissipative nature of the Vlasov-type particle equation and passage to the weak limits in the multilinear coupling terms.
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