相对论性Vlasov-Maxwell系统小数据解的尖锐时间衰减与散射
Sharp Time Decay and Scattering of Small Data Solutions to the Relativistic Vlasov-Maxwell System
- Colorado School of Mines（科罗拉多矿业学院）
- University of Rome Tor Vergata（罗马托尔维加塔大学）
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AI总结：
本文针对相对论性Vlasov-Maxwell系统，在全局电中性假设下构造了具有任意快多项式衰减率的解，建立了可数无穷多个渐近轮廓，并证明了粒子分布函数沿空间特征的线性散射收敛，且可扩展至带对数修正的多齐次展开。
AI中文摘要：
多物种无碰撞等离子体由相对论性Vlasov-Maxwell系统建模。假设等离子体整体呈电中性，我们证明可以构造具有任意快多项式衰减率的解，其衰减率取决于极限空间平均值的抵消性质以及粒子的质量和电荷。在此过程中，我们为电荷密度和电流密度、电场和磁场及其导数建立了可数无穷多个渐近轮廓，每个轮廓都必然由足够光滑的解实现。我们的方法依赖于一种精细的场表示，该表示在光锥边界之外呈现自相似行为。在每种情况下，我们为每个粒子分布函数建立了$L^\infty$中的线性散射结果，即我们证明它们沿输运的空间特征以越来越快的速率在$t \to \infty$时收敛。在电荷抵消不发生的情况下，我们的方法可以进一步扩展以证明具有对数修正的多齐次展开。
英文摘要：
A multispecies, collisionless plasma is modeled by the relativistic Vlasov-Maxwell system. Assuming the plasma is globally neutral, we show that solutions can be constructed with arbitrarily fast, polynomial rates of decay, depending upon the cancellation properties of the limiting spatial averages, as well as the masses and charges of the particles. In doing so, we establish a countably infinite number of asymptotic profiles for the charge and current density, electric and magnetic fields, and their derivatives, each of which is necessarily realized by a sufficiently smooth solution. Our approach relies upon a refined field representation that features self-similar behavior away from the boundary of the light cone. In each case we establish a linear scattering result in $L^\infty$ for every particle distribution function, namely we show that they converge as $t \to \infty$ along the transported spatial characteristics at increasingly faster rates. In the case that charge cancellation does not occur, our methods can be further extended to demonstrate polyhomogeneous expansions with logarithmic corrections.
