Justification of a variational construction of approximate slow manifolds for Hamiltonian two-scale systems with strong gyroscopic forces

两种具有强陀螺力的尺度系统出现在各种应用中：大气或海洋中受科里奥利力作用的大尺度流，受磁场作用的带电粒子，以及在研究辛数值时间积分格式时的抽象意义。这样的系统具有近似不变的慢流形，因此从足够近的点出现的轨迹在很长的时间跨度内保持接近它。慢流形上动力学的描述可以通过范式理论得到；特别是，当整个系统是哈密顿函数时，这个性质通过构造得以保留。当哈密顿二尺度系统是偏微分方程时，经典的范式构造通常会导致“导数损失”：随着构造精度阶数的增加，对初始数据的平滑性要求也会增加。在这个项目中，我们提出可以通过在拉格朗日一侧同时变换辛形式和哈密顿量的特定构造来避免这种导数的损失。我们将使用半线性Klein-Gordon方程的非相对论极限作为原型来证明该原理，努力获得与有限维中经典已知的指数估计一样强的指数估计，并将构造推广到其他系统。我们期望这一项目将为哈密顿偏微分方程的范式理论提供一个新的观点，并将其推广到更大的多尺度系统。两种具有强陀螺力的尺度系统出现在各种应用中：大气或海洋中受柯氏力作用的大尺度流，受磁场作用的带电粒子，以及在研究辛数值时间积分格式时的抽象意义。这样的系统具有近似不变的慢流形，因此从足够近的点出现的轨迹在很长的时间跨度内保持接近它。慢流形上动力学的描述可以通过范式理论得到；特别是，当整个系统是哈密顿函数时，这个性质通过构造得以保留。当哈密顿二尺度系统是偏微分方程时，经典的范式构造通常会导致“导数损失”：随着构造精度阶数的增加，对初始数据的平滑性要求也会增加。在这个项目中，我们打算证明这种导数的损失可以通过在拉格朗日一侧同时变换辛形式和哈密顿量的特定构造来避免。我们将以半线性Klein-Gordon方程的非相对论极限为原型来证明这一原理，努力获得与有限维中经典已知的指数估计一样强的指数估计，并将其构造推广到其他系统。我们期望这一项目将为哈密顿偏微分方程的范式理论提供一个新的观点，并将其推广到更大的多尺度系统。

Two scale systems with strong gyroscopic forces appear in a variety of applications: large-scale flow in atmosphere or ocean subject to the Coriolis force, charged particles subject to a magnetic field, and also in an abstract sense when studying symplectic numerical time integration schemes. Such systems possess an approximately invariant slow manifold, so that trajectories emerging from sufficiently nearby points remain close to it over long time spans. A description of the dynamics on the slow manifold can be obtained via normal form theory; in particular, when the full system is Hamiltonian, this property is preserved through the construction. When the Hamiltonian two scale system is a partial differential equation, a classical normal form construction will typically cause a "loss of derivatives": as the order of accuracy of the construction is increased, the requirements on the smoothness of the initial data will increase, too.In this project, we propose to show that this loss of derivatives can be avoided by a specific construction on the Lagrangian side which transforms the symplectic form and the Hamiltonian simultaneously. We will demonstrate the principle using the non-relativistic limit of the semilinear Klein-Gordon equation as a prototype, work toward obtaining exponential estimates as strong as those classically known in finite dimensions, and generalize the construction to other systems.We expect that this project will lead toward a new view on normal form theory for Hamiltonian partial differential equations which extends to a larger class of multiscale systems.Two scale systems with strong gyroscopic forces appear in a variety of applications: Large-scale flow in atmosphere or ocean subject to theCoriolis force, charged particles subject to a magnetic field, and also in an abstract sense when studying symplectic numerical time integration schemes. Such systems possess an approximately invariant slow manifold, so that trajectories emerging from sufficiently nearby points remain close to it over long time spans. A description of the dynamics on the slow manifold can be obtained via normal form theory; in particular, when the full system is Hamiltonian, this property is preserved through the construction. When the Hamiltonian two scale system is a partial differential equation, a classical normal form construction will typically cause a "loss of derivatives": As the order of accuracy of the construction is increased, the requirements on the smoothness of the initial data will increase, too.In this project, we propose to show that this loss of derivatives can be avoided by a specific construction on the Lagrangian side whichtransforms the symplectic form and the Hamiltonian simultaneously. We will demonstrate the principle using the non-relativistic limit of thesemilinear Klein-Gordon equation as a prototype, work toward obtaining exponential estimates as strong as those classically knownin finite dimensions, and generalize the construction to other systems.We expect that this project will lead toward a new view on normal form theory for Hamiltonian partial differential equations which extends toa larger class of multiscale systems.