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

两个具有强陀螺力的尺度系统出现在各种应用中：科里奥利力作用下的大气或海洋中的大尺度流动，磁场作用下的带电粒子，以及研究辛数值时间积分格式时的抽象意义上的。这样的系统拥有一个近似不变的慢流形，因此从足够近的点出现的轨迹在很长的时间跨度内都保持在它附近。慢流形上的动力学可以通过规范型理论得到描述，特别是当整个系统是哈密顿时，这一性质通过构造得到了保持。当哈密顿双尺度系统是一个偏微分方程组时，经典的规范形构造通常会导致导数的损失：随着构造精度的提高，对初始数据光滑性的要求也会增加。在这个项目中，我们建议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损失。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，并致力于获得与有限维经典已知的指数估计一样强的指数估计，并将其推广到其他系统。我们期望这一项目将导致对哈密顿偏微分方程范式理论的新看法，从而扩展到更大类的多尺度系统。两个具有强陀螺力的尺度系统出现在各种不同的应用中：大气或海洋中受科里奥利力的大尺度流动，带电粒子受磁场的作用，以及在研究辛数值时间积分格式时的抽象意义。这样的系统拥有一个近似不变的慢流形，因此从足够近的点出现的轨迹在很长的时间跨度内都保持在它附近。慢流形上的动力学可以通过规范型理论得到描述，特别是当整个系统是哈密顿时，这一性质通过构造得到了保持。当哈密顿双尺度系统是一个偏微分方程组时，经典的规范形构造通常会导致导数的损失：随着构造精度的提高，对初始数据光滑性的要求也会增加。在这个项目中，我们建议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损失。我们将以非线性Klein-Gordon方程的非相对论极限为原型来证明这一原理，致力于获得与有限维中经典已知的指数估计一样强的指数估计，并将其推广到其他系统。我们期望这一项目将导致对哈密顿偏微分方程范式理论的新看法，从而将其推广到更大类的多尺度系统。

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.