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

• Applicant: Professor Dr. Marcel Oliver
• Subject Area: Mathematics
• Term: from 2015 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 278124199

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.

• DFG Programme: Research Grants