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Control, Stabilization and Propagation of Singularities for dispersive PDEs

  • Zhu, Hui
Publication Date
Mar 27, 2019
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In this thesis, we study the closely related theories of control, stabilization and propagation of singularities for some linear and nonlinear dispersive partial differential equations. Main results come from the author’s works:[1] Zhu, H., 2016. Stabilization of damped waves on spheres and Zoll surfaces of revolution. ESAIM: Control, Optimisation and Calculus of Variations (ESAIM: COCV), to appear.[2] Zhu, H., 2017. Control of three dimensional water waves. arXiv preprint arXiv:1712.06130.[3] Zhu, H., 2018. Propagation of singularities for gravity-capillary water waves. arXiv preprint arXiv:1810.09339.In [1] we studied the stabilization of the damped wave equation on Zoll surfaces of revolution. We gave an example where the region of damping is at the borderline of the geometric control condition, yet the damped waves exhibit a uniform exponential decay of energy, generalizing an example of Lebeau.In [2] we studied the controllability of the gravity-capillary water wave equation. Under the geometric control condition, we proved in arbitrary spatial dimension the exact controllability for spatially periodic small data. This generalizes a result of Alazard, Baldi and Han-Kwan for the 2D gravity-capillary water wave equation.In [3] we studied the propagation of singularities for the gravity-capillary water wave equation. We defined the quasi-homogeneous wavefront set, generalizing the wavefront set of H¨ ormander and the homogeneous wavefront set of Nakamura, and proved propagation results for quasi-homogeneous wavefront sets by the gravity-capillary water wave equation. As corollaries, we obtained local and microlocal smoothing effects for gravity-capillary water waves with sufficient spatial decay.

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