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As a neat application of the ps.d.o. machinery developed thus far, we prove the existence and uniqueness of solutions to hyperbolic evolution equations (Section 7.1); examples include transport and wave equations. We also prove that singularities in the initial data, in the sense of wave front sets, propagate along the Hamiltonian flow for the principal symbol of the operator governing the evolution equation (Section 7.2). We shall state results only on \(\mathbb {R}^n\) and leave the purely notational modifications needed for working on compact manifolds to the reader.

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Hyperbolic Evolution Equations and Egorov’s Theorem

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  • 发表日期：

    2025年

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  • Peter Hintz

摘要

As a neat application of the ps.d.o. machinery developed thus far, we prove the existence and uniqueness of solutions to hyperbolic evolution equations (Section 7.1); examples include transport and wave equations. We also prove that singularities in the initial data, in the sense of wave front sets, propagate along the Hamiltonian flow for the principal symbol of the operator governing the evolution equation (Section 7.2). We shall state results only on \(\mathbb {R}^n\) and leave the purely notational modifications needed for working on compact manifolds to the reader.