Knowlpedia

Theorem

Cauchy problem for normally hyperbolic operators

Compactly supported source and Cauchy data determine a unique smooth solution with finite propagation on a globally hyperbolic spacetime.

Statement

Let P:Γ∞(E)→Γ∞(E)P:\Gamma^\infty(E)\to\Gamma^\infty(E) be a on a vector bundle over a . Let Σ\Sigma be a smooth spacelike with future-directed unit normal ν\nu. For

f∈Γc∞(E),u0,u1∈Γc∞(E∣Σ),f\in\Gamma^\infty_c(E),\qquad u_0,u_1\in\Gamma^\infty_c(E|_\Sigma),

there is a unique u∈Γ∞(E)u\in\Gamma^\infty(E) satisfying

Pu=f,u∣Σ=u0,∇νEu∣Σ=u1,Pu=f,\qquad u|_\Sigma=u_0,\qquad \nabla^E_\nu u|_\Sigma=u_1,

where ∇E\nabla^E is the connection determined by the .

The solution obeys finite propagation:

supp⁡u⊆J ⁣(supp⁡f∪supp⁡u0∪supp⁡u1),J(A)=J+(A)∪J−(A).\operatorname{supp}u \subseteq J\!\left(\operatorname{supp}f\cup \operatorname{supp}u_0\cup \operatorname{supp}u_1\right), \qquad J(A)=J^+(A)\cup J^-(A).

Thus existence, uniqueness, and causal support are global consequences of global hyperbolicity, not of the local principal-symbol condition alone.

References
  1. Christian Bär, Nicolas Ginoux, and Frank Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, European Mathematical Society, 2007. Publisher record. Relevant: Theorem 3.2.11.