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Smooth Cauchy problem on a cylindrical manifold (or how to define the exponential of a...

Discussion in 'Mathematics' started by user209029, Oct 8, 2018.

  1. user209029

    user209029 Guest

    Let $M$ be a manifold, $E \rightarrow M$ be a real or complex smooth vector bundle, and $D: \Gamma_c(M,E) \rightarrow \Gamma_c(M,E)$ be a (first order if necessary) differential operator on smooth sections with compact support. I want to define and study the exponential $e^{tD}$ of $D$.

    To this end, I thought I would study the following Cauchy problem on $\mathbb{R} \times M$: let $\psi_0 \in \Gamma_c(M,E)$, we are looking for a solution $\psi \in \Gamma(\mathbb{R} \times M,E)$ to the equation: $$ \left( \frac{\partial}{\partial t} - D \right) \psi = 0 $$ with the Cauchy boundary condition: $$\psi(t=0, \cdot) = \psi_0.$$

    Although global smooth Cauchy problems are quite complicated (especially on general manifolds), I cannot help but feel that this particular one (i.e. on the cylindrical manifold $\mathbb{R} \times M$) should have a general global solution. This to me is very similar to the existence of flows of vector fields, which are diffeomorphisms. I have sadly not been able to find any theorems pertaining to this very specific problem, with the exception of theorems for the propagation of waves on globally hyperbolic manifolds. But these theorems only apply to hyperbolic differential operators, and not general ones.

    My questions are the following:

    1) Has this problem ever been studied? (especially for $D$ first order)

    2) If so, are there any conditions that must be satisfied so that $\psi$ exists and is unique?

    3) Most importantly, when is the support of $\psi(t,\cdot)$ compact for all $t$?

    Thank you for reading this far!

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