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A completion of our earlier work on the Cauchy problem for non-effectively hyperbolic operators

Tatsuo Nishitani

Abstract

For hyperbolic differential operators $P$ with non-effectively hyperbolic double characteristics, we study the relationship between the Gevrey well-posedness threshold for strong well-posedness and the associated Hamilton map and flow. In our previous work, we showed that if the Hamilton map has a Jordan block of size $4$ on the double characteristic manifold $Σ$ of codimension $3$, then the Cauchy problem for $P$ is well-posed in the Gevrey class $1<s<3$ for all lower-order terms, and that this result is optimal. Moreover, if there are no bicharacterisitcs tangent to $Σ$, then the Cauchy problem is well-posed in the Gevrey class $1<s<3$ for all lower-order terms, and this result is also optimal. In the present paper, we remove the restriction on the codimension of $Σ$, thereby completing the result.

A completion of our earlier work on the Cauchy problem for non-effectively hyperbolic operators

Abstract

For hyperbolic differential operators with non-effectively hyperbolic double characteristics, we study the relationship between the Gevrey well-posedness threshold for strong well-posedness and the associated Hamilton map and flow. In our previous work, we showed that if the Hamilton map has a Jordan block of size on the double characteristic manifold of codimension , then the Cauchy problem for is well-posed in the Gevrey class for all lower-order terms, and that this result is optimal. Moreover, if there are no bicharacterisitcs tangent to , then the Cauchy problem is well-posed in the Gevrey class for all lower-order terms, and this result is also optimal. In the present paper, we remove the restriction on the codimension of , thereby completing the result.
Paper Structure (13 sections, 35 theorems, 195 equations)

This paper contains 13 sections, 35 theorems, 195 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weights $w$ and $\phi$
  4. Composition by PDO with symbol of type $\exp{S_{\rho,\delta}}$
  5. Preliminary composition
  6. Composition $e^{\varphi}\#\lambda\#e^{-\varphi}$
  7. Composition $e^{\varphi}\#q\#e^{-\varphi}$
  8. Energy estimates and existence of solution
  9. Appendix
  10. Oscillatory integral of Gevrey symbol
  11. Pseudodifferential operators with symbol of type $\exp{S^{\kappa}_{\rho,\delta}}$
  12. Composition formula
  13. Proof of Theorem \ref{['thm:matome']}

Key Result

Theorem 1.1

Assume that eq:spaceW holds on $\Sigma$. The Cauchy problem for $P$ is well-posed in the Gevrey class $1< s< 3$ for all lower-order terms, and the Gevrey class $3$ is optimal.

Theorems & Definitions (64)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 54 more