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On Lars Hörmander's remark on the characteristic Cauchy problem

Jean-Philippe Nicolas

TL;DR

This work extends Hörmander's 1990 results on the characteristic Cauchy problem for second-order wave equations to spacetimes with lower regularity, specifically a spatially compact manifold equipped with a Lipschitz metric and locally $L^ ty$ first-order coefficients. Using energy methods, regularization of the metric and coefficients, and trace theory, the authors establish well-posedness and regularity for both spacelike and fully characteristic (Goursat) initial data within Hörmander's framework, including a trace isomorphism to $H^1( abla \,Sigma)$. They prove existence and uniqueness of solutions in the energy class $L^ abla_{ ext{loc}}(\,R_t; H^1(X))$, with time-continuity and $H^2$-regularity for homogeneous equations when data are smoother, and extend the Goursat theory to $C^1$ metrics with appropriate lower-order term regularity. The results apply to scalar, complex, tensor, or spinor-valued fields on space-times admitting a spin structure and provide a robust, regularity-graded picture via an energy $E(t,u)$ and a trace operator $\, ext{T}_ abla_{\, ilde{ ext{X}}}$, contributing to conformal scattering theory in generic non-stationary space-times.

Abstract

We extend the results of a work by L. H\"ormander in 1990 concerning the resolution of the characteristic Cauchy problem for second order wave equations with regular first order potentials. The geometrical background of this work was a spatially compact spacetime with smooth metric. The initial data surface was spacelike or null at each point and merely Lipschitz. We lower the regularity hypotheses on the metric and potential and obtain similar results. The Cauchy problem for a spacelike initial data surface is solved for a Lipschitz metric and coefficients of the first order potential that are $L^\infty_\mathrm{loc}$, with the same finite energy solution space as in the smooth case. We also solve the fully characteristic Cauchy problem with very slightly more regular metric and potential, namely a ${\cal C}^1$ metric and a potential with continuous first order terms and locally $L^\infty$ coefficients for the terms of order 0.

On Lars Hörmander's remark on the characteristic Cauchy problem

TL;DR

This work extends Hörmander's 1990 results on the characteristic Cauchy problem for second-order wave equations to spacetimes with lower regularity, specifically a spatially compact manifold equipped with a Lipschitz metric and locally first-order coefficients. Using energy methods, regularization of the metric and coefficients, and trace theory, the authors establish well-posedness and regularity for both spacelike and fully characteristic (Goursat) initial data within Hörmander's framework, including a trace isomorphism to . They prove existence and uniqueness of solutions in the energy class , with time-continuity and -regularity for homogeneous equations when data are smoother, and extend the Goursat theory to metrics with appropriate lower-order term regularity. The results apply to scalar, complex, tensor, or spinor-valued fields on space-times admitting a spin structure and provide a robust, regularity-graded picture via an energy and a trace operator , contributing to conformal scattering theory in generic non-stationary space-times.

Abstract

We extend the results of a work by L. H\"ormander in 1990 concerning the resolution of the characteristic Cauchy problem for second order wave equations with regular first order potentials. The geometrical background of this work was a spatially compact spacetime with smooth metric. The initial data surface was spacelike or null at each point and merely Lipschitz. We lower the regularity hypotheses on the metric and potential and obtain similar results. The Cauchy problem for a spacelike initial data surface is solved for a Lipschitz metric and coefficients of the first order potential that are , with the same finite energy solution space as in the smooth case. We also solve the fully characteristic Cauchy problem with very slightly more regular metric and potential, namely a metric and a potential with continuous first order terms and locally coefficients for the terms of order 0.

Paper Structure

This paper contains 9 sections, 8 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Geometrical and functional framework and summary of Lars Hörmander's result
  3. Main results
  4. The Cauchy problem
  5. The Goursat problem
  6. Proofs of the main results
  7. Proof of theorem \ref{['Cauchy']}
  8. Proof of corollary \ref{['HomogCauchy']}
  9. Proof of theorem \ref{['Goursat']}.

Key Result

Theorem 1

(Hörmander, 1990) We define on \Sigma the density measure which is positive where \Sigma is spacelike and vanishes where \Sigma is null, and the associated L^2 space L^2 (\Sigma ; \mathrm{d} \nu^0_{_\Sigma} ). The application which is well defined for smooth solutions, extends as an isomorphism. In particular, there exist K_2 (T,g,L_1 ) and K_3 (T,g,L_1 ), two positive continuous functions of T>

Theorems & Definitions (11)

  • Theorem 1
  • Remark 3.1
  • Theorem 2
  • Corollary 3.1
  • Theorem 3
  • Theorem 4
  • Corollary 3.2
  • Proposition 4.1
  • Proposition 4.2
  • Remark 4.1
  • ...and 1 more