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Propagation of singularities for equations with $C^{r}$ coefficients for $r>1$

Jan Rozendaal

TL;DR

This work extends propagation of singularities to linear equations with coefficients of rough spatial regularity $C^{r}_{*}$ for $r>1$ by introducing the rough symbol class $C^{r}_{*}S^{m}_{1,0}$ and using a paradifferential calculus to decompose symbols into a smooth part $p^{\sharp}$ and a rough part $p^{\flat}$. The main theorem shows that if $p(x,D)u$ has $H^{s}$ regularity and $u$ lies in the corresponding lower-regularity space $H^{s+m-r}$, then the $H^{s+m}$-wavefront set of $u$ is contained in the characteristic set of the homogeneous limit $\hat p$, with every point of $\mathrm{WF}^{s+m-1}u$ lying on a null bicharacteristic of $\hat p$ contained in $\mathrm{WF}^{s+m-1}u$. The approach blends Taylor-type symbol decompositions with positive-commutator arguments in the rough-symbol regime and shows endpoint results under additional $\mathcal{H}^{r,\infty}$-type regularity; it also discusses domain and divergence-form extensions and geometric applications to wave equations on manifolds with bounded Ricci curvature. These results unify and extend earlier smooth-coefficient theories (Hörmander) and prior rough-coefficient work (Taylor, Smith), providing a robust propagation framework for rough PDEs with concrete geometric consequences. The article thus broadens the set of PDEs for which propagation of singularities is rigorously understood in the presence of spatial roughness, with potential impacts on observability and control in rough media.

Abstract

We observe that, for $r>1$, $s$ in an $r$-dependent interval, $p$ a homogeneous pseudodifferential symbol of order $m$ having $C^{r}$ regularity in space, and $u\in H^{s+m-r}(\mathbb{R}^{n})$ such that $p(x,D)u\in H^{s}(\mathbb{R}^{n})$, each point in the $H^{s+m-1}$ wavefront set of $u$ lies on a maximally extended null bicharacteristic of $p$ which is contained in the $H^{s+m-1}$ wavefront set of $u$. In fact, for $r=2$ slightly less than $C^{1,1}$ regularity suffices, and here the results apply to manifolds with bounded Ricci curvature.

Propagation of singularities for equations with $C^{r}$ coefficients for $r>1$

TL;DR

This work extends propagation of singularities to linear equations with coefficients of rough spatial regularity for by introducing the rough symbol class and using a paradifferential calculus to decompose symbols into a smooth part and a rough part . The main theorem shows that if has regularity and lies in the corresponding lower-regularity space , then the -wavefront set of is contained in the characteristic set of the homogeneous limit , with every point of lying on a null bicharacteristic of contained in . The approach blends Taylor-type symbol decompositions with positive-commutator arguments in the rough-symbol regime and shows endpoint results under additional -type regularity; it also discusses domain and divergence-form extensions and geometric applications to wave equations on manifolds with bounded Ricci curvature. These results unify and extend earlier smooth-coefficient theories (Hörmander) and prior rough-coefficient work (Taylor, Smith), providing a robust propagation framework for rough PDEs with concrete geometric consequences. The article thus broadens the set of PDEs for which propagation of singularities is rigorously understood in the presence of spatial roughness, with potential impacts on observability and control in rough media.

Abstract

We observe that, for , in an -dependent interval, a homogeneous pseudodifferential symbol of order having regularity in space, and such that , each point in the wavefront set of lies on a maximally extended null bicharacteristic of which is contained in the wavefront set of . In fact, for slightly less than regularity suffices, and here the results apply to manifolds with bounded Ricci curvature.
Paper Structure (17 sections, 10 theorems, 39 equations)

This paper contains 17 sections, 10 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Previous work
  4. Proof of the main result
  5. Organization of this article
  6. Notation
  7. Preliminaries
  8. Microlocal analysis
  9. Function spaces
  10. Rough symbols
  11. Paradifferential calculus
  12. Propagation of singularities
  13. Main result
  14. Auxiliary results
  15. Wave equations
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $r>1$, $m\in\mathbb R$ and $-r<s<r$. Let $p\in C^{r}_{*}S^{m}_{1,0}$ be such that $\mathop{\mathrm{Im}}\nolimits p\in C^{r}_{*}S^{m-1}_{1,0}$, and such that $\hat{p}$ is a well-defined element of $C^{1}(T^{*}\mathbb{R}^{n}\setminus o)$. Set for $\varepsilon>0$. Let $u\in H^{\sigma+m}(\mathbb{R}^{n})$ be such that $p(x,D)u\in H^{s}(\mathbb{R}^{n})$. Then $\operatorname{WF}^{s+m}u\subseteq\Sigm

Theorems & Definitions (30)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Proposition 2.7
  • proof
  • ...and 20 more