Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generalized energy conservation for linear wave equations with time-dependent propagation speed

Marina Ghisi, Massimo Gobbino

Abstract

We consider a wave equation with a time-dependent propagation speed, whose potential oscillations are controlled through bounds on its first and second derivatives and by limiting the integral of the difference with a fixed constant. We investigate when the wave equation exhibits generalized energy conservation (GEC), meaning that the energy of all solutions remains bounded for all times by a multiple of the initial energy. When GEC is not satisfied, we provide upper bounds for the growth of the energy. These upper bounds are derived by analyzing the growth of the Fourier components of the solution. Depending on the frequency and the time interval, different energy inequalities are employed to fully exploit our assumptions on the propagation speed. Finally, we present counterexamples that demonstrate the optimality of our upper bound estimates.

Generalized energy conservation for linear wave equations with time-dependent propagation speed

Abstract

We consider a wave equation with a time-dependent propagation speed, whose potential oscillations are controlled through bounds on its first and second derivatives and by limiting the integral of the difference with a fixed constant. We investigate when the wave equation exhibits generalized energy conservation (GEC), meaning that the energy of all solutions remains bounded for all times by a multiple of the initial energy. When GEC is not satisfied, we provide upper bounds for the growth of the energy. These upper bounds are derived by analyzing the growth of the Fourier components of the solution. Depending on the frequency and the time interval, different energy inequalities are employed to fully exploit our assumptions on the propagation speed. Finally, we present counterexamples that demonstrate the optimality of our upper bound estimates.

Paper Structure

This paper contains 37 sections, 10 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Energy equality and generalized energy conservation
  3. Previous literature
  4. Our contribution
  5. Overview of the technique
  6. A related problem
  7. Structure of the paper
  8. Statements
  9. Notation and definitions
  10. Positive result -- Estimate of energy growth
  11. Negative result -- Counterexamples
  12. Some classical examples
  13. Estimates from above (proof of Theorem \ref{['thm:est-above']})
  14. Classical energy estimates
  15. Kowaleskian estimate
  16. ...and 22 more sections

Key Result

Theorem 2.4

Let $t_0$, $\Lambda_1$, $\Lambda_2$, $S$ and $\gamma$ be as in Definition defn:PS. Let $G$ and $M$ be the functions defined in (defn:G) and (defn:Max), respectively. Then for every Hilbert space $\mathcal{H}$, every multiplication operator $A$ on $\mathcal{H}$, every coefficient $c\in\mathcal{PS}(t_

Theorems & Definitions (22)

  • Definition 2.1: Spectrum of a multiplication operator
  • Definition 2.2: Admissible propagation speeds
  • Definition 2.3: Generalized energy conservation
  • Theorem 2.4: Estimates from above
  • Remark 2.5: Estimates from below
  • Corollary 2.6: Sufficient condition for the generalized energy conservation
  • Example 2.7: Fast growth of derivatives
  • Theorem 2.8: Optimality of the estimates from above
  • Remark 2.9
  • Remark 2.10
  • ...and 12 more