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Resonant solutions and (in)stability of the linear wave equation

Giancarlo Sangalli, Davide Terazzi, Pietro Zanotti

Abstract

We revise the analysis of the acoustic wave equation, addressing the question whether the classical well-posedness implies the existence of an isomorphism between prescribed solution and data spaces. This question is of interest for the design and the analysis of discretization methods. Expanding on existing results, we point out that established choices of solution and data space in terms of classical Bochner spaces must be expected to be incompatible with the existence of such an isomorphism, because of resonant waves. We formulate this observation in the language of the so-called inf-sup theory, with the help of an eigenfunction expansion, which reduces the original partial differential equation to a system of ordinary differential equations. We further verify that an isomorphism can be established, for each equation in the system, upon equipping the data space with a suitable resonance-aware norm. In the appendix, we extend our results to other time-dependent linear PDEs.

Resonant solutions and (in)stability of the linear wave equation

Abstract

We revise the analysis of the acoustic wave equation, addressing the question whether the classical well-posedness implies the existence of an isomorphism between prescribed solution and data spaces. This question is of interest for the design and the analysis of discretization methods. Expanding on existing results, we point out that established choices of solution and data space in terms of classical Bochner spaces must be expected to be incompatible with the existence of such an isomorphism, because of resonant waves. We formulate this observation in the language of the so-called inf-sup theory, with the help of an eigenfunction expansion, which reduces the original partial differential equation to a system of ordinary differential equations. We further verify that an isomorphism can be established, for each equation in the system, upon equipping the data space with a suitable resonance-aware norm. In the appendix, we extend our results to other time-dependent linear PDEs.

Paper Structure

This paper contains 16 sections, 6 theorems, 86 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Model problem statement
  3. Functional setting
  4. Function spaces
  5. Eigenfunctions expansion
  6. Inspecting Zank's counter-example
  7. Resonant waves
  8. Inf-sup instability
  9. Looking for resonance-aware data norms
  10. First approach: Fourier expansion
  11. Second approach: energy balance law
  12. Damped wave equation
  13. Conclusion
  14. Extension to other IBVPs
  15. Heat equation
  16. ...and 1 more sections

Key Result

Proposition 3.1

Let $f_\omega$ and $u_{k,\omega}$ be as in eq:source-term-wave and eq:solution-wave. The respective norms from Section SS:EigenfunctionsExpansion satisfy the relation The constant $C_{k,\omega} \in \mathbb{R}$ is given by otherwise.

Figures (4)

  • Figure 1: Source term $f_\omega$ (first row) and solution $u_{k,\omega}$ (second row) for the IVP \ref{['eq:model-problem-weak-reduced']} in a progression from a non resonant case to the resonant one and back.
  • Figure 2: Plot of the amplification constant in Proposition \ref{['P:amplification-constant']} for $\sqrt{\mu_k} = 2\pi k$ with $k = 1,\dots,15$. The final time is fixed to $T = 1$.
  • Figure 3: Comparing the data norm of $f_\omega$ from Proposition \ref{['P:amplification-constant']} with the $L^2(L^2)-$ and $L^2(H^{-1})-$norm, for $k=50$ and $\sqrt{\mu_k}=2\pi k$.
  • Figure 4: Absolute value of the first coefficients of $\mathcal{W}_{k}(\omega_j,\,\omega_\ell)$ from \ref{['eq:data-norm-fourier-expansion']}, with $\sqrt{\mu_k} =200$, and $T=1$, $c=1$.

Theorems & Definitions (24)

  • Remark 2.1: Embedding theorems
  • Remark 2.2: Data space
  • Remark 2.3: Dimensional analysis
  • Remark 2.4: Operator components
  • Remark 2.5: Boundary conditions
  • Proposition 3.1: Amplification constant
  • proof
  • Proposition 3.2: Frequency dependence
  • proof
  • Corollary 3.3: Inf-sup instability
  • ...and 14 more