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On the Cauchy problem for acoustic waves in hereditary fluids: decay properties and inviscid limits

Wenhui Chen

Abstract

This manuscript considers the viscous/inviscid Moore-Gibson-Thompson (MGT) equations with memory of type I in the whole space $\mathbb{R}^n$. For one thing, associating with a new condition on initial data, we derive the optimal $L^2$ estimates and the optimal leading term of the acoustic velocity potential for large time, in which we analyze different contributions from viscous, thermally relaxing, as well as hereditary fluids on large time asymptotic behavior for the acoustic waves models. For another, via the multi-scale analysis and energy methods in the Fourier space, we demonstrate the $L^{\infty}$ inviscid limits in the sense of the diffusivity of sound tending to zero, which match our WKB expansion of the solution. Finally, we give a further application of our results on large time behavior for the nonlinear Jordan-MGT equation in viscous hereditary fluids.

On the Cauchy problem for acoustic waves in hereditary fluids: decay properties and inviscid limits

Abstract

This manuscript considers the viscous/inviscid Moore-Gibson-Thompson (MGT) equations with memory of type I in the whole space . For one thing, associating with a new condition on initial data, we derive the optimal estimates and the optimal leading term of the acoustic velocity potential for large time, in which we analyze different contributions from viscous, thermally relaxing, as well as hereditary fluids on large time asymptotic behavior for the acoustic waves models. For another, via the multi-scale analysis and energy methods in the Fourier space, we demonstrate the inviscid limits in the sense of the diffusivity of sound tending to zero, which match our WKB expansion of the solution. Finally, we give a further application of our results on large time behavior for the nonlinear Jordan-MGT equation in viscous hereditary fluids.

Paper Structure

This paper contains 16 sections, 9 theorems, 175 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. Result and discussion on the optimal estimates
  4. Result and discussion on the optimal leading term
  5. Result and discussion on the inviscid limits
  6. Large time asymptotic behavior of the solution
  7. Asymptotic analysis for small frequencies
  8. Asymptotic analysis for large frequencies
  9. Exponential stability for bounded frequencies
  10. Optimal growth/decay estimates: Proof of Theorem \ref{['Thm-Optimal-Est']}
  11. Optimal leading term: Proof of Theorem \ref{['Thm-Optimal-Leading']}
  12. Inviscid limits of the solution in the $L^{\infty}$ framework
  13. Formal expansion with the small diffusivity of sound
  14. Rigorous justification of the inviscid limits: Proof of Theorem \ref{['Thm-Inviscid-Limit']}
  15. Final remarks: An application on the nonlinear Jordan-MGT equation in viscous hereditary fluids
  16. ...and 1 more sections

Key Result

Theorem \oldthetheorem

Suppose that initial data $(\psi_0,\psi_1,\psi_2)\in\mathcal{A}_{\delta,\ell}$ with $\ell>\max\{\frac{n}{2}-1,0\}$ and $\delta\geqslant0$. Then, the solution to the MGT equations MGT-Memory with the memory Memory-Kernel satisfies the following optimal growth/decay estimates: for large time $t\gg1$, provided that $P_{\psi_1+\tau\psi_2}\neq0$, in which the time-dependent function $\mathcal{D}_n(t)$

Theorems & Definitions (25)

  • Theorem \oldthetheorem
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Theorem \oldthetheorem
  • Remark 2.8
  • ...and 15 more