Asymptotic behavior of nonlinear sound waves in inviscid media with thermal and molecular relaxation

Abstract

Nonlinear sound propagation through media with thermal and molecular relaxation can be modeled by third-order in time wave-like equations with memory. We investigate the asymptotic behavior of a Cauchy problem for such a model, the nonlocal Jordan–Moore–Gibson–Thompson equation, in the so-called critical case, which corresponds to propagation through inviscid fluids or gases. The memory has an exponentially fading character and type I, meaning that involves only the acoustic velocity potential. A major challenge in studying global behavior is that the linearized equation’s decay estimates are of regularity-loss type. As a result, the classical energy methods fail to work for the nonlinear problem. To overcome this difficulty, we construct appropriate time-weighted norms, where weights can have negative exponents. These problem-tailored norms create artificial damping terms that help control the nonlinearity and the loss of derivatives, and ultimately allow us to discover the model’s asymptotic behavior.

Introduction

Ultrasonic waves traveling through fluids with impurities are known to be influenced by viscoelastic relaxation effects. An example of such a medium is water with micro-bubbles, commonly used as a contrast agent in ultrasonic imaging [1] as well as in improving the speed and efficacy of focused ultrasound treatments [2]. The goal of the present work is to study asymptotic behavior of such ultrasonic waves as modeled by the Jordan–Moore–Gibson–Thompson (JMGT) equation with memory:

$$\tau {\psi }_{ttt}+{\psi }_{tt}-c^2\Delta \psi -b\Delta {\psi }_t+{\int }_0^{t}g\left(r\right)\Delta \psi (t-r)\text{d}r={\left(k{\psi }_t^2+{|\nabla \psi |}^2\right)}_t$$

on the whole space ${\mathbb{R}}^n$, in the so-called critical case when the medium parameters satisfy the relation

$$b=\tau c^2.$$

This wave-like equation models nonlinear sound propagation through inviscid media with thermal and molecular relaxation. The relaxation mechanisms are responsible for the third-order propagation and the viscoelastic term in the equation, whereas the negligible sound diffusivity leads to the critical condition $b=\tau c^2$. This kind of memory acting only on the solution of the equation (and not on its time derivatives) is often referred to in the literature as the memory of type I; cf. [3], [4]. The reader is referred to Section 2 for a more detailed insight into nonlinear acoustic modeling.

To our best knowledge, the present work is the first treating the problem (2.1) in the critical case. A major difficulty in treating the critical case is that the linearized equation’s decay estimates are of regularity-loss type. It is well-known that such a loss of regularity going from the initial data to the solution presents a big obstacle in proving nonlinear stability. The classical energy method fails. To solve this problem, we construct appropriate time-weighted norms:

$${\Vert \mathit{\Psi }\Vert }_{\mathbb{E},t}^2=\sum _{i=0}^{\left[\frac{s-1}{2}\right]}\underset{0\le \sigma \le t}{sup}{(1+\sigma )}^{i-1∕2}{⦀{\nabla }^i\mathit{\Psi }\left(\sigma \right)⦀}_{H^{s-2i}}^2,$$

where $\mathit{\Psi }$ will denote the solution vector after rewriting our problem as a first-order system and $s\ge \left[\frac{3n}{2}\right]+5$; we refer to Section 3 for details. These tailored norms have a weight with a negative exponent, which helps introduce artificial damping to the system. This damping, in turn, allows us to control the nonlinearity and handle the loss of derivatives.

The main result of this work is contained in Theorem 3.1 and concerns global existence and asymptotic decay of solutions in ${\mathbb{R}}^n$, where $n\ge 3$, for smooth and small initial data. The decay estimates hold for a solution with a lower Sobolev regularity than that assumed for the initial data, which is to be expected in the presence of a loss of regularity; see, for instance, [5], [6], [7]. The damping introduced by the memory term plays a key role in stabilizing the solution in the critical case. Without memory, the linearized problem is unstable. Indeed, it has been proven in [8] by relying on the Routh–Hurwitz theorem that the real parts of the eigenvalues associated with the linearized system are negative if and only if $b>\tau c^2$.

We organize the rest of the paper as follows. In Section 2, we discuss the modeling and related work on analyzing third-order acoustic equations. In Section 3, we recall the problem’s local-well posedness in the so-called history framework and then present our main result on the global well-posedness and asymptotic behavior of solutions for small and smooth data. Section 4 deals with the decay estimates for the linear version of the equation, which we will rely on in the decay analysis of the nonlinear problem. We present the proof of the main result in Section 5, up to two crucial energy bounds. Their proof is contained in Sections 6 Energy analysis in the critical case, 7 Introduction of the artificial damping, based on carefully designed time-weighted energies. We conclude the paper with a discussion and an outlook on open problems. Auxiliary technical results are collected in the Appendix.