Asymptotic behavior of nonlinear sound waves in inviscid media with thermal and molecular relaxation
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: on the whole space , in the so-called critical case when the medium parameters satisfy the relation 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 . 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: where will denote the solution vector after rewriting our problem as a first-order system and ; 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 , where , 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 .
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.
Section snippets
Sound waves in media with thermal and molecular relaxation
The Jordan–Moore–Gibson–Thompson (JMGT) equation given by arises in acoustics as a model of nonlinear sound propagation through thermally relaxing fluids and gases; see [9] for its derivation, which builds upon [10], [11], [12]. Here denotes the acoustic velocity potential. The constant is the speed of sound in a given fluid and where represents the so-called sound diffusivity and is the thermal relaxation time. Furthermore,
Theoretical preliminaries and the main result
For future use, in this section we discuss several useful background results and set the notation. Throughout the paper, we thus make the following assumptions on the relaxation kernel, which align with what is to be expected in practice; cf. [4, §1] and [27, §1].
Assumptions on the memory kernel The memory kernel is assumed to satisfy the following conditions: and is absolutely continuous on . for all and There exists , such that the function satisfies the differential
Auxiliary estimates of a regularity-loss type for the corresponding linearization
We intend to use the mild solution (3.11) of the nonlinear problem to establish its asymptotic behavior. Therefore, much of the decay analysis will transfer to the results for the corresponding linear version of the JMGT equation. For this reason, we next recall known decay estimates and also derive new ones for the following linearization: with the same initial data as in (3.3b). To formulate the results, we set where in
Proof of the main result
We present here the proof of Theorem 3.1 up to the following two bounds: and Their proof will be carried out separately in the next section.
Proof of Theorem 3.1 By Lemma 4.1, we have Therefore, if we set the two estimates (5.1), (5.2) yield Provided that is
Energy analysis in the critical case
The remaining part of the paper is devoted to the proof of estimates (5.1), (5.2). To this end, we employ a delicate energy analysis based on time-weighted norms. We begin by proving the following preparatory result.
Lemma 6.1 Let be a given integer. The inequality holds for all such that .
Proof By virtue of the embedding , we have the following bounds: We also note that
Introduction of the artificial damping
We next intend to derive a low-order energy estimate of a regularity-loss type for our problem. To this end, we introduce artificial damping to the system by considering time-weighted energies with a negative exponent.
Proposition 7.1 Let be an integer. The following bound holds: where the energy is defined in (6.5), the dissipative term in (6.6), and see (7.10) for the
Conclusion and outlook
In this work, we have investigated the asymptotic behavior of solutions to the Jordan–Moore–Gibson–Thompson equation in inviscid media with type I memory. Due to the critical condition satisfied by the medium parameters, the linearized equation’s decay estimates are of a regularity-loss type. Such a loss of derivatives prevents the use of classical energy methods in the analysis of the corresponding nonlinear problem. Our approach instead relied on devising appropriate time-weighted
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