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

TL;DR

This paper analyzes the long-term behavior of nonlinear sound waves in inviscid media with thermal and molecular relaxation, using novel energy methods to handle regularity-loss decay estimates.

Contribution

It introduces a new approach with time-weighted norms to study the asymptotic behavior of a complex nonlinear wave model with memory effects.

Findings

Established decay estimates for the nonlinear model

Demonstrated the effectiveness of time-weighted norms in controlling nonlinearity

Characterized the asymptotic behavior of solutions in the critical case

Abstract

Ultrasonic propagation through media with thermal and molecular relaxation can be modeled by third-order in time nonlinear wave-like equations with memory. This paper investigates 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 in inviscid fluids. The memory has an exponentially fading character and type I, meaning that involves only the acoustic velocity potential. A major challenge in the global analysis 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.