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arXiv · 2012.01142

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

Vanja Nikolić·Belkacem Said-Houari
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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.

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Vanja Nikolić, Belkacem Said-Houari. 2020-12-02. Asymptotic behavior of nonlinear sound waves in inviscid media with thermal and molecular relaxation. https://arxiv.org/abs/2012.01142

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