Decay properties of linear thermoelastic plates: Cattaneo versus Fourier law

Abstract

In this article, we investigate the decay properties of the linear thermoelastic plate equations in the whole space for both Fourier and Cattaneo's laws of heat conduction. We point out that while the paradox of infinite propagation speed inherent in Fourier's law is removed by changing to the Cattaneo law, the latter always leads to a loss of regularity of the solution. The main tool used to prove our results is the energy method in the Fourier space together with some integral estimates. We prove the decay estimates for initial data U ₀ ∈ H ^s (ℝ) ∩ L ¹(ℝ). In addition, by restricting the initial data to U ₀ ∈ H ^s (ℝ) ∩ L ^1,γ(ℝ) and γ ∈ [0, 1], we can derive faster decay estimates with the decay rate improvement by a factor of t ^−γ/2.

Keywords:

• thermoelasticity
• second sound
• Fourier's law
• Cattaneo's law
• decay rate
• regularity loss

AMS Subject Classifications::

• 35B37
• 35L55
• 93D15
• 93D20

Acknowledgement

This work has been supported by KAUST.

Notes

Notes

1. The Fourier law was the first constitutive relation of heat flux and was proposed by the French mathematical physicist Joseph Fourier in 1807.

2. The constant τ₀ represents the time lag needed to establish the steady state of the heat conduction in an element of volume when a temperature gradient is suddenly imposed on that element. For most solid materials, τ₀ varies from 10⁻¹⁰ s to 10⁻¹⁴ s. For gases, τ₀ is in the range of 10⁻⁸ s up to 10⁻¹⁰ s. See the survey paper 36 for more details.