[Paper Review] On the Cauchy problem for the standard linear solid model with heat conduction: Fourier versus Cattaneo

Marta Pellicer, Belkacem Said‐Houari|arXiv (Cornell University)|Mar 25, 2019

Advanced Mathematical Physics Problems30 references5 citations

TL;DR

This paper investigates the Cauchy problem for the standard linear solid model coupled with either Fourier or Cattaneo heat conduction laws in $$\mathbb{R}^N$$. Using energy methods in Fourier space and eigenvalue expansion, it establishes optimal decay rates for solutions under the condition $0<\tau<\beta$, showing identical decay behavior regardless of the heat conduction model. Crucially, it identifies that the Cattaneo model introduces a regularity-loss phenomenon, requiring higher initial data regularity for decay, while $\tau=\beta$ leads to slower decay and asymptotic stability only when heat conduction is present.

ABSTRACT

In this paper, we consider the standard linear solid model in $\mathbb{R}^N$ coupled, first, with the Fourier law of heat conduction and, second, with the Cattaneo law. First, we give the appropriate functional setting to prove the well-posedness of both models under certain assumptions on the parameters (that is, $0

Motivation & Objective

• To establish well-posedness of the standard linear solid model coupled with both Fourier and Cattaneo heat conduction laws in $\mathbb{R}^N$.
• To determine the optimal decay rates of solutions using energy methods in Fourier space and eigenvalue expansion.
• To analyze the impact of the heat conduction model (Fourier vs. Cattaneo) on solution decay and regularity, particularly the regularity-loss phenomenon in the Cattaneo case.
• To prove that the condition $0<\tau<\beta$ is both sufficient and necessary for asymptotic stability in the absence of heat conduction.
• To show that asymptotic stability under $\tau=\beta$ is only possible when heat conduction is included, with a slower decay rate.

Proposed method

• Formal functional setting is established to prove well-posedness under the condition $0<\tau\leq\beta$.
• The energy method in Fourier space is applied to derive a Lyapunov functional and estimate decay rates of a solution norm.
• Eigenvalue expansion of the characteristic equation in the Fourier domain is used to verify the optimality of decay rates.
• Asymptotic expansions of eigenvalues are computed for $|\xi|\to 0$ and $|\xi|\to\infty$ to analyze decay behavior and regularity loss.
• The Cattaneo law is modeled via a relaxation term $\tau_0 q_t + q + \kappa \nabla \theta = 0$, leading to a damped wave-type heat equation.
• The analysis distinguishes between the cases $\tau<\beta$ and $\tau=\beta$, showing different decay and stability behaviors.

Experimental results

Research questions

• RQ1 Does the choice between Fourier and Cattaneo heat conduction laws affect the decay rate of solutions in the standard linear solid model?
• RQ2 What is the optimal decay rate of solutions when the model is coupled with either heat conduction law?
• RQ3 Why does the Cattaneo model exhibit a regularity-loss phenomenon, and how does it differ from the Fourier model?
• RQ4 Is the condition $0<\tau<\beta$ necessary for asymptotic stability in the absence of heat conduction?
• RQ5 What happens to the decay rate and stability when $\tau=\beta$, especially in the presence or absence of heat conduction?

Key findings

• For $0<\tau<\beta$, the decay rate of the solution norm is identical under both Fourier and Cattaneo heat conduction, matching the decay rate of the model without heat conduction.
• The Cattaneo model introduces a regularity-loss phenomenon: higher initial data regularity is required for solution decay compared to the Fourier model.
• When $\tau=\beta$, asymptotic stability is achieved only with heat conduction, and the decay rate is slower than in the $\tau<\beta$ case.
• The decay rate of $|\xi|^{-2}$ for large $|\xi|$ in the Cattaneo case indicates a slower decay than the $|\xi|^{-4}$ rate at low frequencies, suggesting potential for improved regularity estimates.
• The condition $0<\tau<\beta$ is proven to be both sufficient and necessary for asymptotic stability in the absence of heat conduction.
• The optimality of the derived decay rates is confirmed via eigenvalue expansion, showing that the decay exponents cannot be improved.