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Rate of convergence to equilibrium for the heated string

Tomasz Cieślak, Jacek Jendrej, Christian Stinner

TL;DR

This work analyzes the rate of convergence to equilibrium for a one-dimensional heated string described by a mixed hyperbolic-parabolic PDE system arising from thermoelasticity. The authors employ a Fourier-space, Kato-style asymptotic analysis of the linearized problem, coupled with careful nonlinear estimates and a Banach fixed-point argument to control the full nonlinear system. They prove that, for initial data with positive temperature and sufficient regularity, the displacement and temperature converge exponentially to the flat equilibrium, with $u\to0$ in $H^{1+s}$ and $(u_t,\theta)\to(0,\theta_{\infty})$ in $H^s$ for $s\in(\tfrac{3}{4},1)$, independent of initial data. The main contribution is a rigorous link between the linear damping observed in high Fourier modes and the nonlinear dynamics, yielding a sharp, data-robust decay rate and an enhanced time decay of the solution.

Abstract

In the present manuscript, we calculate the exponential rate of convergence of the heated string system (a mixed-type hyperbolic-parabolic system of PDEs) towards the equilibrium, independently of the initial data. As a by-product of our analysis, we obtain an enhanced time decay of the solution. The main tool of our reasoning consists of asymptotic analysis at the Fourier side of the linearized problem in the spirit of Kato. The latter method fits very well to the estimates obtained earlier for the system. The matching between the linear problem and the nonlinear original system requires delicate estimates of the nonlinear terms at the Fourier side as well as careful spectral analys

Rate of convergence to equilibrium for the heated string

TL;DR

This work analyzes the rate of convergence to equilibrium for a one-dimensional heated string described by a mixed hyperbolic-parabolic PDE system arising from thermoelasticity. The authors employ a Fourier-space, Kato-style asymptotic analysis of the linearized problem, coupled with careful nonlinear estimates and a Banach fixed-point argument to control the full nonlinear system. They prove that, for initial data with positive temperature and sufficient regularity, the displacement and temperature converge exponentially to the flat equilibrium, with in and in for , independent of initial data. The main contribution is a rigorous link between the linear damping observed in high Fourier modes and the nonlinear dynamics, yielding a sharp, data-robust decay rate and an enhanced time decay of the solution.

Abstract

In the present manuscript, we calculate the exponential rate of convergence of the heated string system (a mixed-type hyperbolic-parabolic system of PDEs) towards the equilibrium, independently of the initial data. As a by-product of our analysis, we obtain an enhanced time decay of the solution. The main tool of our reasoning consists of asymptotic analysis at the Fourier side of the linearized problem in the spirit of Kato. The latter method fits very well to the estimates obtained earlier for the system. The matching between the linear problem and the nonlinear original system requires delicate estimates of the nonlinear terms at the Fourier side as well as careful spectral analys
Paper Structure (7 sections, 9 theorems, 122 equations)

This paper contains 7 sections, 9 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fourier expansion
  4. Perturbation for linear system
  5. Estimates of nonlinear terms
  6. Fixed point argument
  7. Conclusions

Key Result

Theorem 1.1

Let $(u, \theta)$ be the solution to eq1 for initial data satisfying the regularity assumptions regularity, with positive initial temperature $\theta_0$. Assume $s\in (\frac{3}{4},1)$. Then, $u$ converges to $0$ in $H^{1+s}(0,\pi)$, $(u_t,\theta)$ converges to $(0, \theta_\infty)$ in $H^s(0,\pi)$, a

Theorems & Definitions (19)

  • Theorem 1.1
  • Proposition 2.1
  • Remark 2.2
  • Remark 3.1
  • Lemma 4.1
  • proof
  • Remark 4.2
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • ...and 9 more