Optimal semigroup regularity for velocity coupled elastic systems: a degenerate fractional damping case

Kuang, Zhaobin; Liu, Zhuangyi; Tebou, Louis

ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 46

In this note, we consider an abstract system of two damped elastic systems. The damping involves the average velocity and a fractional power of the principal operator, with power θ in [0, 1], The damping matrix is degenerate, which makes the regularity analysis more delicate. First, using a combination of the frequency domain method and multipliers technique, we prove the following regularity for the underlying semigroup:

• The semigroup is of Gevrey class δ for every δ > 1/2θ, for each θ in (0, 1/2).
• The semigroup is analytic for θ = 1/2.
• The semigroup is of Gevrey class δ for every δ > 1/2(1 — θ), for each θ in (1/2, 1).

Next, we analyze the point spectrum, and derive the optimality of our regularity results. We also prove that the semigroup is not differentiable for θ = 0 or θ = 1. Those results strongly improve upon some recent results presented in Ammari et al. [J. Evol. Equ. 21 (2021) 4973-5002].

Classification : 35Q74, 35B35, 35B65, 47D06
Keywords: Regularity, stability, fractional damping, interacting elastic systems

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