The regularity of the coupled system between an electrical network with fractional dissipation and a plate equation with fractional inertial rotational

2023/10/02 by Santos Richard W. Sanguino Bejarano, Bejarano, Santos R. W. S., Filomena Barbosa Rodrigues Mendes +7
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #Thermoelastic and Magnetoelastic Phenomena #Elasticity and Wave Propagation

paper · pdf · doi:10.48550/arxiv.2310.00869

Abstract

In this work we study a strongly coupled system between the equation of plates with fractional rotational inertial force κ(-Δ)^βu_tt where the parameter 0 <β≤ 1 and the equation of an electrical network containing a fractional dissipation term δ(-Δ)^θv_t where the parameter 0≤ θ≤ 1, the strong coupling terms are given by the Laplacian of the displacement speed γΔu_t and the Laplacian electric potential field γΔv_t. When β= 1, we have the Kirchoff-Love plate and when β= 0, we have the Euler-Bernoulli plate recently studied in Suárez-Mendes (2022-Preprinter)\citeSuarez. The contributions of this research are: We prove the semigroup S(t) associated with the system is not analytic in (θ,β)∈ [0,1]×(0,1]-\( 1,1/2)\. We also determine two Gevrey classes: s₁ >\frac12max\ (1-β)/(3-β), \fracθ2+θ-β\ for 2≤ θ+2β and s₂> \frac2(2+θ-β)θ when the parameters θ and β lies in the interval (0, 1) and we finish by proving that at the point (θ,β)=(1,1/2) the semigroup S(t) is analytic and with a note about the asymptotic behavior of S(t). We apply semigroup theory, the frequency domain method together with multipliers and the proper decomposition of the system components and Lions' interpolation inequality.

Related

• A Gevrey class semigroup, exponential decay and Lack of analyticity for a system formed by a Kirchhoff-Love plate equation and the equation of a membrane-like electric network with indirect fractional damping
• Regularity of the semigroups associated with some damped coupled elastic systems II: a nondegenerate fractional damping case
• Regularity of Euler-Bernoulli and Kirchhoff-Love Thermoelastic Plates with Fractional Coupling
• Regularity and stability of the semigroup associated with some interacting elastic systems I: A degenerate damping case
• A Note on the Regularity of Thermoelastic Plates with Fractional Rotational Inertial Force