Title: The regularity of the coupled system between an electrical network with fractional dissipation and a plate equation with fractional inertial rotational Show Chinese title

Title: 具有分数阻尼的电气网络与具有分数惯性旋转的板方程之间的耦合系统的正则性

Abstract: In this work we study a strongly coupled system between the equation of plates with fractional rotational inertial force $\kappa(-\Delta)^\beta u_{tt}$ where the parameter $0 <\beta\leq 1$ and the equation of an electrical network containing a fractional dissipation term $\delta(-\Delta)^\theta v_t$ where the parameter $0\leq \theta\leq 1$, the strong coupling terms are given by the Laplacian of the displacement speed $\gamma \Delta u_t$ and the Laplacian electric potential field $\gamma\Delta v_t$. When $\beta = 1$, we have the Kirchoff-Love plate and when $\beta = 0$, we have the Euler-Bernoulli plate recently studied in Su\'arez-Mendes (2022-Preprinter)\cite{Suarez}. The contributions of this research are: We prove the semigroup $S(t)$ associated with the system is not analytic in $(\theta,\beta)\in [0,1]\times(0,1]-\{( 1,1/2)\}$. We also determine two Gevrey classes: $s_1 >\frac{1}{2\max\{ \frac{1-\beta}{3-\beta}, \frac{\theta}{2+\theta-\beta}\}}$ for $2\leq \theta+2\beta$ and $s_2> \frac{2(2+\theta-\beta)}{\theta}$ when the parameters $\theta$ and $\beta$ lies in the interval $(0, 1)$ and we finish by proving that at the point $(\theta,\beta)=(1,1/2)$ 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. Show Chinese abstract

Abstract: 在本工作中，我们研究了板的方程与分数旋转惯性力 $\kappa(-\Delta)^\beta u_{tt}$之间的强耦合系统，其中参数为 $0 <\beta\leq 1$，以及包含分数耗散项 $\delta(-\Delta)^\theta v_t$ 的电气网络方程，其中参数为 $0\leq \theta\leq 1$，强耦合项由位移速度的拉普拉斯算子 $\gamma \Delta u_t$和电势场的拉普拉斯算子 $\gamma\Delta v_t$给出。 当 $\beta = 1$时，我们得到 Kirchoff-Love 板，当 $\beta = 0$时，我们得到最近在 Suárez-Mendes (2022-Preprinter)\cite{Suarez}中研究的 Euler-Bernoulli 板。 本研究的贡献是：我们证明与系统相关的半群 $S(t)$在 $(\theta,\beta)\in [0,1]\times(0,1]-\{( 1,1/2)\}$中不是解析的。 我们也确定两个Gevrey类：当参数$\theta$和$\beta$位于区间$(0, 1)$时，$s_1 >\frac{1}{2\max\{ \frac{1-\beta}{3-\beta}, \frac{\theta}{2+\theta-\beta}\}}$对于$2\leq \theta+2\beta$和$s_2> \frac{2(2+\theta-\beta)}{\theta}$，我们最后证明在点$(\theta,\beta)=(1,1/2)$时半群$S(t)$是解析的，并对$S(t)$的渐近行为进行说明。 我们应用半群理论、频域方法以及乘子，系统的适当分解和Lions的插值不等式。