We consider the well-posedness of a model for a flow-structure interaction. This model describes the dynamics of an elastic flexible plate with clamped boundary conditions immersed in a supersonic flow. A perturbed wave equation describes the flow potential. The plate's out-of-plane displacement can be modeled by various nonlinear plate equations (including von Karman and Berger). We show that the linearized model is well-posed on the state space (as given by finite energy considerations) and generates a strongly continuous semigroup. We make use of these results to conclude global-in-time well-posedness for the fully nonlinear model.
The proof of generation has two novel features, namely: (1) we introduce a new flow potential velocity-type variable which makes it possible to cover both subsonic and supersonic cases, and to split the dynamics generating operator into a skew-adjoint component and a perturbation acting outside of the state space. Performing semigroup analysis also requires a nontrivial approximation of the domain of the generator. And (2) we make critical use of hidden regularity for the flow component of the model (in the abstract setup for the semigroup problem) which allows us run a fixed point argument and eventually conclude well-posedness. This well-posedness result for supersonic flows (in the absence of rotational inertia) has been hereto open. The use of semigroup methods to obtain well-posedness opens this model to long-time behavior considerations.
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
The regularity of the coupled system between an electrical network with fractional dissipation and a plate equation with fractional inertial rotational
Stability and Finite-Time Blow-Up for a Fractionally Damped Nonlinear Plate Equation: Numerical and Analytical Insights
School of Mathematics and Statistics, Shanxi University（山西大学数学与统计学院）；Department of Mathematics, University of Kotli Azad Jammu and Kashmir(UOKAJK)（阿扎德克什米尔kotli大学数学系）；DIM CIMA Universidad de Concepción（康塞普西翁大学DIM CIMA）
Second order linear evolution equations with general dissipation
分析了由抽象线性耗散发展方程$$ \ddot u + A u + f(A) \dot u=0 $$生成的压缩半群$S(t)={\rm e}^{t\mathbb{A}}$，其中$A$是严格正的自伴算子，$f$是$A$的谱上的任意非负连续函数.提供了$S(t)$的无穷小生成器$\mathbb{A}$的频谱的完整描述。根据$f$的性质及其零集的谱性质，得到了该半群的稳定性、半一致稳定性和指数稳定性的充要条件.文中还讨论了分数阶阻尼在波动方程、梁方程和板方程中的应用。
