本文讨论了生成元$G_{A,R}$在某些Hilbert空间$\mathcal{H}_1$中与$\ddot{u} + R \dot{u} + A^*A u = 0$型抽象阻尼波方程相联系的酉等价性，以及在$\mathcal{H}_1 \oplus \mathcal{H}_2$中与某些非自伴Dirac-type算子$Q_{A,R}$（远离后者的零空间）相联系的酉等价性.算子$Q_{A,R}$表示超对称自伴Dirac型算子的非自伴扰动。特别强调的是专门的情况下，0属于连续谱的$A^*A$。除了关于$G_{A,R}$和$Q_{A,R}$的酉等价结果，我们提供了一个详细的研究域的生成器$G_{A,R}$，考虑频谱特性的基本二次算子铅笔$M(z) = |A|^2 - iz R - z^2 I_{\mathcal{H}_1}$，$z\in\mathbb{C}$，推导出一个家庭的保守量的抽象波方程在没有阻尼，并证明均分的能量超对称自伴Dirac型运营商。$R$代表$|A|$的一个适当的函数的特殊的例子进行了深入的处理，并明确计算了这个例子的半群增长界，并显示出与相应的谱界的基础发电机，也与相应的Dirac型运营商。分别研究了无阻尼（R=0）和有阻尼（$R \neq 0$）抽象波动方程的情形以及某些$ε> 0$和$0 \in σ(A^* A)$（但0不是$A^*A$的特征值）的情形$A^* A \geq εI_{\mathcal{H}_1}$.
英文摘要：
We discuss the unitary equivalence of generators $G_{A,R}$ associated with abstract damped wave equations of the type $\ddot{u} + R \dot{u} + A^*A u = 0$ in some Hilbert space $\mathcal{H}_1$ and certain non-self-adjoint Dirac-type operators $Q_{A,R}$ (away from the nullspace of the latter) in $\mathcal{H}_1 \oplus \mathcal{H}_2$. The operator $Q_{A,R}$ represents a non-self-adjoint perturbation of a supersymmetric self-adjoint Dirac-type operator. Special emphasis is devoted to the case where 0 belongs to the continuous spectrum of $A^*A$.
In addition to the unitary equivalence results concerning $G_{A,R}$ and $Q_{A,R}$, we provide a detailed study of the domain of the generator $G_{A,R}$, consider spectral properties of the underlying quadratic operator pencil $M(z) = |A|^2 - iz R - z^2 I_{\mathcal{H}_1}$, $z\in\mathbb{C}$, derive a family of conserved quantities for abstract wave equations in the absence of damping, and prove equipartition of energy for supersymmetric self-adjoint Dirac-type operators.
The special example where $R$ represents an appropriate function of $|A|$ is treated in depth and the semigroup growth bound for this example is explicitly computed and shown to coincide with the corresponding spectral bound for the underlying generator and also with that of the corresponding Dirac-type operator.
The cases of undamped (R=0) and damped ($R \neq 0$) abstract wave equations as well as the cases $A^* A \geq εI_{\mathcal{H}_1}$ for some $ε> 0$ and $0 \in σ(A^* A)$ (but 0 not an eigenvalue of $A^*A$) are separately studied in detail.
Incompressible Navier-Stokes limit of non-bilinear kinetic equations and application to the BGK, nonlinear Fokker-Planck and Boltzmann-Fermi-Dirac equations
