论文标题
操作员半群的均匀收敛，没有时间规律性
Uniform convergence of operator semigroups without time regularity
论文作者
论文摘要
当我们对线性演化方程的长期行为感兴趣时，我们可以使用$ C_0 $ -Semigroups理论的各种技术。但是，如果我们考虑在$ \ mathbb {r}^d $上使用无界系数的抛物线方程，那么解决方案半群将不会是强烈的。对于此类半群，许多工具可用于研究$ C_0 $ - 元素的渐近行为，不再可用，因此，对它们的长期行为知之甚少。 在这一观察结果的推动下，我们证明了一般半群代表的操作员规范融合的新特征 - 没有任何时间规律性假设 - 通过改编“ Infinity的Semigroup”的概念，该概念最近由M.〜Haase和第二名的作者提出。除了其时间规律性的独立性外，我们的方法还允许我们处理离散的时间案例（即单个操作员的权力），甚至可以在同一统一设置中进行更抽象的半igroup表示。 作为结果的应用，我们证明了与上述属性的抛物线方程系统的解决方案的收敛定理。
When we are interested in the long-term behaviour of solutions to linear evolution equations, a large variety of techniques from the theory of $C_0$-semigroups is at our disposal. However, if we consider for instance parabolic equations with unbounded coefficients on $\mathbb{R}^d$, the solution semigroup will not be strongly continuous, in general. For such semigroups many tools that can be used to investigate the asymptotic behaviour of $C_0$-semigroups are not available anymore and, hence, much less is known about their long-time behaviour. Motivated by this observation, we prove new characterisations of the operator norm convergence of general semigroup representations - without any time regularity assumptions - by adapting the concept of the "semigroup at infinity", recently introduced by M.~Haase and the second named author. Besides its independence of time regularity, our approach also allows us to treat the discrete-time case (i.e., powers of a single operator) and even more abstract semigroup representations within the same unified setting. As an application of our results, we prove a convergence theorem for solutions to systems of parabolic equations with the aforementioned properties.
