Starting from a bi-continuous semigroup in a Banach space X (which might actually be strongly continuous), we investigate continuity properties of the semigroup that is induced in real interpolation spaces between X and the domain D(A) of the generator. Of particular interest is the case \((X,D(A))_{\theta ,\infty }\). We obtain topologies with respect to which the induced semigroup is bi-continuous, among them topologies induced by a variety of norms. We illustrate our results with applications to a nonlinear Schrödinger equation and to the Navier–Stokes equations on \(\mathbb {R}^d\).

中文翻译：

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实插值空间中半群的连续性

从Banach空间X中的双连续半群开始（实际上可能是强连续），我们研究了在X和生成器的域D（A）之间的实际插值空间中诱导的半群的连续性。特别感兴趣的情况是\（（X，D（A））_ {\ theta，\ infty} \）。我们获得关于诱导半群是双连续的拓扑，其中由各种范数诱导的拓扑。我们通过应用非线性Schrödinger方程和\（\ mathbb {R} ^ d \）上的Navier–Stokes方程来说明我们的结果。