We consider evolution equations of the form $$\begin{aligned} \dot{u}(t)+{\mathcal {A}}(t)u(t)=0,\ \ t\in [0,T],\ \ u(0)=u_0, \end{aligned}$$ u ˙ ( t ) + A ( t ) u ( t ) = 0 , t ∈ [ 0 , T ] , u ( 0 ) = u 0 , where $${\mathcal {A}}(t),\ t\in [0,T],$$ A ( t ) , t ∈ [ 0 , T ] , are associated with a non-autonomous sesquilinear form $${\mathfrak {a}}(t,\cdot ,\cdot )$$ a ( t , · , · ) on a Hilbert space H with constant domain $$V\subset H.$$ V ⊂ H . In this note we continue the study of fundamental operator theoretical properties of the solutions. We give a sufficient condition for norm-continuity of evolution families on each spaces V , H and on the dual space $$V'$$ V ′ of V . The abstract results are applied to a class of equations governed by time dependent Robin boundary conditions on exterior domains and by Schrödinger operator with time dependent potentials.

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关于非自主形式进化家族的规范连续性的说明

我们考虑形式为 $$\begin{aligned} \dot{u}(t)+{\mathcal {A}}(t)u(t)=0,\ \ t\in [0,T] ,\ \ u(0)=u_0, \end{aligned}$$ u ˙ ( t ) + A ( t ) u ( t ) = 0 , t ∈ [ 0 , T ] , u ( 0 ) = u 0 ,其中 $${\mathcal {A}}(t),\ t\in [0,T],$$ A ( t ) , t ∈ [ 0 , T ] 与非自治倍半线性形式 $$ 相关联{\mathfrak {a}}(t,\cdot ,\cdot )$$ a ( t , · , · ) 在 Hilbert 空间 H 上，具有恒定域 $$V\subset H.$$ V ⊂ H 。在本笔记中，我们继续研究解的基本算子理论性质。我们给出了每个空间 V 、H 和 V 的对偶空间 $$V'$$V ' 上的演化族范数连续性的充分条件。