Regularity of the Adjoint State for the Instationary Navier-Stokes Equations

In this article, we are considering imbeddings of abstract functions in spaces of func- tions being continuous in time. A family of functions depending on certain parameters is discussed in detail. In particular, this example shows that such functions do not belong to the space C((0,T),H). In the second part, we investigate an optimal control problem for the instationary Navier-Stokes equation. We will answer the question, in which sense the initial value problem for the adjoint equation can be solved. 1. Introduction. In this paper, we will study the regularity of abstract func- tions. The discussed properties are heavily connected to the optimal control of insta- tionary Navier-Stokes equations. Here, the gradient of a given objective functional is evaluated by means of an adjoint state. The adjoint state is itself the solution of an evolution equation. The discussion of abstract functions in the first part of the paper will reflect important properties of the adjoint state. The aim of the present article is two-folded. At first, we want to shed light on imbeddings of abstract functions in spaces of continuous functions. We have to refer to the mostly classical results due to Lions, (4). Given a Gelfand triple V ,! H ,! V 0, the space