ABSTRACT
In the smooth case, to each tube one can associate a non-autonomous velocity field V such that
where T(V) is the flow associated to V. More precisely, the final time T > 0 and the initial domain Q being fixed, consider a velocity field V € C([0,T\;Ck(RN ,1K.N)) (the choice of k depends on the regularity of O(, t € [0,r]) satisfying
V(t).nnt — vv(t) on £ (the lateral boundary of Q) (1.1)
where TIQ, is the unitary outer normal to fit and vv(f) is in connection with the time-component of the unitary outer normal v to Q which can be written in a unique way as, cf. [11],
Then we can define the mapping t — > Tt(V) as the solution of
Tt in(0,r)/ TtT* = \T0 -
Condition (1.1) ensures that, at each time t 6 [0,-r],
Conversely, to any sufficiently smooth non-autonomous vector field V one can associate, in the time interval [0,r], a tube Q(V) (also denoted Qv) by setting Q( = Tt(V)(ty. Obviously (1.1) is satisfied. For a general abstract setting of tube evolution theory we refer to [1]. Taking the previous into consideration, the purpose of our study is to exhibit the structure of the Eulerian derivative of non-cylindrical functionals in the following form,
j(V) = J(V,QV), J being given
In many examples J depends exclusively on Qv-So j is a tube functional. Generally, J is function of Qv through a boundary value problem and of V through its associated boundary conditions. For a bounded tube Q, there exists a bounded open set D (called hold-all) such that Q and its perturbations remain in (0, T) x D. This condition is respected if
V(t).nD=Q on (0 ,T)xd£> (1.2)
since the associated transformations Tj(F) map D onto D. In the computation of the Eulerian derivative we should introduce the transverse field Z solution of
dtZ + [Z,V] = W in (0,r) x D and Z(0, .) = 0 in £>
where V, W are admissible vector fields. The first field is associated to the original tube. The second one represents the direction of the perturbation. For additional properties on the transverse field Z, see [12] where the non-smooth case is considered. In the last section of this paper, we give examples of such derivative in the case of a tube functional and also in the general case.
