288 35. STABILITY OF RICCI FLOWform. For these stability techniques to b e effective, one needs function spaces in
which the linear Cauchy problem
(35.lOa) u' (t) = Au(t) + f (t), 0 :::; t :::; T ,
(35.lOb) u (0) = 0
has a unique solution for which u' and Au possess the same regula rity as f. Maximal
regularity t heory provides a way to do this, through the use of suitable interpolation
spaces. Using some basic ideas from a nalytic semigroup theory, we now show how
the interpolation spaces involved arise and how to use the theory to derive estimates
tha t allow one to move from linear st ability to stability. Parts of our exposition
follow proofs in Lunardi [214], Clement and Heijmans [84], and [9 2 ], to which the
reader is referred for further details. Throughout this section, E 1 and Ea denote
complex Banach spaces with E 1 continuously embedded in Ea, and A : E 1 -t Ea
denotes a densely defined linear operator.
REMARK 35.15. Given a n operator A : f 1 -t Ea between real Banach spaces,
one constructs an analytic semigroup using the complexified spaces Ek = { u +
iv : u , v E f k} (k = 0, 1) and the co mplexified operator A : E 1 -t Ea defined by
A(u+iv) = A(u) +iA(v). Below, we implicitly assume that everything in sight has
been complexified in this way.
Notice that the initial value problem (35.10) h as the formal solution(35.11) u(t) = Sf(t) ~lat e<t-s)A f(s)ds.
Semigroup theory, first introduce d by Hille in 1948, provides a way to interpret etA.If this "variation of constants" formula is a solution of the initial value problem, then
it can b e used together with properties of etA to prove exist ence and uniqueness
of solutions of (35.10), and to obtain the estimates that allow one to determine
stability, instability, and center m anifold b eh avior of solutions of (35.10). (An
excellent referen ce for the calculus of semigroups and for interpolation and maximal
reg ularity theory is found in Chapters 5 and 6 by Clement and Heijmans in [84].
For more detail on a nalytic semigroups, we refer the reader to [214].)
To define etA, one uses the resolvent of A : E 1 -t Ea, given by
R(>-, A)~ (>-j - A)-^1 : Ea -t El,where j : E 1 ~ Ea is the inclusion map. For our purposes, we further wish to
guarantee t hat t -t etA is a n alytic and so restrict ourselves to a certain class of
operators. We say that A is sectorial if the following two conditions are met:
(1) There exists w E IR such that (>-j -A) : E 1 -t Ea is an isomorphism for all
A with Re>. 2 w.
(2) There exists a constant C such that llR(>-, A)JIL~:, :::; l.X~wl llfllt:o.
REMARK 35.16. The t erm "sectorial" comes from the spectrum b eing co ntained
in a wedge in a left (complex) half-plane.
REMARK 35 .17. One can show that condition (2) is satisfied, for example, on
the domain !Rn for an elliptic operator A with bounded coefficient s , via the Agmon-
Douglis- Nirenberg inequalities. For linearized Ricci fl.ow at a fiat metric, A is the
rough Laplacian , and so one can use standard Sch a uder estimates on a compact
manifold (see, for example, Lemma 35.26 b elow).