Gliklikh Y.E. Global and Stochastic Analysis with Applications to Mathematical Physics

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66 2 Connections



∂q



∂q



∂q



∂q



∂q

∂



∂q

. (2.43)

From (2.43) it follows that at every m ∈ M the ﬁrst order tangent space T

is a subspace in τ

M consisting of vectors with zero matrix (β

). However,

if this matrix is not zero, the column (b

) is not a ﬁrst order tangent vector

since it has another transformation rule. On the other hand, by (2.43)the

ﬁeld of matrices (β

) is a symmetric (2, 0)-tensor ﬁeld and it is symmetric

in every coordinate system.

There is an analogous construction of second order diﬀerential forms.

The theory of second order vectors and diﬀerential forms is presented in

detail, for example, in [69, 179, 180, 204, 205]. In these works one can also

ﬁnd an interesting approach to stochastic diﬀerential equations on manifolds.

At every m ∈ M there is a canonical isomorphisms between the space

M  T

M (where  denotes the symmetric tensor product, see Section

1.5) and the quotient space τ

M/T

M, and hence between TM  TM and

τM/TM (see [205]). Taking into account this factorization, we construct the

morphism Q : τM → TM  TM, i.e., the ﬁeld of linear projectors Q

M → T

M  T

M such that

QB(t, m)=Q



∂

∂q

+ β

∂

∂q



= β

∂

∂q

⊗

∂

∂q

. (2.44)

Every connection H on M determines a linear operator from τ

M to T

at any point m ∈ M as follows:



∂

∂q

+ β

∂

∂q





+ Γ



∂

∂q

, (2.45)

where Γ

are the Christoﬀel symbols of the connection H. Thus connections,

and only connections, are smooth cross-sections of the bundle Hom(τM,TM)

of ﬁber-wise linear operators from τM to TM.

Let m(t)=(q

(t),...,q

(t)) be a smooth curve in a chart U. The second

order vector D

m(t)=¨q

∂

∂q

+˙q

˙q

∂

∂q

is called the acceleration of m(t).

Proposition 2.75 For any smooth curve the equality

˙m(t)=HD

m(t)

holds where

is the covariant derivative of a connection H.

Indeed, by formula (2.45) we obtain that HD

m(t)=(¨q

+ Γ

˙q

)

∂

∂q

Corollary 2.76 Acurvem(t) is a geodesic of a connection H if and only if

m(t)=0.

Proof. By Proposition 2.75 the equality HD

m(t) = 0 means that for m(t)

the geodesic equation (2.25) holds. 

Chapter 3

Ordinary Diﬀerential Equations

3.1 Global in Time Existence of Solutions of Ordinary

Diﬀerential Equations

Many criteria for the extendability to (−∞, ∞) of the solutions of diﬀerential

equations in vector spaces are known (see, e.g., the bibliography in [144]). The

main aim of this section is to modify some conditions of this sort in such a

way that they become necessary and suﬃcient. The trick here is the transition

to extended phase spaces and an analysis of the so-called proper functions or

complete Riemannian metrics on manifolds.

3.1.1 A necessary and suﬃcient condition for

completeness of a vector ﬁeld of one-sided type

Here we use the notation and notions from Section 1.1 concerning vector

ﬁelds.

Let M be a smooth manifold with dimension n<∞ and a smooth vector

ﬁeld X be given on M.

Deﬁnition 3.1. A vector ﬁeld X and its ﬂow are called complete if all its

integral curves are well-deﬁned for t ∈ (−∞, +∞).

Denote by m(s):M → M , s ∈ R, the ﬂow of X. For any point m ∈ M

and time t the orbit m(s)(t, m)=m

t,m

(s) of the ﬂow is the solution of the

equation

˙m

t,m

(s)=X(s, m

t,m

(s)) (3.1)

with the initial condition

t,m

(t)=m. (3.2)

Y.E. Gliklikh, Global and Stochastic Analysis with Applications

to Mathematical Physics, Theoretical and Mathematical Physics,

DOI 10.1007/978-0-85729-163-9

3, © Springer-Verlag London Limited 2011

68 3 Ordinary Diﬀerential Equations

If the right-hand side of an equation is a complete vector ﬁeld, we say that

the ﬂow of this equation is complete, i.e., the completeness of vector ﬁelds

and their ﬂows are equivalent.

Deﬁnition 3.2. A mapping F : X → Y of topological spaces is called proper

if the pre-image of every relatively compact set in Y is relatively compact in

X. In particular, a function f : X → R on a topological space X is called

proper if the pre-image of any relatively compact set in R is a relatively

compact set in X.

Recall that in any ﬁnite-dimensional space (in particular, in R)asetis

relatively compact if and only if it is bounded.

Examples of proper functions are the norm in a Euclidean space and the

distance function on a complete Riemannian manifold.

In what follows we shall mainly deal with proper functions on smooth

manifolds.

In questions connected with the completeness of ﬂows in R

the functions

f : R

→ R such that f(x) →∞as x →∞are often considered. Clearly

such functions are proper. The classical completeness theorem with so-called

one-sided estimates says that if there exists a function f as above such that

gradf · X<C(where · denotes the inner product in R

) for a certain

constant C, the ﬂow of the vector ﬁeld X on R

exists up to +∞.Notethat

gradf · X = Xf, the derivative of f in direction of X. In this subsection we

show how to modify this completeness theorem in order to get a necessary

and suﬃcient condition for completeness.

Consider the extended phase space M

= R × M with the natural pro-

jection π

: M

→ M, π

(t, m)=m. Introduce the vector ﬁeld X

on M

given at the point (t, m) ∈ M

as X

(t,m)

=(1,X(t, m)). It is clear that its co-

ordinate representation is given in the form X

∂

∂t

+ X

∂

∂q

+ ... + X

∂

∂q

Hence the corresponding diﬀerential operator on the space of C

-smooth

functions on M

takes the form

∂

∂t

+ X.

Theorem 3.3 A smooth vector ﬁeld X on a ﬁnite-dimensional manifold M

is complete if and only if there exists a smooth proper function ϕ : M

→ R

such that the absolute value of the derivative |X

ϕ| of ϕ in the direction of

is uniformly bounded, i.e., |X

ϕ| = |(

∂

∂t

+X)ϕ|≤C at any (t, m) ∈ M

for some constant C>0 that does not depend on (t, m).

Proof. Suﬃciency.

Consider the ﬂow m

(s):M

→ M

, s ∈ R, with orbits m

(s)(t, m)=

(t,m)

(s) being the solutions of the equation

˙m

(t,m)

(s)=X



(t,m)

(s)



with initial conditions

(t,m)

(t)=(t, m).

3.1 Global in Time Existence of Solutions of Ordinary Diﬀerential Equations 69

Consider the derivative X

ϕ of ϕ along X

.At(t, m) ∈ M

, by deﬁnition

of the derivative in the direction of a vector ﬁeld, we get the equality

ϕ(t, m)=



(t,m)

(s)





s=t

and under the hypothesis of our Theorem





(t,m)

(s)





s=t



≤ C. (3.3)

Represent the values of ϕ along the orbit m

(t,m)

(s) as follows:



(t,x)

(s)



− ϕ((t, m)) =



dτ



(t,m)

(τ)



dτ.

From the last equality and from inequality (3.3) we obtain that if s belongs

to a ﬁnite interval, the values ϕ(m

(t,x)

(s)) are bounded in R.Thensinceϕ is

proper, this means that the set m

(t,m)

(s) is relatively compact in M

Recall that by the classical solution existence theorem the domain of any

solution of an ODE is an open interval in R. In particular for s>tthe

solution of the above Cauchy problem is well-deﬁned for s ∈ [t, t + ε). If ε>0

is ﬁnite, then from the above arguments it follows that the corresponding

values of the solution belong to a relatively compact set in M and so the

solution is well-deﬁned for s ∈ [t, ε]. The same arguments are valid also for

s<t. Thus the domain is both open and closed and so it coincides with the

entire real line (−∞, ∞).

Taking into account the construction of the vector ﬁeld X

,wecanrepre-

sent the integral curves m

(t,m)

(s)intheformm

(t,m)

(s)=(s, m

t,m

(s)). Hence

from the global existence of integral curves of X

we easily obtain the global

existence of integral curves of X. So, the vector ﬁeld X is complete.

Necessity.

Let the vector ﬁeld X be complete. Thus all orbits m

t,m

(s) of the ﬂow

m(s) are well-deﬁned on the entire real line. Let V = {V

}

i∈N

be a countable

locally-ﬁnite cover of M where all V

are open and relatively compact. Such

a cover exists since every ﬁnite-dimensional manifold is locally compact and

by deﬁnition satisﬁes the second countability axiom. Introduce the functions

: M → R by the formula

(m)=



i if m ∈ V

0ifm/∈ V

Denote by {ϕ

}

∞

i=1

the smooth partition of unity subordinate to the above

cover and deﬁne the function ψ on M by:

70 3 Ordinary Diﬀerential Equations

ψ(m)=

∞



i=1

(m)ϕ

(m). (3.4)

It is clear that ψ(m) is smooth and proper by construction. Our construction

of the function ψ(m) is taken from [134].

Introduce the function Φ : M

→ R as follows. For any point (t, m) ∈

set Φ(t, m)=ψ(m

t,m

(0)). By construction the function Φ is constant

along any orbit of the ﬂow m

(s). Indeed, for m

(s)(t, m)=(s, m

t,m

(s)) the

equality m

s,m

t,m

(s)

(0) = m

t,m

(0) holds.

Consider the function ϕ : M

→ R, ϕ(t, m)=Φ(t, m)+t. Clearly ϕ is

smooth and proper. Consider X

ϕ. By the construction of the vector ﬁeld

and of the function ϕ we get

ϕ = X

(Φ(t, m)) + X

t =0+1=1.

Thus we have proven the necessary part of our Theorem for C =1.This

completes the proof. 

3.1.2 A generalization to the inﬁnite-dimensional case

A direct inﬁnite-dimensional generalization of the results of the previous sub-

section cannot be obtained at least because of the absence of proper functions

on inﬁnite dimensional manifolds. To avoid this diﬃculty, here we replace

functions which are proper with respect to the strong topology by functions

which are proper with respect to a weaker topology.

Another diﬀerence is that inﬁnite-dimensional ordinary diﬀerential equa-

tions with smooth right-hand sides very rarely arise in applications. Gener-

ally the right-hand sides are locally Lipschitz continuous and/or completely

continuous (recall that equations with simply continuous right-hand sides in

inﬁnite-dimensional cases may have no solutions at all) or even are given only

on an everywhere dense subset. Examples of the latter are, say, parabolic

equations, considered as ordinary diﬀerential equations on some function

spaces, or equations, close to hydrodynamical ones, on the Sobolev vector

ﬁelds on a compact manifold (see, e.g., [61]). In both cases the diﬀerential

operator on the right-hand side makes functions (vector ﬁelds) less smooth

so that the vectors of the right-hand side that are tangent to the manifold

(phase space) are well-deﬁned only on an everywhere dense subset of “more

smooth” functions (vector ﬁelds).

In this subsection we obtain inﬁnite-dimensional necessary and suﬃcient

conditions for the global existence of solutions both for the right-hand sides

given on the entire manifold and for those given on everywhere dense subsets,

under some additional conditions that seem to be natural.

3.1 Global in Time Existence of Solutions of Ordinary Diﬀerential Equations 71

3.1.2.1 Basic results for linear Banach spaces

Let B be a Banach space and X(t, m) be a continuous non-autonomous vector

ﬁeld on B, t ∈ R. For autonomous vector ﬁelds no simpliﬁcation of the

construction and results occurs and so everywhere below we consider the

general non-autonomous case. For such a vector ﬁeld we consider the following

Cauchy problem

(t,m)

(s)=X(s, m

(t,m)

(s)) (3.5)

(t,m)

(t)=m. (3.6)

Since B is inﬁnite dimensional, without additional conditions (3.5)–(3.6)may

not have solutions. Nevertheless solutions do exist if, say, X(t, m)iscom-

pletely continuous, i.e., it is continuous jointly in t ∈ R and m ∈ B and for

each bounded set A ⊂ B and each interval [a, b] ⊂ R the mapping X sends

the set [a, b] × A into a compact set.

Deﬁnition 3.4. We say that the Cauchy problem for equation (3.5)islocally

well-posed if for each m ∈ B and t ∈ R there exists a unique local solution

of problem (3.5)–(3.6), well-deﬁned on an interval (t −ε



(t,m)

,t+ ε

(t,m)

), that

continuously depends on initial data, i.e., from m

→ m, t

→ t it follows

that m

)

(s) → m

(t,m)

(s)fors ∈



(t − ε



)

,t+ ε

)

), where

(t −ε



)

,t+ ε

)

) is the domain of m

)

(·).

Remark 3.5. On (t − ε



(t,m)

,t] the curve m

(t,m)

(s) may be presented as a

solution of the equation

(t,m)

(s)=−X(s, m

(t,m)

(s)), inverse to (3.5).

This is why instead of saying that the Cauchy problem (3.5)–(3.6) is locally

well-posed one sometimes says both the direct and inverse Cauchy problems

for (3.5) are locally well-posed.

We retain Deﬁnition 3.1 of complete vector ﬁelds for inﬁnite-dimensional

systems.

Deﬁnition 3.6. A function f : E → R,whereE is a Banach space, is called

weakly proper if for any relatively compact set in R its pre-image in E is

bounded.

Remark 3.7. Recall Deﬁnition 3.2 of a proper mapping of topological spaces.

Taking into account the features of weakly compact sets in Banach spaces

(see, e.g., [156, 223, 235]), one can easily see that if B is a reﬂexive Banach

space, Deﬁnition 3.6 means that a weakly proper function is indeed proper

with respect to the weak topology on B.

Consider the extended phase space B

= R × B and the vector ﬁeld

(t,m)

=(1,X(t, m)) on it. From the hypothesis one can easily derive that

(t,m)

is completely continuous. It is clear that the curves

72 3 Ordinary Diﬀerential Equations

(t,m)

(s)=(s, m

(t,m)

(s)) (3.7)

satisfy the following Cauchy problem on B

(t,m)

(s)=X



(t,m)

(s)



(3.8)

(t,m)

(t)=(t, m). (3.9)

Theorem 3.8 Let X(t, m) be a completely continuous vector ﬁeld on a Ba-

nach space B. Let there exist a continuous weakly proper function f : B

→ R

such that for any curve m

(t,m)

(s), as deﬁned by (3.7), there exists a real con-

stant C>0 for which the relation



f(m

(t,m)

)) −f (m

(t,m)

))



<C|s

− s

| (3.10)

holds for every pair s

and s

in the domain of the curve m

(t,m)

(s).Then

X(t, m) is complete.

Proof. Let f be as in the hypothesis and let m

(t,m)

(s) be an arbitrary solution

of (3.5)–(3.6) that is well-deﬁned on some interval s ∈ (t − ε



,t+ ε). The

corresponding curve m

(t,m)

(s) is well-deﬁned on the same interval.

If both ε and ε



are inﬁnite, the Theorem is proved. Let ε be ﬁnite. From

the hypothesis it follows that for s ∈ [t, t + ε) the inequality |f(m

(t,m)

(t)) −

f(m

(t,m)

(s))| <Cεholds. Thus on this interval the values |f (m

t,m

(s))|

are bounded by |f (m

t,m

(t))| + Cε. Then from Deﬁnition 3.6 it follows

that the points of this curve for s ∈ [t, t + ε) belong to a bounded set

Θ = f

−1

([t, t + ε)). Since Θ is bounded, the image of Θ under the com-

pletely continuous mapping X

is compact. Thus the norm of X

as a con-

tinuous function is bounded on Θ and so X

(t,m)

(s) is bounded on

[t, t + ε). Thus the length of m

t,m

(s)fors ∈ [t, t + ε), which is equal to



(t,m)

(s)ds =



X

(t,m)

(s))ds, is bounded. Since B is com-

plete and since m

(t,m)

(s) is continuous in s, one can easily derive from this

fact that the limit lim

s→t+ε

(t,m)

(s) exists, i.e., m

(t,m)

(s) exists on the closed

interval [t, t + ε].

In complete analogy with these arguments we show that if ε



is ﬁnite,

(t,m)

(s) exists on the closed interval [t −ε



,t]. This means that the domain

of m

(t,m)

(s) is both open and closed, hence it is equal to R. Obviously the

same is true for the domain of m

(t,m)

(s). Thus, since the solution m

(t,m)

(s)

was arbitrary, X is complete. 

Corollary 3.9 Let X(t, m) be a completely continuous vector ﬁeld on a

Banach space B. Suppose there exists a continuous weakly proper function

f : B → R such that for any solution m

(t,m)

(s) of (3.5)–(3.6) there exists a

3.1 Global in Time Existence of Solutions of Ordinary Diﬀerential Equations 73

real constant C>0 for which the relation (3.10) holds for every pair s

and

in the domain of m

(t,m)

(s).ThenX(t, m) is complete.

Corollary 3.9 is an analog of a well-known suﬃcient condition for the

completeness of vector ﬁelds in ﬁnite-dimensional spaces (see the previous

subsection). The proof of Corollary 3.9 is the same as that of Theorem 3.8

with a certain simpliﬁcation since here we do not use the extended phase

space. Notice that the conditions of Theorem 3.8 after some modiﬁcation

become necessary and suﬃcient for the completeness of vector ﬁelds (see

Theorem 3.13 below) while Corollary 3.9 gives only suﬃcient conditions for

completeness.

The next statement describes a particular case where (3.10) is fulﬁlled.

The relation obtained here easier to verify in practice.

Theorem 3.10 Let a continuous weakly proper function f : B

→ R be

such that its derivative X

f in the direction of a vector ﬁeld X

(in the

ordinary sense) is well-deﬁned and satisﬁes the estimate |X

f| <C at any

point (t, m) ∈ B

for some constant C>0 that is independent of (t, m).

Then along any curve m

(t,m)

(s), as introduced by (3.7), relation (3.10) is

satisﬁed with this C and so by Theorem 3.8 X(t, m) is complete.

Indeed,





(t,m)

)



− f



(t,m)

)









(t,m)

(s)





≤









(t,m)

(s)







Cds



= C|s

− s

The next statement gives a necessary condition for the completeness of

X(t, m) of the same sort as the suﬃcient condition of Theorem 3.8.

Theorem 3.11 Let X(t, m) be a continuous vector ﬁeld on a Banach space

B such that the Cauchy problem (3.5)–(3.6) is locally well-posed (see Deﬁ-

nition 3.4). If X(t, m) is complete, there exists a continuous weakly proper

function f : B

→ R such that for any curve m

(t,m)

(s), as introduced by

(3.7), there exists a real constant C>0 for which relation (3.10) holds for

every pair s

and s

Proof. Let the solutions m

(t,m)

(s) of the problem (3.5)–(3.6)existonthe

entire line. Introduce the weakly proper continuous function r : B → R,

r(m)=m.

Denote by g

: B → B the ﬂow of the vector ﬁeld X, i.e., g

(m)=

(0,m)

(s). Since X(t, m) is complete, under the hypothesis of our Theo-

rem g

is well-deﬁned and forms a continuous family of homeomorphisms

of B. For the same reason, for every point (t, m) ∈ B

the value m

(t,m)

(0)

of the curve m

(t,m)

(s) is well-deﬁned and continuously depends on (t, m).

Now construct the continuous function Φ : B

→ R by assigning the value

Φ(t, m)=r

t,m

(0)) to the point (t, m) ∈ B

74 3 Ordinary Diﬀerential Equations

Consider the continuous function f : B

→ R, f(t, m)=Φ(t, m)+t. One

can easily see that f is weakly proper.

By construction, Φ takes constant values along the curves m

(t,m)

(s)on

.Indeed,form

(t,m)

(s)=(s, m

(t,m)

(s)) the equality m

(s,m

(t,m)

(s))

(0) =

(t,m)

(0) holds for all s. Hence,





(t,m)

)



− f



(t,m)

)







(t,m)

(0)



+ s

− r



(t,m)

(0)



− s



= |s

− s

Thus, the constant C in the conclusion of the theorem may take any value

greater than or equal to 1. 

For a smooth vector ﬁeld X(t, m), under some additional hypotheses we

obtain a necessary condition for completeness of the same sort as the suﬃcient

condition of Theorem 3.10.

Theorem 3.12 Let a Banach space B have a smooth norm and let X(t, m)

be a smooth and complete vector ﬁeld on B. Then there exist a smooth weakly

proper function f : B

→ R and a real constant C>0 such that for the

derivative X

f of f in the direction of X

the inequality |X

f| <C holds

on B

Proof. Since X(t, m) is smooth, it is locally Lipschitz continuous and so the

Cauchy problem (3.5)–(3.6) is locally well-posed. From the fact that both the

vector ﬁeld X(t, m) and the function r

(m) are smooth it follows that the

function f : B

→ R, constructed in the proof of Theorem 3.11,issmooth.

Since Φ takes constant values along the curves m

(t,m)

(s)onB

, a direct

calculation shows that |X

f| =1. 

Now let a vector ﬁeld X(t, m)onaBanachspaceB be completely contin-

uous and be such that the Cauchy problem (3.5)–(3.6) is locally well-posed.

Then as a corollary to Theorems 3.8 and 3.11 we obtain the following:

Theorem 3.13 Let B be a Banach space. A completely continuous vector

ﬁeld X(t, m) on B for which the Cauchy problem (3.5)–(3.6) is locally well-

posed is complete if and only if there exists a continuous weakly proper func-

tion f : B

→ R such that for any curve m

(t,m)

(s), as introduced by (3.7),

there exists a real constant C>0 for which relation (3.10) holds for every

pair s

and s

in the domain of the curve m

(t,m)

(s).

Corollary 3.14 Let B be a Banach space. Both completely continuous and

locally Lipschitz continuous vector ﬁelds X(t, m) on B are complete if and

only if there exists a continuous weakly proper function f : B

→ R such that

for any curve m

(t,m)

(s), as introduced by (3.7), there exists a real constant

C>0 for which relation (3.10) holds for every pair s

and s

in the domain

of the curve m

(t,m)

(s).

3.1 Global in Time Existence of Solutions of Ordinary Diﬀerential Equations 75

Indeed, for a locally Lipschitz continuous vector ﬁeld the Cauchy problem

(3.5)–(3.6) is locally well-posed.

As a corollary to Theorems 3.10 and 3.12 we obtain the following:

Theorem 3.15 Let a Banach space B have smooth norm and let X(t, m) be

a smooth and completely continuous vector ﬁeld on B. X(t, m) is complete if

and only if there exist a smooth weakly proper function f : B

→ R and a

real constant C>0 such that for the derivative X

f of f in the direction of

the inequality |X

f| <C holds on B

3.1.2.2 The case when the right-hand side of the equation is

deﬁned on an everywhere dense subset

Let D be a Banach space that is embedded into B by a continuous map so

that the image of D is everywhere dense in B. For the sake of simplicity we

do not distinguish between D and its image, so we regard D as a subset of

Let for every m ∈ D and t ∈ R a vector X(t, m) ∈ B be given. Consider

the Cauchy problem

(t,m)

(s)=X



s, m

(t,m)

(s)



(3.11)

(t,m)

(t)=m ∈ D. (3.12)

In analogy with the notions from the theory of partial diﬀerential equations

we give the following:

Deﬁnition 3.16. We say that the Cauchy problem (3.11)–(3.12)islocally

well-posed if for each m ∈ D and t ∈ R there exists a unique local solution,

given on a certain interval (t − ε



,t+ ε), belonging to D and continuously

depending on the initial data, i.e., from m

→ m it follows that m

(t,m

)

(s) →

(t,m)

(s)fors ∈



(t −ε



,t+ ε

), where



(t −ε



,t+ ε

) is the domain of

(t,m

)

(·). Here the convergence is understood to be in the topology of D.

We refer the reader, say, to [163, 164] for examples of conditions that

guarantee for an equation the well-posedness of the Cauchy problem (3.11)–

(3.12).

Deﬁnition 3.17. The Cauchy problem (3.11)–(3.12) is called regular if its

local solutions exist and, for a solution m

(t,m)

(s) with m ∈ D,fromthe

fact that at the time s

∗

the point m

(t,m)

∗

) belongs to B it follows that

(t,m)

∗

) ∈ D.

Examples of regular Cauchy problems can be found, e.g., in [15, 34, 61].

In analogy with the above notation, we introduce D

= R × D and

(t,m)

(s)=(s, m

(t,m)

(s)).