Galdi G.P. An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems

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526 VIII Steady Generalized Oseen Flow in Exterior Domains

VIII.4 On the Unique Sol vability of the Oseen

Initi al-Value Problem

The objective of this section is to study existence a nd uniqueness of solutions

to the Cauchy problem (VIII.3.1), under suitable assumptions on the data

f and u

. Since the problem is linear, we can split it into the following two

separate problems:

∂w

∂t

= ∆w + R

∂w

∂x

−∇φ + f

∇ · w = 0











in R

× (0, T )

w(x, 0) = 0 ,

(VIII.4.1)

and

∂v

∂t

= ∆v + R

∂v

∂x

∇ · v = 0











in R

× (0, T )

v(x, 0) = u

(VIII.4.2)

where T is an arbitrary positive number.

It is useful to introduce some notation. If A is a domain in R

, and t ∈

(0, ∞], we deﬁne

= A × (0, t) .

Moreover, for q ∈ (1, ∞), we set

) =

w : A

→ R

, and φ : A

→ R :

w ,

∂w

∂t

, ∇w , D

w ∈ L

) , φ ∈ L

loc

) , ∇φ ∈ L

)

We also set

kuk

(q,r),A,t

≡













ku(t)k

q,A



1/r

if r ∈ [1, ∞), q ∈ [ 1, ∞] ,

ess sup

t∈[0,t]

ku(t)k

q,A

if r = ∞, q ∈ [1, ∞] ,

and deﬁne

r,q

) = {u : A

→ R , with kuk

(q,r),A,t

< ∞}.

As is customary, whenever confusion does not arise, we shall omit the subscript

A. Clearly, L

q,q

) = L

VIII.4 On the Unique Solvability of the Oseen Initial-Value Problem 527

Exercise VIII.4.1 Show that if w ∈ L

), then, possibly redeﬁned on a set of

zero measure in (0, t), w is continuous at all t ∈ (0, t) in the norm k·k

. Hint: Mollify

w in t, and then use the inequality

kw(t

) − w(t

≤ q

kw(s) − w(t

q−1

‚

∂w

∂s

‚

ds .

We now investigate the solvability of problem (VIII.4.1).

Theorem VIII.4.1 Let f ∈ L

), 1 < q < ∞. Then there exists a pair

(w, φ) ∈ L

) satisfying (VIII.4.1)

1,2

a.e. in R

. If q ∈ (1, 3), then φ is also

in L

q,3q/(3−q)

). Furthermore, w obeys (VIII.4.1)

in the fol lowing sense:

lim

t→0

kw(t)k

= 0 . (VIII.4.3)

In addition, there is C

= C

(q) > 0 such that



|w(t)|

2,q

+ k∇φ(t)k



dt ≤ C

kf(t)k

dt , (VIII.4.4)

so that if R = 0, we also obtain



∂w

∂t



≤ 3

q−1

kf (t)k

dt . (VIII.4 .5)

In addition, if q ∈ (1, 3), we have

kφ(t)k

3q/(3−q)

dt ≤ C

kf (t)k

dt , (VIII.4.6)

for some C

= C

(q) > 0. Finall y, let (w

, φ

) ∈ L

), for some r ∈ (1, ∞),

be a nother solution corresponding to the same f. Then w = w

and φ =

+ h(t), a.e. in R

, where h is a function of time only.

Proof. It suﬃces to prove the result when R = 0, in that the case R 6= 0

follows by the change of variable x → x + Rte

. To show existence, we

observe that we need to consider only the case f ∈ C

∞

), since, as the

reader may wish to show, the general case will then be a consequence of an

elementary density argument. With all the above in mind, we set w and φ

identicall y zero for t < 0, and deﬁne for a ﬁxed λ > 0,

u(x, t) = w(x, t) e

−λt

, Φ(x, t) = φ(x, t) e

−λt

, F (x, t) = f (x, t) e

−λt

(VIII.4.7)

so that u, Φ, and F satisfy the following problem

A more general uniqueness theorem will be given in Lemma VIII.4.2.

528 VIII Steady Generalized Oseen Flow in Exterior Domains

∂u

∂t

= ∆u − λu − ∇Φ + F

∇ · u = 0







in R

× (0, T )

u(x, 0) = 0 .

(VIII.4.8)

A solution to (VIII.4.8) will be found by the Fourier transform method. Thus,

letting

u(x, t) =

(2π)

ix·ξ+it ξ

U(ξ, ξ

)dξdξ

Φ(x, t) =

(2π)

ix·ξ+it ξ

Ψ(ξ, ξ

)dξdξ

F (x, t) =

(2π)

ix·ξ+it ξ

G(ξ, ξ

)dξdξ

(VIII.4.9)

and replacing these expressions back in (VIII.4.8)

1,2

, we obtain

(i ξ

+ ξ

+ λ)U

= −i ξ

Ψ + G

, j = 1, 2, 3 ,

= 0 .

Solving these equations for U and Ψ furnishes

−ξ

(i ξ

+ ξ

+ λ)ξ

≡ M

, j = 1, 2 , 3 ,

Ψ = −i

(VIII.4.10)

It is easy to check that for any ﬁxed j, k, l, m = 1, 2, 3, the functions

, ξ

, satisfy the assumptions of Lizorkin’s theorem, Theo-

rem VII.4 .1 with β = 0 and M = M (q, λ), while i ξ

/ξ

and ξ

satisfy

the same assumption, but with a constant M independent of λ. Consequently,

on the one hand, (u, Φ) is in the class L

), for all q ∈ (1, ∞), and on the

other hand, we can ﬁnd a constant C

= C

(q) > 0 such that u and Φ satisfy

inequality (VIII.4.4) with f ≡ F , which in turn, recalli ng (VIII.4.7), implies



|w(t)|

2,q

+ k∇φ(t)k



dt ≤ C

qλT

kf(t)k

dt . (VIII.4.11)

Also, if q ∈ (1, 3), then it is easy to check that the multiplier i ξ

/ξ

satisﬁes

Theorem VII.4.1 with β = 1/3 and M independent of λ. Therefore, by that

theorem, Φ(t) ∈ L

3q/(3−q)

), for a.a. t ∈ [0, T], and

kΦk

3q/(3−q)

≤ C

kF k

with C

= C

(q) > 0, namely, ag ain by (VIII.4.7),

kφk

3q/(3−q)

≤ C

λT

kf k

. (VIII.4.12)

VIII.4 On the Unique Solvability of the Oseen Initial-Value Problem 529

Since λ is an arbitrary positive number and C

, C

do not depend on it, from

(VIII.4.11) and (VIII.4.12) we recover (VIII.4.4) and (VIII.4.6). Moreover,

recalling that G ∈ S(R

) (see Section VII.4), we prove immediately that u, Φ

are smooth f unctions that satisfy (VIII.4 .8)

1,2

. Summarizing a ll the above, we

thus obtain the existence of a smooth solution (w, φ) to (VIII.4.1 )

1,2

in the

class L

), with φ ∈ L

q,3q/(3−q)

) if q ∈ (1, 3), and satisfying (VIII.4.4),

and (VIII.4.6) as well, if q ∈ (1, 3). We shall next ana lyze the way in which

u (and hence w) tends to zero as t → 0

. To this end, we observe that from

(VIII.4.9) and (VIII.4.10), we have

(x, t) =

(2π)

ix·ξ+it ξ

− ξ

(i ξ

+ ξ

+ λ)ξ

(ξ, ξ

)dξdξ

. (VIII.4.13)

Now, in view of the assumptions on F , it follows that G is analytic i n ξ

Therefore, by a simple application of the Cauchy integral theorem (MacRobert

1966, §27), we can show that the value of the integral

∞

−∞

it ξ

(ξ, ξ

)

(i ξ

+ ξ

+ λ)

dξ

, t ∈ R

does not change (a nd therefore the value of the integral i n (VIII.4.1 3) does

not change) if we perform the integration on a line {α

} in the complex plane

parallel to {ξ

}, provided i ξ

+ ξ

+ λ does not vanish along points of {α

We thus choose α

= ξ

+ i δ, δ > 0, so that (VIII.4.13) furnishes

(x, t) =

δ t

(2π)

ix·ξ+it α

(δ

−ξ

) G

(ξ, α

−i δ)

(i α

+ δ + ξ

+ λ)ξ

dξdα

(VIII.4.14)

Moreover, we obtain

G(ξ, α

− i δ) =

(2π)

F (x, t)e

−δ t

−ix·ξ−it α

dx dt ,

which, by Parseval’s theorem, furnishes

|G(ξ, α

−i δ)|

dξ dα

|F (x, t)|

−δt

dx ,

where we have also employed the fact that F is of compact support in (0, T ).

Using this information in (VIII.4.14) along with the Schwarz i nequality, we

deduce

|u(x, t)| ≤ c

δ t



dζ dξ

+ ξ

+ δ

+ λ



|F (x, t)|

−δt

≤ c

δ t

with c

independent of δ. If we take t < 0 in this latter relation and let δ → ∞,

we conclude that u(x, t) = 0 for all t < 0 and x ∈ R

, and so, by the continuity

530 VIII Steady Generalized Oseen Flow in Exterior Domains

of u, u(x, 0) = w(x, 0) = 0, for all x ∈ R

. Once this property is established,

we notice that for all t ∈ (0, T],

kw(t)k

kw(s)k

ds ≤ q



kw(s)k





∂w

∂s



which, in turn, together with (VII I.4.4), proves (VIII.4.3). It remains to show

uniqueness. Thi s amounts to proving that the problem

∂w

∂t

= ∆w − ∇φ

∇· w = 0







in R

× (0, T )

w(x, 0) = 0 ,

(VIII.4.15)

with (w, φ) ≡ (w

+ w

, φ

+ φ

), (w

, φ

) ∈ L

), for some q

∈ (1, ∞),

i = 1, 2, has only the solution w ≡ ∇φ ≡ 0. In order to prove the above, we

begin by o bserving the following two obvious facts: (a) The method of proof

just described leads to the existence of a solution (v, ζ) ∈ L

) to problem

(VIII.4.1) with R replaced by −R; (b) if f ∈ C

∞

), then (v, ζ) i s in the

class L

) for all q ∈ (1, ∞). With these two remarks in hand, we set

V (x, t) = v(x, T −t) , Z(x, t) = ζ(x, T − t) , H = f(x, T − t) ,

with arbitrary f ∈ C

∞

). Then the ﬁelds V , Z, and H satisfy the following

problem:

∂V

∂t

+∆V = ∇Z − H

∇ · V = 0







in R

×(0, T)

V (x, T ) = 0 ,

(VIII.4.16)

and moreover,

V ,

∂V

∂t

, ∇V , D

V ∈ L

) , Z ∈ L

loc

×[0, T ]) , ∇Z ∈ L

)

for all q ∈ (1, ∞) .

(VIII.4.17)

We next multiply (VIII.4.15)

by V and integrate by parts over B

×[0, T ].

Taking into account the properties (VIII.4.16), (VIII.4.17) of V , the assump-

tion on w, φ, and Exercise II.4.3, we deduce

VIII.4 On the Unique Solvability of the Oseen Initial-Value Problem 531

w(x, t) ·H(x, t)dx dt

i=1

∂B



∂w

∂n

· V −w

∂V

∂n

+ φ

V · n −Zw

· n



dσ dt

i=1

∂B



∂w

∂n

· V −w

∂V

∂n

+ φ

V · n −Zw

· n



dt dσ

≡ I

(R) + I

(R) ,

(VIII.4.18)

where n is the unit outer normal to ∂B

. By choosing q = q

/(q

− 1) in I

i = 1, 2, it i s now readily seen that

+ I

∈ L

(1, ∞) ,

and consequently, that there exists at least a sequence {R

}, with R

→ ∞

as m → ∞, such that

lim

m→∞

) + I

) = 0 .

Plugging this information back into (VIII.4.18), and recalling the deﬁnition

of H, we deduce

w(x, t) · f (x, T − t)dx dt = 0 , for all f ∈ C

∞

) ,

which shows tha t w(x, t) = 0 a.e. in R

. This, in turn, with the help of

(VIII.4.15), implies ∇φ(x, t) = 0 a.e. in R

, and the proof of uniqueness is

complete.

The velocity ﬁeld corresponding to the solution determined in the previous

theorem admits a simpl e representation in terms of the Oseen fundamental so-

lution, under suitable conditions on f . To this end, we introduce the following

lemma .

Lemma VIII.4.1 Let A be a bounded, locally Lipschitz domain of R

, and

let u

, p

, i = 1, 2, be vector and scalar ﬁelds, respectively, with u

, i = 1, 2,

solenoidal, and belonging to the class L

), q

= q, q

= q

. Then the

following Green’s identity holds, for all 0 < t

≤ t

< t:

532 VIII Steady Generalized Oseen Flow in Exterior Domains



·(

∂u

∂τ

+ ∆u

− R

∂u

∂x

− ∇p

)

−u

· (−

∂u

∂τ

+ ∆u

+ R

∂u

∂x

−∇p

)



dx dτ

∂A



·T (u

, p

) − u

·T (u

, p

) + R(u

·u

) e



· N dσdτ

· u



τ=t

where N is the unit outer normal to ∂A, and, we recall, T is the Cauchy

stress tensor deﬁned in (IV.8.6), (IV.8.7).

Proof. The proof is at once achieved by integration by parts, once we take

into account Lemma II.4.1, Exercise II.4.3, (IV.8.9), Exercise VI II.4.1, and

the identity ∇ · T (u, p) = ∆u − ∇p, valid for solenoidal ﬁelds u. ut

We can now prove the following result.

Theorem VIII.4.2 Suppose that

f ∈ L

) , 1 < q < ∞,

and let w, φ be the solution to (VIII.4.1) in the class L

), corresponding

to f and satisfying (VIII.4.3). Then, for a .a. (x, t) ∈ R

, we have the following

volume potential representation for w

w(x, t) =

Γ ( x −y, t −τ; R) ·f(y, τ ) dy dτ . (VIII. 4.19)

Proof. Let {f

} be a sequence of functions from C

∞

) such that

lim

k→∞

− f k

(r,q),T

= 0 .

(VIII.4.20)

We denote by {(w

, φ

)} the sequence of solutions corresponding to f

con-

structed in Theorem VIII.4.1. By the uniqueness properties, the following

facts are at once established:

(a) (w

, φ

) ∈ L

) for all q ∈ (1, ∞) , all k ∈ N ;

(b) lim

k→∞

(t) − w(t)k

dt = 0 .

The functions f

can b e easily constructed by multiplying f by a “cut-oﬀ” func-

tion that i s 1 in B

and 0 in B

, and then mollifying the obtained function

with parameter ε = 1/R

VIII.4 On the Unique Solvability of the Oseen Initial-Value Problem 533

Moreover, since w

is in L

) ∩ D

1,q

) for some q

> 1 a nd q

> 3 for

a.a. t ∈ [0, T], from Theorem II.9.1 we obtain, for all k ∈ N,

(x, t), → 0 uniformly, as |x| → ∞, for a.a. t ∈ [0, T].

Finally, property (a), together with the embedding Theorem II.3.2, ensures

that

(d) w

(x, t) ∈ C(R

) , all k ∈ N .

Set

= (Γ

, Γ

) ,

for ﬁxed i ∈ {1, 2, 3}. We then use Green’s identity in Lemma VIII.4.1 with

A = B

(x), t

= η, t

= t − ε < T , ε, η > 0, u

= w

, and u

= Γ

. Taking

into account tha t Γ satisﬁes (VIII.3.8) for all t > τ , and that w

satisﬁes

(VIII.4.1) with f ≡ f

, we deduce

t−ε

(x)

(x − y, t −τ) ·f

(y, τ) dy dτ

t−ε

∂B

(x)



(y, τ) · T (Γ

, 0)(x − y, t − τ)

−Γ

(x − y, t −τ) ·T (w

, φ

)(y, τ )

+R(w

(y, τ) · Γ

(x − y, t −τ)) e



· N dσ

dτ

(x)

(y, τ) · Γ

(x − y, t −τ)



τ=t−ε

τ=η

We now let η, ε → 0 in this relation. In view of prop erty (d) above, Lemma

VIII.3.1, and (VIII.4.3), we obtain for i = 1, 2, 3 ,

(x, t) =

(x)

(x − y, t −τ) · f

(y, τ) dy dτ

4π

∂B

(x)

−y

|x −y|

(y, t) · N(y)dσ

∂B

(x)



(y, t − τ ) · T (Γ

, 0)(x − y, τ)

−Γ

(x − y, τ) · T (w

, φ

)(y, t − τ)

+R(w

(y, τ) · Γ

(x − y, t −τ)) e



· N dσ

dτ .

(VIII.4.21)

In order to simplify the notation, in the rest of the proof we omit the dependence

of Γ on R.

534 VIII Steady Generalized Oseen Flow in Exterior Domains

We next pass to the li mit as R → ∞ in this latter relatio n. Clearly, since f

is of compact support,

lim

R→∞

(x)

(x −y, t −τ ) ·f

(y, τ) dy =

(x −y, t −τ ) ·f

(y, τ) dy .

(VIII.4.22)

Moreover, in view of the property for w

mentioned in (c) above,

lim

R→∞

4π

∂B

(x)

x − y

|x −y|

(y, t) · N (y)dσ

= 0 , for a.a. t ∈ [0, T].

(VIII.4.23)

Also, from Lemma VIII.3.2 we obtain, for r = |x − y| suﬃciently large and

τ ∈ [0, T]

|Γ (r, τ )| = O(r

−3

) , |∇Γ (r, τ )| = O(r

−4

) .

Thus, denoting by I

(t) the last integral on the right-hand side of (VI II.4.21),

we infer

(t) ≤ c

−1

(y, t − τ )|dσ

−1

(|∇w

(y, t − τ )|+ |φ

(y, t − τ )|



dσ

dτ

≤

(t −τ)k

q,S

+ k∇w

(t − τ )k

q,S

+ kφ

(t − τ)k

q,S

dτ .

(VIII.4.24)

We take q < 3 in (VIII.4.24) (this is allowed by property (a) mentioned

previously) and use Lemma II.6.3. Consequently, by possibly modifying φ

the addition of a function of time only, from (VIII.4.24) it follows that for all

suﬃciently large R,

(t) ≤ c

−3/q

(k∇w

(s)k

1,q

+ k∇φ

(s)k

) ds

≤ c

−3/q





k∇w

(s)k

1,q

+ k∇φ

(s)k





where c

= c

(T ). From this inequality and from (VIII.4.24) we obtain

lim

R→∞

(t) = 0 , for all t ∈ [0, T ] , (VII I.4.25)

and so, collecting (VIII.4.21)–(VIII.4 .25), we may conclude that

(x, t) =

(y, τ) · f

(x − y, t −τ) dy dτ . (VIII.4.26)

We next observe that, by property (b) and Lemma II.2.2, we have

VIII.4 On the Unique Solvability of the Oseen Initial-Value Problem 535

lim

k→∞

(x, t) = w(x, t) for a.a. (x, t) ∈ R

. (VIII.4.27)

Furthermore, from Lemma VIII.3.2, we have, for all r ∈ (1, ∞),

kΓ (t)k

≤ c

(t + |y|

)

3r/2

= c

(

1−

)

. (VIII.4.28)

We also set

V [g](x, t) :=

(y, τ) ·g(x −y, t −τ) dy dτ .

Let B ⊂ R

be an arbitrary ball and s ∈ (1, q). Ta king into account

(VIII.4.26), with the help of the Minkowski inequality we obtain

kw − V [f]k

s,B

≤ kw − w

s,B

+ kV [f

−f]k

s,B

. (VIII.4 .29)

In view of (VIII.4.27 ), we have (along a subsequence, at least)

lim

k→∞

− wk

s,B

. (VII I.4.30)

We shall next prove the inequality

kV [f

−f ]k

s,B

≤ c kf

− fk

q,R

, (VIII.4.31)

so that from (VIII.4.2 0) and (VIII.4.29)–(VI II.4.31), by the arbitrariness of B

we deduce w = V [f ] a.e. in R

, which proves the theorem. In order to prove

(VIII.4.31), we notice that by the Young inequality (II.11.2) and (VII I.4.28),

it follows that

kV [f

−f](t)k

s,B

≤ c

(t −τ)

(

−

)

−f k

q,R

dτ .

Since for any given q ∈ (1, ∞) we can ﬁnd s ∈ (1, q) such that (3/2)(1/s −

1/q) ≤ (1 − 1/q), from Theorem II.11.2 we deduce that the map

t ∈ (0, T ) →

(t − τ)

(

−

)

− fk

q,R

dτ

is bounded from L

(0, T ) into itself, and, consequently, since s < q, from

(0, T ) into L

(0, T ). This shows the validity of (VIII.4.31), a nd the theorem

is thus proved.

We now turn to the unique solvability of problem (VIII.4.2). In this respect,

we begin by proving the following general uniqueness result.