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

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286 IV Steady Stokes Flow in Bounded Domains

On the other hand, integrating by parts the left-hand side of this latter relatio n

we ﬁnd

Ω

∆v·∇ϕ =

Ω

[∇ · (∇v · ∇ϕ) − ∇v : ∇∇ϕ] =

∂Ω

n·∇v·∇ϕ−

Ω

∇v : ∇∇ϕ .

From this equation and (IV.6.16), we may conclude

Ω

p g =

Ω

∇v : ∇∇ϕ −

∂Ω

n · ∇v · ∇ϕ . (IV.6.17)

From the H¨older inequality and (IV.6.15), we have



Ω

∇v : ∇∇ϕ



≤ c |v|

1,q

kgk

. (IV.6.18)

Moreover, again by the H¨older inequality, the trace Theorem II.4.1, and

(IV.6.15), we ﬁnd



∂Ω

n · ∇v · ∇ϕ



≤ k∇vk

r,∂Ω

k∇ϕk

,∂Ω

≤ c k∇vk

r,∂Ω

kϕk

2,q

≤ c k∇vk

r,∂Ω

kgk

where r

∈ [q

, q

(n − 1)/(n − q

)], if q

< n and r

∈ [q

, ∞), if q

≥ n. (These

conditions are equivalent to those given for the exponent r in the statement

of the theorem.) Fina lly, on the ri ght-hand side of this last displayed relation,

we employ the inequality given in Exercise II.4.1, to obtain



∂Ω

n · ∇v ·∇ϕ



≤ (c

|v|

1,q

+ ε kvk

2,r

) kgk

. (IV.6.19)

Since g is an arbitrary element of L

(Ω), the theorem follows from (IV.6.17)–

(IV.6.19) and Theorem II.2.2. ut

By combining the previous result with inequality (IV.6.10), we immedi-

ately obtain the following

Corollary IV.6.1 Let Ω, v, p, f and r be as in Theorem IV.6.2. Then, for

any ε > 0 there exists c = c(Ω, n, q, r, ε) such that

kpk

≤ c |v|

1,q

+ ε



kfk

+ kv

∗

2−1/r,r(∂Ω)



where v

∗

is the trace of v at ∂Ω.

In view of the trace Theorem II.4.4, v

∗

∈ W

2−1/q,q

(∂Ω), as a consequence of the

assumption v ∈ W

2,q

(Ω).

IV.7 Existence and Uniqueness in H¨older Spaces. Schauder Estimates 287

IV.7 Existence and Uniqueness in H ¨ol der Spaces.

Schauder Esti mates

Existence and uniqueness results similar to those proved i n Lemm a IV.6.1 and

Theorem IV.6.1 can also be obtained in H¨older spaces C

k,λ

(Ω), together with

corresponding estimates (Schauder estimates). The procedure is the same as

the one used for Sobolev spaces W

m,q

(Ω); that is, one ﬁrst shows existence,

uniqueness, and the validity of corresponding estimates for solutions in R

and R

and, subsequently, one specializes the results to a (suﬃciently smooth)

bounded domain by means of the “localization procedure” used in the proof

of Lemma IV.6. 1.

However, to obtain existence in R

and R

, instead of the Calder´on–

Zygmund Theorem II.11.4 and Theorem II.11.6, we have to employ their

counterparts in H¨older spaces, namely, the H¨older-Lichtenstein-Giraud the-

orem; see, e.g., Bers, John, & Schechter (1964, pp. 223-224), and Theorem 3.1

of Agmon, Douglis, & Nirenberg (1959), respectively.

Since estimates in H¨older norms will not play any relevant role i n this

book, we shall not give details of their derivation, limiting ourselves to quote

the main results without pro ofs. In this regard, it should b e observed that

they can be obtained, as a particular case, f rom the work of Agmon, Do uglis,

& Nirenberg (1964) a nd Solonnikov (1966) since, as already observed, the

Stokes system is elliptic in the sense of Douglis-Nirenberg. Thus, from the

uniqueness Lemma IV.6.2 and the results of Agmon, Do ugli s, & Nirenberg

(1964, Theorem 9.3 and Rema rks 1 and 2 that follow the theorem) we have

Theorem IV.7.1 Let Ω be a bounded domain in R

, n ≥ 2, of class C

m+2,λ

m ≥ 0, λ ∈ (0, 1), and let v, p be a sol ution to the Stokes problem (IV.0.1),

(IV.0.2) with

v ∈ C

2,λ

(Ω), p ∈ C

1,λ

(Ω).

Then, if

f ∈ C

m,λ

(Ω), v

∗

∈ C

m+2,λ

(∂Ω),

we have

v ∈ C

m+2,λ

(Ω), p ∈ C

m+1,λ

(Ω),

and the following estimate holds:

kvk

m+2,λ

+ inf

c∈R

kp + ck

m+1,λ

≤ c



kfk

m,λ

+ kv

∗

m+2,λ

(∂Ω)



, (IV.7.1)

where c = c(m, λ, Ω, n).

Concerning existence, we have (Solonnikov 1966, Theorem 3.1),

Theorem IV.7.2 Let Ω be a bounded domain of R

, n ≥ 2,

of class C

2,λ

λ ∈ (0, 1). Then, given

Actually, in Solonnikov’s paper the result is proved for n = 3. However, the

technique employed there can be extended to the case where n ≥ 2.

288 IV Steady Stokes Flow in Bounded Domains

f ∈ C

2,λ

(Ω), v

∗

∈ C

2,λ

(∂Ω),

there exists one and only one solution v, p

to (IV.0.1), (IV.0.2) such that

v ∈ C

2,λ

(Ω), p ∈ C

1,λ

(Ω).

Remark IV.7.1 Theorem IV.7.1 and Theorem IV.7 .2 continue to hol d when

g ≡ ∇ · v 6≡ 0. In such a case, one has to assume

g ∈ C

m+1,λ

(Ω)

and to add the term

kgk

m+1,λ

on the right-hand side of (IV.7.1), while for Theorem IV.7.2 we have to take

g ∈ C

1,λ

(Ω). 

IV.8 Green’s Tensor, Green’s Identity and

Representation Formulas

Theorem IV.6.1 allows us, in particular, to construct the Green’s tensor solu-

tion for t he Stokes problem in a bounded domain Ω. Actuall y, for ﬁxed y ∈ Ω,

let us consider the functions A

(x, y), a

(x, y) such that fo r all i, j = 1, . . ., n

∆

(x, y) +

∂a

(x, y)

∂x

= 0, x ∈ Ω,

∂A

(x, y)

∂x

= 0, x ∈ ∂Ω

(x, y) = U

(x − y), x ∈ ∂Ω.

(IV.8.1)

From T heorem IV.6.1, we know that A

(x, y), a

(x, y) exist and, for Ω of

class C

m+2

, they satisfy

m+2,q

+ ka

m+1,q/R

≤ c

(ω, n, m, q),

where c

does not depend on y for all y exceeding a ﬁxed distance d from ∂Ω.

In analogy with the Lapla ce operator, we deﬁne

(x, y) = U

(x − y) − A

(x, y)

(x, y) = q

(x − y) − a

(x, y),

(IV.8.2)

p is determined up to a constant that may be ﬁxed by requiring

Ω

p = 0. In such

a case, the norm involving p on the left-hand side of (IV.7.1) can be replaced by

kpk

m+1,λ

IV.8 Green’s Tensor, Green’s Identity and Representation Formulas 289

which is the Green’s tensor solution for the Stokes problem in the bounded

domain Ω (Odqvist 1930, §5). It i s not diﬃcult to show (Odqvist 1930, p. 358)

that the tensor ﬁeld G satisﬁes the following symmetry condition

(x, y) = G

(y, x). (IV.8.3)

Moreover, from (IV.8.1), (IV.8.2), and the properties of the tensor U it follows,

in particular, for any n > 2, that

(x, y)| ≤ c

|x −y|

2−n

, |D

(x, y)| ≤ c

|x − y|

1−n

(IV.8.4)

for all x ∈ Ω and for all y ∈ Ω with dist (y, ∂Ω) ≥ d > 0, c

→ ∞ as d → 0.

By using (IV.8.3 ) analogous estimates can be obtained interchanging the roles

of x and y. If n = 3 and Ω is of class C

1,λ

, λ ∈ (0, 1), relations (IV.8.4) can

be extended to all x, y ∈ Ω and one has

(x, y)| ≤ c |x − y|

−1

, |D

(x, y)| ≤ c |x − y|

−2

, x, y ∈ Ω, (IV.8.5)

with a constant c = c(Ω); see Odqvist (1930, Satz XVIII) a nd Cattabriga

(1961, pp. 335-3 36). Observe that estimates (IV.8.5) formally coincide wi th

the same estimates in the case of a half-space; see (IV.3. 50). Extension of

(IV.8.5) to higher dimension can be obtained by the results o f Solonnikov

(1970). We al so refer the reader to this paper for further eval uations related

to G, g.

We shall next derive several useful representation formulas for solutions

to the Stokes problem. To this end, we recall that the Cauchy stress tensor

T ≡ {T

= T

(v, p)} associa ted with a ﬂow v, p is given by

= −pδ

+ 2D

, (IV.8.6)

where

= D

(v) ≡



∂v

∂x

∂v

∂x



(IV.8.7)

is the stretching tensor. If u, π are suﬃciently regular vector and scalar ﬁelds,

respectively, and assuming that Ω is a bounded domain of class C

, we may

integrate by parts to obtain the identities

Ω

∇ · T (v, p) · u = −

Ω

T (v, p) : ∇u +

∂Ω

u ·T (v, p) ·n

Ω

∇ · T (u, π) · v = −

Ω

T (u, π) : ∇v +

∂Ω

v · T (u, π) · n,

(IV.8.8)

where n is the unit outer normal at ∂Ω. By the symmetry of T and taking v

and u solenoidal,

Ω

T (v, p) : ∇ u =

Ω

T ( u, π) : ∇v.

290 IV Steady Stokes Flow in Bounded Domains

Therefore, from this relation, (IV.2.2), and the identity

∇ · T = −∇ p + ∆v (IV.8.9)

and the analogous o ne for u and π, we obtain

Ω

[(∆v−∇p)·u−(∆u−∇ π)·v] =

∂Ω

[u·T (v, p)−v·T (u, π)]·n. (IV.8.10)

Relation (IV.8.10) is the Green’s identity for the Stokes system. By using stan-

dard procedures, it is easy to derive from (IV.8.10) a representation formula

for v and p (Odqvist 1930, §2). In fact, we choose for ﬁxed j and x ∈ Ω

u(y) = u

(x − y) ≡ (U

, U

, . . . , U

)

π(y) = q

(x − y),

(IV.8.11)

where U, q is the fundamental solution (IV.2.3), (IV.2.4), and substitute them

into (IV.8.10) with Ω replaced by Ω

≡ Ω −B

(x). Setting f = ∆v −∇p, we

obtain

Ω

f (y) · u

(x − y)dy =

∂Ω

(x − y) · T (v, p)(y)

−v(y) · T (u

, q

)(x − y)] · ndσ

∂B

(x)

(x − y) ·T (v, p)(y)

−v(y) · T (u

, q

)(x − y)] · ndσ

(IV.8.12)

Clearly,

lim

ε→0

Ω

f(y) · u

(x − y)dy =

Ω

f(y) ·u

(x − y)dy

lim

ε→0

∂B

(x − y) · T (v, p)(y) · ndσ

= 0.

(IV.8.13)

Moreover, since

, q

) =

−y

)(x

− y

)(x

−y

)

|x − y|

n+2

, (IV.8.14)

by a simple calcula tion one shows

lim

ε→0

∂B

v(y) · T (u

, q

)(x − y) · n(y)dσ

= −v

(x) (IV.8.15 )

and so from (IV. 8.11)–(IV.8. 15) we ﬁnally deduce the following representation

formula for v

, j = 1, . . . , n, valid for all x ∈ Ω:

IV.8 Green’s Tensor, Green’s Identity and Representation Formulas 291

(x) =

Ω

(x − y)f

(y)dy −

∂Ω

(x − y)T

(v, p)(y)

−v

(y)T

, q

)(x −y)]n

(y)dσ

(IV.8.16)

To give a sim ilar representation for the pressure p, we begin to observe that

for f smooth enough (e.g., H¨older continuous) the volume potentials

(x) =

Ω

(x − y)f

(y)dy,

S(x) = −

Ω

(x − y)f

(y)dy

are (at least) of class C

(Ω) and C

(Ω), respectively (Odqvist 1 930, Satz 1;

see also Section IV.7) a nd that, moreover, it is (see Exercise IV.8.1)

∆W (x) − ∇S(x) = f (x), x ∈ Ω. (IV.8.17)

We next observe that from (IV.8.16)

∂p

∂x

= ∆v

= ∆W

∂Ω

∆T

, q

)−(∆U

(v, p)]n

, (IV.8.18)

which, by (IV.8.17) and (IV.2.5)

in turn implies

∂p

∂x

∂S

∂x

∂Ω



∂q

∂x

(v, p) + v

∆T

, q

)



Observing that q

is harmo nic (for x 6= y) we also have

∆T

, q

) = −δ

∆q

∂

∂x

∆U

∂

∂x

∆U

= −2

∂

∂x

for x 6= y, which, once substituted into (IV.8.18) and upon using the relation

∂q

/∂x

= ∂q

/∂x

, yields for all x ∈ Ω

p(x) = −

Ω

(x − y)f

(y)dy +

∂Ω

(x −y)T

(v, p)(y)

−2v

(y)

∂q

(x − y)

∂x

(y)dσ

(IV.8.19)

Identity (IV.8.19) gives the representation formula for the pressure.

Exercise IV.8.1 Prove the validity of equation (IV.8.17). Hint: Set w = E ∗ f

(f ≡ 0 in Ω

). From potential theory it is well known that, for f H¨older continuous,

it is (at least) w ∈ C

(Ω) and, moreover, ∆w = f in Ω, see Kellog (1929, Chapter

VI, §3).

292 IV Steady Stokes Flow in Bounded Domains

Formulas (IV.8.16) and (IV.8.19) can be easily extended to derivatives of

arbitrary order. Actually, observing that for any multi-index α

∆(D

v) −∇(D

p) = D

one readily shows, for all x ∈ Ω,

(x) =

Ω

(x − y)D

(y)dy −

∂Ω

(x − y)T

v, D

p)(y)

−D

(y)T

, q

)(x − y)]n

(y)dσ

(IV.8.20)

and

p(x) = −

Ω

(x − y)D

(y)dy+

∂Ω

(x − y)T

v, D

p)(y)

−2D

(y)

∂q

(x − y)

∂x

(y)]dσ

(IV.8.21)

Relations (IV.8.20) and (IV.8.21) were obtained under the assumption of

suitable regularity on v and p. Nevertheless, it i s not diﬃcult to extend them

to the case when velocity and pressure ﬁelds belong to suitable Sobolev spaces.

Precisely, we have

Theorem IV.8.1 Let Ω be a bounded domain of R

, n ≥ 2, of class C

m+2

m ≥ 0, and let v ∈ W

m+2,q

(Ω), p ∈ W

m+1,q

(Ω) be a solution to (IV.0.1 )

corresponding to f ∈ W

m,q

(Ω), 1 < q < ∞. Then, v a nd p obey (IV.8.20)

and (IV.8.21), respectively, for al l |α| ∈ [0, m] and almost all x ∈ Ω.

Proof. We prove (IV.8.20), the proo f of (IV.8.21) being entirely analogous.

Let v

∗

be the trace of v on ∂Ω. From Theorem II.4.4 we then have v

∗

∈

m+2−1/q,q

(∂Ω). Denote by {f

}, {v

∗k

} two sequences of smooth functions

approximating f and v

∗

, respectively, in the spaces to which they belong, and

by {v

, p

} the corresponding solutions to (IV.0.1) and (IV.0.2). By what we

have seen, for all |α| ∈ [0, m], these solutions obey (IV.8.20) and (IV.8.21)

with v

in place of v and with f

in place o f f. Denote by (IV.8.20)

these

relations and let k → ∞. In this limit, from Lemma IV.6.1 we obtain

→ v in W

m+2,q

(Ω)

→ p in W

m+1,q

(Ω).

(IV.8.22)

Set

(b) =

Ω

(x − y)D

(y)dy

(b, s) =

∂Ω

(y) T

, q

)(x − y)

−U

(x − y)T

b, D

s)(y)]n

(y)dσ

IV.8 Green’s Tensor, Green’s Identity and Representation Formulas 293

and

P (b) =

Ω

(x −y)D

(y)dy

β(b, s) =

∂Ω

(x − y) T

b, D

s)(y)

−2

∂Ω

(y)

∂q

(x − y)

∂x

(y)dσ

and assume ﬁrst q > n/2. From the estimates (IV.2.6) for U we have, if n > 2,

≡ |V

) −V

(f)| ≤ ck|x − y|

2−n

q/(q−1)

−f k

m,q

≤ c

− fk

m,q

(IV.8.23)

while, if n = 2,

≤ cklog |x − y|k

q/( q−1)

− fk

m,q

≤ c

− fk

m,q

. (IV.8.24)

Moreover, for any ﬁxed x ∈ Ω, from Theorem II.4.1 it follows for all q > 1

that

, p

) − B

(v, p) | ≤ c



− vk

m+1,q(∂Ω)

+ kp

−pk

m,q(∂Ω)



≤ c

(kv

− vk

m+2,q,Ω

+ kp

− pk

m+1,q,Ω

)

(IV.8.25)

where c

depends o n dist (x, ∂Ω) ≡ d (c

→ ∞ a s d → 0). Also, from the

embedding Theorem II.3.4 , we have D

v ∈ C(Ω) and, as k → ∞,

→ D

v in C(Ω). (IV.8.26)

Relations (IV.8.22)–(IV.8.26) show (IV.8.20) if q > n/2, and then for all q > 1

if n = 2. Assume now 1 < q ≤ n/2, n > 2. L et Ω

be any subdomain of Ω

with Ω

⊂ Ω. Using the Minkowski inequal ity several times we obtain

−V

(f) −B

(v, p)k

q,Ω

≤ kv

−vk

m,q,Ω

+ kV

) − V

(f )k

q,Ω

+kB

, p

) − B

(v, p)k

q,Ω

+ kD

) −V

) −B

, p

q,Ω

(IV.8.27)

If q = n/2, from (IV.2.6) and Theorem II.11.2 we derive that V

is a bounded

transformation of L

n/2

(Ω) into L

(Ω), for all r ∈ (1, ∞), while, if q ∈ (1, n/2),

again from (IV.2. 6) and Sobolev Theorem II.11.3, V

is a bounded transforma-

tion of L

(Ω) into L

nq/(n−2q)

(Ω). Therefore, observing that q < nq/(n − 2q),

in either case we derive

) − V

(f)k

q,Ω

≤ c

−fk

m,q,Ω

. (IV.8.28)

Thus, recalling that v

satisﬁes (IV.8.20)

identically, from (IV.8.22), (IV.8.25),

(IV.8.27), and (IV.8.28) we conclude the validity of (IV.8.20) also for q ∈

(1, n/2]. The proo f of (IV.8.21) is entirely analogous, provided we distinguish

the two cases n ≥ q and n < q. We leave details to the reader. The proof of

the theorem is complete. ut

294 IV Steady Stokes Flow in Bounded Domains

Representation formulas that involve only the body fo rce and the velocity

at the boundary can be obtained if we make use of the Green’s tensor solution

(IV.8.2). Actually, applying (IV.8.10) with

u(y) = A

(x, y) ≡ (A

, A

, . . . , A

)

π(y) = a

(x, y)

and taking into account that A, a a re smooth ﬁelds solving (IV.8.1), we ﬁnd

Ω

f (y)·A

(x, y)dy =

∂Ω

(x−y)·T (v, p)(y)−v(y)·T (A

, a

)(x, y)]·ndσ

(IV.8.29)

where, we recall, u

≡ (U

, U

, . . . , U

). Subtracting (IV.8.29) from (IV.8.16)

and bearing in mind the deﬁnition of G given in (IV.8.2) we then conclude

(x) =

Ω

(x, y)f

(y)dy −

∂Ω

(y)T

, g

)(x, y)]n

(y)dσ

, (IV.8.30)

where G

≡ (G

, G

, . . . , G

). Finally, along the same lines leading to

(IV.8.19), one proves the f ollowing formula:

p(x) = −

Ω

(x − y)f

(y)dy − 2

∂Ω

(y)n

(y)dσ

. (IV.8.31)

Exercise IV.8.2 Let v be a q-generalized solution to the Stokes problem (IV.0.1),

(IV.0.2) in a half-space, corresponding to smo oth data of bounded support. Show

that v and the corresponding pressure p satisfy the representation (IV.8.30),

(IV.8.31) with G and g given in (IV.3.46).

IV.9 Notes for the Chapter

Section IV.1. The ﬁrst existence and uniqueness result for the Stokes prob-

lem in a bounded domain is due to Korn (1908), under the restriction ∇·f = 0.

The problem of existence with no restriction on the body force was solved for Ω

a ball by Boggio (1910), Crudeli (1925a, 1925b), and Oseen (19 27, §§9.1,9.2).

In particular, Oseen determines explicitly the Green’s tensor for the Stokes

problem in a ball . Existence and uniqueness in full generality, with no restric-

tion on f or the shape of Ω, was provided by Lichtenstein (1928), in the wake

of the work of Umberto Crudeli.

The existence of a pressure ﬁeld associated to a q-generalized solution along

with the validity of the corresponding estimate was ﬁrst established by Cat-

tabriga (19 61) for space dimension n = 3. The same result was obtained, with

a much simpler proof, in the case q = 2 (generalized solutions) by Solonnikov

Sˇcadilov (1973) and it was successively rediscovered, essentiall y along the

IV.9 Notes for the Chapter 295

same methods, by Amick (1976); see also Temam (1977, Chapter I, L emma

2.1).

Section IV.2. The material contained in this section is taken, basically, from

Galdi & Simader (1990). However, the uniqueness part of Theorem IV.2.2 i s

due to me. Simil ar results can be found in C attabriga (1961), Borchers &

Miyakawa (1990, Proposition 3.7 (iii)), and Kozono & Sohr (1991, §2.2).

Existence and uniqueness of solutions in weighted Lebesgue and homoge-

neous Sobolev spaces can be immediately obtained by using, in the proofs

of Theorem IV.2.1 and Theorem IV.2.2, Stein’s Theorem II.11.5 in place of

Calder´on–Zygmund Theorem II.11.4; see Pulidori (1993). For similar results

in the two-dimensional case, we also refer to Dur´an & L´opez Garc´ıa (2010).

Section IV.3. The guiding ideas are taken from the work of Cattabriga (1961,

§§2,3). However, all theorems in this section are due to me. In this respect, I

am grateful to the late Professor Lamberto Cattabriga for the inspiring and

enjoyable conversations I had with him, in the wi nter of 1987, on the existence

part of Theorem IV.3.3.

A weaker version of the estimates contained in Theorem IV.3.2 and The-

orem IV.3.3 is given by Borchers & Miyakawa (1988, Theorem 3.6) and by

Maslennikova & Timoshin (1990, Theorem 1). Results in weighted L

spaces

can b e found i n Borchers & Pil eckas (1992).

The special case o f Theorem IV.3.2 corresponding to f ≡ 0, g ≡ 0 and

q = 2 is proved by Tanaka (1995), by the Fourier transform method. More

general (slip) boundary conditions are also considered.

The Green’s tensor for a three-dimensional half-space was determined for

the ﬁrst time by Lorentz (1896); see also Oseen (1927, §9.7).

Section IV.4. Theorem IV.4.1 (for n = 3) is essentially due to Cattabriga

(1961, §5), while Theorem I V.4.2, Theorem IV.4.4, and Theorem IV.4.5 are

due to me.

A result similar to that proved in Remark IV.4.2 was ﬁrst shown by

Sver´ak

& Tsai (2000, Theorem 2.2). In f act, Remark IV.4.2 is motivated by their work.

Section IV.5. The results contained in this section are a generalization to

n ≥ 2 of those proved by Cattabriga (1961, §5) for n = 3.

An improved version of the results stated in Remark IV.5.1 and Remark

IV.5. 2 can be found in Kang (2004).

Section IV.6. Theorem IV.6 .1 plays a central role in the mathematical theory

of the Navier–Stokes equations. In the case where n = 3 it was shown for

the ﬁrst time by Cattabriga (1961, Teorema at p.311). The same result of

Cattabriga for m ≥ 0 was announced by Solonni kov (1960) and a full proof,

based on the theory of hydrodynamical potentials, appeared later in 1963 in

the ﬁrst edition of the book by Ladyzhenskaya (1969) (in this regard, see

also Deuring, von Wahl, & Weidemaier (19 88) and the book of Varnhorn

(1994)). Sobolevski (1960) proved a weaker result in the special case m = 0 a nd

q = 2. In their study on the unique solvability of steady-state Navier–Stokes