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208 J. Escher, M. Kohlmann and B. Kolev

Acknowledgment

The authors would like to thank Mats Ehrnstr¨om and Jonatan Lenells for several

fruitful discussions as well as the referee for helpful remarks.

References

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Funct. Anal., 12 (2002), 1080–1104.

Joachim Escher and Martin Kohlmann

Institute for Applied Mathematics

Leibniz University Hanover

Welfengarten 1

D-30167 Hanover, Germany

e-mail: escher@ifam.uni-hannover.de

kohlmann@ifam.uni-hannover.de

Boris Kolev

CMI

39, rue F. Joliot-Curie

F-13453 Marseille Cedex 13, France

e-mail: kolev@cmi.univ-mrs.fr

Progress in Nonlinear Diﬀerential Equations

and Their Applications, Vol. 80, 211–232

 2011 Springer Basel AG

Global Leray-Hopf Weak Solutions

of the Navier-Stokes Equations with

Nonzero Time-dependent Boundary Values

R. Farwig, H. Kozono and H. Sohr

Abstract. In a bounded smooth domain Ω ⊂ R

and a time interval [0,T),

0 <T ≤∞, consider the instationary Navier-Stokes equations with initial

value u

∈ L

(Ω) and external force f =divF , F ∈ L

(0,T; L

(Ω)). As is

well known there exists at least one weak solution in the sense of J. Leray

and E. Hopf with vanishing boundary values satisfying the strong energy

inequality. In this paper, we extend the class of global in time Leray-Hopf

weak solutions to the case when u

∂Ω

= g with non-zero time-dependent

boundary values g. Although there is no uniqueness result for these solutions,

they satisfy a strong energy inequality and an energy estimate. In particular,

the long-time behavior of energies will be analyzed.

Mathematics Subject Classiﬁcation (2000). 35Q30; 35J65; 76D05.

Keywords. Instationary Navier-Stokes equations; weak solutions; energy in-

equality; non-zero boundary values; time-dependent data; long-time behavior.

1. Introduction and main results

Let Ω ⊂ R

be a bounded domain with boundary of class C

1,1

,andlet[0,T),

0 <T ≤∞, be a time interval. In Ω × [0,T) we consider the instationary Navier-

Stokes system in the form

− Δu + u ·∇u + ∇p = f, div u =0

∂Ω

= g, u(0) = u

(1.1)

with viscosity ν = 1 and data f, g,u

212 R. Farwig, H. Kozono and H. Sohr

For the data we assume the following:

f =divF, F ∈ L



0,T; L

(Ω)



∈ L

(Ω),

g ∈ L



0,T; W

−

(∂Ω)



∩ L



0,T; W

−

(∂Ω)



=1, 2 <s<∞, 3 <q<∞;

(1.2)

the initial data u

has to satisfy further assumptions to be introduced later, see

Section 5.

First we recall the well-known deﬁnition of a weak solution u in the sense

of Leray-Hopf with u

∂Ω

= 0 and initial value u

∈ L

(Ω) = C

∞

0,σ

(Ω)

·

,and

describe their main properties.

Deﬁnition 1.1. Let f =divF , F ∈ L

(0,T; L

(Ω)), 0 <T ≤∞,andu

∈ L

(Ω)

be given. Then the vector ﬁeld u is called a Leray-Hopf weak solution of the system

(1.1) with data f, u

and g = 0 if the following conditions are satisﬁed:

(i) u ∈ L

∞



0,T; L

(Ω)



∩ L



0,T; W

1,2

(Ω)



(ii) for each test function w ∈ C

∞



[0,T); C

∞

0,σ

(Ω)



−u, w



Ω,T

+ ∇u, ∇w

Ω,T

−uu, ∇w

Ω,T

= u

,w(0)

−F, ∇w

Ω,T

(iii) for 0 ≤ t<T

u(t)



∇u

dτ ≤

u



−



F, ∇u

dτ.

As is well known [20] there exists at least one weak solution u of (1.1) in the

sense of Deﬁnition 1.1; moreover, u maybechosentobeweaklycontinuousfrom

[0,T)toL

(Ω) and to satisfy beyond the energy inequality (iii) in Deﬁnition 1.1

the so-called strong energy inequality

u(t)



∇u

dτ ≤

u(t

)

−



(F, ∇u)

dτ (1.3)

for almost all t

∈ [0,T) including t

= 0 and for all t ∈ [t

,T). We note that the

strong energy inequality (1.3) yields the energy estimate

u(t)



∇u

dτ ≤u(t

)



F 

dτ

for a.a. t

∈ (0,T), for t

=0andallt ∈ [t

,T). An application of H¨older’s

inequality implies that

u ∈ L



0,T; L

(Ω)



, 2 ≤ s ≤∞, 2 ≤ q ≤ 6.

In order to extend Deﬁnition 1.1 to the more general case of time-dependent

boundary data u

∂Ω

= g we will reduce the system (1.1) to a perturbed Navier-

Stokes system with g = 0. For this purpose we have to ﬁnd ﬁrst of all a (so-called)

Global Leray-Hopf Weak Solutions 213

very weak solution of the inhomogeneous Stokes system

− ΔE + ∇h = f

, div E =0

∂Ω

= g, E(0) = E

(1.4)

see [2]–[5], in Ω ×[0,T) with suitable data f

=divF

and E

;here∇h means the

associated pressure. At ﬁrst sight, it seems to suﬃce to choose f

=0,F

= 0, but

for later application it will be helpful to consider general data f

, see Corollary

1.5 and (1.15) below. Setting

v = u − E, ˜p = p −h, f

= f − f

= u

−E

(1.5)

we write (1.1) in the form

− Δv +(v + E) ·∇(v + E)+∇˜p = f

, div v =0,

∂Ω

=0,v(0) = v

(1.6)

which is a perturbed Navier-Stokes system with homogeneous data v

∂Ω

= 0, but

with the new perturbation terms

(v + E) ·∇(v + E)=div



vv +(Ev + vE)+EE



; (1.7)

here Ev = E ⊗ v =(E

)

i,j=1,2,3

denotes the dyadic product of the vector ﬁelds

E and v, and the divergence is taken columnwise.

To deal with Leray-Hopf type weak solutions v of (1.6), see Deﬁnition 1.2

below, we need that E in (1.4) has the following properties:

E ∈ L



0,T; L

(Ω)



∩L



0,T; L

(Ω)



=1, 2 <s≤∞, 3 ≤ q<∞.

(1.8)

Actually, the condition E ∈ L

(0,T; L

(Ω)) in (1.8) is needed for estimates of

the term EE in (1.7) in the space L

(0,T; L

(Ω)), whereas the second condition

E ∈ L

(0,T; L

(Ω)) will help to estimate the terms vE and Ev.NotethatE ∈

(0,T; L

(Ω)) is the classical condition on weak solutions of the Navier-Stokes

system to satisfy the energy identity, and it automatically holds true when 4 ≤ s ≤

8, 4 ≤ q ≤ 6. To guarantee (1.8) for the solution E of (1.4) the data f

,g,E

have to

satisfy certain assumptions known from the theory of the very weak Stokes system.

However, looking at (1.6), it suﬃces to assume (1.8) and v

∈ L

(Ω), f

=divF

∈ L

(0,T; L

(Ω)), in order to deﬁne Leray-Hopf type weak solutions of (1.6);

later concrete conditions on g, u

will be described to satisfy these assumptions,

see Section 5.

In this respect, this paper mainly deals with the perturbed Navier-Stokes

system (1.6) rather than with (1.1).

Deﬁnition 1.2. Let E satisfy (1.8) and assume v

∈ L

(Ω), f

=divF

, F

∈

(0,T; L

(Ω)). Then a vector ﬁeld v on Ω × [0,T)isaLeray-Hopf type weak

solution of the perturbed Navier-Stokes system (1.6) if the following conditions

are satisﬁed:

214 R. Farwig, H. Kozono and H. Sohr

(i) v ∈ L

∞

loc



[0,T); L

(Ω)



∩ L

loc



[0,T); W

1,2

(Ω)



(ii) for each test function w ∈ C

∞



[0,T); C

∞

0,σ

(Ω)



v, w



Ω,T

−∇v, ∇w

Ω,T

−(v + E)(v + E), ∇w

Ω,T

= v

,w(0)

−F

, ∇w

Ω,T

(1.9)

(iii) the energy inequality

v(t)



∇v

dτ ≤

v



−



F

− (v + E)E,∇v

dτ (1.10)

holds for 0 <t<T.

Now our main theorem reads as follows:

Theorem 1.3. Let Ω ⊂ R

be a bounded domain with ∂Ω ∈ C

1,1

,andletf

div F

, F

∈ L

(0,T; L

(Ω)), v

∈ L

(Ω). Assume that E satisﬁes (1.8) where in

the case s = ∞, q =3this condition is replaced by

E ∈ C



[0,T); L

(Ω)



, E

([0,T );L

(Ω))

<α

(1.11)

for a suﬃciently small constant α

= α

(Ω) > 0. Then the perturbed Navier-Stokes

system

− Δv +(v + E) ·∇(v + E)+∇˜p = f

, div v =0,

∂Ω

=0,v(0) = v

(1.12)

has at least one Leray-Hopf type weak solution v in the sense of Deﬁnition 1.2.In

addition to the energy inequality (1.10) v satisﬁes the strong energy inequality

v(t)



∇v

dτ ≤

v(t

)

−



F

−(v + E)E, ∇vdτ (1.13)

for a.a. t

∈ [0,T) including t

=0and for all t

<t<T. Moreover, for these

,t the energy estimate

v(t)



∇v

dτ

≤ c



v(t

)



(F



+ E

) dτ



exp





E

dτ



(1.14)

holds, where c = c(Ω,q) > 0 means a constant.

In view of (1.1), (1.4)–(1.6) and Deﬁnition 1.2 u = v + E is called a Leray-

Hopf type weak solution of (1.1) with boundary data u

∂Ω

= g = E

∂Ω

and initial

value u(0) = v

. In the most general setting of very weak solutions, cf. [19, 18],

these terms are not well deﬁned separately from each other and from f, but have

to be interpreted in the generalized sense that v = u − E satisﬁes v

∂Ω

=0and

v(0) = v

. For a more concrete situation and precise assumptions on g, u

we refer

to Section 5.

Global Leray-Hopf Weak Solutions 215

Remark 1.4. (i) The weak solution v in Theorem 1.3 may be modiﬁed on a null

set of (0,T) such that v :[0,T) → L

(Ω) is weakly continuous. Hence v(0) = v

well deﬁned, v

∂Ω

= 0 is well deﬁned for a.a. t ∈ [0,T) in the sense of traces, and

there exists a distribution ˜p on Ω × (0,T) such that

− Δv +(v + E) ·∇(v + E)+∇˜p = f

(ii) Assume T = ∞ and

E ∈ L



0, ∞; L

(Ω)



∩ L

loc



[0, ∞); L

(Ω)



, 2 <s<∞,

such that E

q,s;t



E

dτ is increasing at least linearly as t →∞.Wenote

that in this case the proof in Section 4 will easily show the existence of a weak

solution v in (0, ∞). Then the energy estimate (1.14) (with t

= 0) yields for the

kinetic energy

v(t)

only an exponentially increasing bound as t →∞.This

worst case estimate reﬂects the fact that nonzero boundary values could imply a

permanent ﬂux of energy through the boundary into the domain.

(iii) If s = ∞ and E

([0,∞);L

(Ω))

is suﬃciently small, cf. (1.11), then due

to energy dissipation the scenario of (ii) will not occur, and (1.14) is replaced by

the better estimate

v(t)



∇v

dτ ≤v(t

)





F



+ E



dτ

for a.a. t

∈ [0, ∞) including t

= 0 and for all t ∈ (t

, ∞). The proof of this

estimate is based on the energy estimate (3.12) below and arguments in Sections

3–4.

The smallness assumptions in Remark 1.4 (iii), see also (1.11), is too re-

strictive for further applications. Next we consider an assumption on E known as

Leray’s inequality in the context of stationary Navier-Stokes equations in multiply

connected domains: Let E satisfy the conditions

E ∈ L

∞



0, ∞; L

(Ω)



and



E(t) ·∇w

≤

∇w



∇w



for all w

∈ W

1,2

(Ω) ∩ L

(Ω) and a.a. t ∈ (0, ∞).

(1.15)

We recall that in a (multiply connected) bounded domain Ω ⊂ R

with ∂Ω=

∪

j=0

∈ C

1,1

(L ∈ N) to any boundary data g ∈ W

1/2,2

(∂Ω) satisfying the

restricted ﬂux condition



g · Ndo= 0 for each boundary component Γ

⊂ ∂Ω,j=1,...,L,

and any ε>0 there exists an extension E

∈ W

1,2

(Ω) with following properties:

∈ W

1,2

(Ω), div E

=0,E

∂Ω

= g



·∇wdx

≤ ε ∇w

for all w ∈ W

1,2

(Ω) ∩ L

(Ω).

(1.16)

216 R. Farwig, H. Kozono and H. Sohr

In particular this result holds for a simply connected domain (having only one

boundary component Γ = ∂Ω). In our case, it suﬃces to consider ε =

in (1.15)

since the viscosity ν =1.

Corollary 1.5. Let Ω ⊂ R

be a bounded domain with ∂Ω ∈ C

1,1

,letv

∈ L

(Ω)

and let

=divF

∈ L

∞



0, ∞; L

(Ω)



. (1.17)

Furthermore, assume that E satisﬁes (1.15). Then the perturbed Navier-Stokes

system (1.12) has a global in time Leray-Hopf type weak solution v satisfying the

energy estimate

v(t)



∇v

dτ ≤v



+ c





F



+ E



dτ , (1.18)

where c = c(Ω) > 0 is a constant. To be more precise, there exists a bound κ

∗

(depending on the norms F



2,∞;∞

and E

4,∞;∞

,butnotanv



) andatime

= T

(v



) such that

v(t)

≤ κ

∗

for all t ≥ T

Corollary 1.5, see also Corollary 5.6, strictly exploits the assumption (1.15)

on E and the dissipativity of the Navier-Stokes system (1.1), (1.12). It is closely

related to the work [14] of E. Hopf where a similar result is proved in the context

of moving bodies for given smooth solutions. The impact of the ﬁnal result of

Corollary 1.5 is the fact that all solutions are bounded after a ﬁnite time by a

bound independent of the initial value.

There are many applications of the two-dimensional Navier-Stokes system

with nonhomogeneous boundary values in optimal control theory since the 2D-

system admits global smooth solutions and uniqueness. For the three-dimensional

case there are only few results on the existence of global or weak solutions. We

mention the existence of local in time strong solutions by A.V. Fursikov, M.D. Gun-

zburger and L.S. Hou [7], and results in a scale of Besov spaces by G. Grubb [12].

The existence of global in time weak solutions is proved by J.-P. Raymond [17]

for boundary data in a fractional Sobolev space on ∂Ω ×(0,T) with derivatives in

space and time of fractional order 3/4 for domains with boundary ∂Ω ∈ C

2. Preliminaries

Let Ω ⊂ R

be a bounded domain with ∂Ω ∈ C

1,1

,let0<T≤∞and 1 ≤ q,

s ≤∞with conjugate exponents 1 ≤ q



, s



≤∞. We will use standard no-

tation for Lebesgue spaces (L

(Ω), ·

(Ω)

= ·

) and for Bochner spaces

(0,T; L

(Ω)), ·

(0,T ;L

(Ω))

= ·

q,s;T

). Here v ∈ L

loc

([0,T); L

(Ω)) means

that v ∈ L

(0,T



; L

(Ω)) for each ﬁnite 0 <T



≤ T . The pairing of functions (or

vector ﬁelds) in Ω and Ω × (0,T) is denoted by ·, ·

and ·, ·

Ω,T

, respectively.

Sobolev spaces are denoted by (W

m,q

(Ω), ·

m,q

),m ∈ N, the corresponding

trace space by (W

m−1/q,q

(∂Ω), ·

m−1/q,q

)when1<q<∞. The dual space to

Global Leray-Hopf Weak Solutions 217

1−1/q



(∂Ω) is denoted by W

−1/q,q

(∂Ω), the corresponding pairing is ·, ·

∂Ω

Concerning smooth functions we need the spaces C

∞

(Ω),C

∞

0,σ

(Ω) = {v ∈ C

∞

(Ω) :

div v =0}, and in the context of very weak solutions

0,σ

(Ω) = {w ∈ C

(Ω) : w =0on∂Ω, div w =0},

seeSection5.NotethatW

1,q

(Ω) := C

∞

(Ω)

·

1,q

and L

(Ω) := C

∞

0,σ

(Ω)

·

.By

w ∈ C

∞

([0,T); C

∞

0,σ

(Ω)), the space of test functions in Deﬁnitions 1.1 and 1.2, we

mean that w ∈ C

∞

([0,T) × Ω) satisﬁes div

w =0forallt ∈ [0,T) (taking the

divergence with respect to x ∈ Ω).

For 1 <q<∞ let P

: L

(Ω) → L

(Ω) be the Helmholtz projection and let

= −P

Δ:D(A

)=W

2,q

(Ω) ∩W

1,q

(Ω) ∩ L

(Ω) ⊂ L

(Ω) → L

(Ω)

denote the Stokes operator. For its fractional powers A

: D(A

) → L

(Ω), −1 ≤

α ≤ 1, we know that D(A

) ⊆D(A

) ⊆ L

(Ω) and R(A

)=L

(Ω) for 0 ≤ α ≤ 1;

ﬁnally, (A

)

−1

= A

−α

for −1 ≤ α ≤ 1. In particular, for A = A

, one has

A

v

= ∇v

for v ∈D(A

). Moreover, we note the following embedding

estimates (with constants c = c(q, Ω) > 0):

v

≤ cA

v

, 0 ≤ α ≤

, 2 ≤ q<∞, 2α +

,v∈D(A

) (2.1)

A

v

≤Av

v

1−α

, 0 ≤ α ≤ 1,v∈D(A) (2.2)

v

≤ cv

1,2

v

1−β

, 2 ≤ q ≤ 6,β =3(

−

),v ∈ W

1,2

(Ω). (2.3)

Finally we note that −A

generates a bounded, exponentially decaying ana-

lytic semigroup {e

−tA

: t ≥ 0} on L

(Ω) satisfying the estimate

A

−tA

v

≤ ct

−α

v

, 0 ≤ α ≤ 1,t>0,v ∈ L

(Ω) (2.4)

with a constant c = c (q, Ω) > 0andc =1ifq =2.

To ﬁnd approximate solutions of the Navier-Stokes system in Section 3 we

need Yosida’s approximation operators

=(I +

)

−1

,m∈ N,

where I denotes the identity on L

(Ω). The following properties are well known:

J

v

≤v

, 

v

≤v

lim

m→∞

v = v for all v ∈ L

(Ω),

∇J

v

≤∇v

for all v ∈ W

1,2

(Ω) ∩ L

(Ω) = D(A

);

(2.5)

for the proof of the last inequality we use that A

v

= ∇v

and the commu-

tativity of J

with A

For these and further properties of the Stokes operator and Yosida’ s approx-

imation we refer, e.g., to [9], [10] and [20].