Digital Object Identifier (DOI) 10.1007/s002080000101

Math. Ann. 317, 635–675 (2000) Mathematische Annalen

The Navier-Stokes equations in the weak-Ln spacewith time-dependent external force

Masao Yamazaki�

Received: 30 March 1999 / Accepted: 21 September 1999 /Published online: 28 June 2000 –c© Springer-Verlag 2000Dedicated to Professor Daisuke Fujiwara on the Occasion of his Sixtieth Birthday

Abstract. We consider the Navier-Stokes equations with time-dependent external force, eitherin the whole time or in positive time with initial data, with domain either the whole space, thehalf space or an exterior domain of dimensionn ≥ 3. We give conditions on the external forcesufficient for the unique existence of small solutions in the weak-Ln space bounded for all time.In particular, this result gives sufficient conditions for the unique existence and the stability ofsmall time-periodic solutions or almost periodic solutions. This result generalizes the previousresult on the unique existence and the stability of small stationary solutions in the weak-Ln spacewith time-independent external force.

Mathematics Subject Classification (2000):35Q30, 35B10, 35B15, 46G10

Introduction

Let Ω be the whole spaceRn, the half spaceRn+ = Rn−1 × (0,+∞), or anexterior domain inRn with smooth boundary∂Ω, where the space dimensionnsatisfiesn ≥ 3.We are concerned with the nonstationary Navier-Stokes equationin the whole time with the Dirichlet boundary condition inΩ as follows:

∂u

∂t(t, x) = ∆xu(t, x) − (u · ∇x) u(t, x)−∇xπ(t, x) + f (t, x)

in R × Ω, (0.1)∇x · u(t, x) = 0 inR × Ω, (0.2)

u(t, x) = 0 onR × ∂Ω, (0.3)u(t, x) → 0 as|x| → +∞, (0.4)

and the Cauchy problem for the same equations as follows:

M. YamazakiDepartment of Mathematics, Hitotsubashi University, Kunitachi, Tokyo 186-8601, Japan(e-mail: cc00053@srv.cc.hit-u.ac.jp)

� Partly supportedbyGrant-in-Aid forScientificResearch (C)11640156,MinistryofEducation,Sciences, Sports and Culture, Japan

636 M.Yamazaki

∂u

∂t(t, x) = ∆xu(t, x) − (u · ∇x) u(t, x)−∇xπ(t, x) + f (t, x)

in R+ × Ω, (0.5)∇x · u(t, x) = 0 inR+ × Ω, (0.6)

u(t, x) = 0 onR+ × ∂Ω, (0.7)u(t, x) → 0 as|x| → +∞, (0.8)u(0, x) = a(x) onΩ, (0.9)

whereR+ = (0,+∞), u(t, x) = (u1(t, x), . . . , un(t, x)) denotes the unknownvector function standing for the velocity at(t, x), π(t, x) denotes the unknownscalar function standing for the pressure at(t, x), andf (t, x) = (f1(t, x), . . . ,fn(t, x)) denotes a given function standing for the external force at(t, x). Weare interested in the case where the external forcef (t, ·) does not necessarilydecay ast tends to+∞, and consider the unique existence of small boundedcontinuous solutions of the problem (0.1)–(0.4) and the continuous dependenceof the solutions on the data. We are also interested in the same problem on theCauchy problem (0.5)–(0.9), which implies the stability of the solutions of theoriginal problem (0.1)–(0.4) ast → +∞. For example, we are interested in theunique existence and the stability of time-periodic and almost periodic solutionsu(t, x).

In this paper we introduce the notion of mild solutions of the systems above,and show that, iff (t, x) can bewritten asP∇xF (t, x)with a small, bounded andcontinuous functionF(t, ·) with values in the spaceLn/2,∞(Ω), and if a(x) ∈Ln,∞σ (Ω) is sufficiently small, then the systems (0.1)–(0.4) and (0.5)–(0.9) admitone and only one mild solution bounded and continuous in the spaceLn,∞(Ω)with norm bounded by a definite constant. HereP denotes the projection ontothe space of solenoidal vector fields. We also show that the mild solution is asolution of the Navier-Stokes equation in the sense of distributions.

We moreover show that the mild solutions depend continuously on the dataF(t, x) anda(x). This fact implies that, if the functionF(t, x) is periodic withrespect tot with periodT or almost periodic with respect tot , then so is thesolution of the system (0.1)–(0.4). The result on (0.5)–(0.9) implies that thesolution of (0.1)–(0.4) is stable ast → +∞ under small initial perturbation att = 0 and small perturbation onF(t, x).

We also consider the asymptotic stability of the time-global solution abovewith respect to theLq,1(Ω) norm for q > n under small initial perturbation.Namely, suppose that the time-global mild solution above is sufficiently small.Then there exists another mild solution with the same external force and initialdata sufficiently close to the original one, and theLq,1(Ω)-norm of the differencebetween these solutions decays with definite rate. As we shall see in the next

Navier-Stokes equations in the weak-Ln space 637

section, we cannot expect the asymptotic stability in the spaceLn,∞σ (Ω) itself,even in the trivial caseu(t, x) ≡ F(t, x) ≡ 0.

We review previous results on the problems above. Among a number ofliteratures concerning time-global solutions of the Navier-Stokes equations withthe external force as above, Maremonti [26], [27] was the first in which theequations in unbounded domains are treated. In fact, [26], [27] considered thecaseΩ = R3 andΩ = R3+ respectively, and proved the unique existenceof the time-global solution of problems (0.1)–(0.4) and (0.5)–(0.9) under theassumption

f (t, ·) is small inL∞ (I : L3(Ω) ∩ Ḣ−1p (Ω)) (0.10)with somep < 3/2, whereI denotes eitherR orR+, and proved the stability ofthe solution above under a small perturbation. As a result, forf (t, x) periodicin time, he showed the unique existence of time-periodic solution with the sameperiod.

ThenKozonoandNakao [15] considered theproblem (0.1)–(0.4) onΩ,whereΩ is the whole spaceRn or the half spaceRn+ for n ≥ 3 or an exterior domain inRn for n ≥ 4, and constructed time-periodic solutions for time-periodicf (t, x)

satisfying the assumption

f (t, ·) is small inL∞ (R : Lr(Ω) ∩ Ḣ−1p (Ω)) (0.11)with somep < n/2 andr > n/3. The condition in (0.10) corresponds to thecaser = n = 3. Taniuchi [34] proved the stability of the periodic solutionsconstructed in [15] in the spaceLn(Ω). These works treated solutions belongingto suitableLp spaces. Yamazaki [37] considered the problem onRn for n ≥ 3,and generalized the results of [15], [34] for Morrey spaces.

On the other hand, Salvi [32] considered the problem (0.1)–(0.4) on three-dimensional exterior domainsΩ, and proved the existence of a time-periodicweak solution with periodT for time-periodicf (t, x) with periodT satisfyingthe assumption

f (t, ·) ∈ L2 ([0, T ];L2(Ω) ∩ Ḣ−12 (Ω)) . (0.12)He also showed the existence of a time-periodic strong solution with periodT under the assumption thatf (t, x) is small in the class above. Actually heconsidered a more general situation; he solved the problem above on three-dimensional exterior domains with boundary moving periodically with periodT . For the time being, this seems to the the only result for three-dimensionalexterior domains. However, the uniqueness of the periodic solution is not known.

More detailed references, including results for bounded domains, are foundin [26], [27], [15], [32].

638 M.Yamazaki

If the external forcef (t, x) is independent oft , the problem (0.1)–(0.4)reduces to the stationary problem

∆xu(x) − (u · ∇x) u(x) − ∇xπ(x) + f (x) = 0 inΩ, (0.13)∇x · u(x) = 0 inΩ, (0.14)

u(x) = 0 on∂Ω, (0.15)u(x) → 0 as|x| → +∞, (0.16)

and the problem (0.5)–(0.9) witha(x) nearu(x) above concerns the stability ofthe stationary solutionu(x). This problem was studied in more detail, including3-dimensional exterior domains. In fact, whenΩ is ann-dimensional exteriordomain, Novotny and Padula [31], Galdi and Simader [10] and Borchers andMiyakawa [5] proved the following: Ifm is an integer such that 1≤ m ≤ n− 2and iff (x)enjoys thecondition|f (x)| ≤ c|x|−m−2with sufficiently smallc, thenthere exists a unique solutionu(x) of (0.13)–(0.16) such that|u(x)| ≤ C|x|−mand that|∇xu(x)| ≤ C|x|−m−1. In particular, forn = 3, they proved the uniqueexistence of physically reasonable solutions in the sense of Finn [6], and obtainedsharp estimates of the solutions and their derivatives.

On the other hand, forn ≥ 4, Kozono and Sohr [17] proved the following: Iff (x) can be written in the formP∇xF (x) such thatF(x) is sufficiently small inLn/2(Ω), then there exists a unique solutionu(x) of the problem (0.13)–(0.16)such thatu(x) ∈ Ln(Ω) and that∇xu(x) ∈ Ln/2(Ω) hold with norms boundedby a certain constant.

Later on, forn ≥ 3, Kozono andYamazaki [21] unified the results above asfollows: If f (x) can be written in the form∇xF (x) such thatF(x) is sufficientlysmall in Ln/2,∞(Ω), then there exists a unique solutionu(x) of the problem(0.13)–(0.16) such thatu(x) ∈ Ln,∞(Ω) and that∇xu(x) ∈ Ln/2,∞(Ω) withnorms bounded by a definite constant, whereLp,∞(Ω) denotes the weak-Lpspace onΩ. The assumption on the external force in this result generalizes theassumption of [31], [10], [5] as well as that of [17]. Hence this result impliesthat, in the casen = 3, the classL3,∞ is a natural generalization of the class ofphysically reasonable solutions satisfyingu(x) → 0 as|x| → ∞.

The stability of the stationary solutions obtained in the results above is alsostudied by Borchers and Miyakawa [5], Kozono and Ogawa [16] and KozonoandYamazaki [22].

It is to be noted that, in the casen = 3, we cannot replace the spacesL3,∞(Ω) andL3/2,∞(Ω) byL3(Ω) andL3/2(Ω) respectively. In fact, Borchersand Miyakawa [5, Theorem 2.4], Kozono and Sohr [18, Theorem C] and Ko-zono, Sohr and Yamazaki [19, Theorem 2, (1)] showed that the solutionu(x)of (0.13)–(0.16) belongs toLn(Ω) only under very restricted situations. Moredetailed references are found in [21], [22].

All the results above on the stationary solutions give the condition onf (x)in terms of norms invariant under the scaling(u, π, f ) → (uλ, πλ, fλ) such

Navier-Stokes equations in the weak-Ln space 639

thatuλ(t, x) = λu(λ2t, λx), πλ(t, x) = λ2π(λ2t, λx), fλ(t, x) = λ3f (λ2t, λx);while theconditions (0.10)and (0.11)arenot in the formaboveandmuchstrongerthan those for the stationary solutions.

On the other hand, for the existence of weak solutions of the problem (0.13)–(0.16) in the sense of Leray [24], it suffices to assumef (x) = ∇xF (x)with someF(x) ∈ L2(Ω), and no smallness is necessary. The condition in Salvi [32] seemsto be the composition of this one (condition for the existence of stationary weaksolution) and the condition for the existence of non-stationary weak solution.(See Leray [24], [25].)

For the stationary problem (0.13)–(0.16), Galdi [9, Chapter 9, Theorem 9.4]and Miyakawa [29] showed that, ifu(x) is a weak solution and if supx∈Ω

(|x| +1)|u(x)| is sufficiently small, thenu(x)enjoys theenergy identity, andeveryweaksolution enjoying the energy inequality coincides withu(x) above. Kozono andYamazaki [23] proved the same result under the more general assumption that‖u‖n,∞ is sufficiently small. In other words, the uniqueness of weak solutionsis proved only for small physically reasonable solutions, or small solutions inthe class generalizing physically reasonable solutions. Hence it seems to be verydifficult to prove the uniqueness of the solutions given by Salvi [32] withoutassuming conditions as above.

In order to describe the idea of the proof of this paper, let us recall the outlineof the proof of Kozono and Nakao [15]. They modified the method of Fujitaand Kato [7], in which they transformed the original equation into an integralequation on the interval[0, t) for everyt ∈ R. Kozono and Nakao [15] showedthe unique solvability of the integral equation on the infinite interval(−∞, t)for every

t ∈ R under appropriate assumptions. Iff (x) is independent oft , it sufficesto consider the linearization of this system around the stationary solutionu(x)in the study of (0.5)–(0.9). However, iff (t, x) depends ont , the linearization ofthis system around the solution of (0.1)–(0.4) depends ont , and hence the lin-earization as above becomes difficult to handle. Instead, they solved the integralequation by regarding the Stokes operator as the principal part and everythingelse as the perturbation. However, for the integral on the infinite interval shouldconverge so that the iteration scheme associated with the viewpoint above shouldwork, the external force must enjoy decay property and regularity stronger thanthose in the case (0.13)–(0.16). Namely, under our weaker assumption, the con-vergence is difficult to prove.

Moreover, for three-dimensional exterior domains, the integral in questiondoes not converge inL3(Ω) in general even under the stronger condition in [15],as is understood from the results of [5], [18], [19]. Hence we must work on thespaceL3,∞(Ω) instead, as is stated in [5], [23]. But the weak-Lp spaces containnontrivial homogeneous functions, and the integral in question fails to converge

640 M.Yamazaki

in the strong topology in any of the weak-Lp spaces when the integrand containssuch homogeneous functions, as we shall see in Remark 1.2.

Our method is similar to the one in [15] in spirit, but in order to get aroundthe difficulty above, we show that the integral in question does converge in theweak-∗ topology of certain weak-Lp spaces. For this purpose we employ du-ality argument, which leads naturally to the notion of mild solutions. Roughlyspeaking, amild solution is a function, bounded in an appropriate function space,which solves the integral equation associated with the Navier-Stokes equation inthe sense of distributions. As in Kozono andYamazaki [21], [22], the duality be-tween the Lorentz spacesLn/(n−1),1σ (Ω) andLn/2,∞σ (Ω) plays themost importantrole. In order to employ the duality argument above, we prove a sharp version oftheLp-Lq estimates of the Stokes semigroup formulated in the Lorentz spaces.This estimate itself seems to be of interest.

As a result, for three-dimensional exterior domains as well, we can constructbounded solutions in thewhole time, including time-periodic andalmost periodicsolutions under an appropriate assumptions onf (t, x), which is unique in asmall ball inLn,∞(Ω) and depending continuously onf (t, x). We can alsoshow their stability under small initial perturbation in the same classLn,∞(Ω),which is exactly the same as the unique existence and the stability class ofstationary solutions. Our class of time-dependent solutions is equipped with anorm invariant under the scaling above, and is a natural generalization of theclass of stationary solutions introduced in [21], [22], [23], and hence of the classof reasonable stationary solutions satisfyingu(x) → 0 as|x| → ∞.

As is seen above, our assumption is more general than those in [15] possiblyexcept the smallness. On the other hand, neither of our assumption or the as-sumption of [32] implies the other. In particular, we need not assume the squaresummability off (t, x).

The outline of this paper is as follows. In Sect.1 we introduce necessary no-tations, introduce the notion of mild solutions and state main results. In Sect.2we derive a Lorentz space version of theLp-Lq estimates of the Stokes semi-group. In Sect.3 we construct an operator which gives the solution of the integralequations in question. We then construct a mild solution in Sect.4, and verifythe necessary properties in Sect.5. We study the asymptotic behavior of the dif-ference between two mild solutions associated with the same external force inSect.6. Finally, in Sect.7, we study the regularity of the mild solutions.

1. Main results

Before stating our result, we introduce some function spaces. For 1< p < ∞and 1≤ q ≤ ∞, letLp,q(Ω) denote the Lorentz space onΩ defined by

Lp,q(Ω) = {u(x) ∈ L1loc(Ω) | ‖u‖p,q < +∞} ,

Navier-Stokes equations in the weak-Ln space 641

where

‖u‖p,q =(∫ +∞

0

(sµ ({x ∈ Ω | |u(x)| > s})1/p)q ds

s

)1/q

for 1≤ q < ∞ and‖u‖p,q = sup

s>0sµ ({x ∈ Ω | |u(x)| > s})1/p .

Although the function‖u‖p,q above does not satisfy the triangle inequality, thereexists a norm equivalent to‖u‖p,q , and with this norm the spaceLp,q(Ω) be-comes a Banach space. Note that the spaceLp,p(Ω) is equivalent to the stan-dard spaceLp(Ω). For these spaces, real interpolation yields the equivalence(Lp0(Ω), Lp1(Ω))θ,q = Lp,q(Ω), where 1 < p0 < p < p1 < ∞ and0 < θ < 1 satisfy 1/p = (1− θ)/p0 + θ/p1 and 1≤ q ≤ ∞. Note that thisspace is determined independently of the choice ofp0 andp1 up to equivalentnorms. (See Bergh and L¨ofström [2] or Triebel [35] for example.) We remarkthat, if 1 ≤ q < ∞, the dual of the spaceLp,q(Ω) coincides with the spaceLp/(p−1),q/(q−1)(Ω). Note furthermore that, for 1≤ q < ∞, the spaceC∞0 (Ω)is dense inLp,q(Ω), while it is not so forLp,∞(Ω). The dual of the closure ofC∞0 (Ω) in Lp,∞(Ω) coincides with the spaceLp/(p−1),1(Ω).

We next recall that, for every 1< p < ∞, we have the Helmholtz decompo-sition (Lp(Ω))n = Lpσ (Ω) ⊕ Gp(Ω), whereLpσ (Ω)

= {u(x) ∈ (Lp(Ω))n | div u(x) ≡ 0 inΩ andν · u(x) ≡ 0 on∂Ω}and

Gp(Ω) = {u(x) = gradf (x) ∈ (Lp(Ω))n for somef (x) ∈ Lploc(Ω)} .For the proof, see Fujiwara and Morimoto [8], Miyakawa [28] and Simader andSohr [33]. LetPp denote the projection operator from(Lp(Ω))

n ontoLpσ (Ω)alongGp(Ω). Then the dual of the operatorPp coincideswithPp/(p−1). In partic-ular, the operatorP2 is an orthogonal projection in the Hilbert space

(L2(Ω)

)n.

We next generalize the Helmholtz decomposition to the Lorentz spaces fol-lowingMiyakawa andYamada [30].We havePp = Pq on(Lp(Ω))n∩ (Lq(Ω))nand hencewe can extendPp as a projection operatorP in

(∑1

642 M.Yamazaki

In order to introduce the notion of mild solution, we define some function

classes. PutK = BUC (R, Ln,∞σ (Ω)) andL = BUC(R,

(Ln/2,∞(Ω)

)n2),

whereBUC(R, X) denotes the set of bounded and uniformly continuous func-tions onR, equipped with the norm‖f |BUC(R, X)‖ = supt∈R ‖f (t, ·) |X‖.Next, putK+ = BC

(R+, Ln,∞σ (Ω)

)andL+ = BC

(R+,

(Ln/2,∞(Ω)

)n2),

whereBC(R+, X) denotes the set of bounded continuous functions onR+ withvalues in the Banach spaceX, equipped with the norm‖f |BC(R+, X)‖ =supt∈R+ ‖f (t, ·) |X‖.Definition 1 A functionu(t, x) ∈ K is said to be amild solutionof the system(0.1)–(0.4) if the identity

(u(t, ·), ϕ) =n∑

j,k=1

∫ +∞0(

uj (t − τ, ·)uk(t − τ, ·) − Fjk(t − τ, ·), ∂∂xj

(exp(−τA)ϕ)

k

)dτ (1.1)

holds for everyϕ ∈ Ln/(n−1),1σ (Ω) and everyt ∈ R.Definition 2 A functionu(t, x) ∈ K+ is said to be amild solutionof the system(0.5)–(0.9) if the identity

(u(t, ·), ϕ) = (a,exp(−tA)ϕ) +n∑

j,k=1

∫ t0(

uj (t − τ, ·)uk(t − τ, ·) − Fjk(t − τ, ·), ∂∂xj

(exp(−τA)ϕ)

k

)dτ (1.2)

holds for everyϕ ∈ Ln/(n−1),1σ (Ω) and everyt > 0.Remark 1.1As is explained in the introduction, the relations (1.1) and (1.2) arethe weak form of the integral equations

u(t) =∫ +∞0

exp(−τA)[−P [(u · ∇)u](t − τ) + P∇F(t − τ)]dτ (1.3)and

u(t) = exp(−tA)a +∫ t0exp(−τA)[−P [(u · ∇)u](t − τ) + P∇F(t − τ)]dτ (1.4)

respectively. if we regard the term(u · ∇)u as an element of the space above byway the duality pairing

((u · ∇)u, ϕ) = −(u ⊗ u,∇ϕ) for ϕ ∈ C∞0,σ (Ω).

Navier-Stokes equations in the weak-Ln space 643

Then our main result is the following:

Theorem 1.1. There exist positive numbersA, ε andC0 depending onn andΩsuch that the following holds:

1. For everyF(t, x) ∈ L such that‖F |L‖ < ε, there exists one and only onemild solutionu(t, x) ∈ K of the system(0.1)–(0.4)withf (t, x) = ∇xF (t, x)such that‖u |K‖ < A. Moreover, for everyδ ∈ (0, ε), the mappingT fromthe closed ball inL centered at the origin with radiusδ to K defined byT (F ) = u is uniformly continuous. Furthermore, the functionu(t, x) is theonly solution of(0.1)–(0.4) in the sense of distributions inR × Ω such thatu(t, x) ∈ K with ‖u |K‖ < A. Namely, the functionu(t, x) is the only onesatisfying the estimate‖u |K‖ < A and the identity

d

dt

(u(t, ·), ϕ)

= (u(t, ·),∆ϕ) +n∑

j,k=1

(uj (t, ·)uk(t, ·) − Fjk(t, ·), ∂

∂xjϕk

)(1.5)

for everyϕ(x) ∈ C∞0,σ (Ω) and everyt ∈ R.2. For everyF(t, x) ∈ L+ and everya(x) ∈ Ln,∞σ such thatC0‖a‖n,∞ +‖F |L+ ‖ < ε, there exists one and only one mild solutionu(t, x) ∈ K+ of

the system(0.5)–(0.9)with f (t, x) = ∇xF (t, x) such that‖u |K+ ‖ < A.Moreover, for everyδ ∈ (0, ε), the mappingT+ from the set

{(F(t, x), a

) |‖F |L+ ‖ + C0‖a‖n,∞ ≤ δ

}to K+ defined byT+(F, a) = u is uniformly

continuous. Furthermore, the functionu(t, x) is the only solution of the(0.5)–(0.9) in the sense of distributions inR+ × Ω such thatu(t, x) ∈ K+ with‖u |K+ ‖ < A. Namely, the functionu(t, x) is the only one satisfying theestimate‖u |K+ ‖ < A, the identity(1.5) for everyϕ(x) ∈ C∞0,σ (Ω) andeveryt > 0, and (

u(t, ·), ϕ) → (a, ϕ) ast → +0 (1.6)for everyϕ(x) ∈ C∞0,σ (Ω).

As an application to the unique existence of time periodic and almost periodicsolutions, we have the following.

Corollary 1.2. Suppose thatu(t, x) ∈ K is the unique mild solution such that‖u |K‖ < A. Then we have the following:1. If F(t, ·) is time-periodicwith periodT , then the uniquemild solutionu(t, x)

such that‖u |K‖ < A is also time-periodic with periodT .2. If F(t, ·) is almost periodic with respect tot ∈ R, then the unique mild

solutionu(t, x) such that‖u |K‖ < A is also almost periodic with respectto t ∈ R.

644 M.Yamazaki

Remark 1.2For three-dimensional exterior domains, the best spatial decay con-dition expected in general isu(t, ·) ∈ L3,∞(Ω). On the other hand, if weput u(t, x) = U(x) with some homogeneous functionU(x) of degree−1 onR3 such thatU(x) ∈ L3,∞(R3), the functionV (τ, x) defined by the formula

V (τ, ·) = exp(−τA)∇(U ⊗ U) is forward self-similar; namely, it enjoys theequalityV (λ2τ, λx) = λ−3V (τ, x) for everyλ, τ > 0 andx ∈ R3. It followsthat

‖V (τ, ·)‖q,∞ =(

1√τ

)3 ∥∥∥∥V(1,

·√τ

)∥∥∥∥q,∞

= Cτ 3/2q−3/2

for everyq ∈ (1,∞). We thus conclude that the right-hand side of (1.3) inRemark 1.1 is not Bochner integrable inLq,∞ for anyq ∈ (1,∞).Remark 1.3Assertion 2 of Theorem 1.1 implies the Lyapunov stability of the so-lution given inAssertion 1 ofTheorem 1.1. In particular, ifF(t, x) is independentof t , then by the same reasoning as in Corollary 1.2, the solution given in Asser-tion 1 becomes the stationary solution given in Kozono andYamazaki [21], andAssertion 2 implies the stability of this stationary solution under small initial per-turbation. This result removes the technical assumption∇u(x) ∈ Lq,∞(Ω) withsomeq > n on the stationary solutionu(x) posed in Kozono andYamazaki [22].

We also have the following theorem, which shows the asymptotic behavior ofthe difference of two solutions with the same external force.

Theorem 1.3. For everyp ∈ (n,∞) there exists a positive numberε(p) ≤ εsuch that the following holds. Suppose thata(x) ∈ Ln,∞σ (Ω) and F(t, x) ∈L+ satisfy‖F |L+ ‖ + C0‖a‖n,∞ < ε(p), and let u(t, x) ∈ K+ denote theunique mild solution of(0.5)–(0.9) such that‖u |K+ ‖ < A. Then there existsa positive constantC = C(p) such that, for everyb(x) ∈ Ln,∞σ (Ω) such thatC0‖b(x) − a(x)‖n,∞ < ε(p), there uniquely exists a mild solutionv(t, x) ∈ K+of (0.5)–(0.9)with a(x) replaced byb(x) such that‖v |K+ ‖ < A and that

‖v(t, ·) − u(t, ·)‖p,∞ ≤ Ctn/2p−1/2 (1.7)holds for everyt > 0. Moreover, for everyq ∈ (n, p), there exists a positiveconstantC ′ = C ′(p, q) such that the estimate

‖v(t, ·) − u(t, ·)‖q,1 ≤ C ′tn/2q−1/2 (1.8)holds for everyt > 0.

Remark 1.4Even in the trivial caseF(t, x) ≡ u(t, x) ≡ 0, we cannot expect theasymptotic stability in the spaceLn,∞σ itself. This is observed in the followingfact. Suppose thatΩ = R3, and putb(x) = (0,0, log |x|) and

a(x) = c rotb(x) = c(

x2

|x|2 ,−x1|x|2 ,0

).

Navier-Stokes equations in the weak-Ln space 645

Thena(x) ∈ L3,∞σ (R3). Hence Kozono andYamazaki [20] implies that, if|c| issufficiently small, there exists a solutionu(t, x) ∈ BC ((0,+∞), L3,∞σ (R3)) ofthe evolution equation

du

dt(t, x) = −Au(t, x) − P [(u(t, ·) · ∇)u(t, ·)] (x) + f (t, x) (1.9)

with f (t, x) ≡ 0 on (0,+∞), satisfying a kind of boundedness property andthe initial conditionu(0, x) = a in a suitable sense. Since the initial dataa(x)is homogeneous of order−1, it follows that the solutionu(t, x) is forward self-similar; namely,u(t, x) enjoys the scaling propertyu(λ2t, λx) = λ−1u(t, x).for everyλ, t > 0 andx ∈ Rn. From this fact we see that∥∥u(λ2t, ·)∥∥3,∞ =‖u(t, ·)‖3,∞. It follows that‖u(t, ·)‖3,∞ is independent oft > 0. This impliesthat even the trivial solution 0 is not asymptotically stable in the spaceL3,∞σ (R3),in contrast to the spaceL3σ (R

3).

As for the regularity of the mild solutions, we have only modest results underour general assumptions.

Theorem 1.4. Suppose thatn/2< p < n.

1. Letu(t, x) ∈ K be a mild solution of(0.1)–(0.4). Then, for everys, t ∈ Rsuch thats < t , we haveu(t, ·)− u(s, ·) ∈ Lp,1 and‖u(t, ·) − u(s, ·)‖p,1 ≤C(t − s)n/2p−1/2 with a constantC independent ofs and t .

2. Suppose thata(x) enjoys the conditionAn/2p−1/2a(x) ∈ Lp,∞(Ω) as well,and letu(t, x) ∈ K+ beamild solutionof(0.5)–(0.9). Then, for everys, t ∈ Rsuch that0< s < t , we have the same conclusion as in Assertion 1.

Remark 1.5Theorem 1.4 implies that the functionu(t, ·) is Hölder continuouswith respect tot with values in the spaceLn,∞σ (Ω) + Lp,1σ (Ω). On the otherhand, as we have seen in Remark 1.1, mild solutionsu(t, x) of (0.1)–(0.4) and(0.5)–(0.9) can be regarded as formal solutions of the integral equations (1.3)and (1.4) respectively. Hence, if the external forcef (t) enjoys some suitableHölder continuity, it may be possible to deduce that the equality

∂u

∂t= Au − P ((u · ∇)u) + f

holds as functions with values iṅH−1np/(n+p),∞(Ω) + Ḣ−1n/2,∞(Ω), if we regardthe Stokes operator as a densely defined closed linear operator in the spaceḢ−1q ′ (Ω)+ Ḣ−1r ′ (Ω) with someq ′ < np/(n+p) andr ′ > n/2, which is strictlylarger thanḢ−1n/2(Ω). However we do not enter this problem in detail. Amann [1]extensively studied such relationship betweenmild solutions andweak solutions,but his study seems to be limited to the solutions strongly continuous att = 0and not to be applicable to our solutions.

646 M.Yamazaki

For the time being we can prove better results except for three-dimensionalexterior domain. Namely, we have the following:

Theorem 1.5. Suppose either thatn ≥ 4,Ω = R3 or Ω = R3+, and letq be anumber such thatn/3< q < n/2.

1. Suppose thatF(t, x) ∈ L enjoys the condition

‖F(t, ·) − F(s, ·)‖q,∞ ≤ C|t − s|n/2q−1 (1.10)

for everys, t ∈ R such thats < t with some positive constantC, and letu(t, x) be a mild solution of(0.1)–(0.4). Then{∇u(t, ·)}t∈R is bounded inLn/2,∞(Ω), and we also have

‖∇u(t, ·) − ∇u(s, ·)‖q,∞ ≤ C|t − s|n/2q−1 (1.11)

for everys, t ∈ R such thats < t .2. Suppose thatF(t, x) enjoys the condition(1.10)for everys, t ∈ R such that

0 < s < t with some positive constantC, and thata ∈ Ln,∞σ (Ω) and∇a ∈Ln/2,∞(Ω). Letu(t, x) be a mild solution of(0.5)–(0.9). Then{∇u(t, ·)}t>0is bounded inLn/2,∞(Ω). Moreover, if we assumeAn/2q−1/2u ∈ Lq,∞(Ω),we also have(1.11)for everys, t ∈ R such that0< s < t .

From the theoremabovewecan show that ourmild solutions coincidewith strongsolutions under additional assumptions when the dimension is 4 or higher.

Corollary 1.6. Suppose thatn ≥ 4, and letr be a number such thatn/4< r <n/3.

1. Suppose thatf (t, ·) is a bounded uniformly continuous function with respectto t ∈ R with values inLn/3,∞σ (Ω) satisfying the inequality

‖f (t, ·) − f (s, ·)‖r,∞ ≤ C(t − s)n/2r−3/2 (1.12)

for everys, r ∈ R such thats < t with some positive constantC. Letu(t, x)beamild solutionof(0.1)–(0.4)withF(t, x) such thatf (t, x) = P∇F(t, x).Thenu(t, x) is a strong solution of the evolution equation(1.9)onR.

2. Suppose thatf (t, ·) is a bounded continuous function with respect tot > 0with values inLn/3,∞σ (Ω) satisfying the inequality(1.12)for everys, r ∈ Rsuch that0 < s < t with some positive constantC. Suppose moreover thata(x) ∈ Ln,∞σ (Ω) and thatAn/2r−1a(x) ∈ Lrn/(n−r),∞(Ω). Let u(t, x) be amild solution of (0.5)–(0.8)with F(t, x) such thatf (t, x) = P∇F(t, x).Thenu(t, x) is a strong solution of the evolution equation(1.9)on (0,+∞)and enjoys the conditionu(t, ·) → a in the sense of distributions ast → +0.

Navier-Stokes equations in the weak-Ln space 647

2. Lp-Lq estimates

In this section we prove a sharp version formulated in the Lorentz spaces of theLp-Lq estimates of the Stokes semigroup. StandardLp-Lq estimates are provedby Kato [13], Ukai [36], Iwashita [12], Giga and Sohr [11] and Borchers andMiyakawa [3], [4]. We start with a fact which corresponds to [4, Theorem 4.4].

Theorem 2.1. Suppose thatΩ is an exterior domain inRn. Then we have thefollowing:

1. For everyp ∈ (1,∞), there exists a positive constantC such that the in-equalities

∥∥A1/2u∥∥p,1 ≤ C‖∇u‖p,1 and

∥∥A1/2u∥∥p,∞ ≤ C‖∇u‖p,∞ always

hold.2. For everyp ∈ (1, n), thereexists apositive constantC such that the inequality

‖∇u‖p,∞ ≤ C∥∥A1/2u∥∥

p,∞ always holds.3. For everyp ∈ (1, n], thereexists apositive constantC such that the inequality

‖∇u‖p,1 ≤ C∥∥A1/2u∥∥

p,1 (2.1)

always holds.

Proof. By real interpolation,wecanderiveAssertion1 from [4,Theorem4.4, (i)],and Assertion 2 from [4, Theorem 4.4, (ii)].

It remains to prove Assertion 3. Suppose thatu(x) ∈ C∞0,σ (Ω), and putf (x) = −∆u(x). Then Kozono and Yamazaki [21, Theorem 2.1, (i)] impliesthatu(x) is the only solution of the equation

−∆u(x) = f (x) and divu(x) = 0 onΩsuch that∇u(x) ∈ Lp,1(Ω), and thatu(x) is subject to the estimate

‖∇u‖p,1 ≤ C∥∥∥f ∣∣∣Ĥ 1p/(p−1),∞(Ω)∗

∥∥∥. (2.2)HereĤ 1p/(p−1),∞(Ω) denotes the completion of the spaceC

∞0,σ (Ω) with respect

to the norm‖∇ϕ‖p/(p−1),∞.In view of (2.2), it suffices to show the estimate∥∥∥f ∣∣∣Ĥ 1p/(p−1),∞(Ω)∗

∥∥∥ ≤ C∥∥A1/2u∥∥p,1. (2.3)

Letϕ(x) be an arbitrary element ofC∞0,σ (Ω) such that‖∇ϕ‖p/(p−1),∞ ≤ 1. Thenwe have

|(f, ϕ)| = |(Au, ϕ)| = ∣∣(A1/2u,A1/2ϕ)∣∣≤ C∥∥A1/2u∥∥

p,1

∥∥A1/2ϕ∥∥p/(p−1),∞ ≤ C

∥∥A1/2u∥∥p,1

‖∇ϕ‖p/(p−1),∞,where the last inequality follows form Assertion 1. Now (2.3) follows immedi-ately from this inequality.

648 M.Yamazaki

From the theoremabovewecanprove the following version in the Lorentz spacesof theLp-Lq inequality. ��Theorem 2.2. For everyp ∈ (1,∞), the operator−A generates a bounded an-alytic semigroupexp(−tA) onLp,1(Ω), and this semigroup enjoys the followingestimates forp, q such that1< p ≤ q < ∞:1. There exists a positive constantC such that the estimates

‖exp(−tA)u‖q,1 ≤ Ctn/2q−n/2p‖u‖p,1 (2.4)and ∥∥A1/2 exp(−tA)u∥∥

q,1 ≤ Ctn/2q−n/2p−1/2‖u‖p,1 (2.5)hold for everyu ∈ Lp,1σ (Ω) and everyt > 0.

2. Suppose thatq ≤ n in the case whereΩ is an exterior domain. Then thereexists a positive constantC such that the estimates

‖∇ exp(−tA)u‖q,1 ≤ Ctn/2q−n/2p−1/2‖u‖p,1 (2.6)and ∥∥∇A1/2 exp(−tA)u∥∥

q,1 ≤ Ctn/2q−n/2p−1‖u‖p,1 (2.7)hold for everyu ∈ Lp,1σ (Ω) and everyt > 0.

Proof. Applying real interpolation to the results of Iwashita [12] or Borchersand Miyakawa [4, Corollary 4.6, (i)], we see that the operator−A generates abounded analytic semigroup exp(−tA) onLp,1(Ω) and that the estimates (2.4)and (2.5) hold. Next, sincep ≤ q ≤ n, Theorem 2.1 and the estimates (2.4) and(2.5) imply that

‖∇ exp(−tA)u‖q,1 ≤ C∥∥∥∥A1/2 exp

(− t2A

)exp

(− t2A

)u

∥∥∥∥q,1

≤ Ct−1/2∥∥∥∥exp

(− t2A

)u

∥∥∥∥q,1

≤ Ct−1/2+n/2q−n/2p‖u‖p,1and

∥∥∇A1/2 exp(−tA)u∥∥q,1 ≤ C

∥∥∥∥Aexp(

− t2A

)exp

(− t2A

)u

∥∥∥∥q,1

≤ Ct−1∥∥∥∥exp

(− t2A

)u

∥∥∥∥q,1

≤ Ct−1+n/2q−n/2p‖u‖p,1,which imply (2.6) and (2.7) respectively. ��

Navier-Stokes equations in the weak-Ln space 649

Remark 2.1The estimate (2.6) for 1< p ≤ q < n follows immediately fromthe result of Iwashita [12] and real interpolation. In order to treat the caseq = nwe need Theorem 2.1.

The next corollary gives a sharp version of theLp-Lq estimates formulated inthe Lorentz spaces.

Corollary 2.3. Suppose that1< p < q < ∞.1. There exists a constantC such that we have the estimates

tn/2p−n/2q‖exp(−tA)u‖q,1 ≤ C‖u‖p,∞ (2.8)and

tn/2p−n/2q+1/2∥∥A1/2 exp(−tA)u∥∥

q,1 ≤ C‖u‖p,∞ (2.9)for everyu ∈ Lp,∞σ (Ω) and everyt > 0, and∫ +∞

0tn/2p−n/2q−1‖exp(−tA)u‖q,1 dt ≤ C‖u‖p,1 (2.10)

and ∫ +∞0

tn/2p−n/2q−1/2∥∥A1/2 exp(−tA)u∥∥

q,1 dt ≤ C‖u‖p,1 (2.11)

for everyu ∈ Lp,1σ (Ω).2. Suppose thatq ≤ n in the case thatΩ is an exterior domain. Then the

constantC can be taken so that we also have the estimates

tn/2p−n/2q+1/2‖∇ exp(−tA)u‖q,1 ≤ C‖u‖p,∞ (2.12)and

tn/2p−n/2q+1∥∥∇A1/2 exp(−tA)u∥∥

q,1 ≤ C‖u‖p,∞ (2.13)for everyu ∈ Lp,∞σ (Ω) and everyt > 0, and∫ +∞

0tn/2p−n/2q−1/2‖∇ exp(−tA)u‖q,1 dt ≤ C‖u‖p,1 (2.14)

and ∫ +∞0

tn/2p−n/2q∥∥∇A1/2 exp(−tA)u∥∥

q,1 dt ≤ C‖u‖p,1. (2.15)

for everyu ∈ Lp,1σ (Ω).

650 M.Yamazaki

Proof. Fix q and apply real interpolation with the parameterp ∈ (1, q) tothe inequalities (2.4), (2.5), (2.6) and (2.7), taking into account of the fact(Lp1,1, Lp2,1

)θ,∞ = Lp,∞ with 1/p = (1 − θ)/p1 + θ/p2. Then we see that

the estimates (2.8) and (2.9) hold for everyt > 0, and that the estimate (2.12)and (2.13) hold for everyt > 0 as well ifq ≤ n in the case thatΩ is an exteriordomain.

In order to prove (2.10), fixp andq and choosep1 andp2 such that 1<p1 < p < p2 < q and that 1/p − 1/p2 < 2/n. Then we putρ = n/2p −n/2q − 1, and consider the sublinear operatorT which maps a functionu(x) ∈Lp1,∞σ (Ω) + Lp2,∞σ (Ω) to a functionv(t) on (0,+∞) defined by the formulav(t) = tρ‖exp(−tA)u‖q,1. Then (2.8) implies that, forj = 1, 2, we havev(t) ≤Ctn/2p−n/2pj−1‖u‖pj ,∞ for everyu ∈ Lpj ,∞σ (Ω). Nowchoosepositivenumberss1ands2 such that 1/sj = 1−n/2p+n/2pj . Thenwe havev(t) ∈ Lsj ,∞

((0,+∞))

and‖v‖sj ,∞ ≤ C‖u‖pj ,∞. Furthermore, since the constantθ ∈ (0,1) such that1/p = (1−θ)/p1+θ/p2 enjoys the equality 1= (1−θ)/s1+θ/s2, we have thefollowing real interpolation relations among quasi-Banach spaces as follows:(

Lp1,∞σ (Ω), Lp2,∞σ (Ω)

)θ,1 = Lp,1σ (Ω)

and (Ls1,∞

((0,+∞)), Ls2,∞((0,+∞)))

θ,1 = L1((0,+∞)).

Hence, by applying real interpolation for the operatorT ,(see Bergh and L¨ofström [2, Sect.5.3] or Komatsu [14],) we conclude thatv(t) ∈ L1((0,+∞)) for u(x) ∈ Lp,1σ (Ω) and that the estimate∫ +∞

0v(t) dt ≤ C‖u‖p,1

holds with a constantC independent ofu(x). Hence we have (2.10).The estimates (2.11), (2.14) and (2.15) can be proved in the same way.��

At the end of this section we prove the following fact.

Corollary 2.4. Suppose thatp ≤ n. Then there exists a positive constantC suchthat, for everyLp,1σ (Ω) and everyT ∈ (0,+∞], we have the estimate∥∥∥∥∇A1/2

∫ T0

exp(−tA)ϕ dt∥∥∥∥p,1

≤ C‖ϕ‖p,1. (2.16)

Proof. Sincep ≤ n, Theorem 2.1, 3 implies that the left-hand side of (2.16) isdominated from above by

C

∥∥∥∥A∫ T0

exp(−τA)∇ϕ dτ∥∥∥∥p,1

= C‖ϕ − exp(−TA)ϕ‖p,1.

This completes the proof. ��

Navier-Stokes equations in the weak-Ln space 651

3. Integral equations

In this section we show a theorem on the solvability of the integral equationsequivalent to the original problems. Foru(t, ·), v(t, ·) ∈ K andF(t, ·) ∈ L, wedefineΦ[u, v, F ](t, x) so that the formula

(Φ[u, v, F ](t, ·), ϕ) =n∑

j,k=1

∫ +∞0(

uj (t − τ, ·)vk(t − τ, ·) − Fjk(t − τ, ·), ∂∂xj

(exp(−τA)ϕ)

k

)dτ (3.1)

holds for everyϕ ∈ Ln/(n−1),1σ (Ω), and that(Φ[u, v, F ](t, ·),∇xψ) = 0 (3.2)

for every scalar functionψ such that∇xψ ∈(Ln/(n−1),1(Ω)

)n. On the other

hand, foru(t, ·), v(t, ·) ∈ K+, F(t, ·) ∈ L+ anda(x) ∈ Ln,∞(Ω), we defineΨ [u, v, F, a](t, x) so that the formula

(Ψ [u, v, F, a](t, ·), ϕ) = (a,exp(−tA)ϕ) +n∑

j,k=1∫ t0

(uj (t − τ, ·)vk(t − τ, ·) − Fjk(t − τ, ·), ∂

∂xj

(exp(−τA)ϕ)

k

)dτ (3.3)

holds for everyϕ ∈ Ln/(n−1),1σ (Ω), and that(Ψ [u, v, F, a](t, ·),∇xψ) = 0 (3.4)

for every scalar functionψ such that∇xψ ∈(Ln/(n−1),1(Ω)

)n. Then, by virtue

of Corollary 2.3, we can prove the following theorem.

Theorem 3.1. There exist positive constantsC1, C2 and C3 such that the fol-lowing holds:

1. For everyu(t, x), v(t, x) ∈ K and everyF(t, x) ∈ L, the functionΦ[u, v, F ](t, x) is well-defined inK, and satisfies the estimate

‖Φ[u, v, F ] |K‖ ≤ C1‖F |L‖ + C2‖u |K‖‖v |K‖.2. For everyu(t, x), v(t, x) ∈ K+, everyF(t, x) ∈ L+ and everya(x) ∈

Ln,∞σ (Ω), the functionΨ [u, v, F, a](t, x) is well-defined inK+, and satisfiesthe estimate

‖Ψ [u, v, F, a] |K+ ‖ ≤ C1‖F |L+ ‖ + C2‖u |K+ ‖‖v |K+ ‖ + C3‖a‖n,∞.

652 M.Yamazaki

Proof. For every fixedt andϕ ∈ Ln/(n−1),1σ (Ω), Corollary 2.3, 1 implies that themodulus of the first term of the right-hand side of (3.3) is dominated by

C‖a‖n,∞‖exp(−tA)ϕ‖n/(n−1),1 ≤ C‖a‖n,∞‖ϕ‖n/(n−1),1. (3.5)

On the other hand, sincen ≥ 3, we haven/(n − 2) ≤ n. Hence, for every fixedt andϕ ∈ Ln/(n−1),1σ (Ω), we can apply Corollary 2.3, 2 withp = n/(n− 1) andq = n/(n− 2) to see that the modulus of the right-hand side of (3.1) and that ofthe second term of the right-hand side of (3.3) is dominated by

n∑j,k=1

∫ +∞0∣∣∣∣

(uj (t − τ, ·)vk(t − τ, ·) − Fjk(t − τ, ·), ∂

∂xj

(exp(−τA)ϕ)

k

)∣∣∣∣ dτ

≤ Cn∑

j,k=1

∫ +∞0

‖∇x exp(−τA)ϕ‖n/(n−2),1(∥∥uj (t − τ, ·)∥∥n,∞‖vk(t − τ, ·)‖n,∞ +

∥∥Fjk(t − τ, ·)∥∥n/2,∞)dτ

≤ C (‖F |L‖ + ‖u |K‖‖v |K‖) ‖ϕ‖n/(n−1),1. (3.6)

The estimates (3.5) and (3.6), together with the facts (3.2) and (3.4), imply thatΦ[u, v, F ](t) ∈ (Ln,∞(Ω))n andΨ [u, v, F, a](t) ∈ (Ln,∞(Ω))n the estimates

‖Φ[u, v, F ](t)‖n,∞ ≤ C1‖F |L‖ + C2‖u |K‖‖v |K‖

and

‖Ψ [u, v, F, a](t)‖n,∞ ≤ C1‖F |L+ ‖ + C2‖u |K+ ‖‖v |K+ ‖ + C3‖a‖n,∞

hold for everyt . Moreover, in view of (3.2) and (3.4), we see thatΦ[u, v, F ](t)andΨ [u, v, F, a](t) belong toLn,∞σ (Ω) for everyt . SinceC1,C2 andC3 are in-dependent oft , we conclude thatΦ[u, v, F ](t) andΨ [u, v, F, a](t) are boundedfunctions oft with values inLn,∞σ (Ω), and that they enjoy the required estimates.

It remains to show the continuity. We first considerΦ[u, v, F ](t). For everypositive number ε, we can take a positive numberδ so that supt‖u(t + s, ·) − u(t, ·)‖n,∞ < ε, supt ‖v(t + s, ·) − v(t, ·)‖n,∞ < ε andsupt ‖F(t + s, ·) − F(t, ·)‖n/2,∞ < ε holds for everys ∈ (0, δ). For suchswe putũ(t) = u(t + s), ṽ(t) = v(t + s) andF̃ (t) = F(t + s). Then we have

Navier-Stokes equations in the weak-Ln space 653

Φ[ũ, ṽ, F̃ ](t) = Φ[u, v, F ](t + s). It follows that

‖Φ[u, v, F ](t + s) − Φ[u, v, F ](t)‖n,∞=

∥∥∥Φ[ũ, ṽ, F̃ ](t) − Φ[u, v, F ](t)∥∥∥n,∞

=∥∥∥Φ[ũ, ṽ − v,0](t) + Φ[ũ − u, v,0](t) + Φ[0,0, F̃ − F ](t)

∥∥∥n,∞

≤ C2‖ũ |K‖‖ṽ − v |K‖ + C2‖ũ − u |K‖‖v |K‖ + C1∥∥∥F̃ − F |L

∥∥∥< {C2 (‖u |K‖ + ‖v |K‖) + C1} ε

for everys such that 0< s < δ. Since the right-hand side can be taken arbitrarilysmall, this implies the required uniform continuity ofΦ[u, v, F ](t) with valuesin Ln,∞σ (Ω).

Wenext considerΨ [u, v, F, a](t). Lett andε bepositivenumbers.Then thereexists a positive numberδ < min{ε, t} such that, for everyϕ(x) ∈ Ln/(n−1),1σ (Ω),we have the inequality

∫ t+δt−δ

‖∇x exp(−τA)ϕ‖n/(n−2),1 dτ ≤ C∫ t+δt−δ

1

τdτ‖ϕ‖n/(n−1),1

≤ ε‖ϕ‖n/(n−1),1.

We now takes ∈ [t − δ/2, t + δ/2]. Then we have(Ψ [u, v, F, a](t) − Ψ [u, v, F, a](s), ϕ)

= (a,exp(−tA)ϕ − exp(−sA)ϕ)

+n∑

j,k=1

∫ t−δ0

(uj (t − τ, ·)vk(t − τ, ·) − Fjk(t − τ, ·)

− uj (s − τ, ·)vk(s − τ, ·) + Fjk(s − τ, ·),

∂

∂xj

(exp(−τA)ϕ)

k

)dτ

+n∑

j,k=1

∫ tt−δ

(uj (t − τ, ·)vk(t − τ, ·) − Fjk(t − τ, ·),

∂

∂xj

(exp(−τA)ϕ)

k

)d

654 M.Yamazaki

+n∑

j,k=1

∫ st−δ

(Fjk(s − τ, ·) − uj (s − τ, ·)vk(s − τ, ·),

∂

∂xj

(exp(−τA)ϕ)

k

)dτ

= I1 + I2 + I3 + I4. (3.7)Then we have

|I1| ≤ C‖a‖n,∞|t − s| sup0≤θ≤1

∥∥Aexp(−((1− θ)s + θt)A)ϕ∥∥n/(n−1),1

≤ C‖a‖n,∞|t − s|∥∥Aexp(−(t − δ)A)ϕ∥∥

n/(n−1),1

≤ C‖a‖n,∞‖ϕ‖n/(n−1),1 εt − δ . (3.8)

We next have

|I3| , |I4|≤ C (‖u |K+ ‖‖v |K+ ‖ + ‖F |L+ ‖)

∫ t+δt−δ

‖∇x exp(−τA)ϕ‖n/(n−2),1dτ≤ C (‖u |K+ ‖‖v |K+ ‖ + ‖F |L+ ‖) ε‖ϕ‖n/(n−1),1. (3.9)

We finally estimateI2. Sincet − δ/2 ≤ s ≤ t + δ/2 and since 0≤ τ ≤ t − δ, wehaves− τ , t − τ ∈ [δ/2, t + δ/2]. Sinceu(t, ·), v(t, ·) andF(t, ·) are uniformlycontinuous on the interval[δ/2, t + δ/2], we can takeγ ∈ (0, δ/2) so small that|t − s| < γ implies‖u(t, ·) − u(s, ·)‖n,∞ < ε, ‖v(t, ·) − v(s, ·)‖n,∞ < ε and‖F(t, ·) − F(s, ·)‖n/2,∞ < ε. It follows that

|I2|≤ C (‖u |K+ ‖ + ‖v |K+ ‖ + 1) ε

∫ t−δ0

‖∇x exp(−τA)ϕ‖n/(n−2),1 dτ≤ C (‖u |K+ ‖ + ‖v |K+ ‖ + 1) ε‖ϕ‖n/(n−1),1. (3.10)

Combining (3.7), (3.8), (3.9) and (3.10) we conclude that|t − s| < γ implies∣∣(Ψ [u, v, F, a](t) − Ψ [u, v, F, a](s), ϕ)∣∣ ≤ C‖ϕ‖n/(n−1),1 εt − δ

for everyϕ(x) ∈ Ln/(n−1),1σ (Ω). Since the dual space ofLn/(n−1),1σ (Ω) is identicalwith Ln,∞σ (Ω), this implies

‖Ψ [u, v, F, a](t) − Ψ [u, v, F, a](s)‖n,∞ ≤ C εt − δ .

Sinceε > 0 is arbitrary, this implies thatΨ [u, v, F, a](t) is continuous att > 0as anLn,∞σ -valued function. ��

Navier-Stokes equations in the weak-Ln space 655

4. Construction of mild solutions

In this section we prove the existence part of the proof of Theorem 1.1. Supposethat eitherF(t, x) ∈ L. [resp.F(t, x) ∈ L+ anda(x) ∈ Ln,∞σ (Ω).] Then weconstruct a sequence

{u(j)(t, x)

}∞j=0 =

{{u(j)

; (t, x)}n;=1

}∞j=0

inductively byu(0)(t, x) ≡ 0 andu(j+1)(t, x) = Φ [u(j), u(j), F ] (t, x). [resp.u(j+1)(t, x) = Ψ [u(j), u(j), F, a] (t, x).] Then Theorem 3.1 immediately yieldsthe following lemma.

Lemma 4.1. Wehaveu(j)(t, x) ∈ K [resp.u(j)(t, x) ∈ K+] for everyj ∈ N, andputtingAj =

∥∥u(j)∣∣ K∥∥, [resp.Aj = ∥∥u(j)∣∣ K+∥∥, ] we haveAj+1 ≤ C1‖F |L‖+C2A

2j . [resp.Aj+1 ≤ C1‖F |L+ ‖ + C3‖a‖n,∞ + C2A2j .]

Suppose that‖F |L‖ < 1/4C1C2, [resp.C1‖F |L‖ +C3‖a‖n,∞ < 1/4C2,] andlet A∞ denote the smaller root of the equation 4C2x2 − x + C1‖F |L‖ = 0;[resp. 4C2x2 − x + C1‖F |L+ ‖ + C3‖a‖n,∞ = 0;] namely,

A∞ = 1−√1− 4C1C2‖F |L‖

2C2.

[resp.A∞ = 1−

√1− 4C1C2‖F |L+ ‖ − 4C3C2‖a‖n,∞

2C2.

]

Then it is easily seen by induction onj that

Aj < A∞. (4.1)

Next, in view of Lemma 4.1, we can putBj =∥∥u(j+1) − u(j)∣∣ K∥∥. Then we have

the following lemma.

Lemma 4.2. We haveBj+1 ≤ 2BjC2A∞ for everyj ∈ N.Proof. Since

u(j+2) − u(j+1)= Φ [u(j+1), u(j+1), F ] − Φ [u(j), u(j), F ]= Φ [u(j+1), u(j+1) − u(j),0] + Φ [u(j+1) − u(j), u(j),0]

or

u(j+2) − u(j+1)= Ψ [u(j+1), u(j+1), F, a] − Ψ [u(j), u(j), F, a]= Ψ [u(j+1), u(j+1) − u(j),0,0] + Ψ [u(j+1) − u(j), u(j),0,0] ,

656 M.Yamazaki

Theorem 3.1 yields the inequality

Bj+1 =∥∥u(j+2) − u(j+1)∣∣ K∥∥

≤ C2∥∥u(j+1)∣∣ K∥∥∥∥u(j+1) − u(j)∣∣ K∥∥

+ C2∥∥u(j+1) − u(j)∣∣ K∥∥∥∥u(j)∣∣ K∥∥

≤ C2Bj(Aj+1 + Aj

)≤ 2BjC2A∞

or

Bj+1 =∥∥u(j+2) − u(j+1)∣∣ K+∥∥

≤ C2∥∥u(j+1)∣∣ K+∥∥∥∥u(j+1) − u(j)∣∣ K+∥∥

+ C2∥∥u(j+1) − u(j)∣∣ K+∥∥∥∥u(j)∣∣ K+∥∥

≤ 2BjC2A∞.This completes the proof. ��In view of Lemma 4.2, we have

∥∥u(k) − u(j)∣∣ K∥∥ ≤k−1∑;=j

B; ≤k−1∑;=j

B0 (2C2A∞); ≤ B0 (2C2A∞)j

1− 2C2A∞or

∥∥u(k) − u(j)∣∣ K+∥∥ ≤ B0 (2C2A∞)j

1− 2C2A∞for everyj , k ∈ N such thatj < k. Since

2C2A∞ = 1−√1− 4C1C2‖F |L‖ < 1

or

2C2A∞ = 1−√1− 4C1C2‖F |L‖ − 4C3C2‖a‖n,∞ < 1,

we see that∥∥u(k) − u(j)∣∣ K∥∥ → 0 or∥∥u(k) − u(j)∣∣ K+∥∥ → 0 asj , k → +∞. It

follows that, for everyt and every; = 1, . . . , n, the functionu(j); (t, ·) convergesto an element ofLn,∞σ (Ω) as j → +∞. Let u(t, ·) = {u;(t, x)}n;=1 denotethe limit. Then the formula (4.1) implies that‖u(t, ·)‖n,∞ ≤ A∞ for everyt ∈ R. It follows that u(t, x) ∈ K or u(t, x) ∈ K+ with ‖u |K‖ ≤ A∞ or‖u |K+ ‖ ≤ A∞. On the other hand, puttingu(t, x) = v(t, x) = u(;)(t, x)

Navier-Stokes equations in the weak-Ln space 657

in (3.1) or (3.3) and making use of the fact thatu(;+1) = Φ[u(;), u(;), F ] oru(;+1) = Ψ [u(;), u(;), F, a], we obtain(

u(;+1)(t, ·), ϕ)

=n∑

j,k=1

∫ +∞0

(u(;)j (t − τ, ·)u(;)k (t − τ, ·) − Fjk(t − τ, ·),

∂

∂xj

(exp(−τA)ϕ)

k

)dτ

or (u(;+1)(t, ·), ϕ) = (a,exp(−tA)ϕ) +

n∑j,k=1

∫ t0

(u(;)j (t − τ, ·)u(;)k (t − τ, ·) − Fjk(t − τ, ·),

∂

∂xj

(exp(−τA)ϕ)

k

)dτ.

Letting; → +∞, we obtain (1.1) or (1.2) for everyϕ ∈ Ln/(n−1),1σ (Ω) andt . Itfollows thatu(t, x) is the desired mild solution.

Thus the proof of the existence of mild solution is complete if we putA =1/2C2, ε = 1/4C1C2 andC0 = C3/C1.

5. Properties of mild solutions

In this section we first prove the following theorem.

Theorem 5.1.

1. For everyF(t, x) ∈ L, there exists at most one mild solutionu(t, x) ∈ K ofthe system(0.1)–(0.4)satisfying the condition‖u |K‖ < A. Furthermore, themapping fromF(t, x) ∈ L to u(t, x) ∈ K, whereu(t, x) is the mild solutionabove, is uniformly continuous on the closed ballB(0, δ) in L toK for everyδ < ε.

2. For everya(x) ∈ Ln,∞σ (Ω) and F(t, x) ∈ L+, there exists at most onemild solutionu(t, x) ∈ K+ of the system(0.5)–(0.9)satisfying the conditionC0‖a‖n,∞+‖u |K+ ‖ < A. Furthermore, themapping from

(a(x), F (t, x)

) ∈Ln,∞σ (Ω) × L+ to u(t, x) ∈ K+, whereu(t, x) is the mild solution above, isuniformly continuous on the set{(

a(x), F (t, x)) | C0‖a‖n,∞ + ‖F |L‖ ≤ δ} ⊂ Ln,∞σ (Ω) × L+

toK+ for everyδ > 0.

658 M.Yamazaki

Proof. We proveAssertion 1 only, sinceAssertion 2 can be proved exactly in thesame way.

Suppose thatu(t, x) andv(t, x) are mild solutions withF(t, x) ∈ L andG(t, x) ∈ L satisfying the estimates‖u |K‖, ‖v |K‖ ≤ A − γ for someγ > 0.Then, from the equalitiesu = Φ[u, u, F ] andv = Φ[v, v,G], we obtain

u − v = Φ[u, u, F ] − Φ[v, v,G]= Φ[u, u − v,0] + Φ[u − v, v,0] + Φ[0,0, F − G].

It follows that

‖u − v |K‖ ≤ C2 (‖u |K‖ + ‖v |K‖) ‖u − v |K‖ + C1‖F − G |L‖≤ (1− 2C2γ )‖u − v |K‖ + C1‖F − G |L‖.

This implies

‖u − v |K‖ ≤ AC1γ

‖F − G |L‖. (5.1)

In particular, ifF = G, then we haveu = v. Next, suppose that‖F |L‖ ≤ δ forsomeδ < ε. Then any mild solutionu(t, x) satisfying‖u |K‖ < A coincideswith the mild solution constructed in the previous section. Henceu(t, x) enjoysthe estimate

‖u |K‖ ≤ 1−√1− 4C1C2‖F |L‖

2C2≤ A

(1− √1− 4C1C2δ

).

Now put γ = A√4C1C2δ. Then, for everyF(t, x), G(t, x) ∈ L such that‖F |L‖, ‖G |L‖ ≤ δ, the corresponding mild solutionu(t, x), v(t, x) ∈ K suchthat‖u |K‖,‖v |K‖ < A enjoys the estimate‖u |K‖,‖v |K‖ ≤ A−γ . It followsthat (5.1) holds, and this implies the required uniform continuity. ��We next prove the following theorem, which implies the equivalence betweenmild solutions and solutions in the sense of distributions. This completes theproof of Theorem 1.1.

Theorem 5.2.

1. Suppose thatu(t, x) ∈ K. Thenu(t, x) is a mild solution of(0.1)–(0.4) ifand only if u(t, x) is a solution of(0.1)–(0.4) in the sense of distributions inR × Ω.

2. Suppose thatu(t, x) ∈ K+. Thenu(t, x) is a mild solution of(0.5)–(0.9) ifand only if u(t, x) is a solution of(0.5)–(0.9) in the sense of distributions inR+ × Ω.

Navier-Stokes equations in the weak-Ln space 659

Proof. We first introduce a notation. In the sequel we writeHjk(t, x)= uj (t, x)uk(t, x) − Fjk(t, x) for everyj , k = 1, . . . , n. Then (1.1) and (1.2)are equivalent to

(u(t, ·), ϕ) =n∑

j,k=1

∫ +∞0

(Hjk(t − τ, ·), ∂

∂xj

(exp(−τA)ϕ)

k

)dτ

and

(u(t, ·), ϕ) = (a,exp(−tA)ϕ)

+n∑

j,k=1

∫ t0

(Hjk(t − τ, ·), ∂

∂xj

(exp(−τA)ϕ)

k

)dτ

respectively.We begin with Assertion 1. We first show that a mild solutionu(t, x) enjoys

(1.5) att = t0 ∈ R for everyϕ(x) ∈ C∞0,σ (Ω). For this purpose it suffices toshow that the term(

u(t, ·) − u(s, ·)t − s , ϕ

)

= 1t − s

∫ +∞0

(Hjk(t − τ, ·) − Hjk(s − τ, ·), ∂

∂xj

(exp(−τA)ϕ)

k

)dτ

= I1 + I2,where

I1 = 1t − s

n∑j,k=1

∫ t−s0

(Hjk(τ, ·), ∂

∂xj

(exp

(−(t − τ)A)ϕ)k

)dτ

and

I2 =n∑

j,k=1

∫ +∞0

(Hjk(s − τ, ·),

∂

∂xj

(exp

(−(t − s + τ)A)ϕ)k− (exp(−τA)ϕ)

k

t − s)dτ,

converges ass → t0 − 0 andt → t0 + 0, and the limit coincides with(Hjk(t0, ·), ∂

∂xjϕk

)+ (u(t0, ·),∆ϕ) .

660 M.Yamazaki

Sinceu(τ, ·) andF(τ, ·) are continuous onR as anLn,∞(Ω)-valued functionand as anLn/2,∞(Ω)-valued function respectively and since

∂

∂xj

(exp

(−τA)ϕ)k

→ ∂∂xj

ϕk

in(Ln/(n−2),1(Ω)

)n2asτ → +0 for ϕ ∈ C∞0,σ (Ω), it follows that

I1 →(Hjk(t0, ·), ∂

∂xjϕk

). (5.2)

On the other hand, putting

ψ(x) =(exp

(−(t − s)A) − I)ϕt − s , (5.3)

we see thatPψ(x) ∈ Ln/(n−1),1σ (Ω) and henceI2 = (u(s, ·), Pψ). Lettings → t0 − 0, we see thatu(s, ·) approaches tou(t0, ·) in Ln,∞σ (Ω). On the otherhand,Pψ(x) approaches toP

(−Aϕ(x)) = ∆ϕ(x). It follows thatI2 →

(u(t0, ·),∆ϕ

). (5.4)

Now the conclusion follows immediately from (5.2) and (5.4).Conversely, suppose thatu(t, x) ∈ K is a solution of (0.1)–(0.4) in the sense

of distributions. Then we have

d

dt

(u(t, ·), ϕ) = (u(t, ·),∆ϕ) +

n∑j,k=1

(Hjk(t, ·), ∂ϕk

∂xj

)

for everyϕ(x) ∈ C∞0,σ (Ω). Now suppose thatψ(x) ∈ C∞0,σ (Ω) and t0 < t1.Then, by approximating exp

(−(t1− t0)A)ψ by functions inC∞0,σ (Ω), we obtaind

dt

(u(t, ·),exp(−(t1 − t0)A)ψ)

= (u(t, ·),∆exp(−(t1 − t0)A)ψ)

+n∑

j,k=1

(Hjk(t, ·), ∂

∂xj

(exp

(−(t1 − t0)A)ψ)k). (5.5)

Here we remark that

∆exp(−(t1 − t0)A)ψ) = −Aexp(−(t1 − t0)A)ψ)

= − ddt0

exp(−(t1 − t0)A)ψ).

Navier-Stokes equations in the weak-Ln space 661

Now putt = t0 in (5.5) and making use of the equality above, we obtaind

dt

(u(t, ·),exp(−(t1 − t)A)ψ)

=n∑

j,k=1

(Hjk(t, ·), ∂

∂xj

(exp

(−(t1 − t)A)ψ)k).

Integrating this formula on the interval[t0, t1] with respect tot , we conclude(u(t1, ·), ψ

) − (u(t0, ·),exp(−(t1 − t0)A)ψ)

=n∑

j,k=1

∫ t1−t00

(Hjk(t1 − τ, ·), ∂

∂xj

(exp

(−τA)ψ)k

). (5.6)

Letting t0 → −∞ and observing the facts supt0∈R ‖u(t0, ·)‖n,∞ < +∞ andlim t→+∞ ‖exp(−tA)ψ‖n/(n−1),1 = 0, we conclude that

(u(t1, ·), ψ

) =n∑

j,k=1

∫ +∞0

(Hjk(t1 − τ, ·), ∂

∂xj

(exp

(−τA)ψ)k

),

which implies thatu(t, x) is a mild solution of (0.1)–(0.4).We turn to the proof ofAssertion 2. Suppose thatϕ(x) ∈ C∞0,σ (Ω) andt0 > 0.

In this case we have (u(t, ·) − u(s, ·)

t − s , ϕ)

= I1 + I3,where

I3 =(a,exp(−sA)(exp(−(t − s)A) − 1)ϕ)

+n∑

j,k=1

∫ s0

(Hjk(s − τ, ·),

∂

∂xj

(exp

(−(t − s + τ)A)ϕ)k− (exp(−τA)ϕ)

k

t − s)dτ.

Puttingψ(x) as in (5.3), we see thatI3 = (u(s, ·), Pψ) also in this case. Nowwe can prove (1.5) in the same way as Assertion 1.

Next, for everyϕ(x) ∈ C∞0,σ (Ω) and everyt > 0, we have(u(t, ·), ϕ) − (a, ϕ)

= (a,exp(−tA)ϕ − ϕ)

+n∑

j,k=1

∫ t0

(Hjk(t − τ, ·), ∂

∂xj

(exp(−τA)ϕ)

k

)dτ

= I4 + I5.

662 M.Yamazaki

We can estimate these terms as follows:

|I4| ≤ ‖u‖n,∞‖exp(−tA)ϕ − ϕ‖n/(n−1),1 → 0 ast → +0and

|I5| ≤ C(‖u |K‖2 + ‖F |L‖)

∫ t0

‖∇x exp(−τA)ϕ‖n/(n−2),1dτ→ 0 ast → +0,

which follows from Corollary 2.3, 2. We thus conclude (1.6).Conversely, suppose thatu(t, x) ∈ K+ is a solution of (0.5)–(0.9) in the sense

of distributions. Then, for everyδ > 0, there exists a functionϕ(x) ∈ C∞0,σ (Ω)such that‖exp(−t1A)ψ − ϕ‖n/(n−1),1 < δ. Then we have∣∣(a,exp(−t1A)ψ) − (u(t0, ·),exp(−(t1 − t0)A)ψ)∣∣

≤ ∣∣(a,exp(−t1A) − ϕ)∣∣ + ∣∣(a − u(t0, ·), ϕ)∣∣+ ∣∣(u(t0, ·), ϕ − exp(t1A)ψ)∣∣+ ∣∣(u(t0, ·), {exp(−t1A) − exp(−(t1 − t0)A)}ψ)∣∣

≤ ‖a‖n,∞δ +∣∣(a − u(t0, ·), ϕ)∣∣ + ‖u(t0, ·)‖n,∞δ

+ ‖u(t0, ·)‖n,∞∥∥{exp(−t1A) − exp(−(t1 − t0)A)}ψ∥∥n/(n−1),1.

Sinceδ can be taken arbitrarily small and sinceu(t0, ·) → a in the sense ofdistributions, together with the fact that exp(−tA)ψ is strongly continuous inLn/(n−1),1(Ω), we see that(

a,exp(−t1A)ψ) − (u(t0, ·),exp(−(t1 − t0)A)ψ) → 0

ast0 → +0. In view of this fact, we lett0 → +0 in (5.6) to conclude that(u(t1, ·), ψ

) − (a,exp(−t1A)ψ)

=n∑

j,k=1

∫ t10

(Hjk(t1 − τ, ·), ∂

∂xj

(exp

(−τA)ψ)k

),

which implies thatu(t, x) is a mild solution of (0.5)–(0.9). ��At the end of this section we prove Corollary 1.2.

Proof (of Corollary 1.2.). Assertion 1 follows immediately from theuniqueness in Theorem 5.1 and the fact thatF(t + T ) ≡ F(t). Hence it sufficesto show Assertion 2. Suppose thatu(t, x) is the unique mild solution of (0.1)–(0.4) such that‖u |K‖ < A. Then, for everyγ > 0, we can takeδ > 0 so smallthat‖G − F |L‖ < δ implies‖v − u |K‖ < γ , wherev(t, x) is the unique mildsolution satisfying‖v |K‖ < A of (0.1)–(0.4) withF(t, x) replaced byG(t, x).

Navier-Stokes equations in the weak-Ln space 663

From the assumption we can findL > 0 such that, for everya ∈ R, we can findT ∈ [a, a+L] such that supt∈R ‖F(t + T ) − F(t)‖n/2,∞ < δ. This implies thatG(t) = F(t + T ) enjoysG ∈ L and‖G − F |L‖ < δ. Thenv(t) = u(t + T ) isthe unique mild solution satisfying‖v |K‖ = ‖u |K‖ < A of (0.1)–(0.4) withF(t) replaced byF(t + T ). It follows that

supt∈R

‖u(t + T ) − u(t)‖n,∞ = ‖v − u |K‖ < γ,

which implies the required almost periodicity of the mild solutionu(t, x). ��

6. Asymptotic behavior

In this section we fixp ∈ (n,∞), and prove Theorem 1.3. We first choose thatε(p) ≤ ε/2. Then, for everyb(x) ∈ Ln,∞σ (Ω) satisfying the assumption, wehaveC0‖b‖n,∞ + ‖F |L+ ‖ < ε, and hence there uniquely exists a mild solutionv(t, x) ∈ K+ of the problem (0.5)–(0.9) witha(x) replaced byb(x) such that‖v |K+ ‖ < A. We next introduce the function class

M+ = {u(t, x) ∈ K+ | supt>0

tn/2p−1/2‖u(t, ·)‖p,∞ < ∞}

with the norm

‖u |M+ ‖ = ‖u |K+ ‖ + supt>0

tn/2p−1/2‖u(t, ·)‖p,∞.

Then we have the following lemma.

Lemma 6.1. There exists a positive numberCp such that, for everyu(t, x) ∈ K+and v(t, x) ∈ M+, we haveΨ [u, v,0,0], Ψ [v, u,0,0] ∈ M+, and the enjoythe estimate

‖Ψ [u, v,0,0] |M+ ‖, ‖Ψ [v, u,0,0] |M+ ‖ ≤ Cp‖u |K+ ‖‖v |M+ ‖.Proof. We consider the functionΨ [u, v,0,0], since the functionΨ [v, u,0,0]can be treated exactly in the same way. First we observe that the inequality

‖Ψ [u, v,0,0] |K+ ‖ ≤ C2‖u |K+ ‖‖v |K+ ‖is already proved in Theorem 3.1, 2. Hence it suffices to show that the estimate

supt>0

tn/2p−1/2‖Ψ [u, v,0,0](t)‖p,1 ≤ Cp‖u |K+ ‖‖v |M+ ‖ (6.1)

holds with some positive constantCp ≥ C2.

664 M.Yamazaki

Suppose thatϕ(x) ∈ Lp/(p−1),1σ (Ω) enjoys‖ϕ‖p/(p−1),1 ≤ 1. Then, from thedefinition

(Ψ [u, v,0,0](t, ·), ϕ)

=∫ t0

(uj (t − τ, ·)vk(t − τ, ·), ∂

∂xk

(exp(−τA)ϕ)

k

)dτ,

we obtain

∣∣(Ψ [u, v,0,0](t, ·), ϕ)∣∣≤

∫ t0

∣∣∣∣(uj (t − τ, ·)vk(t − τ, ·), ∂

∂xk

(exp(−τA)ϕ)

k

)∣∣∣∣ dτ = I1 + I2, (6.2)where

I1 =∫ t/20

∣∣∣∣(uj (t − τ, ·)vk(t − τ, ·), ∂

∂xk

(exp(−τA)ϕ)

k

)∣∣∣∣ dτ

and

I2 =∫ tt/2

∣∣∣∣(uj (t − τ, ·)vk(t − τ, ·), ∂

∂xk

(exp(−τA)ϕ)

k

)∣∣∣∣ dτ.On the other hand, the inequality

1> 1− 1p> 1− 1

p− 1

n> 1− 2

n≥ 1

n

implies

1<p

p − 1 <np

np − n − p <n

n − 2 ≤ n.

Hence we can apply Corollary 2.3, 2 to obtain the estimates

I1 ≤ C supτ>0

‖u(τ, ·)‖n,∞ supτ>t/2

‖v(τ, ·)‖p,∞

×∫ t/20

∥∥∥∥ ∂∂xk(exp(−τA)ϕ)

k

∥∥∥∥np/(np−n−p),1

dτ

≤ C‖u |K+ ‖(t

2

)n/2p−1/2‖v |M+ ‖‖ϕ‖p/(p−1),1, (6.3)

Navier-Stokes equations in the weak-Ln space 665

and

I2 ≤ C supτ>0

‖u(τ, ·)‖n,∞ supτ>0

‖u(τ, ·)‖n,∞

×∫ tt/2

∥∥∥∥ ∂∂xk(exp(−τA)ϕ)

k

∥∥∥∥n/(n−2),1

dτ

≤ C‖u |K+ ‖‖v |M+ ‖ (6.4)×

∫ +∞t/2

τn/2{−(p−1)/p+(n−2)/n}−1/2dτ‖ϕ‖p/(p−1),1≤ C‖u |K+ ‖‖v |M+ ‖tn/2p−1/2‖ϕ‖p/(p−1),1. (6.5)

Now taking the supremum with respect toϕ(x) in (6.2) and making use of (6.3)and (6.5), we conclude (6.1). ��Proof (of Theorem1.3.). Since the estimate (1.8) follows from the factu(t, x),v(t, x) ∈ K+ and the estimate (1.7) by real interpolation, it suffices to prove(1.7). We return to the iteration scheme. In Sect.3 we constructed the mild so-lutionsu(t, x) andv(t, x) as the limit of the sequence

{u(j)(t, x)

}∞j=1 and that

of{v(j)(t, x)

}∞j=1 respectively, whereu

(0)(t, x) ≡ v(0)(t, x) ≡ 0, u(j+1(t, x) =Ψ

[u(j), u(j), F, a

](t, x) andv(j+1(t, x) = Ψ [v(j), v(j), F, b ] (t, x) for j =

0,1, . . . . Hence, puttingw(j)(t, x) = v(j)(t, x)−u(j)(t, x), wehavew(0)(t, x) ≡0 and

w(j+1)(t, x) = Ψ [u(j), w(j),0,0](t, x)+ Ψ [w(j), v(j),0,0](t, x) + Ψ [0,0,0, b − a](t, x) (6.6)

for j = 0,1, . . . . Corollary 2.3, 1 implies thatw(1)(t, x) ∈ M+ and that theestimate

∥∥w(1)∣∣ M+∥∥ ≤ C1(p)‖b − a‖n,∞ holds with some constantC1(p). Onthe other hand, applying Lemma 6.1 to (6.6), we obtain the estimate

∥∥w(j+1)∣∣ M+∥∥≤ ∥∥w(1)∣∣ M+∥∥ + Cp (∥∥u(j)∣∣ K+∥∥ + ∥∥v(j)∣∣ K+∥∥) ∥∥w(j)∣∣ M+∥∥. (6.7)

From the inequalitiesC0‖a‖n,∞+‖F |L+ ‖ < ε(p) andC0‖b‖n,∞+‖F |L+ ‖ <2ε(p), we deduce that the inequality

∥∥u(j)∣∣ K+∥∥, ∥∥v(j)∣∣ K+∥∥ ≤ 14Cp

holds if we chooseε(p) > 0 so small. Substituting this inequality into (6.7) wecan deduce ∥∥w(j)∣∣ M+∥∥ < 2∥∥w(1)∣∣ M+∥∥ ≤ 2C1(p)‖b − a‖n,∞

666 M.Yamazaki

by induction onj .Sincew(j)(t, x) = v(j)(t, x)−u(j)(t, x) converges tov(t, x)−u(t, x) inK+

asj → ∞,wesee thatv(t, x)−u(t, x) ∈ M+with theestimate‖v − u |M+ ‖ ≤2Cp, which yields (1.7). ��

7. Regularity

In this section we prove Theorem 1.4, Theorem 1.5 and Corollary 1.6. For tech-nical reasons, the proof of Theorem 1.5 is divided into two subsections. In theformer we prove the boundedness of∇u(t, ·) and in the latter the H¨older conti-nuity.

7.1. Proof of Theorem1.4

We first prove Assertion 1. It suffices to show the estimate∣∣(u(t, ·) − u(s, ·), ϕ)∣∣ ≤ C(t − s)n/2p−1/2‖ϕ‖p/(p−1),∞ (7.1)for ϕ ∈ C∞0,σ (Ω). We have∣∣(u(t, ·) − u(s, ·), ϕ)∣∣

≤∣∣∣∣∣∣∫ t−s0

n∑j,k=1

(Hjk(t − τ, ·), ∂

∂xj

(exp(−τA)ϕ)

k

)dτ

∣∣∣∣∣∣+

∣∣∣∣∫ +∞0

n∑j,k=1

(Hjk(s − τ, ·),

∂

∂xj

(exp(−(t − s + τ)A)ϕ)

k− ∂

∂xj

(exp(−τA)ϕ)

k

)dτ

∣∣∣∣= I1 + I2, (7.2)

where the functionsHjk(t, ·) are introduced in the proof of Theorem 5.2.Sincen ≥ 3, it follows thatp/(p − 1) < n/(n − 2) ≤ n. Hence Corol-

lary 2.3, 2 implies that the termI1 can be estimated as follows:

I1 ≤ C supτ∈R

(‖u(τ, ·)‖n,∞2 + ‖F(τ, ·)‖n/2,∞)∫ t−s0

‖∇x exp(−τA)ϕ‖n/(n−2),1dτ

≤ C∫ t−s0

τn/2p−1−1/2‖ϕ‖p/(p−1),∞dτ≤ C(t − s)n/2p−1/2‖ϕ‖p/(p−1),∞. (7.3)

Navier-Stokes equations in the weak-Ln space 667

On the other hand, by using Corollary 2.3, 2 again, we obtain

I2

≤ C supτ∈R

(‖u(τ, ·)‖n,∞2 + ‖F(τ, ·)‖n/2,∞)∫ +∞0

‖∇x exp(−(t − s + τ)A)ϕ − ∇x exp(−τA)ϕ‖n/(n−2),∞dτ

≤ C∫ t−s0

∥∥∇x exp(−τA)(exp(−(t − s)A)ϕ − ϕ)∥∥n/(n−2),∞dτ+ C

∫ +∞t−s

∥∥∥∥∇x exp(−τ2A

) (exp(−(t − s)A) − 1)

exp(−τ2A

)ϕ

∥∥∥∥n/(n−2),∞

dτ

≤ C∫ t−s0

τn/2p−1−1/2∥∥exp(−(t − s)A)ϕ − ϕ∥∥

p/(p−1),∞dτ

+ C∫ +∞t−s

τ n/2p−1−1/2

∥∥∥∥∫ t−s0

Aexp(−

(θ + τ

2

)A

)ϕ dθ

∥∥∥∥p/(p−1),∞

dτ

≤ C(t − s)n/2p−1/2‖ϕ‖p/(p−1),∞+ C

∫ +∞t−s

τ n/2p−1−1/2(t − s)∥∥∥Aexp(−τ

2A

)ϕ

∥∥∥p/(p−1),∞

dτ

≤ C(t − s)n/2p−1/2‖ϕ‖p/(p−1),∞+ C(t − s)

∫ +∞t−s

τ n/2p−2−1/2‖ϕ‖p/(p−1),∞dτ≤ C(t − s)n/2p−1/2‖ϕ‖p/(p−1),∞. (7.4)

Substituting (7.3) and (7.4) into (7.2), we obtain Assertion 1.We turn to Assertion 2. It suffices to show the estimate∣∣(u(t, ·) − u(s, ·), ϕ)∣∣ ≤ C(t − s)n/2p−1/2‖ϕ‖p/(p−1),∞

668 M.Yamazaki

for ϕ ∈ C∞0,σ (Ω). We have

∣∣(u(t, ·) − u(s, ·), ϕ)∣∣

≤∣∣∣∣∣∣∫ t−s0

n∑j,k=1

(Hjk(t − τ, ·), ∂

∂xj

(exp(−τA)ϕ)

k

)dτ

∣∣∣∣∣∣+

∣∣∣∣∫ s0

n∑j,k=1

(Hjk(s − τ, ·),

∂

∂xj

(exp(−(t − s + τ)A)ϕ)

k− ∂

∂xj

(exp(−τA)ϕ)

k

)dτ

∣∣∣∣+ ∣∣(a, {exp(−(t − s)A) − 1}exp(−sA)ϕ)∣∣

= I1 + I2 + I3.

The termsI1 andI2 are estimated exactly in the same way as in the proof ofAssertion 1. The termI3 can be estimated as

I3 =∣∣({exp(−(t − s)A) − 1} a,exp(−sA)ϕ)∣∣

=∣∣∣∣∫ t−s0

(Aexp(−τA)a,exp(−sA)ϕ) dτ∣∣∣∣

≤ C∫ t−s0

‖Aexp(−τA)a‖p,∞‖exp(−sA)ϕ‖p/(p−1),1

≤ C∫ t−s0

τ−1+(n/2p−1/2)∥∥An/2p−1/2a∥∥

p,∞‖ϕ‖p/(p−1),1= C(t − s)n/2p−1/2∥∥An/2p−1/2a∥∥

p,∞‖ϕ‖p/(p−1),1.

This estimate completes the proof of Assertion 2. ��

7.2. Proof of Theorem1.5: The boundedness

For technical reasons. we first show the boundedness of∇u first.We first prove Assertion 1. Sincen/2 < n, it suffices to show∥∥A1/2u(t, ·)∥∥

n/2,∞ ≤ C in view of Theorem 2.1, 2. Letϕ be an element ofC∞0,σ (Ω), and suppose thatu(t, x) is a mild solution of (0.1)–(0.4). Then we

Navier-Stokes equations in the weak-Ln space 669

have

∣∣(A1/2u(t, ·), ϕ)∣∣≤

n∑j,k=1

∣∣∣∣∫ +∞0

(Hjk(t − τ, ·), ∂

∂xj

(exp(−τA)A1/2ϕ)

k

)dτ

∣∣∣∣

≤n∑

j,k=1

∣∣∣∣∫ +∞0

(Hjk(t − τ, ·) − Hjk(t, ·), ∂

∂xj

(exp(−τA)A1/2ϕ)

k

)dτ

∣∣∣∣

+n∑

j,k=1

∣∣∣∣(Hjk(t, ·),

∫ +∞0

∂

∂xj

(exp(−τA)A1/2ϕ)

kdτ

)∣∣∣∣= I1 + I2. (7.5)

Now putp = nq/(n − q). Then we haven/2 < p < n. Hence Proposition 1.4implies that

∥∥uj (t − τ, ·)uk(t − τ.·) − uj (t, ·)uk(t, ·)∥∥q,∞≤ ∥∥uj (t − τ, ·)∥∥n,∞‖uk(t − τ, ·) − uk(t − τ, ·)‖p,∞

+ ∥∥uj (t − τ, ·) − uj (t − τ, ·)∥∥p,∞‖uk(t, ·)‖n,∞≤ Cτn/2p−1/2 = Cτn/2q−1. (7.6)

From this fact and (1.10) we have

∥∥Hjk(t − τ, ·) − Hjk(t, ·)∥∥q,∞ ≤ τn/2q−1. (7.7)for j , k = 1, . . . , n. From this estimate and the inequalitiesn/(n−2) < q/(q−1) < n/(n−3) ≤ n in the case whereΩ is an exterior domain, we can dominatethe termI1 from above by

n∑j,k=1

∫ +∞0

∥∥Hjk(t − τ, ·) − Hjk(t, ·)∥∥q,∞× ∥∥∇ exp(−τA)A1/2ϕ∥∥

q/(q−1),1dτ

≤ C∫ +∞0

τn/2q−1∥∥∇A1/2 exp(−τA)ϕ∥∥

q/(q−1),1dτ

≤ C‖ϕ‖n/(n−2),1 (7.8)

by using Corollary 2.3, 2.

670 M.Yamazaki

On the other hand, sincen/(n − 2) ≤ n, the termI2 can be dominated fromabove by

C(‖F |L‖ + ‖u‖K2)

∥∥∥∥∇A1/2∫ +∞0

exp(−τA)ϕdτ∥∥∥∥n/(n−2),1

.

Applying Corollary 2.4 we see that the right-hand side of this formula is domi-nated from above byC‖ϕ‖n/(n−2),1. Now we can easily see from (7.5), (7.8) andthe fact above that the family{∇u(t, ·)}t∈R is bounded inLn/2,∞(Ω).

We next prove Assertion 2. Letϕ be an element ofC∞0,σ (Ω) such that‖ϕ‖n/(n−2),1 ≤ 1, and suppose thatu(t, x) is a mild solution of (0.5)–(0.9).Then we have

∣∣(A1/2u(t, ·), ϕ)∣∣≤

n∑j,k=1

∣∣∣∣∫ t0

(Hjk(t − τ, ·), ∂

∂xj

(exp(−τA)A1/2ϕ)

k

)dτ

∣∣∣∣

≤n∑

j,k=1

∣∣∣∣∫ t0

(Hjk(t − τ, ·) − Hjk(t, ·), ∂

∂xj

(exp(−τA)A1/2ϕ)

k

)dτ

∣∣∣∣

+n∑

j,k=1

∣∣∣∣(Hjk(t, ·),

∫ +∞0

∂

∂xj

(exp(−τA)A1/2ϕ)

kdτ

)∣∣∣∣+ ∣∣(a,exp(−tA)A1/2ϕ)∣∣

= I1 + I2 + I3.

The termsI1 andI2 can be dominated in the same way as in the proof of Asser-tion 1, andI3 can be dominated from above by

∣∣(A1/2a,exp(−tA)ϕ)∣∣ ≤ C∥∥A1/2a∥∥n/2,∞‖exp(−tA)ϕ‖n/(n−2),1

≤ C‖∇a‖n/2,∞‖ϕ‖n/(n−2),1by Theorem 2.1, 1. This completes the proof. ��

7.3. Proof of Theorem1.5: The Hölder continuity

In this subsection we prove the H¨older continuity of∇u(t, ·). We first proveAssertion 1. In view of Theorem 2.1, 3, it suffices to show the estimate

∣∣(A1/2u(t, ·) − A1/2u(s, ·), ϕ)∣∣ ≤ C(t − s)n/2q−1‖ϕ‖q/(q−1),1 (7.9)

Navier-Stokes equations in the weak-Ln space 671

for ϕ ∈ Lq/(q−1),1σ (Ω). We have(A1/2u(t, ·), ϕ)

=n∑

j,k=1

(∫ t−s0

(Hjk(t − τ, ·) − Hjk(t, ·), ∂

∂xj

(exp(−τA)A1/2ϕ)

k

)dτ

+(Hjk(t, ·),

∫ t−s0

∂

∂xj

(exp(−τA)A1/2ϕ)

kdτ

)

+∫ +∞t−s

(Hjk(t − τ, ·) − Hjk(s, ·), ∂

∂xj

(exp(−τA)A1/2ϕ)

k

)dτ

+(Hjk(s, ·),

∫ +∞t−s

∂

∂xj

(exp(−τA)A1/2ϕ)

kdτ

)).

It follows that

∣∣(A1/2u(t, ·) − A1/2u(s, ·), ϕ)∣∣≤

∣∣∣∣∫ t−s0

n∑j,k=1

(Hjk(t − τ, ·) − Hjk(t, ·), ∂

∂xj

(A1/2 exp(−τA)ϕ)

k

)dτ

∣∣∣∣

+∣∣∣∣

n∑j,k=1

(Hjk(t, ·) − Hjk(s, ·),

∫ t−s0

∂

∂xj

(A1/2 exp(−τA)ϕ)

kdτ

)∣∣∣∣

+∣∣∣∣∫ +∞0

n∑j,k=1

(Hjk(s − τ, ·) − Hjk(s, ·),

∂

∂xj

(A1/2 exp(−(t − s + τ)A)ϕ)

k

− ∂∂xj

(A1/2 exp(−τA)ϕ)

k

)dτ

∣∣∣∣= I1 + I2 + I3. (7.10)

If Ω is an exterior domain, we haveq/(q − 1) < n/(n− 3) ≤ n. Hence, we cansubstitute (7.7) and apply Corollary 2.3, 2 to estimate the termI1 as follows:

I1 ≤ C∫ t−s0

τn/2q−1∥∥∇A1/2 exp(−τA)ϕ∥∥

q/(q−1),1dτ

≤ C∫ t−s0

τn/2q−2‖ϕ‖q/(q−1),1dτ≤ C(t − s)n/2q−1‖ϕ‖q/(q−1),1. (7.11)

672 M.Yamazaki

Next, substituting (7.7) and applying Corollary 2.4, we can estimate the termI2as follows:

I2 ≤ C(t − s)n/2q−1∥∥∥∥∫ t−s0

∇A1/2 exp(−τA)ϕdτ∥∥∥∥q/(q−1),1

≤ C(t − s)n/2q−1‖ϕ‖q/(q−1),1. (7.12)

Finally, in the same way as in (7.11), the termI3 is estimated as follows:

I3

≤ C∫ +∞0

τn/2q−1

∥∥∇A1/2 (exp(−(t − s + τ)A) − exp(−τA))ϕ∥∥q/(q−1),1dτ

≤ C∫ t−s0

τn/2q−1(∥∥∇A1/2 exp(−(t − s + τ)A)ϕ∥∥

q/(q−1),1

+ ∥∥∇A1/2 exp(−τA)ϕ∥∥q/(q−1),1

)dτ

+ C∫ +∞t−s

τ n/2q−1∥∥∥∥∇A1/2 exp

(−τ2A

)(exp

(−

(t − s + τ

2

)A

)− exp

(−τ2A

))ϕ

∥∥∥∥q/(q−1),1

dτ

≤ C∫ t−s0

τn/2q−1(1

τ+ 1

t − s + τ)

‖ϕ‖q/(q−1),1dτ

+ C∫ +∞t−s

τ n/2q−1(τ2

)−1∫ t−s0

∥∥∥Aexp(− (θ + τ2

)A

)ϕ

∥∥∥q/(q−1),1

dθ dτ

≤ C(t − s)n/2q−1‖ϕ‖q/(q−1),1+ C(t − s)

∫ +∞t−s

τ n/2q−2τ−1‖ϕ‖q/(q−1),1dτ≤ C(t − s)n/2q−1‖ϕ‖q/(q−1),1. (7.13)

Combining (7.10), (7.11), (7.12) and (7.13), we obtain (7.9). This completes theproof of Assertion 1.

Navier-Stokes equations in the weak-Ln space 673

We turn to Assertion 2. In the same way as (7.10), we have∣∣(A1/2u(t, ·) − A1/2u(s, ·), ϕ)∣∣≤

∣∣∣∣∫ t−s0

n∑j,k=1

(Hjk(t − τ, ·) − Hjk(t, ·), ∂

∂xj

(A1/2 exp(−τA)ϕ)

k

)dτ

∣∣∣∣

+∣∣∣∣

n∑j,k=1

(Hjk(t, ·) − Hjk(s, ·),

∫ t−s0

∂

∂xj

(A1/2 exp(−τA)ϕ)

kdτ

)∣∣∣∣

+∣∣∣∣∫ s0

n∑j,k=1

(Hjk(s − τ, ·) − Hjk(s, ·),

∂

∂xj

(A1/2 exp(−(t − s + τ)A)ϕ)

k

− ∂∂xj

(A1/2 exp(−τA)ϕ)

k

)dτ

∣∣∣∣+ ∣∣(a,A1/2 exp(−tA)ϕ − A1/2 exp(−sA)ϕ)∣∣

= I1 + I2 + I3 + I4. (7.14)The termsI1, I2 and I3 can be estimated in the same way as in the proof ofAssertion 1.I4 is estimated as follows:

I4 =∣∣({exp(−(t − s)A) − 1}A1/2a,exp(−sA)ϕ)∣∣

=∣∣∣∣∫ t−s0

(A3/2 exp(−τA)a,exp(−sA)ϕ) dτ

∣∣∣∣≤ C

∫ t−s0

∥∥A3/2 exp(−τA)a∥∥q,∞dτ‖exp(−sA)ϕ‖q/(q−1),1

≤ C∫ t−s0

τ−3/2+(n/2q−1/2)∥∥An/2q−1/2u∥∥

q,∞‖ϕ‖q/(q−1),1≤ C(t − s)n/2q−1∥∥An/2q−1/2u∥∥

q,∞‖ϕ‖q/(q−1),1.Substituting this estimate into (7.14) we obtain the desired estimate. ��

7.4. Proof of Corollary1.6

We first considerAssertion 1. Putq = nr/(n−r). Thenwe haven/3< q < n/2,and the Biot-Savard law implies that there exists a functionF(t, x) ∈ L enjoying(1.10). Hence we can apply Theorem 1.11, 1. It follows that

(u(t, ·) · ∇)u(t, ·)

is uniformly continuous with respect tot and enjoys the estimate∥∥(u(t, ·) · ∇)u(t, ·) − (u(s, ·) · ∇)u(s, ·)∥∥r,∞ ≤ (t − s)n/2r−3/2

674 M.Yamazaki

for everys, t such thats < t . It follows from this fact and (1.12) that, if we put

v(t, x) =∫ tt0

exp(−(t − τ)A) [−P(u(τ, ·) · ∇)u(τ, ·) + f (τ, ·)] dτ,

we see that the equality

∂v

∂t(t, x) = −Av(t, x) − P(u(t, x) · ∇)u(t, x) + f (t, x)

holds on(t0,+∞). Hence the differencew(t, x) = u(t, x) − v(t, x) enjoys thelinear Stokes equation on(t0,+∞). This implies thatu(t, x) solves (1.9) on(t0,+∞). Sincet0 ∈ R is arbitrary, this implies thatu(t, x) solves (1.9) onR.

Assertion 2 can be proved exactly in the same way. ��

Acknowledgements.The author expresses his sincere gratitude to the referee for valuable sugges-tions.

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