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A Blow-Up Criterion for the Nonhomogeneous Incompressible Navier--Stokes Equations

SIAM J.M ATH.A NAL.c 2006Society for Industrial and Applied Mathematics Vol.37,No.5,pp.1417–1434

A BLOW-UP CRITERION FOR THE NONHOMOGENEOUS

INCOMPRESSIBLE NA VIER–STOKES EQUATIONS?

HYUNSEOK KIM?

Abstract.Let(ρ,u)be a strong or smooth solution of the nonhomogeneous incompressible Navier–Stokes equations in(0,T?)×Ω,where T?is a?nite positive time andΩis a bounded domain in R3with smooth boundary or the whole space R3.We show that if(ρ,u)blows up at T?,then

T?0|u(t)|s

L r w(Ω)

dt=∞for any(r,s)with2

s

+3

r

=1and3

we obtain a regularity theorem and a global existence theorem for strong solutions.

Key words.blow-up criterion,nonhomogeneous incompressible Navier–Stokes equations AMS subject classi?cations.35Q30,76D05

DOI.10.1137/S0036141004442197

1.Introduction.The motion of a nonhomogeneous incompressible viscous?uid in a domainΩof R3is governed by the Navier–Stokes equations

(ρu)t+div(ρu?u)?Δu+?p=ρf,

(1.1)

ρt+div(ρu)=0in(0,∞)×Ω,

(1.2)

div u=0

(1.3)

and the initial and boundary conditions

(ρ,ρu)|t=0=(ρ0,ρ0u0)inΩ,u=0on(0,T)×?Ω,

(1.4)

ρ(t,x)→0,u(t,x)→0as|x|→∞,(t,x)∈(0,T)×Ω.

Here we denote byρ,u,and p the unknown density,velocity,and pressure?elds of the?uid,respectively.f is a given external force driving the motion.Ωis either a bounded domain in R3with smooth boundary or the whole space R3.Throughout this paper,we adopt the following simpli?ed notation for standard homogeneous and inhomogeneous Sobolev spaces:

L r=L r(Ω),D k,r={v∈L1loc(Ω):|v|D k,r<∞};

H k,r=L r∩D k,r,D k=D k,2,H k=H k,2;

D10={v∈L6:|v|D1

<∞and v=0on?Ω};

H10=L2∩D10,,D10,σ={v∈D10:div v=0inΩ};

|v|D k,r=|?k v|L r and|v|D1

0=|v|D1

0,σ

=|?v|L2.

Note that the space D10is the completion of C∞c(Ω)in D1,and thus there holds the following Sobolev inequality:

|v|L6≤2√

3|v|D1

for all v∈D10.

(1.5)

?Received by the editors March17,2004;accepted for publication(in revised form)May9,2005; published electronically January10,2006.This work was supported by the Japan Society for the Promotion of Science under the JSPS Postdoctoral Fellowship for Foreign Researchers.

http://www.doczj.com/doc/5461a11e5f0e7cd18425367f.html/journals/sima/37-5/44219.html

?School of Mathematics,Korea Institute for Advanced Study207-43Cheongnyangni2-dong, Dongdaemun-gu,Seoul,130-722,Korea(khs319@postech.ac.kr,khs319@kias.re.kr).

1417

1418

HYUNSEOK KIM

For a proof of (1.5),see sections II.5and II.6in the book by Galdi [11].

The global existence of weak solutions has been established by Antontsev and Kazhikov [1],Fernandez-Cara and Guillen [10],Kazhikov [13],Lions [21],and Simon [26,27].From these results (see [21]in particular),it follows that for any data (ρ0,u 0,f )with the regularity

0≤ρ0∈L 3

2∩L ∞,

u 0∈L 6,

and f ∈L 2(0,∞;L 2),

there exists at least one weak solution (ρ,u )to the initial boundary value problem (1.1)–(1.4)satisfying the regularity ρ∈L ∞(0,∞;L 3

2∩L ∞),

ρu ∈L ∞(0,∞;L 2),

and u ∈L 2(0,∞;D 1

0,σ)

(1.6)as well as the natural energy inequality.Then an associated pressure p is determined

as a distribution in (0,∞)×Ω.

But the global existence of strong or smooth solutions is still an open problem and only local existence results have been obtained for su?ciently regular data satisfying some compatibility conditions.For details,we refer to the papers by Choe and the author [6],Kim [15],Ladyzhenskaya and Solonnikov [19],Okamoto [22],Padula [23],and Salvi [24].In particular,it is shown in [6](see also [7])that if the data ρ0,u 0,and f satisfy 0≤ρ0∈L 3

2∩H 2,

u 0∈D 10,σ∩D 2

,

?Δu 0+?p 0=ρ1

2

0g,

(1.7)

f ∈L 2(0,∞;H 1),

and f t ∈L 2(0,∞;L 2)

for some (p 0,g )∈D 1×L 2,then there exist a positive time T and a unique strong

solution (ρ,u )to the problem (1.1)–(1.4)such that ρ∈C ([0,T ];L 3

2∩H 2),

u ∈C ([0,T ];D 10,σ∩D 2)∩L 2(0,T ;D 3

),

(1.8)

u t ∈L 2(0,T ;D 1

0,σ),

and √ρu t ∈L ∞(0,T ;L 2).Moreover,the existence of a pressure p in C ([0,T ];D 1)∩L 2(0,T ;D 2)can be deduced

from (1.1)–(1.3).See [5]for a detailed argument.

Let (ρ,u )be a global weak solution to the problem (1.1)–(1.4)with the data (ρ0,u 0,f )satisfying condition (1.7).Then from the above local existence result and weak-strong uniqueness results in [6]and [21],we conclude that the solution (ρ,u )satis?es the regularity (1.8)for some positive time T .One fundamental problem in mathematical ?uid mechanics is to determine whether or not (ρ,u )satis?es (1.8)for all time T .As an equivalent formulation,we may ask the following.

Fundamental question 1.1.Does the solution (ρ,u )blow up at some ?nite time T ??Such a time T ?,if it exists,is called the ?nite blow-up time of the solution (ρ,u )in the class H 2.

In spite of great e?orts since the pioneering works by Leray [20]in 1930s,there have been no de?nite answers to the fundamental question even for the case of the homogenous Navier–Stokes equations with only some blow-up criteria available.The ?rst criterion is due to Leray [20]who proved,among other things,that if T ?is the ?nite blow-up time of a strong solution u to the Cauchy problem for the homogeneous Navier–Stokes equations,then for each r with 30such that |u (t )|L r ≥C (T ??t )?1

2(1?3

r )

for all near t

(1.9)

NONHOMOGENEOUS NAVIER–STOKES EQUATIONS

1419

This estimate near the blow-up time was extended by Giga [12]to the case of bounded domains.An immediate consequence of (1.9)is the following well-known blow-up criterion in terms of the so-called Serrin class (see [9,25,29]):

T ?

0|u (t )|s L r dt =∞for any (r,s )with 2s +3

r =1,3

T ?

|?u (t )|4L 2dt =∞.

(1.11)Further generalizations of (1.10)and (1.11)have been obtained by Beirao da Veiga [2],Berselli [3],Chae and Choe [4],and Kozono and Taniuchi [17].

The major purpose of this paper is to prove the blow-up criterion (1.10)for strong solutions of the nonhomogeneous Navier–Stokes equations (1.1)–(1.3).In fact,we establish a more general result.To state our main result precisely,we ?rst introduce the notion of the blow-up time of solutions in the class H 2m with m ≥1.Let (ρ,u )be a strong solution to the initial boundary value problem (1.1)–(1.4)with the regularity

ρ∈C ([0,T ];L 3

2∩H 2m ),

u ∈C ([0,T ];D 10,σ∩D

2m

)∩L 2(0,T ;D 2m +1),?j t u ∈C ([0,T ];D 10,σ∩D

2m ?2j

)∩L 2(0,T ;D 2m ?2j +1)for 1≤j

?m t u ∈L 2(0,T ;D 1

0,σ

),and √ρ?m t u ∈L ∞(0,T ;L 2)for any T

Φm (T )=1+sup 0≤t ≤T

|?ρ(t )|H 2m ?1+|u (t )|D 10

∩D 2m + T

|u (t )|2D 2m +1dt

+sup

1≤j

sup 0≤t ≤T

|?j

t u (t )|D 10

∩D 2m ?2j +

T

|?j

t u (t )|2D 2m ?2j +1dt

(1.13)+ess sup 0≤t ≤T

|√ρ?m

t

u (t )|L 2+

T

|?m

t u (t )|2D 10

dt

for any T

|·|X ∩Y =|·|X +|·|Y

for (semi-)normed spaces X,Y.

Definition 1.2.A ?nite positive number T ?is called the ?nite blow-up time of

the solution (ρ,u )in the class H 2m ,provided that

Φm (T )<∞

for

0

and

lim T →T ?

Φm (T )=∞.

We are now ready to state the main result of this paper.Theorem 1.3.For a given integer m ≥1,we assume that

?m

t

f ∈L 2(0,∞;L 2)and

?j

t f ∈L 2(0,∞;H 2m ?2j ?1)

for 0≤j

Let (ρ,u )be a strong solution of the nonhomogeneous Navier–Stokes equations (1.1)–

(1.3)satisfying the regularity (1.12)for any T

T ?

|u (t )|s L r w

dt =∞for any (r,s )with 2s +3

r =1,3

1420

HYUNSEOK KIM

Here L r w denotes the weak L r

-space,that is,the space consisting of all vector ?elds v ∈L 1loc (Ω)such that |v |L r w

=sup α>0α|{x ∈Ω:|v (x )|>α}|1

r <∞for 3

=|v |L ∞<∞.In the case when 3

is a proper subspace of L r w (|x |?3/r ∈L r w (R 3

)for instance)and so Theorem 1.3is in fact a generalization

of the blow-up criterion (1.10)due to Leray and Giga even for the homogeneous Navier–Stokes equations.

Theorem 1.3is proved in the next two sections.In section 2,we provide a proof of the theorem for the very special case m =1.Our method of the proof is quite well known in the case of the homogeneous Navier–Stokes equations and was also applied in an earlier paper [6]by Choe and the author to the nonhomogeneous case:combining classical regularity results on the Stokes equations with H¨o lder and Sobolev inequal-ities,we show that Φ1(T )is bounded in a double exponential way by T

0|u (t )|s L r w

dt for any T less than the blow-up time T ?

.But the use of weak Lebesgue spaces in space variables makes it more di?cult to estimate the nonlinear convection term.To overcome this technical di?culty,we utilize some basic theory of the Lorenz spaces developed in [18]and [30].See the derivations of (2.6)and (2.7)for details.Concern-ing the proof for the general case m ≥2,the basic idea is also to show that Φm (T )is

bounded in some speci?c way by T

0|u (t )|s L r w

dt for any T

Some corollaries of Theorem 1.3can be deduced from a local existence result on strong solutions in the class H 2m .For instance,as an immediate consequence of Theorem 1.3,the local existence result in the class H 2,and the weak-strong uniqueness result,we obtain the following regularity result whose obvious proof may be omitted.

Corollary 1.4.Let (ρ,u )be a global weak solution to the initial boundary value problem (1.1)–(1.4)with the data satisfying condition (1.7).If there exists a ?nite positive time T ?such that

u ∈L s (0,T ?;L r w )

for some (r,s )with

2s +3

r

=1,3

(1.15)

then the solution (ρ,u )satis?es regularity (1.8)for some T >T ?.

A similar result was obtained by Choe and the author [6]assuming,however,a

stronger condition on u ,that is,u ∈L 4(0,T ?;D 1

0).By virtue of Corollary 1.4,we may conclude that the class (which we call a weak Serrin class )in (1.15)is a regularity class for weak solutions of the nonhomogeneous Navier–Stokes equations,which was already proved by Sohr [28]for the homogeneous case.See also a local version of Sohr’s result in [14].Moreover,thanks to a recent result by Dubois [8],the weak Serrin class is a uniqueness class for the homogeneous Navier–Stokes equations.It is also noticed that the same results can be easily derived from regularity and uniqueness results due to Kozono by adapting the arguments in the remarks of Theorem 3in [16].

Theorem 1.3and its proof can be used to obtain a global existence result on solutions in the class H 2under some smallness condition on u 0and f (but not on ρ0).

Theorem 1.5.For each K >1,there exists a small constant ε>0,depending only on K and the domain Ω,with the following property:if the data ρ0,u 0,and f satisfy

|ρ0|L 32

∩L

∞≤K,|u 0|D 10≤ε,and ∞

|f (t )|2L 2dt ≤ε2

(1.16)

NONHOMOGENEOUS NAVIER–STOKES EQUATIONS

1421

in addition to condition (1.7),then there exists a unique global strong solution (ρ,u )to problem (1.1)–(1.4)satisfying regularity (1.8)for any T <∞.

A rather simple proof of Theorem 1.5is provided in section 4.Finally,we recall that in the case when ρ0is bounded away from zero and Ωis a bounded domain in R 3with smooth boundary,Salvi [24]proved the local existence of strong solutions in the class H 2m for every m ≥1.Hence adapting the proofs of Corollary 1.4and Theorem 1.5,we can also prove analogous regularity and global existence results on strong solutions in every class H 2m provided that Ω??R 3and ρ0>0on Ω.2.Proof of Theorem 1.3with m =1.In this section,we prove Theorem 1.3for the special case m =1.Let t 0be a ?xed time with 0

Φ0(T )= T

|u (t )|s L r w

dt for t 0≤T

only to show that Φ1(T )≤C exp (C exp (C Φ0(T )))

for t 0≤T

(2.1)

Throughout this paper,we denote by C a generic positive constant depending only

on r ,m ,Φm (t 0),T ?,Ω,|ρ(0)|L 3

2∩L ∞

,|u (0)|L 6,and the norm of f ,but independent of T in particular.To begin with,we recall from (1.6)that sup

0≤t ≤T

|ρ(t )|

L 3

2∩L ∞

+|√

ρu (t )|L 2 +

T

|u (t )|2D 10

dt ≤C

(2.2)

for t 0≤T

2.1.Estimates for T 0|√

ρu t (t )|2L 2dt and sup 0≤t ≤T |u (t )|D 10.Next,we will

show that

T 0

|√ρu t (t )|2L 2+|u (t )|2

D 10

∩D 2 dt +sup 0≤t ≤T

|u (t )|2D 10

≤C exp (C Φ0(T ))

(2.3)for t 0≤T

integrate over Ω.Then using (1.2)and (1.3),we easily derive

ρ|u t |2dx +

1

2d dt |?u |2dx = ρ(f ?u ·?u )·u t dx and

ρ|u t |2dx +

d

dt

|?u |2dx ≤2

ρ|f |2dx +2

ρ|u ·?u |2dx.

(2.4)

On the other hand,since (u,p )is a solution of the stationary Stokes equations

?Δu +?p =F

and

div u =0

in

Ω,

where F =ρ(f ?u ·?u ?u t ),it follows from the classical regularity theory that |?u |H 1≤C (|F |L 2+|?u |L 2)

(2.5)

≤C (|f |L 2+|√

ρu t |L 2+|u ·?u |L 2+|?u |L 2).

1422HYUNSEOK KIM

To estimate the right-hand sides of(2.4)and(2.5),we?rst observe that

|u·?u|L2=|u·?u|L2,2≤C|u|L r

w |?u|

L

2r

r?2

,2

,

(2.6)

which follows from H¨o lder inequality in the Lorenz spaces.See Proposition2.1in[18]. Next,we will show that

|?u|

L

2r

r?2

,2≤

C|?u|1?3r

L2

|?u|3r

H1

.

(2.7)

If r=∞,then(2.7)is obvious.Assuming that3

that3

r =1

r1

+1

r2

.Then in view of H¨o lder and Sobolev

inequalities,we have

|?u|

L

2r i

r i?2

≤|?u|1?3r i

L2

|?u|3r i

L6

≤|?u|1?3r i

L2

(C|?u|H1)3r i

for each i=1,2.Since L2r r?2,2is a real interpolation space of L2r2

r2?2and L

2r1

r1?2,more

precisely,L2r r?2,2=(L2r2

r2?2,L

2r1

r1?2)1

2

,2

,it thus follows that

|?u|

L

2r

r?2

,2≤

C|?u|12

L

2r2

r2?2

|?u|12

L

2r1

r1?2

≤C

|?u|1?3r2

L2

(C|?u|H1)3r2

1

2

|?u|1?3r1

L2

(C|?u|H1)3r1

1

2

,

which proves(2.7).For some facts on the real interpolation theory and Lorenz spaces used above,we refer to sections1.3.3and1.18.6in Triebel’s book[30].

The estimates(2.6)and(2.7)yield

|u·?u|L2≤C|u|L r

w |?u|2s

L2

|?u|3r

H1

≤η?3s2r C|u|s2L r

w

|?u|L2+η|?u|H1

for any small numberη∈(0,1).Substituting this into(2.5),we obtain

|?u|H1≤C

|f|L2+|

ρu t|L2+|u|s2L r

w

|?u|L2+|?u|L2

,

(2.8)

and thus

|u·?u|L2≤η?3s2r C

|u|s2L r

w

+1

|?u|L2+C|f|L2+η|

ρu t|L2.

Therefore,substituting this estimate into(2.4)and choosing a su?ciently smallη>0, we conclude that

1 2|

ρu t(t)|2L2+

d

dt

|?u(t)|2L2≤C|f(t)|2L2+C

|u(t)|s L r

w

+1

|?u(t)|2L2

(2.9)

for t0≤t

T

0|

ρu t(t)|2L2dt+sup

0≤t≤T

|?u(t)|2L2≤C exp(CΦ0(T))

for any T with t0≤T

NONHOMOGENEOUS NAVIER–STOKES EQUATIONS

1423

2.2.Estimates for ess sup 0≤t ≤T |√

ρu t (t )|2L 2and T 0|u t (t )|2D 10

dt .To de-rive these estimates,we di?erentiate the momentum equation (1.1)with respect to time t and obtain

ρu tt +ρu ·?u t ?Δu t +?p t =ρt (f ?u t ?u ·?u )+ρ(f t ?u t ·?u ).

Then multiplying this by u t ,integrating over Ω,and using (1.2)and (1.3),we have

12d dt

ρ|u t |2

dx + |?u t |2dx

(2.10)=

(ρt (f ?u t ?u ·?u )+ρ(f t ?u t ·?u ))·u t dx.

Note that since ρ∈C ([0,T ];L 3

2∩L ∞),ρt ∈C ([0,T ];L 3

2),and u t ∈L 2(0,T ;D 1

0)for

any T

,the right-hand side of (2.10)is well de?ned for almost all t ∈(0,T ?).Hence using ?nite di?erences in time,we can easily show that the identity (2.10)holds for almost all t ∈(0,T ?).

In view of the continuity equation (1.2)again,we deduce from (2.10)that

12d

dt

ρ|u t |2dx + |?u t |2dx

2ρ|u ||u t ||?u t |+ρ|u ||u t ||?u |2+ρ|u |2|u t ||?2u |

(2.11)

+ρ|u |2|?u ||?u t |+ρ|u t |2|?u |+ρ|u ||u t ||?f |

+ρ|u ||f ||?u t |+ρ|f t ||u t |dx ≡

8 j =1

I j .

Following the arguments in [6],we can estimate each term I j :

I 1,I 5≤C |ρ|12L ∞|?u |L 2|√ρu t |L 3|?u t |L 2≤C |ρ|34L ∞|?u |L 2|√ρu t |12L 2|?u t |32

L 2

≤C |?u |4L 2|√ρu t |2

L 2+116

|?u t |2L 2,I 2,I 3,I 4≤C |ρ|L ∞|?u |2L 2|?u t |L 2|?u |H 1≤C |?u |4L 2|?u |2

H 1+

1

16

|?u t |2L 2,I 6,I 7≤C |ρ|L 6|?u |L 2|f |H 1|?u t |L 2≤C |?u |2L 2|f |2

H 1+

1

16

|?u t |2L 2,and ?nally

I 8≤C |ρ|L 3|f t |L 2|?u t |L 2≤C |f t |2L 2+

1

16

|?u t |2L 2.Substitution of these estimates into (2.11)yields

d dt |√ρu t |2L 2+|?u t |2

L 2≤C |?u |4L 2 |√ρu t |2L 2+|?u |2H 1+|f |2H 1 +C |f |2H 1

+|f t |2

L 2 .

1424

HYUNSEOK KIM

Therefore,by virtue of estimate (2.3),we conclude that ess sup 0≤t ≤T

|√

ρu t (t )|2L 2+

T

|?u t (t )|2L 2dt ≤C exp (C Φ0(T ))

(2.12)

for t 0≤T

equations again,we have

|?u |H 1≤C (|f |L 2+|√

ρu t |L 2+|u ·?u |L 2+|?u |L 2)

≤C |f |L 2+|√ρu t |L 2+|?u |32L 2|?u |12

H 1+|?u |L 2

and

|?u |H 1,6≤C (|u t |L 6+|u ·?u |L 6+|f |L 6+|?u |L 6)

≤C

|?u t |L 2+|?u |2H

1+|f |H 1+|?u |H 1 .Hence it follows immediately from (2.3)and (2.12)that sup

0≤t ≤T

|u (t )|2D 10

∩D 2+

T

|?u (t )|2H 1,6dt ≤C exp (C Φ0(T ))

(2.13)

for t 0≤T

2.3.Estimates for sup 0≤t ≤T |?ρ(t )|H 1and T

0|u (t )|2D 3dt .To derive these,we ?rst observe that each ρx j (j =1,2,3)satis?es

ρx j

t +u ·?ρx j =?u x j ·?ρ.Then multiplying this by ρx j ,integrating over Ω,and summing up,we obtain

d dt

|?ρ|2dx ≤C

|?u ||?ρ|2dx ≤C |?u |L ∞|?ρ|2L 2.

A similar argument shows that

d

dt

|?2ρ|2dx ≤C

|?u ||?2ρ|2+|?2u ||?ρ||?2ρ| dx ≤C |?u |L ∞|?2ρ|2L 2+|?2u |L 6|?ρ|L 3|?2

ρ|L 2.

Hence using Sobolev embedding results and then Gronwall’s inequality,we derive the

well-known estimate

|?ρ(t )|H 1≤C exp C t 0

|?u (τ)|H 1,6dτ

≤C exp t

C |?u (τ)|2H 1,6+1 dτ .

Therefore by virtue of (2.13),we conclude that sup 0≤t ≤T

|?ρ(t )|H 1≤C exp (C exp (C Φ0(T )))

(2.14)

NONHOMOGENEOUS NAVIER–STOKES EQUATIONS

1425

for t 0≤T

|u |D 3≤C (|ρu t |H 1+|ρu ·?u |H 1+|ρf |H 1)

≤C (|?ρ|L 3+1) |?u t |L 2+|?u |2

H 1+|f |H 1 ,

we easily deduce from (2.3),(2.12),(2.13),and (2.14)that

T

|u (t )|2D 3dt ≤C exp (C exp (C Φ0(T )))

(2.15)for t 0≤T

3.Proof of Theorem 1.3with m ≥ 2.Assume that m ≥ 2.Then to prove Theorem 1.3,it su?ces to show that the following estimate holds for each k ,0≤k

Φk +1(T )≤C exp C exp

C Φk (T )10m for t 0≤T

the case 1≤k

Let k be a ?xed integer with 1≤k

Φk (T )=1+sup

0≤j

sup

0≤t ≤T

|?j

t u (t )|D 10

∩D 2k ?2j +

T

|?j

t u (t )|2D 2k ?2j +1dt

(3.2)

+sup 0≤t ≤T

|?ρ(t )|H 2k ?1

+ess sup 0≤t ≤T

|√ρ?k

t u (t )|L 2+

T

|?k

t u (t )|2D 10

dt

for any T

3.1.Estimates for ?j t

(u ·?u ),?j +1t ρ,and ?j

t (ρu )with 0≤j ≤k .To estimate nonlinear terms,we will make repeated use of the following simple lemma whose proof is omitted.

Lemma 3.1.If g ∈D 1

0∩D j ,h ∈H i ,0≤i ≤j ,and j ≥2,then

gh ∈H i

and

|gh |H i ≤C |g |D 10

∩D j |h |H i for some constant C >0depending only on j and Ω.Using this lemma together with the fact that

?j t (u

·?u )=

j i =0

j !i !(j ?i )!

?i

t

u ·??j ?i t u,we can estimate ?j

t

(u ·?u )as follows:for 0≤j

t

(u ·?u )|H 2k ?2j ?1≤C

j i =0|?i

t u ·??j ?i t u |H 2k ?2j ?1

≤C j i =0|?i

t u |D 10∩D 2k ?2j |??j ?i t

u |H 2k ?2j ?1≤C

j i =0

|?i

t u |D 10∩D 2k ?2i |?j ?i t

u |D 10∩D 2k ?2(j ?i )

1426HYUNSEOK KIM and

|?k t(u·?u)|L2≤C k?1

i=0

|?i t u·??k?i

t

u|L2+|?k t u·?u|L2

≤C k?1

i=0

|?i t u|D1

∩D2|??

k?i

t

u|L2+|?k t u|D1

|?u|H1

≤C k?1

i=0

|?i t u|D1

∩D2k?2i|?

k?i

t

u|D1

+|?k t u|D1

|u|D1

∩D2.

Hence it follows from(3.2)that

sup 0≤j

sup

0≤t≤T

|?j t(u·?u)(t)|H2k?2j?1+

T

|?k t(u·?u)(t)|2L2dt≤CΦk(T)4

(3.3)

for t0≤T

ρt=?div(ρu)=?u·?ρ,

(3.4)

we also deduce that

sup 0≤t≤T |ρt(t)|H2k?1≤CΦk(T)2for t0≤T

(3.5)

Using(3.4)and(3.5),we can show that

sup 1≤j

sup

0≤t≤T

|?j+1

t

ρ(t)|H2k?2j+

T

|?k+1

t

ρ(t)|2L2dt≤CΦk(T)2k+4

(3.6)

for t0≤T

|?j+1

t

ρ|H2k?2j=|??j t(u·?ρ)|H2k?2j

≤C

j

i=0

|?j?i

t

u·??i tρ|H2k?2j

≤C

j

i=0

|?j?i

t

u|D1

∩D2k?2j|??

i

t

ρ|H2k?2j

≤C

j

i=0

|?j?i

t

u|D1

∩D2k?2(j?i)|?

i

t

ρ|H2k?2(i?1)

and

|?k+1

t ρ|L2≤C

k

i=0

|?k?i

t

u·??i tρ|L2≤C

k

i=0

|?k?i

t

u|D1

|?i tρ|H2.

Moreover,it follows easily from(3.6)that

sup 0≤j

sup

0≤t≤T

|?j t(ρu)(t)|H2k?2j+

T

|?k t(ρu)(t)|2H1dt≤CΦk(T)4k+10

(3.7)

NONHOMOGENEOUS NAVIER–STOKES EQUATIONS

1427

for t 0≤T

?k +1t f ∈L 2(0,∞;L 2)

and ?j

t f ∈L 2(0,∞;H 2k ?2j +1)

for 0≤j ≤k,

we deduce from standard embedding results that

?j

t f ∈C ([0,∞);H 2k ?2j )for

0≤j ≤k.

3.2.Estimates for T 0|√ρ?k +1

t u (t )|2L 2dt and sup 0≤t ≤T |?k t u (t )|D 10.From

the momentum equation (1.1),we derive

ρ ?k t u t ?Δ?k t u +??k t p =?k t (ρf ?ρu ·?u )+ ρ?k t u t ??k t (ρu t ) .

Hence multiplying this by ?k +1

t u and integrating over Ω,we have

ρ|?k +1t u |2dx +12d dt

|??k t u |2

dx

=

?k t (ρf ?ρu ·?u )+ ρ?k t u t ??k

t (ρu t ) ·?k +1

t

u dx (3.8)=I 0,1+

k j =1

k !

j !(k ?j )!

(I j,1+I j,2),

where

I j,1=

?j t ρ?k ?j t (f ?u ·?u )·?k +1t u dx,I j,2=?

?j t ρ?k ?j t u t ·?k +1

t u dx.

We easily estimate I 0,1as follows:

I 0,1≤|ρ|12

L ∞ |?k t f |L 2+|?k t (u ·?u )|L 2 |

√ρ?k +1

t u |L 2≤C |?k t f |2

L 2+|?k

t (u ·?u )|2L

2 +12

|√ρ?k +1t u |2L 2.To estimate I j,1for 1≤j ≤k ,we rewrite it as

I j,1=d dt

?j t ρ?k ?j t (f ?u ·?u )·?k t u dx ? ?j +1t ρ?k ?j t (f ?u ·?u )·?k

t u dx ?

?j t ρ?k ?j +1t (f ?u ·?u )·?k

t u dx and observe that ?

?j +1t

ρ?k ?j t (f ?u ·?u )·?k

t u dx ≤C |?k ?j t f |2H 1+|?k ?j t (u ·?u )|2

H 1 |?j +1t ρ|2L 2+|??k t u |2L 2and

?

?j t

ρ?k ?j +1t (f ?u ·?u )·?k t u dx ≤C |?j t ρ|2H 1 |?k ?j +1t f |2L 2+|?k ?j +1t (u ·?u )|2L 2 +|??k t u |2

L 2.

1428

HYUNSEOK KIM

Using the continuity equation (1.2),we can also estimate I j,2as follows:

I 1,2=? ρt 12|?k t u |

2 t dx =?d dt ρt 12|?k t u |2dx + ?2t ρ12

|?k t u |2

dx =?d dt ρu ·? 12|?k t u |2 dx + ?t (ρu )·? 12|?k t

u |2

dx ≤?d dt (ρu ·??k t u )·?k t u dx +C |?t (ρu )|H 1|??k t u |2

L 2and similarly

I j,2=?d dt ?j t ρ?k ?j t u t ·?k t u dx + ?j +1t ρ?k ?j t u t +?j t ρ?k ?j +1t u t ·?k

t u dx ≤?d dt ?j ?1t (ρu )·? ?k ?j +1t u ·?k

t u dx +C |?j t (ρu )|H 1|??k ?j +1t u |L 2+|?j ?1t (ρu )|H 1|??k ?j +2t u |L 2 |??k

t u |L 2

for 2≤j ≤k .Substituting all the estimates into (3.8),we have

12 ρ|?k +1t u |2dx +12d dt

|??k t u |2

dx ≤d dt ??k j =1

k !j !(k ?j )!?j t ρ?k ?j t (f ?u ·?u )·?k t u ?(ρu ·??k t u )·?k t u ??dx ?d dt k

j =2

k !j !(k ?j )!?j ?1t

(ρu )·? ?k ?j +1t u ·?k

t u dx

+C

k ?1 j =1

|?j

t (ρu )|2H 1

+

|?j +1t ρ|2H 1

|?k ?j t f |2

H 1

+

|?k ?j

t (u

·

?u )|2H 1

+

|?k ?j +1t u |2

D 1

+C |?k +1t ρ|2

L 2

|f |2H 1

+|u |2D 10

∩D 2

+C 1+|?t ρ|2H 1 |?k t f |2L 2+|?k t (u ·?u )|2

L

2 +C |?k t (ρu )|2H 1

+C 1+|?t (ρu )|2H 1+|??t u |2L 2 |??k t u |2

L 2.Hence,integrating this in time over (t 0,T )and using (3.3),(3.5),(3.6),and (3.7)

together with the estimates

|?j t ρ||?k ?j t (f ?u ·?u )||?k t u |dx

≤η?1|?j t ρ|2H 1|?k ?j t (f ?u ·?u )|2L 2+η|??k t u |L 2,

ρ|u ||??k t u ||?k t u |dx ≤η?3C |ρ|3L ∞|?u |4L 2|√ρ?k t u |2L 2+η|??k t u |2L 2

and

|?j ?1t (ρu )|

|??k ?j +1t u ||?k

t u |

+

|?k ?j +1t u ||??k

t u |

dx

≤η?1C |?j ?1t (ρu )|2H 1|??k ?j +1t

u |2L 2+η|??k

t u |L 2,

NONHOMOGENEOUS NAVIER–STOKES EQUATIONS1429 whereηis any small positive number,we deduce that

T

t0|

ρ?k+1

t

u(t)|2L2dt+|??k t u(T)|2L2

≤CΦk(T)20m+C

T

t0

1+|?t(ρu)(t)|2H1+|u t(t)|2

D1

|??k t u(t)|2L2dt

for t0≤T

T t0

1+|?t(ρu)(t)|2H1+|u t(t)|2

D1

dt≤CΦk(T)10m.

Therefore,in view of Gronwall’s inequality,we conclude that T

0|

ρ?k+1

t

u(t)|2L2dt+sup

0≤t≤T

|?k t u(t)|2

D1

≤C exp

CΦk(T)10m

(3.9)

for any T with t0≤T

3.3.Estimates for ess sup0≤t≤T|√

ρ?k+1

t

u(t)|L2and

T

|?k+1

t

u(t)|2

D1

dt.

From the momentum equation(1.1),it follows that

ρ

?k+1

t

u

t

+ρu·??k+1

t

u?Δ?k+1

t

u+??k+1

t

p

=?k+1

t

(ρf)+

ρ?k+1

t

u t??k+1

t

(ρu t)

+

ρu·??k+1

t

u??k+1

t

(ρu·?u)

.

Multiplying this by?k+1

t

u and integrating overΩ,we have

1 2d

dt

ρ|?k+1

t

u|2dx+

|??k+1

t

u|2dx

=

?k+1

t

(ρf)·?k+1

t

u dx+

ρ?k+1

t

u t??k+1

t

(ρu t)

·?k+1

t

u dx

(3.10)

+

ρu·??k+1

t

u??k+1

t

(ρu·?u)

·?k+1

t

u dx.

This identity can be derived rigorously by using a standard?nite di?erence method because if0

C

k

j=0

|?j tρ||?k?j+1

t

f||?k+1

t

u|dx+

|?k+1

t

ρ||f||?k+1

t

u|dx

≤C

k

j=0

|?j tρ|2H1|?k?j+1

t

f|2L2+C|?k+1

t

ρ|2L2|f|2H1+

1

6

|??k+1

t

u|2L2.

In view of the continuity equation(1.2),we can rewrite the second term as

?k+1

j=1

(k+1)!

j!(k?j+1)!

?j tρ?k?j+1

t

u t·?k+1

t

u dx

=?

k+1

j=1

(k+1)!

j!(k?j+1)!

?j?1

t

(ρu)·?

?k?j+2

t

u·?k+1

t

u

dx,

1430HYUNSEOK KIM which is bounded by

C|ρ|L∞|u|2D1

0∩D2

|

ρ?k+1

t

u|2L2+C

k

j=1

|?j t(ρu)|2H1|?k?j+1

t

u|2

D1

+

1

6

|??k+1

t

u|2L2.

Finally,the last term is bounded by

C

k

j=1

|?j t(ρu)||??k?j+1

t

u||?k+1

t

u|dx+

|?k+1

t

(ρu)||?u||?k+1

t

u|dx

≤C

k

j=1

|?j t(ρu)|2H1+|?j tρ|2H1|u|2

D1

∩D2

|?k?j+1

t

u|2

D1

+C|?k+1

t

ρ|2L2|u|4

D1

∩D2

+C|ρ|L∞|u|2D1

∩D2

|

ρ?k+1

t

u|2L2+

1

|??k+1

t

u|2L2.

Hence substituting these estimates into(3.10),we have

d dt

ρ|?k+1

t

u|2dx+

|??k+1

t

u|2dx

≤C

1+|u|2

D1

∩D2

|

ρ?k+1

t

u|2L2+C

k

j=0

|?j tρ|2H1|?k?j+1

t

f|2L2+C|?k+1

t

ρ|2L2|f|2H1 +C

k

j=1

|?j t(ρu)|2H1+|?j tρ|2H1|u|2

D1

∩D2

|?k?j+1

t

u|2

D1

+C|?k+1

t

ρ|2L2|u|4

D1

∩D2

.

Therefore,by virtue of(3.5),(3.6),(3.7),and(3.9),we conclude that

ess sup

0≤t≤T |

ρ?k+1

t

u(t)|L2+

T

|?k+1

t

u(t)|2

D1

dt≤C exp

CΦk(T)10m

(3.11)

for any T with t0≤T

3.4.Estimates for sup0≤t≤T|?j t u(t)|D1

0∩D2k?2j+2with0≤j≤k.To

derive these estimates,we observe that

?j t u∈C([0,T?);D10,σ)and?Δ?j t u+??j t p=?j t(ρf?ρu·?u?ρu t) (3.12)

for each j≤k.From(3.5),(3.6),(3.9),and(3.11),it follows easily that

ess sup

0≤t≤T

|?k t(ρu·?u)(t)|L2+|?k t(ρu t)(t)|L2

≤C exp

CΦk(T)10m

for t0≤T

sup 0≤t≤T |?k t u(t)|D1

∩D2≤C exp

CΦk(T)10m

for t0≤T

It also follows from the Stokes regularity theory that for0≤j

|?j t u|D1

0∩D2k?2j+2≤C|?

j

t

(ρf?ρu·?u?ρu t)|H2k?2j+C|?j t u|D1

.

(3.13)

NONHOMOGENEOUS NAVIER–STOKES EQUATIONS1431 Using the estimates in section3.1,we can estimate the?rst term of the right-hand side in(3.13)as follows:

C|?j t(ρf?ρu·?u?ρu t)(t)+ρ?j t(u·?u+u t)(t)|H2k?2j

≤C|ρ?j t f(t)|H2k?2j+C

j

i=1

|?i tρ?j?i

t

(f?u·?u?u t)(t)|H2k?2j

≤C exp

CΦk(T)10m

,

C|ρ?j t(u·?u)(t)|H2k?2j

≤C

j

i=1

|ρ?i t u·??j?i

t

u(t)|H2k?2j+C|ρu·??j t u(t)|H2k?2j

≤C exp

CΦk(T)10m

+C|ρ(t)|H2k|u(t)|D1

∩D2k|?

j

t

u(t)|D1

∩D2k?2j+1

≤C exp

CΦk(T)10m

+

1

2

|?j t u(t)|D1

∩D2k?2j+2

and

C|ρ?j t u t(t)|H2k?2j≤C|ρ(t)|H2k|?j+1

t u(t)|D1

∩D2k?2j.

Substituting these into(3.13),we deduce that

|?j t u(t)|D1

0∩D2k?2j+2≤C exp

CΦk(T)10m

1+|?j+1

t

u(t)|D1

∩D2k?2j

.

Therefore,by a backward induction on j,we conclude that

sup 0≤j

sup

0≤t≤T

|?j t u(t)|D1

∩D2k?2j+2≤C exp

CΦk(T)10m

.

(3.14)

3.5.Estimates for sup0≤t≤T|?ρ(t)|H2k+1and T

|?j t u(t)|2

D2k?2j+3

dt with

j≤k.Letαbe a multi-index with1≤|α|≤2k+2.Then taking the di?erential operator Dαto the continuity equation(1.2),we have

(Dαρ)t+u·?(Dαρ)=u·?(Dαρ)?Dα(u·?ρ).

Multiplying this by Dαρand integrating overΩ,we obtain

d dt

|Dαρ|2dx≤C

|u·?(Dαρ)?Dα(u·?ρ)||Dαρ|dx.

(3.15) But since

|u·?(Dαρ)?Dα(u·?ρ)|≤C |α|

l=1

?|α|+1?l u

?lρ

,

H¨o lder and Sobolev inequalities yield

|u·?(Dαρ)?Dα(u·?ρ)|L2≤C(|?u|H2k+1|?ρ|H2k+1+|?u|L∞|?ρ|H2k+1)

≤C|?u|H2k+1|?ρ|H2k+1.

1432

HYUNSEOK KIM

Hence from (3.15),we derive the standard estimate

d dt

|?ρ|2H 2k +1≤C |u |D 10∩D 2k +2|?ρ|2

H

2k +1,which implies then that

sup 0≤t ≤T |?ρ(t )|H 2k +1

≤|?ρ0|H 2k +1exp C

T

|u (t )|D 10∩D 2k +2dt

.

Therefore,by virtue of (3.14),we conclude that

sup 0≤t ≤T

|?ρ(t )|H 2k +1≤C exp C exp

C Φk (T )10m

for any T with t 0≤T

sup 0≤j

T

|?j t u (t )|2D

2k ?2j +3dt ≤C exp C exp C Φk (T )10m

for any T with t 0≤T

completed the proof of Theorem 1.3.

4.Proof of Theorem 1.

5.Let (ρ,u )be a strong solution satisfying the regu-larity (1.8)for some T >0.Assume that |ρ0|L 32∩L ∞

≤K for some constant K >1.

Then it follows easily from (1.2)and (1.3)that

|ρ(t )|

L 3

2∩L ∞

=|ρ0|

L 3

2∩L ∞

≤K for 0≤t ≤T.

Moreover,from the energy equality

12d

dt ρ|u |2dx + |?u |2dx = ρf ·u dx,

we derive

|√

ρu (t )|2L 2+

t

|?u (τ)|2L 2

dτ≤|√

ρ0u 0|2L 2+C

t

|ρ(τ)|2L 3|f (τ)|2

L 2dτ,

and thus

T

0|?u (τ)|2

dτ≤C K

|?u 0|2L 2

+

|f (τ)|2L 2

.

(4.1)

Throughout the proof,we denote by C K >1a generic constant dependent only on K

and Ωbut independent of time T .On the other hand,from the estimate (2.9)with (r,s )=(6,4),it follows that

|?u (t )|2L 2≤|?u 0|2L 2+C K t 0

|f (τ)|2L 2dτ+C K t 0

|u (τ)|4

L 6w

+1

|?u (τ)|2L 2dτfor 0≤t ≤T .Hence by virtue of the estimate (4.1)and Sobolev inequality (1.5),we have

|?u (t )|2L 2≤C K sup 0<τ

|?u (τ)|4L 2+1 |?u 0|2L 2+ ∞

|f (τ)|2L 2dτ

(4.2)

NONHOMOGENEOUS NAVIER–STOKES EQUATIONS

1433

for 0

0<ε<1

and

6C K ε2<1.

We now prove the global existence of a strong solution under the assumption that |?u 0|L 2≤εand ∞0|f (τ)|2L 2dτ≤ε2.The local existence of a unique solution (ρ,u )was already proved in [6](see also [7]).To prove the global existence,we argue by contradiction.Assume that (ρ,u )blows up at some ?nite time T ?,0

|?u (t )|2L 2≤

13 sup 0<τ

|?u (τ)|4

L 2+1 for 0

0).Hence from (4.3),we easily deduce

that |?u (t )|2L 2<1for any 0≤t

.Therefore,in view of Sobolev embedding again,we conclude that

T ?

|u (t )|4L 6w

dt ≤C

T ?

|?u (t )|4L 2dt <∞,

which contradicts Theorem 1.3.This completes the proof of Theorem 1.5.

Acknowledgments.The author would like to express his deep gratitude to

Professor Hideo Kozono for very helpful discussion on the weak spaces.The author is also grateful to the referees for several valuable comments on the paper.

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1434HYUNSEOK KIM

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