Global weak solutions of the Navier-Stokes equations with nonhomogeneous boundary data and divergence

Global Weak Solutions of the Navier-Stokes Equations with Nonhomogeneous Boundary Data and Divergence

R. FARWIG(*) - H. KOZONO(**) - H. SOHR(***)

ABSTRACT- Consider a smooth bounded domainVR³with boundary@V, a time interval [0;T),05T 1, and the Navier-Stokes system in [0;T)V, with in- itial valueu02L²_s(V) and external forcef ˆdivF,F2L²(0;T;L²(V)). Our aim is to extend the well-known class of Leray-Hopf weak solutions usatisfying uj_@Vˆ0, divuˆ0to the more general class of Leray-Hopf type weak solutions uwith general datauj_@Vˆg, divuˆksatisfying a certain energy inequality.

Our method rests on a perturbation argument writinguin the formuˆv‡E with some vector fieldEin [0;T)Vsatisfying the (linear) Stokes system with f ˆ0 and nonhomogeneous data. This reduces the general system to a per- turbed Navier-Stokes system with homogeneous data, containing an additional perturbation term. Using arguments as for the usual Navier-Stokes system we get the existence of global weak solutions for the more general system.

1.Introduction and main results.

LetVR³be a bounded domain with boundary@Vof classC^2;1, and let [0;T), 05T 1, be a time interval. We consider in [0;T)V, together with an associated pressurep, the following general Navier-Stokes system

ut Du‡u ru‡ rp ˆ f; divuˆk uj_@V ˆ g; uj_tˆ0ˆu₀ (1:1)

with given dataf,k,g,u₀.

(*) Indirizzo dell'A.: Department of Mathematics, Darmstadt University of Technology, 64289 Darmstadt, Germany,andCenter of Smart Interfaces (CSI), 64287 Darmstadt, Germany.

E-mail: farwig@mathematik.tu-darmstadt.de

(**) Indirizzo dell'A.: Mathematical Institute, ToÃhoku University, Sendai, 980-8578 Japan.

E-mail: kozono@math.tohoku.ac.jp

(***) Indirizzo dell'A.: Faculty of Electrical Engineering, Informatics and Mathematics, University of Paderborn, 33098 Paderborn, Germany.

E-mail: hsohr@math.uni-paderborn.de

First we have to give a precise characterization of this general system.

To this aim, we shortly discuss our arguments to solve this system in the weak sense (without any smallness assumption on the data). Using a perturbation argument we writeuin the form

uˆv‡E;

(1:2)

and the initial valueu0 at timetˆ0 in the form u0ˆv0‡E0: (1:3)

HereEis the solution of the (linear) Stokes system Et DE‡ rhˆ0; divEˆk

Ej_@Vˆg; Ej_tˆ0ˆE₀ (1:4)

with some associated pressureh, andvhas the properties v2L¹_loc [0;T);L²_s(V)

\L²_loc [0;T);W₀^1;2(V)

; v:[0;T)7!L²_s(V) is weakly continuous; vj_tˆ0ˆv₀: (1:5)

Inserting (1.2), (1.3) into the system (1.1) we obtain the modified system v_t Dv‡(v‡E) r(v‡E)‡ rpˆf; divvˆ0

vj_@V ˆ0; vj_tˆ0ˆv0

(1:6)

with associated pressure p ˆp h and homogeneous conditions for v.

Thus (1.6) can be called aperturbed Navier-Stokes system in [0;T)V.

This system reduces the general system (1.1) to a certain homogeneous system which contains an additional perturbation term in the form

(v‡E) r(v‡E)ˆv rv‡v rE‡E r(v‡E):

Therefore, the perturbed system (1.6) can be treated similarly as the usual Navier-Stokes system obtained from (1.6) withE0.

In order to give a precise definition of the general system (1.1) we need the following steps:

First we develop the theory for the perturbed system (1.6) for dataf,v0

and a given vector fieldE, as general as possible. In the second step we consider the system (1.4) for general given datak,g,E0to obtain a vector fieldEin such a way thatuˆv‡Ewithvfrom (1.6) yields a well-defined solution of the general system (1.1) in the (Leray-Hopf type) weak sense.

Thus we start with the definition of a weak solution v of (1.6) under rather weak assumptions onEneeded for the existence of such solutions.

DEFINITION1.1. (Perturbed system). Suppose

f ˆdivF with Fˆ(F_i;j)³_i;jˆ12L² 0;T;L²(V)

; v02L²_s(V);

E2L^s 0;T;L^q(V)

; divEˆk2L⁴ 0;T;L²(V)

; (1:7)

with4s51,4q51,2 s‡3

qˆ1.

Then a vector field v is called a weak solution of the perturbed system (1.6)in[0;T)V with data f , v0 if the following conditions are satisfied:

a) For each finite T;05TT, v2L¹ 0;T;L²_s(V)

\L² 0;T;W₀^1;2(V)

; (1:8)

b) for each test function w2C¹₀ ([0;T);C¹_0;s(V));

hv;w_ti_V;T‡ hrv;rwi_V;T

(v‡E)(v‡E);rw V;T

k(v‡E);w

V;T ˆ

v0;w(0)

V hF;rwi_V;T; (1:9)

c) for0t5T, 1

2kv(t)k²₂‡ Z^t

0

krvk²₂dt 1 2kv0k²₂

Z^t

0

hF;rvi_Vdt

‡ Z^t

0

(v‡E)E;rv

V dt‡1 2

Z^t

0

k(v‡2E);v

V dt;

(1:10)

d) and

v:[0;T)!L²_s(V)is weakly continuous and v(0)ˆv0: (1:11)

In the classical caseE0 we obtain with (1.8)-(1.11) the usual (Leray- Hopf) weak solutionv. As in this case the condition (1.11) already follows from the other conditions (1.8)-(1.10), after possibly a modification on a null set of [0;T), see, e.g., [16, V, 1.6]. Here (1.11) is included for simplicity. The relation (1.9) and the energy inequality (1.10) are based on formal calcu- lations as forE0. The existence of an associated pressurepsuch that

vt Dv‡(v‡E) r(v‡E)‡ rpˆf (1:12)

in the sense of distributions in (0;T)V follows in the same way as for E0.

In the next step we consider the linear system (1.4). A very general solution class for this system, sufficient for our purpose, has been devel-

oped by the theory of so-called very weak solutions, see [1], [3, Sect. 4]. In particular, the boundary values g are given in a general sense of dis- tributions on@V.

LEMMA1.2(Linear system forE, [3]). Suppose k2L^s 0;T;L^q(V)

; g2L^s 0;T;W ^1q;q(@V)

; E02L^q(V);

4s51; 4q51; 2 s‡3

qˆ1; 1 qˆ 1

q 1 3; (1:13)

satisfying the compatibility condition Z

V

k(t)dxˆ Z

@V

Ng(t)dS for almost all t2[0;T);

(1:14)

where NˆN(x)means the exterior normal vector at x2@V, and R

@V. . .dS the surface integral (in a generalized sense of distributions on@V).

Then there exists a uniquely determined (very) weak solution E2L^s 0;T;L^q(V)

(1:15)

of the system(1.4)in[0;T)Vwith data k, g, E₀defined by the conditions:

a) For each w2C¹₀([0;T);C_0;s² (V)),

hE;w_ti_V;T hE;Dwi_V;T‡ hg;N rwi_V;Tˆ

E₀;w(0)

V; (1:16)

b) for almost all t2[0;T),

divEˆk; NEj_@V ˆNg:

(1:17)

Moreover, E satisfies the estimate

kA_q¹PqEtk_q;s;V;T‡kEk_q;s;V;TC kE0k_q‡kkk_q;s;V;T‡kgk ¹

q;q;s;@V;T

(1:18)

with constant CˆC(V;T;q)>0.

The trace Ej_@V ˆg is well-defined at@V for almost all t2[0;T), and the initial value condition Ej_tˆ0ˆE₀is well-defined (modulo gradients) in the sense that P_qE:[0;T)!L^q_s(V)is weakly continuous satisfying

P_qEj_tˆ0ˆP_qE₀: (1:19)

Finally, there exists an associated pressure h such that E_t DE‡ rhˆ0

(1:20)

holds in the sense of distributions in(0;T)V.

To obtain a precise definition for the general system (1.1) we have to combine Definition 1.1 and Lemma 1.2 as follows:

DEFINITION1.3. (General system). Let k2L^s 0;T;L^q(V)

\L⁴ 0;T;L²(V) with s;qas in(1.13)and suppose that

E is a very weak solution of the linear system…1:4†in [0;T)Vwith data k;g;E0in the sense of Lemma1:2;

(1:21) and

v is a weak solution of the perturbed system1:6in [0;T)Vin the sense of Definition1:1with data f;v0

as in1:7:

(1:22)

Then the vector field uˆv‡E is called a weak solution of the general system(1.1)in[0;T)V with data f , k, g and initial value u0ˆv0‡E0. Thus it holds

u_t Du‡u ru‡ rpˆf; divuˆk (1:23)

in the sense of distributions in (0;T)V with associated pressure pˆp‡h, pas in(1.12),h as in(1.20).Further,

uj_@V ˆvj_@V‡Ej_@Vˆg (1:24)

is well-defined by Ej_@V ˆg, and the condition

uj_tˆ0ˆvj_tˆ0‡Ej_tˆ0ˆv₀‡E₀ˆu₀ (1:25)

is well-defined in the generalized sense modulo gradients by(1.19).

Therefore the general system (1.1) has a well-defined meaning for weak solutionsuin a generalized sense.

However, if we suppose in Definition 1.3 additionally the regularity properties

k2L^s 0;T;W^1;q(V)

; kt2L^s 0;T;L²(V)

; g2L^s 0;T;W^{2 1=q;q}(@V)

; gt2L^s 0;T;W ^1q;q(@V)

; E₀2W^2;q(V);

(1:26)

and the compatibility conditions u₀j_@Vˆgj_tˆ0, divu₀ˆkj_tˆ0, then the so- lutionEin Lemma 1.2 satisfies the regularity properties

E2L^s 0;T;W^2;q(V)

; Et 2L^s 0;T;L^q(V)

; E2C [0;T);L^q(V)

;

and Ej_@Vˆg; Ej_tˆ0ˆE0 are well-defined in the usual sense, see [3, Corollary 5]. Further it holds rh2L^s(0;T;L^q(V)) for the associated pressure h in (1.20). Therefore, uˆv‡E satisfies in this case the boundary conditionuj_@V ˆgand the initial conditionuj_tˆ0ˆv₀‡E₀in the usual (strong) sense.

The most difficult problem is the existence of a weak solutionv of the perturbed system (1.6). For this purpose we have to introduce, see (2.12) in Sect. 2, an approximate system of (1.6) for eachm2Nwhich yields such a weak solution when passing to the limitm! 1. Then the existence of a weak solutionuˆv‡Eof the general system (1.6) is an easy consequence.

This yields the following main result.

THEOREM1.4 (Existence of general weak solutions).

a) Suppose

f ˆdivF; F2L² 0;T;L²(V)

; v02L²_s(V);

E2L^s 0;T;L^q(V)

; divEˆk2L⁴ 0;T;L²(V)

; 4s51; 4q51; 2

s‡3 qˆ1: (1:27)

Then there exists at least one weak solution v of the perturbed system(1.6) in[0;T)Vwith data f , v0in the sense of Definition1.1.The solution v satisfies with some constant CˆC(V)>0the energy estimate

kv(t)k²₂‡ Z^t

0

krvk²₂dtC

kv0k²₂‡ Z^t

0

kFk²₂dt‡ Z^t

0

kkk⁴₂dt

‡ Z^t

0

kEk⁴₄dt

exp Ckkk⁴_2;4;t‡CkEk^s_q;s;t (1:28)

for each0t5T.

b) Suppose additionally k2L^s 0;T;L^q(V)

; g2L^s 0;T;W ^1q;q(@V)

; E02L^q(V);

Z

V

k dxˆ Z

@V

Ng dS for a:a:t2[0;T);

(1:29)

and let E be the very weak solution of the linear system(1.4)in[0;T)V with data k, g, E₀as in Lemma1.2.Then uˆv‡E is a weak solution of

the general system(1.1)with data f , k, g and initial value u0ˆv0‡E0in the sense of Definition1.3.

There are some partial results with nonhomogeneous smooth boundary conditions uj_@V ˆg6ˆ0 based on an independent approach by Raymond [15]. For the case of weak solutions with constant in time nonzero boundary conditions g see [4]. Further there are several independent results for smooth boundary valuesuj_@V ˆg6ˆ0 in the context of strong solutionsuif g or (equivalently) the time interval [0;T) satisfy certain smallness con- ditions, see [1], [3], [6], [10]. Our existence result for weak solutions in Theorem 1.4 does not need any smallness condition, like for usual Leray- Hopf weak solutions. But, on the other hand, there is no uniqueness result as for local strong solutions.

A first result on global weak solutions with time-dependent boundary data (and kˆdivuˆ0) can be found in [5]. In that paper, the authors consider generals>2,q>3 with2

s‡3

qˆ1; however, in that case,Ehas to satisfy the assumptions

E2L^s 0;T;L^q(V)

\L⁴ 0;T;L⁴(V)

;

which is automatically fulfilled in the present article, see Theorem 1.4.

Moreover, in simply connected domains or under a further assumption on the boundary data g, the energy estimate (1.28) can be improved con- siderably.

2.Preliminaries.

First we recall some standard notations. Let C_0;s¹(V)ˆ fw2C₀¹(V);

divwˆ0g be the space of smooth, solenoidal and compactly supported vector fields. Then letL^q_s(V)ˆC_0;s¹(V)^kkq, 15q51, where in generalk k_q denotes the norm of the Lebesgue spaceL^q(V), 1q 1. Sobolev spaces are denoted byW^m;q(V) with normk k_Wm;q ˆ k k_m;q,m2N, 1q 1, and W₀^m;q(V)ˆC¹₀ (V)^kkm;q, 1q51. The trace space to W^1;q(V) is W^{1 1=q;q}(@V), 15q51, with norm k k_{1 1=q;q}. Then the dual space to W^{1 1=q0;q0}(@V), where 1

q⁰‡1

qˆ1, isW ^1=q;q(@V); the corresponding pairing is denoted byh;i_@V.

As spaces of test functions we need in the context of very weak solutions the space C²_0;s(V)ˆ fw2C²(V); wj_@V ˆ0; divwˆ0g; for weak insta- tionary solutions let the space C₀¹([0;T);C¹_0;s(V)) denote vector fields

w2C₀¹([0;T)V) such that divxwˆ0 for allt2[0;T) taking the diver- gence div_xwith respect toxˆ(x₁;x₂;x₃)2V. The pairing of functions on V and (0;T)Vis denoted byh;i_V andh;i_V;T, respectively.

For 1q, s 1 the usual Bochner space L^s(0;T;L^q(V)) is equip- ped with the norm k k_q;s;Tˆ(R^T

0 k k^s_qdt)^1=s whens51and k k_q;1;Tˆ ess sup_(0;T)k k_q when sˆ 1.

LetPq:L^q(V)!L^q_s(V), 15q51, be the Helmholtz projection, and let A_qˆ P_qD with domain D(A_q)ˆW^2;q(V)\W₀^1;q(V)\L^q_s(V) and range R(A_q)ˆL^q_s(V) denote the Stokes operator. We writePˆP_qandAˆA_qif there is no misunderstanding. For 1a1 the fractional powers A^a_q:D(A^a_q)!L^q_s(V) are well-defined closed operators with (A^a_q) ¹ˆA_q^a. For 0a1 we haveD(Aq)D(A^a_q)L^q_s(V) andR(A^a_q)ˆL^q_s(V). Then there holds the embedding estimate

kvk_qCkA^a_qvk_g; 0a1; 2a‡3 qˆ3

g; 15gq;

(2:1)

for allv2D(A^a_q). Further, we need the Stokes semigroupe ^tAq:L^q_s(V)! L^q_s(V),t0, satisfying the estimate

kA^a_qe ^tAqvk_qCt ^ae ^btkvk_q; 0a1; t>0;

(2:2)

forv2L^q_s(V) with constantsCˆC(V;q;a)>0,bˆb(V;q)>0; for details see [2, 7, 8, 9, 11].

In order to solve the perturbed system (1.6) we use an approximation procedure based on Yosida's smoothing operators

J_m ˆ I‡ 1

mA¹⁼² ₁

and J_mˆ I‡1

m( D)¹⁼²

₁

; m2N;

(2:3)

where I denotes the identity and D the Dirichlet Laplacian on V. In particular, we need the properties

kJ_mvk_q Ckvk_q; kA¹⁼²J_mvk_qmCkvk_q; m2N;

m!1lim J_mvˆv for allv2L^q_s(V);

(2:4)

and analogous results forJ_mv,v2L^q(V); see [8, 9, 16].

To solve the instationary Stokes system in [0;T)V, cf. [1, 13, 16, 17, 18], let us recall some properties for the special system

V_t DV‡ rH ˆ f₀‡divF₀; divV ˆ 0 V ˆ 0 on@V; V(0) ˆ V0

(2:5) with data

f02L¹ 0;T;L²(V)

; F02L² 0;T;L²(V)

; V02L²_s(V);

hereF₀ ˆ(F_0;ij)³_i;jˆ1and divF₀ˆ P³

iˆ1

@

@x_iF_0;ij3

jˆ1. The linear system (2.5) admits a unique weak solution

V2L¹ 0;T;L²_s(V)

\L² 0;T;W₀^1;2(V)

; (2:6)

satisfying the variational formulation

hV;w_ti_V;T‡hrV;rwi_V;Tˆ hV₀;w(0)i_V‡hf₀;wi_V;T hF₀;rwi_V;T (2:7)

for allw2C₀¹([0;T);C¹_0;s(V)), and the energy equality 1

2kV(t)k²₂‡ Z^t

0

krVk²₂dtˆ1

2kV₀k²₂‡ Z^t

0

hf₀;Vi_Vdt Z^t

0

hF₀;rVi_Vdt (2:8)

for 0t5T. As a consequence of (2.8) we get the energy estimate 1

2kVk²_2;1;T‡ krVk²_2;2;T8 kV₀k²₂‡ kf₀k²_2;1;T‡ kF₀k²_2;2;T

; (2:9)

and see thatV:[0;T)!L²_s(V) is continuous withV(0)ˆV0:Moreover, it holds the well-defined representation formula

V(t)ˆe ^tAV₀‡ Z^t

0

e ^{(t t)A}Pf₀dt‡ Z^t

0

A¹⁼²e ^{(t t)A}A ¹⁼²PdivF₀dt;

(2:10)

0t5T; see [16, Theorems IV.2.3.1 and 2.4.1, Lemma IV.2.4.2], and, concerning the operatorA ¹⁼²Pdiv , [16, Ch. III.2.6].

Consider the perturbed system (1.6) withf ˆdivF,v0,kandEas in Definition 1.1, here written in the form

vt Dv‡div (v‡E)(v‡E) k(v‡E)‡ rpˆf; divvˆ0 (2:11)

together with the initial-boundary conditionsvˆ0 on@Vandv(0)ˆv0. In order to obtain the following approximate system, see [16, V, 2.2] for the known caseE0, we insert the Yosida operators (2.3) into (2.11) as follows:

vt Dv‡div (Jmv‡E)(v‡E) (Jmk)(v‡E)‡rpˆf; divvˆ0 vj_@Vˆ0; vj_tˆ0ˆv0

(2:12)

withvˆvm,m2N. Setting

Fm(v)ˆ(Jmv‡E)(v‡E); fm(v)ˆ(Jmk)(v‡E) (2:13)

we write the approximate system (2.12) in the form vt Dv‡ rp ˆfm(v)‡div F Fm(v)

; divvˆ0;

vj_@V ˆ0; vj_tˆ0ˆv₀; (2:14)

as a linear system, see (2.5), with right-hand side depending onv. In this form we use the properties (2.6)-(2.10) of the linear system (2.5).

The following definition for (2.12) is obtained similarly as Definition 1.1.

DEFINITION2.1. (Approximate system). Suppose f ˆdivF; F2L² 0;T;L²(V)

; v₀2L²_s(V);

E2L^s 0;T;L^q(V)

; divEˆk2L⁴ 0;T;L²(V)

; 4s51; 4q51; 2

s‡3 qˆ1:

(2:15)

Then a vector field vˆvm, m2N, is called a weak solution of the approximate system (2.12) in [0;T)V with data f , v₀ if the following conditions are satisfied:

a)

v2L¹_loc [0;T);L²_s(V)

\L²_loc [0;T);W₀^1;2(V)

; (2:16)

b) for each w2C¹₀ ([0;T);C_0;s¹(V)), hv;wti_V;T‡ hrv;rwi_V;T

(Jmv‡E)(v‡E);rw V;T

(J_mk)(v‡E);w

V;Tˆ

v₀;w(0)

V hF;rwi_V;T; (2:17)

c) for0t5T;

1

2kv(t)k²₂‡ Z^t

0

krvk²₂dt1 2kv₀k²₂

Z^t

0

F (J_mv‡E)E;rv

V dt

‡ Z^t

0

(J_mk 1 2k)v;v

V dt‡ Z^t

0

(J_mk)E;v

V dt; (2:18)

d) v:[0;T)!L²_s(V)is continuous satisfying v(0)ˆv₀:

3.The approximate system.

The following existence result yields a weak solutionvˆvmof (2.12) first of all only in an interval [0;T⁰) whereT⁰ˆT⁰(m)>0 is sufficiently small.

LEMMA3.1. Let f , k, E, v₀be as in Definition2.1and let m2N. Then there exists some T⁰ˆT⁰(f;k;E;v0;m), 05T⁰min (1;T), such that the approximate system(2.12)has a unique weak solution vˆvmin[0;T⁰)V with data f , v₀in the sense of Definition2.1with T replaced by T⁰.

PROOF. First we consider a given weak solution vˆv_m of (2.12) in [0;T⁰)Vwith any 05T⁰1. Hence it holds

v2X_T⁰ :ˆL¹ 0;T⁰;L²_s(V)

\L² 0;T⁰;W₀^1;2(V) with

kvk_XT0 :ˆ kvk_2;1;T0‡ kA¹²vk_2;2;T051:

(3:1)

Using HoÈlder's inequality and several embedding estimates, see [16, Ch. V.1.2], we obtain with some constantCˆC(V)>0 the estimates

k(Jmv)vk_2;2;T0 CkJmvk_6;4;T0 kvk_3;4;T0

CkA¹⁼²Jmvk_2;4;T0kvk_XT0

Cmkvk_2;4;T0Cm(T⁰)¹⁼⁴kvk²_XT0; (3:2)

and

k(J_mv)Ek_2;2;T0 CkJ_mvk_4;4;T0kEk_4;4;T0 CkJ_mvk_6;4;T0kEk_4;4;T0

(3:3)

Cm(T⁰)¹⁼⁴kvk_XT0 kEk_4;4;T0; kEvk_2;2;T0 CkEk_q;s;T0kvk₍1

2 1

q) ¹;(¹₂ ¹_s) ¹;T⁰ CkEk_q;s;T0kvk_XT0; (3:4)

of course,kEEk_2;2;T0CkEk²_4;4;T0. Moreover,

k(Jmk)vk_2;1;T0 CkJmkk_3;2;T0kvk_6;2;T0 Ck( D)¹²Jmkk_2;2;T0kvk_XT0 (3:5)

Cmkkk_2;2;T0kvk_XT0 Cm(T⁰)¹⁴kkk_2;4;T0kvk_XT0;

k(J_mk)Ek_2;1;T0 CkJ_mkk_4;2;T0kEk_4;2;T0 Ck( D)¹²J_mkk_2;2;T0kEk_4;4;T0

(3:6)

Cmkkk_2;2;T0kEk_4;4;T0 Cm(T⁰)¹⁴kkk_2;4;T0kEk_4;4;T0:

Using (2.14) and the energy estimate (2.9) withf0,F0replaced byfm(v), F Fm(v) we get from (3.2)-(3.5) the estimate

kvk_XT0 C kv0k₂‡ kFk_2;2;T0‡ kEk²_4;4;T0‡m(T⁰)¹⁴kvk²_XT0‡

‡m(T⁰)¹⁴kvk_XT0kEk_4;4;T0‡ kvk_XT0kEk_q;s;T0‡

‡m(T⁰)¹⁴kkk_2;4;T0(kEk_4;4;T0‡ kvk_XT0) (3:7)

withCˆC(V)>0.

Applying (2.10) to (2.14) we obtain the equation vˆ F_T⁰(v)

(3:8) where

F_T⁰(v)

(t)ˆ e ^tAv₀‡ Z^t

0

e ^{(t t)A}Pf_m(v)dt

‡ Z^t

0

A¹²e ^{(t t)A}A ¹²Pdiv F F_m(v) dt:

Let

aˆCm(T⁰)¹⁴; bˆCkEk_q;s;T0‡Cm(T⁰)¹⁴kEk_4;4;T0‡Cm(T⁰)¹⁴kkk_2;4;T0; dˆC(kv₀k₂‡ kEk²_4;4;T0‡ kFk_2;2;T0‡m(T⁰)¹⁴kkk_2;4;T0kEk_4;4;T0) (3:9)

withCas in (3.7). Then (3.7) may be rewritten in the form kF_T⁰(v)k_XT0 akvk²_XT0‡bkvk_XT0 ‡d:

(3:10)

Up to nowvˆvmwas a given solution as desired in Lemma 3.1. In the next step we treat (3.8) as a fixed point equation in XT⁰ and show with Banach's fixed point principle that (3.8) has a solutionvˆvmifT⁰>0 is sufficiently small.

Thus letv2X_T⁰ and choose 05T⁰min (1;T) such that the smallness condition

4ad‡2b51 (3:11)

is satisfied. Then the quadratic equationyˆay²‡by‡d has a minimal positive root given by

05y1ˆ2d

1 b‡ 

b²‡1 (4ad‡2b)

p ₁

52d

and, since y1ˆay²₁‡by1‡d>d, we conclude thatFT⁰ maps the closed ballBT⁰ ˆ fv2XT⁰ :kvk_XT0 y1ginto itself.

Further letv1,v22B_T⁰. Then we obtain similarly as in (3.10) the esti- mate

kF_T⁰(v₁) F_T⁰(v₂)k_XT0 Cm(T⁰)¹⁴kv₁ v₂k_XT0 kv₁k_XT0‡ kv₂k_XT0

‡Ckv₁ v₂k_XT0 kEk_q;s;T0‡m(T⁰)¹⁴kkk_2;4;T0‡m(T⁰)¹⁴kEk_4;4;T0

kv1 v2k_XT0 a(kv1k_XT0‡ kv2k_XT0)‡b

(3:12)

where

a kv1k_XT0‡ kv2k_XT0

‡b2ay1‡b54ad‡2b51:

(3:13)

This means thatFT⁰is a strict contraction onBT⁰. Now Banach's fixed point principle yields a solutionvˆvm2BT⁰of (3.8) which is unique inBT⁰.

Using (2.6)-(2.10) with f0‡divF0 replaced by fm(v)‡div (F Fm(v)) we conclude from (3.8) thatvˆv_mis a solution of the approximate system (2.12) in the sense of Definition 2.1.

Finally we show thatvis unique not only inB_T⁰, but even in the whole spaceXT⁰. Indeed, consider any solution~v2XT⁰of (2.12). Then there exists some 05Tmin (1;T⁰) such that k~vk_XT y1, and using (3.12), (3.13) withv₁,v₂replaced byv,~vwe conclude thatvˆ~von [0;T]. WhenT5T⁰ we repeat this step finitely many times and obtain thatvˆ~von [0;T⁰). This

completes the proof of Lemma 3.1. p

The next preliminary result yields an energy estimate for the approx- imate solutionvˆvm of (2.12). It is important that the right-hand side of this estimate does not depend onm2N. This will enable us to treat the limitm! 1and to get the desired solution in Theorem 1.4, a).

LEMMA 3.2. Consider any weak solution vˆvm, m2N, of the ap- proximate system (2.12) in the sense of Definition 2.1. Then there is a constant CˆC(V)>0such that the energy estimate

kv(t)k²₂‡ Z^t

0

krvk²₂dt

C kv0k²₂‡kFk²_2;2;t‡kkk⁴_2;4;t‡kEk⁴_4;4;t

exp Ckkk⁴_2;4;t‡CkEk^s_q;s;t (3:14)

holds for0t5T.

PROOF. The proof of (3.14) is based on the energy inequality (2.18).

Using similar arguments as in (3.2)-(3.6) we obtain the following estimates of the right-hand side terms in (2.18); here e>0 means an absolute con- stant, C₀ˆC₀(V)>0 and CˆC(e;V)>0 do not depend on m, and aˆ2

sˆ1 3

q. First of all

Z^t

0

h(Jmv)E;rvi_VdtC0

Z^t

0

kJmvk₍1 2 1

q) ¹kEk_qkrvk₂dt

C0

Z^t

0

kvk₍1 2 1

q) ¹kEk_qkrvk₂dt

C₀ Z^t

0

kvk^a₂kEk_qkrvk²₂ ^adt

ekrvk²_2;2;t‡C Z^t

0

kEk^s_qkvk²₂dt;

(3:15)

and

Z^t

0

hEE;rvi_V dtC₀ Z^t

0

kEk²₄krvk₂dtekrvk²_2;2;t‡CkEk⁴_4;4;t;

Z^t

0

hF;rvi_V dtekrvk²_2;2;t‡CkFk²_2;2;t:

Moreover, sincekvk₄C0krvk¹⁼⁴₂ krvk³⁼⁴₂ ,

Z^t

0

hJmkv;vi_V dtekrvk²_2;2;t‡C Z^t

0

kkk⁴₂kvk²₂dt;

Z^t

0

h(Jmk)E;vi_V dtC0

Z^t

0

k(Jmk)Ek⁶₅kvk₆ dt

C0

Z^t

0

kkk₂kEk₃krvk₂dt

ekrvk²_2;2;t‡C kkk⁴_2;4;t‡ kEk⁴_4;4;t :

A similar estimate as forR^t

0hJmkv;vi_Vdtalso holds forR^t

0hkv;vi_Vdt.

Choosinge>0 sufficiently small we apply these inequalities to (2.18) and obtain that

kv(t)k²₂‡ krvk²_2;2;tC kv0k²₂‡ kFk²_2;2;t‡ kEk⁴_4;4;t‡ kkk⁴_2;4;t

‡C Z^t

0

kkk⁴₂‡ kEk^s_q kvk²₂ dt

for 0t5T. Then Gronwall's lemma implies that

kv(t)k²₂‡ Z^t

0

krvk²₂dtC kv₀k²₂‡ kFk²_2;2;t‡ kEk⁴_4;4;t‡ kkk⁴_2;4;t

exp Ckkk⁴_2;4;t‡CkEk^s_q;s;t (3:16)

for 0t5T. This yields the estimate (3.14). p

The next result proves the existence of a unique approximate solution vˆvmfor the given interval [0;T).

LEMMA3.3. Let f , k, E, v₀be given as in Definition2.1and let m2N.

Then there exists a unique weak solution vˆv_mof the approximate system (2.12)in[0;T)Vwith data f , v0.

PROOF. Lemma 3.1 yields such a solution if 05T1 is sufficiently small. Let [0;T)[0;T),T>0, be the largest interval of existence of such a solutionvˆvmin [0;T)V, and assume thatT5T. Further we choose some finite T>T with TT, and some T0 satisfying 05T₀5T. Then we apply Lemma 3.1 with [0;T⁰) replaced by [T₀;T₀‡d) whered>0,T₀‡dT, and find a unique weak solutionvˆv_mof the system (2.12) in [T₀;T₀‡d)V with initial value vj_tˆT0ˆv(T₀). The lengthdof the existence interval [T0;T0‡d), see the proof of Lemma 3.1, only depends on kv(T0)k₂ kvk_2;1;T51 and on kFk_2;2;T, kEk_q;s;T, kkk_2;4;T, and can be chosen independently ofT0. Therefore, we can choose T₀close toTin such a way thatT5T₀‡dT. Thenvyields a unique extension ofvfrom [0;T) to [0;T₀‡d) which is a contradiction. This proves

the lemma. p

In the next step, see §4 below, we are able to letm! 1similarly as in the classical caseE0. This will yield a solution of the perturbed system (1.6).

4.Proof of Theorem 1.4.

It is sufficient to prove Theorem 1.4, a). For this purpose we start with the sequence (v_m) of solutions of the approximate system (2.12) constructed in Lemma 3.3. Then, using Lemma 3.2, we find for each finite T, 05TT, some constant C_T >0 not depending on m such that

kv_mk²_2;1;T‡ krv_mk²_2;2;TC_T : (4:1)

Hence there exists a vector field v2L¹ 0;T;L²_s(V)

\L² 0;T;W₀^1;2(V)

; (4:2)

and a subsequence of (vm), for simplicity again denoted by (vm), with the following properties, see, e.g. [16, Ch. V.3.3]:

vm * vinL² 0;T;W₀^1;2(V)

(weakly) vm!vin L² 0;T;L²(V)

(strongly) v_m(t)!v(t) in L²(V) for a:a:t2[0;T):

(4:3)

Moreover, for allt2[0;T) we obtain that krvk²_2;2;t lim inf

m!1 krv_mk²_2;2;t; kv(t)k²₂ lim inf

m!1 kv_m(t)k²₂: (4:4)

Further, using HoÈlder's inequality and (4.2)-(4.4) we get with some further subsequence, again denoted by (vm), that

vm*v in L^s1 0;T;L^q1(V)

; 2 s1‡ 3

q1ˆ3

2; 2s1;q151;

v_mv_m*vv in L^s2 0;T;L^q2(V)

; 2 s₂‡ 3

q₂ˆ3; 1s₂;q₂51;

vm rvm*v rv in L^s3 0;T;L^q3(V)

; 2 s₃‡ 3

q₃ˆ4; 1s3;q351; (4:5)

and that with some constantCˆCT>0:

k(J_mv_m)v_mk_q2;s2;T Ckv_mk²_q1;s1;T

(4:6)

k(Jmvm)Ek₍1

q‡_q¹₁) ¹;(¹_s‡_s¹₁) ¹;T Ckvmk_q1;s1;TkEk_q;s;T

(4:7)

kEvmk₍1

q‡_q¹₁) ¹;(¹_s‡_s¹₁) ¹;T Ckvmk_q1;s1;TkEk_q;s;T

(4:8)

(J_mv_m)E;rv_m

V;TCkv_mk_q1;s1;TkEk_q;s;Tkrv_mk_2;2;T

(4:9) as well as

kv_m;v_mi_V;TCkkk_2;4;Tkv_mk²_q1;s1;T

(Jmk)vm;vm

V;TCkkk_2;4;Tkvmk²_q1;s1;T

(Jmk)E;vm

V;TCkkk_2;4;TkEk_q;s;Tkvmk_q1;s1;T: (4:10)

The theorem is proved when we show that (2.16)-(2.18) imply letting m! 1the properties (1.8)-(1.10) and the estimate (1.28). This proof rests on the above arguments (4.1)-(4.10).

Obviously, (1.8) follows from (4.1), lettingm! 1. Further, the relation (1.9) follows from (2.17) and (2.4) using that

hvm;wti_V;T ! hv;wti_V;T

hrv_m;rwi_V;T ! hrv;rwi_V;T

(J_mv_m‡E)(v_m‡E);rw

V;T !

(v‡E)(v‡E);rw

V;T

(Jmk)(vm‡E);w

V;T !

k(v‡E);w

V;T: (4:11)

To prove the energy inequality (1.10) we need in (2.18), lettingm! 1, the following arguments.

The left-hand side of (1.10) follows obviously from (4.4). To prove the right-hand side limitm! 1in (2.18) we first show that

(Jmvm)E;rvm

V;T ! hvE;rvi_V;T: (4:12)

It is sufficient to prove (4.12) withEreplaced by some smooth vector fieldE~ such thatkE Ek~ _q;s;Tis sufficiently small. This follows using (4.9) withEreplaced byE E. Thus we may assume in the following that~ Ein (4.12) is a smooth function E2C¹₀ ([0;T);C₀¹(V)). Using (4.1)-(4.4) and (2.4), we conclude that

(Jmvm)E vE;rvm

V;T

k(Jmvm)E vEk_2;2;T krvmk_2;2;T

C(E)kJ_mv_m vk_2;2;T

C(E) kJm(vm v)k_2;2;T‡ k(Jm I)vk_2;2;T

C(E) kv_m vk_2;2;T‡ k(Jm I)vk_2;2;T

!0 asm! 1whereC(E)>0 is a constant. This yields (4.12).

Similarly, approximatingkby a smooth functionk2C₀¹([0;T);C¹₀ (V)), we obtain the convergence properties

hkvm;vmi_V;T ! hkv;vi_V;T; (Jmk)vm;vm

V;T ! hkv;vi_V;T; (Jmk)E;vm

V;T ! hkE;vi_V;T:

SinceE2L⁴(0;T;L⁴(V)), the convergencehEE;rvmi_V;T! hEE;rvi_V;T

is obvious.

This proves thatvis a weak solution in the sense of Definition 1.1.

To prove the energy estimate (1.28) we apply (4.4) to (3.14). This com-

pletes the proof. p

5.More general weak solutions.

The existence of a weak solutionvfor the perturbed system (1.6) under the general assumption onEin Theorem 1.4 a) enables us to extend the solution class of the Navier-Stokes system (1.1) using certain generalized data. For simplicity we only consider the casekˆ0.

THEOREM5.1 (More general weak solutions). Consider f ˆdivF; F2L² 0;T;L²(V)

; v₀2L²_s(V);

(5:1)

E2L^s 0;T;L^q(V)

; 4s51; 4q51; 2

s‡3 qˆ1;

(5:2) satisfying

E_t DE‡ rhˆ0; divEˆ0 (5:3)

in(0;T)Vin the sense of distributions with an associated pressure h.

Let v be a weak solution of the perturbed system(1.6)in[0;T)V in the sense of Definition1.1with E;f;v0from(5.1)-(5.3).

Then the vector field uˆv‡E is a solution of the Navier-Stokes system

ut Du‡u ru‡ rpˆf; divuˆ0 (5:4)

uj_@V ˆg; uj_tˆ0ˆu₀ (5:5)

in[0;T)Vwith external force f and (formally) given data g:ˆEj_@V ; u₀:ˆv₀‡Ej_tˆ0 ;

(5:6)

in the generalized (well-defined) sense that

(u E)j_@V ˆ0; (u E)j_tˆ0ˆv₀;

and (5.4) is satisfied in the sense of distributions with an associated pressure p.

REMARK5.2. (Regularity properties)

a) Let E in(5.2)be regular in the sense that g and E0ˆEj_tˆ0in(5.6) have the properties in Lemma 1.2.Then the solution uˆv‡E has the properties in Theorem1.4, b).

b) Let E in(5.2)be regular in the sense that g and E0ˆEj_tˆ0in(5.6) have the properties in(1.26).Then the solution uˆv‡E is correspond- ingly regular and(5.5)is well-defined in the usual strong sense.

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Manoscritto pervenuto in redazione l'8 luglio 2010.