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Transactions on Engineering and Computing Sciences - Vol. 11, No. 1

Publication Date: February 25, 2023

DOI:10.14738/tecs.111.13785.

Tong, M. (2023). Strong Maximal Lq

- L

-estimates for the Solution of the Stokes Equation. Transactions on Engineering and

Computing Sciences, 11(1). 132-142.

Services for Science and Education – United Kingdom

Strong Maximal L

− L

− estimates for the Solution

of the Stokes Equation

Maoting Tong

Department of Mathematical Science, The University of Liverpool, U.K.,

Public Basic Department,Nanjing Vocational University of Industry Technology,

Nanjing, China

Abstract

Suppose that  is a bounded domain in R

with smooth boundary  of class C

. In

this paper we prove that if the initial value u0

D((−)

), then there are strong

maximal L

− L

-estimates for the solution of the Stokes equation (1) for any 1<q<

,2  p<. Mathematics Subject Classification (2010). Primary 35Q30, 76D07;

Secondary76N10, 47D06.

Keywords. Navier-Stokes equation, Existence and uniqueness, Global solution,

Semigroup of operators, Invariance, Banach lattice, Fractional powers.

The Navier-Stokes initial value problem can be written in its classical form as

where  is a bounded domain in R

with smooth boundary  of class C

, u = u(t, x) =

(u1

(t,x),u2

(t,x),u3

(t,x)) is the velocity field, u0

=u0

(x) is the initial velocity,p = p(t, x) is the

pressure, f = f (t, x) is the external force,

In these four equations u, p are unknown and f , u0

are given. The boundary condition imposed

on the velocity at  is homogeneous. If the nonlinear term (u•)u=0 we get the following Stokes

equation with the classical Dirichlet boundary condition

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133

Tong, M. (2023). Strong Maximal Lq

- L

-estimates for the Solution of the Stokes Equation. Transactions on Engineering and Computing Sciences, 11(1).

132-142.

URL: http://dx.doi.org/10.14738/tecs.111.13785

The existence,uniqueness and regularity properties of solutions for the Stokes equations are

extensively studied. There is an extensive literature on the solvability of the initial value

problem for Stokes equations.

Let Lp

() ( 2 p ˂) be the Banach space of real vector functions in LP () . That is

For u=(u1

,u2

,u3

)LP

(), we define the norm

then LP

() is a Banach space. The set of all real vector functions u such that div u=0and

is denoted by Let DL () be the closure of in L().If then

uDL() implies divu=0.(seep.270in [3]). Similarly if u  C  () then

uDLp

()DL2

()impli div u=0. (2)

In this paper we always consider the spaces of vector value functions on  . Some authors den

ote DLp () by L

 (). From [3],[7] and [10,p.129] we have the Helmholtz decomposition

() = DLp

()  DLp

()

⊥

where

Let P be the orthogonal projection from Lp

() onto DLp

(). By applying

P to the first equation of (1) and taking account of the other equations , we are let the following

abstract initial value problem, Pr . II

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We consider equation(3)in integral form Pr.III

The projection operator P is called the Helmholtz projection. An important tool in order to

handle the incompressible condition div u = 0 is the Helmholtz decomposition. In the simplest

case, where   R

is a domain with compact and smooth boundary, the operator Ap

given by

is called the Stokes operator.([10.p.119]).

By maximal L

− L

− estimates for the solution of the Stokes equation , one understands the

following: given1<p,q< and fL

((0,T);DLp

()) does there exist a unique uL

((0,T);D(Ap

))

with t

uL

((0,T);DLp

()) satisfying

and a constant C >0, independent of f , such that

It was shown by Geissert, Heck, Hieber,and Sawada [4] that the existence of the Helmholtz

decomposition implies the existence of the Stokes semigroup on L

 () and that the Stokes

operator admits maximal regularity for domains  with uniform C

− boundaries   (but

possible non-compact) and u0

= 0 . It is known that the Helmholtz decomposition exists for p =

2 for arbitrary open sets . But note that due to the results by Maslennikova and Bogovoskii

[14], the Helmholtz decomposition for Lp

() does not exist in general if p  2. Hence, a

characterization of those domains, even with smooth boundaries , for which the solution of the

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135

Tong, M. (2023). Strong Maximal Lq

- L

-estimates for the Solution of the Stokes Equation. Transactions on Engineering and Computing Sciences, 11(1).

132-142.

URL: http://dx.doi.org/10.14738/tecs.111.13785

Stokes equation is governed by an analytic semigroup on DLp

() remains a challenging open

problem until now.

The existence of the Helmholtz decomposition is not necessary for the existence of the Stokes

semigroup by Bolkart, Giga, Miura, Suzuki, and Tsutsui [2] shows. Maximal L

−L

− estimates

for bounded or exterior domains with smooth boundaries were proved first by Solonnikov [21],

while maximal mixed L

− L

− estimates were obtained first by Giga and Sohr [9] for bounded

or exterior  and . In [5] M.Geissert, M.Hess, M.Hieber, C.Schwarz, K.Stavrakidis

gave a short and new proof for maximal L

− L

-estimates for bounded or exterior  with

compact C 3

-boundary ∂Ω and u0

= 0. Nowweintroduceanewconcept.Let C



((0,T);X) denote the

space of Hölder continuous functions on (0,T ) with exponent  and with values in a Banach

space X . By strong maximal L

− L

− estimates for the solution of the Stokes equation ,one under

stands the following:given1<p,q< and fC



((0,T);DLp

()) does there exist a unique strong

solution uL

((0,T);D(Ap

))of the Stokes equation

with t

uL

((0,T);DLp

()), and a constant C>0,independent of f, such that

In this paper we will prove that the solution of the Stokes equation (1) is governed by an

analytic semigroup on L

 () for bounded domain  in R

with smooth boundary  of class

C 3

by using the theory of semigroup of operators. If the initial value then there

exists a strong maximal L − L − estimates for the solution of the Stokes equation for bounded

domains  with smooth boundary. Although f C



((0,T);DLp

()) implies f  L

((0 T);DLp

()),

our result is always different from the previous result: u is the strong solution of (3), ∂Ω is

smooth but does not need to be compact and u0

does not need to be zero.

For we define u = (u1

, u2

, u3

) and Since the

operator is strongly elliptic of order 2.

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Theorem7.3.6 in [17] implies that  is the infinitesimal generator of an analytic semigroup of

contractions on L

() with D() = W

2, p

() W

p () . Hence  is also the infinitesimal

generator of an analytic semigroup of contraction on Lp

() with D() = W2p () W1

p () ,

where W2p() and W1

p () are the Sobolev spaces of vector value in W

2,p

()and W0

1,p

()

respectively.We will prove that  is also the infinitesimal generator of an analytic semigroup of

contraction on DLp

().

A operator A is called preserving divergence-free on a vector value functions space X if A map

severy uX with div u=0 toan Au with div Au=0.

Lemma 1.(Lemma 1 in [22]) For every uLp

() , div u = 0 if and only if div(I−)u=0.

for



=:− <arg<−,r

where 0˂˂

Lemma2.(1.5.12in[5])Let T(t):t0 bea C0

-semi group on a Banach space X.If Y is a closed

subspace of X such that T(t)YY for all t0,i.e.,if

Yis T(t)

t0

-invariant,then the restrictions

form a C0

-semigroup T (t) : t  0, called the subspace semigroup, on the Banach space Y.

Lemma 3. (Proposition 2.2.3 in [ 5 ]) Let ( A, D( A)) be the generator of a C0

-semigroup T (t) :

t  0on a Banach space X and assume that the restricted semigroup (subspace semigroup)

T (t) : t  0 is a C0

-semigroup on some (T (t))t0

− invariant Banach space Y → X . Then the

generator of T (t) : t  0 is thepart (A,D(A)) of A in Y.

Lemma 4. The operator  with D()  DLp

() is the infinitesimal generator of an analytic

semigroup of contractions on DLp

().

Proof. From Theorem 7.3.6 in [17]  is the infinitesimal generator of an analytic semigroup of

contractions on L

(). Then  is also the infinitesimal generator of an analytic semigoup of

contractions on Lp

(). Let T (t) t  0 be the restriction of the analytic semigroup generated

by  on Lp

() to the real axis . T (t) t  0is a C0

semigroup of contractions by Theorem 7.2.5

and Theorem 3.1.1 in [17]. We have already noted that DLp

() is a closed subspace of Lp

()

and is also a Banach space. We want to show that DLp

() is T (t)t0 − invariant. For every

uLp

()with div u=0 and()

=:−<arg<−,rwe have (I − )R( : )u= u

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137

Tong, M. (2023). Strong Maximal Lq

- L

-estimates for the Solution of the Stokes Equation. Transactions on Engineering and Computing Sciences, 11(1).

132-142.

URL: http://dx.doi.org/10.14738/tecs.111.13785

where 

is the same as in the proof of lemma 1. From Lemma 1 it follows that div R(:)u=0.

That is to say that R(:) is preserving divergence-free for    ()  

. From Theorem 2.5.2

, and so () :r. Hence R(:) is preserving

divergence free for every   r. Let u  DLp

() then there exists a sequence un

such that

and div un

=0 for n =1,2,... . Since R( : ) is bounded and so is continuous . Hence

and div R( :)un

= 0 for every   r. Therefore R( :)uDLp

() for every

  r. It follows that DLp

() is R( : ) -invariant for every   r. Now the Theorem 4.5.1 in [17]

implies that DLp

() is T (t)t0 − invariant. From Lemma 2 and Lemma 3 it follows that  DLp

()with D(DLp())=D()DLp() is the infinit esimal generator of the C0

semigroup T(t)

DLp():t0of contractions on DLp

().

We will prove that T(t) DLp():t0 can also be extended to an analytic semigroup on

on DLp(). Suppose that (),i.e. there exists R(:) from Lp

() intoD(). Then for

any uDLp

()Lp

(), R(:)uDLp

(). We have

(I−)R(:)u=u

R(:)(I−)u=u. (5)

Since R( : ) is preserving divergence-free we have

R(:)uDLp

() and (I−)uDLp

().

Thus the formula (5) becomes

From the formula (6) and Theorem 2.5.2(c) in [17] we have

( DLp())() =(:arg

2+0)

=1

=:arg˂1

,r

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invertible. From Theorem 2.5.2(c) in [17] we have

Now Theorem 2.5.2(c) in [17] implies that can also be extended to an analytic

semigroup on DLp

() . Therefor  DLp

() is a infinitesimal generator of an analytic

semigroup of contraction on DLp

() .

Lemma5.Ap

= DL2() on DLp

().

Proof. We have

D(DLp())=D()DLp

() and D(Ap

):=W

2,p

()W0

1,p

()DLp

().

Hence

D(DLp ()) = D(Ap

On the other hand, we proved in lemma 4 that  DLp () with is the infinitesimal generator of

the C0 semigroup T (t)

DLp(): t  0 of contractions on DLp

(). That is, DLp

()is a invariant

subspace under / DLp(). And Ap

u:=Pu. So for any u D(DLp()) =D(Ap

) uDLp

(). But P is

the orthogonal projection from Lp

() onto DLp

(). So

Pu = u.

Finally we have

Ap =

DLp()



Theorem 2 in [6] is similar the above Lemma 5. Sobolevskii proved the fact that Ap = P

generates an analytic semigroup on L

()= DLp ()in [20 ] . Giga gave a different proof in [6].

Our proof using the theory of semigroups of bounded operators is more simple.

Suppose that − A is the infinitesimal generator of an analytic semigroup T (t) on a Banach space

X . From the results of section 2.6 in [17] we can define the fraction powers A



for 0    1 and



is a closed linear invertible operator with domain D(A



) dense in X. D(A



) equipped with the

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139

Tong, M. (2023). Strong Maximal Lq

- L

-estimates for the Solution of the Stokes Equation. Transactions on Engineering and Computing Sciences, 11(1).

132-142.

URL: http://dx.doi.org/10.14738/tecs.111.13785

norm is a Banach space denoted by X

. It is clear that 0< <  implies X  X

and

that the embedding of X

into X

is continuous.

Assumption (F). Let X = DLp

() and U be an open subset in R

 X

(0˂ ˂1). The function

f :U → X satisfies the assumption (F) if for every (t,u)U there is a neighborhood VU and

constants L0, 0<1 such that for all (t

,ui

)V(i=1,2)

Lemma 6. ( Theorem 6.3.1 in [17]) Let − A be the generator of an analytic semigroup T(t)

satisfying and assume that 0(−A). If, 0<<1 and f satisfies the assumption (F)

then for every initial date (t

, u0

) U the initial value problem

has a unique local solution uC(t

): X)C

((t

): X) where t

,u0

)˃t

Now we study the Stokes equation (1).

A function u which is differentiable almost everywhere on 0, T  such that u  L1 0,T : DL2

() is called a strong solution of the initial value problem (1) if u(0) = u0

and u satisfies (1)

a.e. on 0,T .

Theorem 1. Suppose that  is a bounded domain in R

with smooth boundary  of class C

The Stokes equation (1) has a unique local strong solution if the initial value u0 D((−)

), f

C



((0,T);DLp

()), i.e. there exist constant C and 0 ˂  ˂1 such that

for s,t(0,T) and x.

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Proof. We will find that by incorporating the divergence-free condition, we can remove the

pressure term from our equation. (see p. 271 in [3], p. 2346

and p. 2399

in (9). So first we can

rewrite (1) into a abstract initial value problem on DLp

()

where F(t,u(t)) = f is a abstract function. From Lemma 4  is the generator of an analytic

semigroup T (t) of contraction on DLp

() and //T (t)  1//. From Theorem 2.5.2(c) in [17] 0

  ().

Let U=R

DLp

()1/2

.Then U is a open set and F(t,u(t))=f(t)is a function: U→DLp

(). Since

u0D((−)

)=DLp

()1/

, (0,u0

)U.

From(8)itfollowsthatforall (t

,ui

)U(i=1,2)

Hence F(t,u(t)) satisfies the assumption (F) , then by Lemma 6 for every initial data (0,u0 )U

the initial value problem (9) has a unique local solution

wheret

(u0

In a similar induction way as Theorem 3.9 in [2] or as Theorem 5.1 in [19] we can prove that

the solution We can also prove directly that u(t,x) is smooth. In fact, the

solution (10) of (9) is also the solution of (4). The Theorem 3.4 in [2] mean that as long as the

solution of (4) exists , this solution is smooth. From Theorem 3.4 in [2] we have the solution

Substituting u(t,x)into(1)we get the solution p(t,x).We also have p(t,x)

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141

Tong, M. (2023). Strong Maximal Lq

- L

-estimates for the Solution of the Stokes Equation. Transactions on Engineering and Computing Sciences, 11(1).

132-142.

URL: http://dx.doi.org/10.14738/tecs.111.13785



((0,t

)). It follows from the formula (2) and u,p DLp

() that the solution u(t, x) and

p(t, x) are divergence-free. Changing the value of u on  to zero we get a unique local strong

solution u = u(t, x),p(t, x) of the Stokes equation(1) 

From(1) we see that u(t)D()=W

2,p

()W0

1,p

()DLp

(). From(10) u(t) C(0,t

):DLp

())

and u

(t)C((0,t

):DLp

())forany 1<q<.Hence u(t) L

((0,t

):D(Ap

)) and u

, L

((0,t

): DLp

())

for any 1<q<. From(9)we have

−u= f.

Hence we get

Therefore we have

Theorem 2. Suppose that  is a bounded domain in R

with smooth boundary  of class C

If the initial value u0  D((−)

), then there are strong maximal L

− L

-estimates for the

solution of the Stokes equation (1) for any 1 < q < , 2 p<.

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