the navier-stokes equations for the motion of compressible, viscous fluid flows with the no-slip ...

7/29/2019 The Navier-Stokes equations for the motion of compressible, viscous fluid flows with the no-slip boundary condition

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The Navier-Stokes equations for the motion of

compressible, viscous fluid flows with the no-slip

boundary condition

Mikhail Perepelitsa

Abstract

The Navier-Stokes equations for the motion of compressible, viscous fluids

in the half-space R3+ with the no-slip boundary condition are studied. Given a

constant equilibrium state (, 0), we construct a global in time, regular weak

solution, provided that the initial data 0, u0 are close to the equilibrium state

when measured by the norm

|0

|L +

|u0

|H1

and discontinuities of 0 decay near the boundary ofR3+.

0.1 Introduction

We consider a model for the motion of a compressible, isothermal, viscous

flow based on the Navier-Stokes equations. With (t, x) and u(t, x) being the

density and the velocity of the fluid, the model consist of the equations:

t + div (u) = 0, (1)

t(u) + div (u u) ( + )div u u + P = f, (2)

3 + 2 0, > 0,(t, x) R+ , P() = A, A > 0,

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and a set of initial and boundary conditions:

((0, x), u(0, x)) = (0(x), u0(x)) , x , (3)u(t, x) = 0, (t, x) R+ . (4)

There is an extensive literature concerning different aspects of the problem

(1) (4). For the detailed discussion of the results we refer the reader to the

recent monographs [3, 13]. We shortly mention some of them. It is known that

if the initial data of the problem are smooth then the problem is well-posed.

Moreover, a unique, global solution exists if the initial data are close to a

static equilibrium state, measured in strong norms, for example in H3 R3+ ,

see [11, 15]. On the other hand, there is a well-developed theory of weaksolutions of the problem (1) (4) and other related problems, see [8, 3]. A

typical result is contained in the following theorem.

Theorem (P.-L. Lions, [8]). Suppose that 95 and C2+, > 0. Sup-pose that the initial data(0, m0) satisfy L() , |m0|2/0 L1 () , wherewe agree that m0 = 0 on {0() = 0} . Then there is a global weak solution ofthe problem (1)(4), (, u), such that (0, ) = 0() and (0, )u(t, ) = m0Moreover, for any t > 0 the energy inequality holds.

12 (t, )|u(t, )|2 + A(t, )

1

+

t0

|u|2 + ( + )| div u|2

1

2|u0|2 + A

0

1

.

Solutions constructed in the above theorem have somewhat limited reg-

ularity properties: L (R+ : L()) and u L2R+ : W

1,2 ()3

, and

thus, may incorporate some non-physical phenomena.

For the Cauchy problem, i.e. when the flow occupies the whole space

R3, global existence of weak solutions that remain near a static equilibrium

sate, (, 0), was proved in [4], see also [5] for related results. In contrast

with the result of [11], solutions built in [4] are essentially weak; the density

is an element of L. On the other hand, solutions possess many favorable

properties, such as, impossibility of spontaneous formation of vacuum, the

fact which is being implicitly assumed when equations (1) (2) are used to

model a motion of real fluids.

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Theorem (D. Hoff, [4, 5]). Let = RN, N = 2, 3. Let > 0 and L > 0 be

given. There is a positive number c = c(N) and a pair of positive numbersA, C depending on (,, ,L,N,c), with the property that if

+ c (5)

and the initial data (0, u0) satisfy bounds

0 0 , a.e. R3+,RN

|u0(y)|2 + (0(y) )2 dy A

and |u0|L2N(RN)N L,then, a global weak solution (, u) of the problem (1)(3) exists for which

||L(R+RN) C,

u L ({t : t > } R)N , > 0.(We refer the reader to [5] for the complete statement of the Theorem.)

The analogous result was obtained for flows in domains with boundaries

under the Navier boundary condition, i.e. the condition that tangential veloc-ity at the boundary is proportional to the tangential component of the stress,

see [5].

In this work we present a development of the existence theory of the near

equilibrium weak solutions initiated in [4] to the problems with the no-slip

boundary condition (4). The density component of the weak solution that we

construct is L away form the boundaries and such that discontinuities in

(t, ) decay near the boundary. Specifically, we measure (t, ) by the norm

(t,

)

+

|(t,

)

|LL2 ,

]0, 1[,

where is defined in (13). The localization of discontinuities in (t, ) insidethe domain corresponds to a physical situation when motion of a fluid results

from disturbances that occur in the interior of the domain. At the level of

technical description of the proof, the introduction of the above functional

to measure the density is dictated by the fact that the L norm along is

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then, there exists a weak solution, (, u), of the problem (1) (4), defined

for all times t > 0. Moreover, for t R+, the following estimates hold withindependent of time c2 > 0.

osc (t, ) + (t, ) < c2(osc (t, ) + (t, ) + |u0|H10 ),

|u(t, )|H10 (R3+) + |(t, ) |L2(R3+) c2(|u0|H10 (R3+) + |0 |L2(R3+)),

L6 R+ R3+ ,xu L2(R+ : L6 R3+),

(t)(ut + u u) L(R+ : L2R3+

) L2(R+ : H10

R3+

),

(8)

where (t) = min{1, t}.

Remark 1. Since the solution, constructed in theorem 2, is such that the

oscillations in density are small, there is no loss of generality in assuming

the pressure law P = A instead of the isentropic law, P = A, 1.Indeed, the derivation of a priori estimates in case > 1 is identical to case

= 1. Moreover, the strong convergence of the sequence of the approximate,

classical solutions for > 1 is established by the Lions-Feireisl theory, see [3].

Remark 2. We will prove theorem 2. Theorem 1 follows by a rather straight-

forward argument as all apriori estimates that we establish for theorem 2 can

be obtained locally in time for solutions with arbitrary large initial data.

The framework of the analysis was established in works [4, 8]. We shortly

describe the new issues appearing in the problem with the no-slip boundary

conditions. First, we notice that unlike the situation for the Cauchy problem,

L norm alone is not well-suited to measure the density of a solution. Indeed,

the Navier-Stokes equations (2) can be written as a problem

( + ) div u + u = a + ( ),u = 0, R3+,

(9)

where a = Dtu the acceleration. Assuming for a moment that a = 0, the

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Lemma 2. Let u be a locally integrable function with u Lp R3+ , p > 3.

Then, there is c = c(p) such that for a.e. x, y R3+ it holds

|u(x) u(y)| c|x y||u|Lp ,

where = 1 3p1.

Definition 2. A pair of functions

(, u) = ((t, x), u1(t, x), u2(t, x), u3(t, x))

is called a weak solution of (1)-(4) if

, ui, ui L1locR+ R3+

, i = 1, 2, 3,

uk ul L1locR+ R3+

, i, k,l = 1..3,

u L2 R+ R3+ ,u = 0, on R3+,

and for all test functions , i C

[t, T] : C0R3+

, i = 1, 2, 3, with

0 t < T < + it holds (summation over the repeated indexes is assumed)

R+R

3+

t + u

R3+(,

)(,

)

T

t

= 0,

R+R3+

uktk + ukujkj

R+R3+

( + )div u div + kulkl + (P P)kk

R3+

(, )uk(, )(, )Tt

= 0.

To simplify the presentation we assume that the constant A = 1 in (2). It is

always possible to reduced to this case through the substitution (t,x,, u) (a2t, ax, , au), a = A

12 , without changing the viscosity coefficients.

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0.5 1st energy estimate

In all subsequent estimate we assume

Hypothesis H0. For all (t, x),

(t, x) < M := 10.

Lemma 6. Let

() =

s2(s ) ds, 0

and

E(t) = R3+

(t,

)

|u(t,

)

|2/2 + ((t,

)).

Then, for any smooth solution (, u) of the problem (1)(4) the following

equality holds.

E(t) +

t0

R3+

( + 2)| div u(t, )|2 + | curl u(t, )|2 = E(0). (15)

The proof of this Lemma is well-known and can be found, for example, in

[4].

0.6 Bogovski operator in a half-spaceThe estimates of lemma 8 of this subsection can be found in [?]. We give here

an alternative, short proof of these estimates. Consider a problem

v = P, vR3+

= 0.

If H = |x y|1 a fundamental solution for the Laplaces equation, thenthere is a representation formula for

div v = P

R3+ y xH(x, y)P(y) dy,

where y = (y1, y2, y3). Let

AP :=R3+

y xH(x, y)P(y) dy.

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On the other hand,

t0

R3+

|u||v| ct0

|u|2|v|6/5 ct0

|u|2| |56

Ct0

|u|22 + t0

| |66. (21)

Combining above estimate and choosing the appropriate value of we obtain

the next lemma.

Lemma 9. There is c > 0 independent of t > 0 such that

t0

H

| |6 cE(0) + E2(0) + t0

|u|22

. (22)

0.8 Energy estimates of higher order

The following Lemmas are proved in exactly the same way as Lemma ?? from

[4]

Lemma 10. Under hypothesisH0, for t > 0 it holds:

sups[0,t] R3

+

|u(s, )|2 + t

0 R3+

|u(s, )|2 c(|u0|2, |u0|2, |0 |2)

+

t0

R3+

|u|3.

Corollary 1. For any > 0 there is a C such that

sups[0,t]

R3+

|u(s, )|2 +t0

R3+

|u(s, )|2 c(|u0|2, |u0|2, |0 |2)

+

t0

|u(s, )|22 + |(s, ) |66 + Ct0

|u(s, )|62 + |u(s, )|22 . (23)Proof. Indeed, by the Holder inequality and elliptic estimates of lemmas 3, 5

we have:

|u|36 |u|3/22 |u|3/26 c|u|3/22 (|u|2 + | |6)3/2

|u|22 + | |66 + C|u|62 + |u|22 (24)

and the corollary follows from the previous lemma and lemma 9.

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0.9 Representation formulas for the solution of (14)

Now, we derive a representation formula for u. Let G be a Greens matrix for

the problem (14), i.e.

u(x) =

R3+

G(x, y)f(y) dy.

The explicit expression for the Greens matrix can be found, for example, in

[14]. Let

A = +

2( + 2), B =

+ 3

+

and ik be the Kronecker symbol. Then,

Gki (x, y) = A

Bik + (xi yi)

yk

1

4|x y| 1

4|x y|

+ x1

ik B1y1

yk

1

2

xi

1

|x y| , i, k = 1, 2, 3. (28)

Let v, w be two vector fields defined for any t > 0 as the solutions of the

following problems.

Lv = u, x H, vH

= 0

and

Lw = , x H, wH

= 0,

where L = div + . It follows from the above considerations that

u(t, x) = v + w (29)

and

u =

R3+

G(x, y)u(t, y) dy, w =

R3+

G(x, y)((t, y) ) dy. (30)

Now, we compute a representation formula for div w.

For x = (x1, x2, x3) R3+, let x = (x1, x2, x3). Let

H(x, y) =1

4|x y| ,

and denote by

G(x, y) = H(x, y) + H(x, y) (31)

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and

|D2

v|4,B1(x)H c (|u|4 + |u|2 + |u|2 + | |6) .The first statement of the lemma follows. To prove the second we notice that

|u|4 |u|1/42 |u|3/46

and for s < t

ts

|u|4 t0

|u|1/42 |u|3/46 3/83/8 ts

1|u|22 + 1s|u|26 + 3/4

(41)

and ts

|u|2 + | |6 ts

1|u|22 + 5| |66 + .

We conclude by applying estimates (25) and (27) and a trivial estimatets

3/4 4 + (t s).

We will work out the regularity of w later, after developing some potential

estimates in the next subsection.

0.10 Some potential estimates

Let P1(x) be the function from (37). We will need the following lemma.

Lemma 13. For any ]0, 1[ and > 0 there arec > 0, c > 0, independentof (, , , t), such, that

P1 c (|() |2 + ) , (42)

and for any x R3+,

|P1(x)| + c|() |2. (43)

Proof. We proof only the first part of the Lemma. The proof for the second

part goes along the same line of arguments. Let x1, x2 R3+, |x1 x2| < and set B2 = B(x1, 2), B1 = B(x1, 4), B = B(x1, 2|x1 x2|),

S2 = R3+ B2

\ R3+,

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and

S = R

3+ B \ R3+.

We can write the following representation for

P1(x1) P1(x2) = ((x1) (x2))

R3+\B2

{ div yxG(x1, y) div yxG(x2, y)} ((y) )

+

S2

{xG(x1, y) xG(x2, y)} ny((y) ) dSy

B2\B1 { div yxG(x1, y) div yxG(x2, y)} ((y) (x1))B1

{ div yxG(x1, y) div yxG(x2, y)} ((y) (x1))

+

S2

{xG(x1, y) xG(x2, y)} (ny)((y) (x1)) dSy. (44)

We set x2 to be the projection of point x2 onto R3+. and consequently,

P1(x1) P1(x2) = ((x1) (x2))

R3+\B2

{ div yxG(x1, y) div yxG(x2, y)} ((y) )

+S2

{xG(x1, y) xG(x2, y)} ny((x1) ) dSy

B2\B1


B1\B


B

div yxG(x1, y)((y) (x1))

+

B

div yxG(x2, y)((y) (x2))

+ ((x2) (x1))S

div yxG(x2, y) ny dSy. (45)

Finally, we split G = H + H as in (31) and use the fact that H is a fun-

damental solution of the Laplaces equation. With the notation K(x, y) :=

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which approximates the given initial data in the space L6loc(R3+) L6 R

3+

3.

Moreover, we require that M > n0 > m > 0, and smallness condition (7) as

required by analysis of the previous sections. Such a sequence, clearly exists.

We can take n0 (x) = (0(x) + n1) n1(x), un0 = (u0(x)) n1(x), where

is the standard mollifier. Accordingly, let n, un be the sequence of smooth

solutions of the problem with n0 , un0 as the initial data. The existence of such

solutions follows from the local existence result [11] and a priori estimates

we obtained in the previous sections. In particular, we established that the

following norms are bounded with bounds independent of n.

{n} bounded in L R+ R3+ , (56){nun} bounded in L R+ : L2(R3+) , (57)

{un} bounded in L2 R+ R3+ , (58){D2un} bounded in L2 R+ R3+ . (59)

By the weak stability result of P.-L. Lions, see Theorem 5.1 of [8], bounds

(56)(58) imply the existence of an accumulation point (, u) of the sequence

(n, un) in the weak topology of L6loc(R3+) L6(R3+) which is a weak solution

of (1)(4). Moreover, the bounds in the spaces from (56)(59) hold for this

(, u).

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