10 The Incompressible Navier-Stokes Equations Under Initial and Boundary Conditions

10

The Incompressible Navier-S tokes Equations under Initial and Boundary Conditions

In two space dimensions the nonlinear equations have the form (10.1.1)

ut

+ uu, + vuY + gradp = V A U+ F,

u,

+ ul/ = 0,

u = ( u ,v). F = ( F ,G), x = (z,y). We consider the equations in the region

OlZLl,

-co
t>o,

and prescribe initial conditions

u ( r ,Y,0 ) = WAZ.Y). d r ,y. 0 ) = V O ( T , y),

mr

+ uol, = 0 ,

and boundary conditions at z = 0, T = 1. All data are assumed to be Cm-smooth and 2n-periodic in y; we seek a C"-solution which is 2n-periodic in y.

10.1. The Linearized Equations in 2D In this section we start our discussion of the linearized equations under boundary conditions. To this end, let

345

346

Initial-Boundary Value Problems and the Navier-Stokes Equations

u, v, P, u, + v,

= 0,

denote smooth functions of x, y, t which are 2r-periodic in y, and substitute

u=U+u’, v = V + v ’ , p = P + p ’ into the N-S equations. Neglecting terms quadratic in the corrections u’, etc. and dropping the prime ’ in our notation, we find the linear equations (10.1.2)

ut

+ Uu, + Vu, + Au + gradp = vAu + F,

u,

+ vy = 0,

where

The forcing function F in (10.1.2) is the defect of (U. V , P ) in (10.1.1). It is convenient to study (10.1.2) for a general smooth matrix function A(z.y. t) which is 2r-periodic in y. Our plan is to approximate (10.1.2) by the evolutionary system (10.I .3a) (10.1.3b)

ut

+ Uu, + Vu, + Au + gradp = vAu + F, t p t + u, + uy = 0 , E > 0.

v. p) in such a way that our theory We shall choose boundary conditions for (u, for mixed parabolic-hyperbolic equations applies for every fixed E > 0. Then, if we can estimate the solutions of (10.1.3) and their derivatives independently of E , we obtain a smooth solution of (10.1.2) in the limit E 0. We start with the basic estimate. ---f

Lemma 10.1.1. Suppose that u, p solve (10.1.3). There is a constant c > 0 , depending on the coefficients U , V , in the differential equation (10.1.3) but independent oft and F, with (10.1.4)

(Here we use the notation

for boundary terms.)

The Incompressible Navier-Stokes Equations under Initial and Boundary Conditions

347

Proof. The differential equations (10.1.3) yield

= - (u, Uu,) - (u, Vu,) - (u, Au) - (u, gradp)

+ V(U, Au) + (u,F) - @, + vy). U,

The result follows from integration by parts and estimates of zero-order terms. The boundary contributions appear from integration by parts in the x-variable.

Boundary conditions providing the basic estimate. The boundary conditions

- together with the term

on the right-hand side of (10.1.4) - have to provide bounds for the boundary term (10.1.5)

) - UP

+

V(UU,

+ VV,)

x=l

We consider different possibilities.

Case 1. The lines x = 0, x = 1 represent walls. In this case it is reasonable to prescribe the boundary velocities u=v=O

at x = O , x = 1

for the nonlinear system (10.1.1). (This is obvious for u and confirmed by observation for v.) Thus one chooses a base flow with

U=V=O

at x = O , x = 1

u=v=O

at r = O , x = 1

and obtains (10.1.6)

for the linearized system. Clearly, the boundary term (10.1.5) vanishes, and the basic estimate follows.

Case 2.

Inflow at x = 0, outflow at x = 1. Now one chooses

U(O,y, t)

> 0, U(1,y, t) > 0.

Again, we can use the Dirichlet conditions (10.1.6). Also, the following conditions of Neumann type may be used:

348

Initial-Boundary Value Problems and the Navier-Stokes Equations

I

x= I

+ v 2 )- u p + u ( p - au - 9 ) - v(pv + h ) } x=O Here

etc. Using a (one-dimensional) Sobolev inequality for each fixed y and integrating over y, we have (10.1.8) Thus the basic estimate follows from Lemma 10.1.1.

Remarks. 1. The above boundary conditions are of the general form discussed for mixed parabolic-hyperbolic systems in Section 8.5. Therefore, if smooth initial data are given for u , z1, p and provided that suitable compatibility conditions are fulfilled at t = 0, the equations (10.1.3) determine a unique smooth solution satisfying these boundary conditions. 2. In many applications the viscosity constant v > 0 is very small. Then, if estimate (10.1.8) is employed, rapid exponential growth of the energy is allowed. However, if the coefficients a j , pj in the boundary conditions (10.1.7) are suitably chosen, one obtains an estimate of the boundary term (10.1.5) by c(11g11; llh11$}. In this case, inequality (10.1.8) need not be employed, and the rate of exponential growth becomes independent of v. The same is true, of course, for Dirichlet boundary conditions. 3. It is no restriction to assume the Dirichlet conditions (10.1.6) to be homogeneous. If inhomogeneous data are given, we can introduce new variables ii = u - q5 which satisfy the homogeneous condition, and if we differentiate with respect to y and t, the condition remains homogeneous. In contrast to this, consider (10.1.7). Of course we could also transform to the homogeneous case, but if we want to estimate y- and t-derivatives, new inhomogeneous terms are introduced, and nothing is gained.

+

The Incompressible Navier-Stokes Equations under Initial and Boundary Conditions

349

4. We have assumed conditions of the same type at x = 0 and x = 1, either of Dirichlet or of Neumann type. Clearly, this assumption is not necessary. In applications one often specifies u = w = 0 at inflow and

(10.1.7) at outflow. If v > 0 is small and (Y,p are suitably chosen (see Remark 2), such a Neumann condition at outflow leads to a “smoother” solution than the simple requirement u = z1 = 0 at outflow.

10.2. Auxiliary Results for Poisson’s Equation In this section we prove some elementary estimates for solutions of Laplace’s equation and Poisson’s equation in the strip

0 5 2 5 1,

-03


<

03.

As in the last section, all functions are assumed to be C”-smooth and 27rperiodic in y. We apply Fourier expansion in y-direction and use the notation m

u(x,y) =

&1

2s

G(x,k)eikYl G(x,k ) = k=-m

u(2, y)ePikydy.

Then Parseval’s relation reads

and integration over x gives us (10.2.1) k=-m

Lemma 10.2.1.

Suppose that u E C” solves

and u is 27r-periodic in y. There is a constant KI independent of u with

r f the spatial average of u vanishes, i.e.,

350

Initial-Boundary Value Problems and the Navier-Stokes Equations

then

11412I K211UZII 2 with K2 independent of u.

Proof. Since Au = 0 it follows that

CZZ(x,k) - k2C(x,k) = 0, k

= 0, f l , .

..

Thus we have a representation

(10.2.2)

C(x,k) =

ake-lklZ + bkelkl(Z-l)

Using (10.2.1) for u y , we find that

Similarly,

Elementary computation shows that (for k

Using

for all integers k

# 0.

# 0)

k # 0, k = 0.

The Incompressible Navier-Stokes Equations under Initial and Boundary Conditions

To show the estimate for

1 ) 2 ~ 1 1 ~we ,

I

0=

35 1

first observe that

C(x,O)dx = ao

1 + -bo, 2

and therefore C(X,O)

= bo(x

Thus (10.2.1)yields ll.1L1I2

I

lIuy112

I

lluy112

+ 27T

1

-

1 2

-).

I

0

lCiL(x,0"

+ 2TlboI2 I

dx

lluy112

+ ll~z112.

This proves the lemma.

An easy generalization of the last lemma is

Lemma 10.2.2.

Suppose that u E C" solves

Au=O

in O I x I l ,

and u is 2r-periodic in y. For every j of u such that

--oo
2 1 there is a constant h; independent

Proof. The equation Au = 0 allows us to express even order s-derivatives in terms of y-derivatives:

d2'u 8x2'

-= ( - 1 )

'a2'u I3p

Thus, any derivative term appearing in 1uIHJreduces to

The estimate of the last lemma finishes the proof. Next we want to show estimates of u in terms of boundary data on the lines x = 0. z = 1.

352

Initial-Boundary Value Problems and the Navier-Stokes Equations

Lemma 10.2.3.

Let 2r-periodic C"-functions go(y), gI(y) be given. The boundary value problem

Au = 0, 4 0 , Y) = go(y),

4 1 , y) = 91 (y)

has a unique C"-solution u(x,y) which is 2~-periodicin y. For any j = 0, 1 , ... there is Ki independent of g with 2 llullkJ 5 KjllgllHJ(r)'

Here

11911kJ(r)=

119011kJ

+ (19111kJ

denotes the Hi-norm of the boundary data.

Proof. First assume that there is a smooth solution u. We use (10.2.1) and the representation (10.2.2) to obtain

On the other hand,

k=-cc

The formula (10.2.2) gives us the relations

Thus we find that

The Incompressible Navier-Stokes Equations under Initial and Boundary Conditions

353

If we apply this estimate to @-Iu. -1

instead of u, and use Lemma 10.2.2, then we obtain the desired bound of 11~11%~. The bound clearly implies uniqueness. Existence of a solution follows , because we can define u in terms of its Fourier expansion. Now consider the inhomogeneous differential equation (10.2.3)

Au = F ( x , y)

where F E C” is 2r-periodic in y. We ask for estimates of a Cm-solution with homogeneous boundary data (10.2.4)

u(O,y)=u(l,y)=O,

-co
and require, as before, that u is 2r-periodic in y. We show The above strip problem has a unique solution u. For any Lemma 10.2.4. j = 0, 1 , . . ., there is a constant Kj independent of F with

In this sense, we “gain” two derivatives. Proof. First assume that the problem has a solution u. Fourier expansion of (10.2.3), (10.2.4) yields (10.254

G z z ( x ,k ) - k2G(x, k ) = E ( x , k ) !

(10.2.5b)

G(0,

k) = a( 1, k ) = 0.

Thus each function G ( . , k ) , k = 0 , f l , ...,

is determined by an ordinary boundary value problem. By

we denote the L2-norm of such a function of x , and obtain through integration by parts (for k # 0)

354

Initial-Boundary Value Problems and the Navier-Stokes Equations

Thus

k 4 i i w ,k)ii2

I IIPC,k)it2;

and with (10.2.5~) IIGxA., k)ll L k2IlW, k)ll

+ lip(.,QII

L 2 1 1 k k)ll. For k = 0 an explicit integration of the boundary value problem (10.2.5) shows that

Il l ~ ( . ~ O ) I l .

llfi(.,O)Il

Thus we have obtained that (1

+ rc4)tta(..k)ti2+ (1 + I ~ ~ ) I I Q , ( .IC)II’ , + IIM.,

k)ii2

I c t ~ ~ ( . ,IC)II~.

and Parseval’s relation (10.2.1) yields ll.lL11~2

I h61tF112.

This shows the desired bound for j = 0. There are no difficulties in estimating higher y-derivatives of 21.

ux,

uxx

because (10.2.3), (10.2.4) can be differentiated with respect to g . If we want to estimate higher 2-derivatives, we use (10.2.3) and replace them by y-derivatives, e.g., uxxx

= Fr

- uxyy.

Thus we obtain the estimate of the lemma. Existence of a Cm-solution follows as before since we can define u by its Fourier expansion. If both - the boundary data and the differential equation - are inhomogeneous, i.e.,

AIL= F ( r . y).

4 0 ,.v) = go(.y),

~ ( 1 Y). = 91(Y).

then we decompose the problem in an obvious way, and obtain the solution as a sum,

The Incompressible Navier-Stokes Equations under Initial and Boundary Conditions

355

10.3. The Linearized Navier-Stokes Equations under Boundary Conditions The linearized system (10.1.2) is not of the standard form of an evolution equation since we do not have an equation for p t . To show the existence of a solution of (10.1.2) under initial and boundary conditions, we go over to (10.1.3) for E > 0 and send E + 0.

10.3.1. Convergence Theorem We consider the system (10.1.3) and restrict ourselves, for simplicity, to the no-slip condition (10.3.1)

u=u=O

at z = O , z = l

and the initial condition (10.3.2)

u(z,y, 0) = 4x7 y, 0 ) = P ( Z , Y,0 ) = 0.

As previously, the forcing function F and the coefficients U, V, A in (10.1.3) are assumed to be Cm-smooth and 27r-periodic in y. Also, we require that (10.3.3)

@F(z,y,O)/dtj = @G(z, y , O ) / d t j = 0, j = 0, 1,2, .. ..

This ensures compatibility of the data at t = 0. For each E > 0 the initialboundary value problem (10.1.3), (10.3.1), (10.3.2) has a unique Cm-solution

u = u%, y, t ) , p = p C ( z Y, , t) which is 27r-periodic in y. To begin with, note that at t = all derivatives of u, p are = 0, as follows from (10.3.3). Our aim is to show that all derivatives of u, p can be bounded independently of E in any interval 0 5 t 5 T. Then, as E + 0,

we obtain a smooth solution of the linearized Navier-Stokes equations.

As remarked earlier, the solution is only unique if one fixes a (time-dependent)

constant to determine the pressure, e.g., (10.3.4)

(1 P(., t ) ) = 0. 1

356

Initial-Boundary Value Problems and the Navier-Stokes Equations

We will show

Theorem 10.3.1. (10.1.2) in

Consider the linearized viscous incompressible equations O
--oo
t2o

with boundary conditions u = u = 0 at x = 0, x = 1, initial condition u = u = 0 at t = 0, and side-condition (10.3.4). Assume that the data and coejjicients are Cm-smooth, 27r-periodic in y, and fulfill (10.3.3). Then there is a unique C"-solution which is 27r-periodic in y. In any interval 0 5 t 5 T, the solution is the limit of the corresponding solutions of (10.1.3) as E -t 0. The assumptions of the theorem can be relaxed: One can allow, for example, certain inhomogeneous initial conditions and different boundary conditions; see Section 10.3.2. The proof of the estimates of all derivatives of u = u', p = p c proceeds in several steps. We fix a time interval 0 5 t 5 T. For brevity, we will call a function w = w ' ( x , y, t) estimated, if we have established a bound for which is uniform in 0 < E 5 1. As previously, the main difficulty is estimating x-derivatives, because one cannot differentiate the boundary conditions in xdirection.

Step I . The basic estimate (see Lemma 10.1.1) gives us bounds for u, ~ p We . can differentiate the differential equation and the boundary conditions arbitrarily often w.r.t. y and t and obtain systems of the same form where the lower-order terms are already estimated. Thus we get bounds for all y, t-derivatives of u7 ~ p . In particular, we have bounded d

-{lbIl2 dt + EllPIl2), and the basic estimate (10.1.4) gives us a bound for u,. From the differentiated systems we obtain, with the same argument, bounds for all y,t-derivatives of u,. To summarize, we have estimates for

and all y, t-derivatives of these functions.

+

Step 2. We differentiate the equation Ept u, + zly = 0 w.r.t. x and replace ux, using the first equation (10.1.3~).Then we obtain

The Incompressible Navier-Stokes Equations under Initial and Boundary Conditions EVPtX

+ Px = H I ,

H I = F - U t - U U , - V U ,- uwxy - allu - ~

1

357

+ 2u u Y~ y ,

where the function H I is already estimated. Integrating the above ordinary differential equation for p x ( x , y, t) in time and observing that

we have bounded p,. Also, since all y,t-derivatives of H I are estimated, we have bounds for all y, t-derivatives of p,.

Step 3. Differentiating the u-equation (10.1.34 w.r.t. z,the w-equation w.r.t. y, and summing the two results, we obtain that (21,

+ u y ) t + U ( U , + uy)x + V ( U ,+ vy)y + A p + L(u, uz1uy)

= uA(u,

+ v,) + F, + G,,

where L(u, u,, uy) is a linear expression in u,u,, uy. We replace u, -tpt and find that tvApt H2 =

Fx+ G,

+

E (ptt

+ vy by

+ A p = H2,

+ Up,, + V P , ~ )- L(u, u,, u,).

The function H2 is already estimated, and using the same argument as in Step 2, we have established a bound for the function A p =H

and all its y, t-derivatives.

Step 4.

At each time t

2 0 the pressure p can be written as a sum P=Pl+P2

where Apl = O

1

P l ( 0 , Y l t ) = P(OlY,t)l

Pl(l,Y,t) =P(l,Y,t),

and (10.3.5)

AP2 = H ,

P2(0,y, t ) = P2(1, Y, t ) = 0.

From Lemma 10.2.4 we obtain bounds for p z , p2,, m ,, and all y, t-derivatives of these functions. In Step 2 we have bounded the function p , and all its y l tderivatives; thus we can estimate the difference p l , and all its y, t-derivatives.

Initial-Boundary Value Problems and the Navier-Stokes Equations

358 Step 5.

We want to derive a bound for pl itself. To this end note that (1,p(., t ) )= 0,

+ + vy = 0 and the boundary conditions. Therefore, the

as follows from ~ p t u, function

1 PI(z,Ytt)= m ( z , y , t ) + 2.1,(1’P2) has spatial average zero and, using Lemma 10.2.1, we can bound PI. (Note that = PI,.) Since p 2 is already bounded, we obtain an estimate for pl itself. In the same way we obtain also a bound for each time derivative pit, p l t t , etc., and Lemma 10.2.2 yields bounds for all other derivatives of p l . Thus we have established estimates for all derivatives of p l .

PI,

Step 6. We have shown estimates for p , p,, p,,, u, u, and all y, t-derivatives of these functions. It remains to bound higher z-derivatives (and their y, tderivatives); this can now be done recursively as follows: The equations (10.1.3a) can be solved for u,,, v,,, and therefore these functions are also bounded. Thus all first derivatives of H2 are estimated, and (10.3.5) gives us bounds for all third derivatives of p. Then (10.1.3~)yields bounds for u,,,, u,,,,. A simple induction argument finishes the proof.

10.3.2. Remark on Initialization In the last section we have assumed that the initial conditions are homogeneous. Let us explain the reason for this. Suppose that we would give general nonhomogeneous initial data (10.3.6)

4x1

Y10) = f(z,Y),

4 2 ,Y,0 ) =

dz,Y),

P(Z,

Y,0 ) = N z , Y).

For every fixed E > 0 the corresponding initial-boundary value problem (10.1.3) has a solution. However, we need bounds of the derivatives which are independent of E . Our estimates of the previous section show these bounds if the y, t-derivatives are bounded independently of E at t = 0. For the y-derivatives we have no difficulties, provided the boundary conditions are compatible with the initial conditions. Therefore, the crucial question is if one can bound the time derivatives independently of E at t = 0. By (10.1.3b) the first time derivatives are bounded independently of E at t = 0 if and only if (10.3.7)

u,(., 0 )

+ vy(., 0 ) = Edl,

dl = O( 1).

The Incompressible Navier-Stokes Equations under Initial and Boundary Conditions

359

Thus, to first approximation, the initial data have to satisfy the divergence relation. Differentiation of (10.1.36) with respect to t gives us Eptt = - ( u x t

+ v,t)

= Ap

+ H ( u , V) - F, - G,.

Here H ( u ,v) is an expression in u,v and its space derivatives. Thus, the second time derivatives are bounded independently of E if we have, at t = 0, (10.3.8)

Ap

+ H ( u , t ~ -) Fx - G,

= € p i . pl = O(1).

We can consider (10.3.8) as an equation for the pressure. For the third time derivative of p we obtain 2

(10.3.9)

+ ~ { H t ( u , v-) Fxt - G , t } = - A ( u , + v,) + E { Ht(u,V) - F,t - G , t }

pttt = &t

= E { -Ad,

+ H ~ ( u , - F,t V)

- G,t}.

Thus we have to choose dl so that

-Ad,

+ Ht(u,V)

-

F,, - G,t = Ed*,

d2 = O(1).

Note that we can use (10.1.3~)to express Ht(u,v)in terms of u , v and their space derivatives. Further differentiation of (10.3.9) with respect to t gives us an equation for pl . This process can be continued. We can ensure that more and more time derivatives are bounded independently of E by choosing the initial data so that the above relations are satisfied. The easiest way to comply with all requirements is to give homogeneous initial data and to request (10.3.3). This case has been treated above. However, more general data are allowed as long as they satisfy the above requirements and lead to bounded time derivatives. To choose initial data in this way is called initialization by the bounded derivative principle. In applications, one often solves compressible equations which are almost incompressible. Here a similar problem occurs: To ensure that the solution is close to a solution of the incompressible equation, one has to “prepare” the initial data. The principle is the same, one chooses the initial data so that a number of time derivatives at t = 0 remain bounded independent of the compressibility. For details, see Kreiss (1985).

10.4. Remarks on the Passage from the Compressible to the Incompressible Equations Consider the nonlinear system (10.1.1) and choose functions (an approximate solution)

360

Initial-Boundary Value Problems and the Navier-Stokes Equations

u, v,P, u, + v, = 0. For example, the functions may satisfy inhomogeneous boundary and initial conditions. We substitute u=U+u',

v=V+v',

p=P+p'

into (10.1.1) and obtain equivalent (nonlinear) equations for the corrections u', v', p'. If we drop the prime ' in our notations, the equations read (10.4.1)

ut + ( U +u)u, + ( V +v)u, +gradp = v A u + F ,

u,

+ v, = 0.

Here F is a new forcing which is determined by U , V, P. This system can be approximated by (10.4.1) together with (10.4.2)

Ept

+ u, + vy = 0,

E

> 0.

If we have boundary conditions (10.3.1), (10.3.2) and assume (10.3.3), then we can proceed in the same way as above for the linearized equation. As E 0, we obtain convergence to a solution of the nonlinear incompressible equations. The process which has been chosen here to obtain a solution of the incompressible equations does not describe the passage of the compressible to the incompressible N-S equations. If one wants to study this limit, one should replace (10.4.2) by --f

(10.4.3)

€{Pt

+ (U + u ) p , + (V + v)p,} + + wy = 0. 1 '1,

+

(We recall the continuity equation pt div(pu) = 0 and formally substitute p = 1 ~ p . )In both cases, (10.4.1) with (10.4.2) or (10.4.1) with (10.4.3), we have a coupled hyperbolic-parabolic system. To illustrate the differences between (10.4.2) and (10.4.3), we assume inflow (U > 0) at x = 0 and require the boundary condition u = v = 0 at x = 0. Whereas the boundary x = 0 is characteristic for the equation (10.4.2), the variable p is an ingoing characteristic variable for (10.4.3). Therefore, no boundary condition for p is needed with (10.4.2), but (10.4.3) requires a boundary condition. For example, one can prescribe

+

P(0,Y7 t ) = 0. For the limit-equations ( E = 0), no boundary condition for p is allowed. In Kreiss, Lorenz, and Naughton (1988) it is shown that convergence for E -+ 0 is also obtained with (10.4.3), at least away from a boundary layer at x = 0.