Pergamon Nonlinear Anulysk Theory, Metho& d Applications. Vol. 30. No. I,

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NAVIER- STOKES APPROXIMATIONS IN EXTERIOR

DOMAINS

NICOLE ROSS

Fachbereich Mathematik- Informatik, Universit;it Paderbom

Warburger Str.lOO

D-33098 Paderbom, Germany

Key words and phrases: Nonstationary Navier- Stokes approximation problem in an exterior domain, local smooth

solutions, first order convergence of Rothe’s scheme in L*, boundedness in HZ, convergence in Lm

1. INTRODUCTION

Let us consider a moving body B = n in a viscous incompressible fluid filling the whole space R3. Here C’n is the exterior to the body B with the boundary 80 which we assume to be a compact 2- dimensional C”- submanifold of R3. The velocity vector ‘(I (t, z) = (~1, us, us) and the kinematic pressure p (t, z) 1 0 of the flow fulfil the Navier- Stokes equations

$J-Au+u.VU+V~=F

v.u=o, t>o,xEm

up = 0, u(O, .) = uo (x)

u-+u,= 0 for 1x1 + 00.

(1.1)

Here F = F (t, 3) and u0 = u0 (z) denote, respectively, the given external force and initial velocity. Our main aim is to prove convergence of Navier- Stokes approximations in high order norms. The- refore we need solutions as smooth as possible under assumptions which are as strong as possible but also avoiding the non realistic compatibility condition [l]. In distinction from Miyakawa [2] we work with a smooth solution u E C” ([0, t], DA) for u,, E DA. Here C” ([0, t] , DA) denotes the set of all uniformly bounded and continuous functions f defined on [0, t] with values in C” ([0, t] , DA), where DA denotes the domain of the Stokes operator. We will approximate this solution by Rothe’s method, that is by a first order semidiscrete approximation scheme. Until now for my knowledge we have convergence results for Rothe’s method for the Navier- Stokes equations in high order norms in bounded domains only. Indeed Rautmann proved in [3] uniform first order convergence in L2 (0) of Rothe’s scheme for the Navier- Stokes initial- boundary value problem in a 3- dimensional bounded domain 0 and the uniform boundedness of Rothe’s approximations in H2(Q) by energy methods. In

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[4] he even showed H2(0)- convergence of Rothe’s approximations. Now we could also prove similar results to [3] for problem (1.1) in the case u0 E DA. The proof is based on estimates established by Masuda in [5] and Miyakawa [3]. From this by the momentum inequality we get convergence of Rothe’s scheme in DAM for a < 1 with explicit convergence rates.

2. EXISTENCE OF A UNIQUE AND SMOOTH LOCAL SOLUTION

In this paragraph our aim is to sketch the proof of the existence of a local- in- time solution u E C” ([0, T] , DA) of (1.1) for an initial velocity u0 E DA. In the following by H” we denote the space of measurable real vector functions, which are square integrable on CB together with their spatial derivatives up to,the order m 2 ]a] = oi +. . . + (Y,,, oj = 0, 1, . . . , m. Further H denotes the closure with respect to the space L2 of the space of real test functions on Co. The latter functions are divergence free, have compact support in CD and spatial derivatives of any order. Using H. Weyl’s orthogonal projection P : Ho -+ H, which sends into zero exactly the generalized gradients Vq E Ho, from (1.1) we get the initial value problem of the evolution Navier- Stokes equation

$+Au+PuVu = PF

u(0) = 210

(2.1)

for the function u : [0, T] -+ DA. For simplicity in the following we will always assume PF = 0. This implies that the density of the outer force in the Navier- Stokes equation is the gradient of a scalar function. Instead of the evolution equation we consider the following integral equation

21 (t) = eetAuo - J ’ e-(t-“)APuVu (s) ds. (2.2) 0

In the following we will sketch the proof of the existence of a local solution u E C” ([0, t] , DA) of (2.2) . Hence we define the operator

(Su) (t) := eetAuo - 0t e-(t-“)APuVu (s) ds. J (2.3)

In our existence proof we also get the strong Holder continuity of the convective term as a result. Now from [6,p.128] it is well known that then the solution u(t) of the fixpoint equation (2.2) is the unique solution of (2.1). To fulfil the necessary conditions of the contracting mapping principle we have to introduce some function spaces, see [l].

DEFINITION 2.1. For 0 < Q 5 4 let H(O) be the completion of DA= with respect to the norm

M(a) = II ‘4-412.

DEFINITION 2.2. By Hcma) we denote the dual space of H(*), 0 5 (Y < 3.

Finally we want to define the spaces H(“) for i < cr 5 1.

DEFINITION 2.3. By H(“) , 4 < (Y 5 1, we denote the completion of DA” with respect

to the norm ]w](~) = (I] Anwlli+ II A~wli~)‘.

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The following Proposition is a slight generalization of one of Miyakawa’s results.

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PROPOSITION’ 2.1. The estimate

I PuVv I(-r)5 M Ia lel v I,, , 21 E H e, v E HP

holds for all 0 5 y 5 4, 0 5 8 < 4, f 5 p 5 1, (3 + p + y = 5. Here M > 0 is a constant only dependent of y, 0, p.

By a bootstrap argument from this Proposition we get the following theorem:

THEOREM 2.1. For the initial value u0 let us assume u0 E DA. Then there exists a unique solution u E C” (IO, T] , DA) of the exterior problem for the Navier- Stokes equations (I) for sufficiently small

T.

3. SEMIDISCRETE APPROXIMATI’ON iN TIME BY MEANS OF ROTHE’S SCHEME

Starting from the Navier- Stokes equation of evolution we will make a semidiscrete approximation in time of the solution u E Co ([0, I’], DA) of (1.1). Th ere ore we consider the time grid {tk) with f tk = k . h for k = 0, ... , K and v,” = uO, where h = $ denotes the length of the time step. The approximations ut of the values of the solution u of the Navier- Stokes equation of evolution in tk will be calculated with the help of Rothe’s scheme in L2. We get this scheme replacing the time derivative by the difference quotient and taking u$-i as the first factor on the right side. By induction and again by Proposition 2.1 with the help of the regularizing properties of the Stokes resolvent, from the contracting mapping principle we get the following theorem.

THEOREM 3.1. We assume the initial value ut E DA. Then a unique solution uk = (l+ hA)-‘{u&, - hP&, * Vu:}, ug E DA, exists far all k 2 1.

4. CONVERGENCE RATES FOR ROTHE’S SCHEME

For some h = 8 2 0 beginning with ut E DA we compute the Rothe approximations ZL%, k = 1,. . . , I< successively. In virtue of Theorem 2 all ut E DA exist. Let u E Co ([O,T] , DA) be a solution of (1) with PF = 0. For k = 1,. . ., I<, i!k = kh, by integrating the Navier- Stokes equation of evolution (2.1) from t&r to tk and adding Au(tk) on both sides, we

get

U(tk) -hu’tk-l) $ AU(&) = ; 111, A{u(t) - ‘Il(tk)}dt - fk

P tk where fk = h

J 11. Vu&.

tk-1

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Now we consider the differences Wk = t& - ‘Il(tk).

First writing fk in terms of u(i!k), u(rk-r) and their differences, and secondly introducing the diffe- rences 2~; - U(i&) in Rothe’s scheme we get

wk - wk-I

h + A% = Sk,

where

Sk = ; I

tk {A(+) - u(tk)) + (u(t) - U(tk-1)) * V(‘ll(t) - +k)) tk-I

+(@) - ‘Il(tk-1)) . V+k) + @k-l) . V(u(t) - @k))} dt

-P{wk-1 . vU(tk) + Wk-1 ’ vwk + U(tk-1) . VU&}.

Similar methods have been used by Varnhorn [7] in the case of the linear Stokes problem and by Raut- mann [3] for the Navier- Stokes problem in bounded domains. By some further technical calculations we get the formula

11 wk1t2+ 11 wk - wk-d12 + 2h 11 vwk112 =[I wk-1112 + e Tjr j=l

where

‘&T~ j=l

= 2 1 (111, A(+ - +k)) dt 7 wk) 1

+2 1 i:r, ( (U(t) - u(tk-1)) . vwk , u(t) - U(tk) > dt I

+21qk P(u(t) - ‘Il(tk-1)) * V’ll(tk) dt, wk) 1 tk-1

+2 I (1, P+k-1) * v(@) - ‘Ll(tk)) dt , wk) 1

+2h 1 ( pwk-1 * vU(tk), wk) I .

From this by energy methods, interpolation theory and induction we get the following estimate

11 wk112 + h 11 vwkll’ i qk { 11 ~011~ + c,h2 (Jd’ 11 Vdt4t)l12dt + 3kh)) .

Now we will use the following inequality from [8,p. 661- 6661

I o*k 11 V&u(t)l12 dt 5 G

which holds uniformly in t E [O,T]. By means of this inequality assuming ]I u~-uO]] 5 ch and further by an estimate for q” = (1 + 2c,h)k we get the following error bounds

llwk112 5 CA2 , 11 vwk112 5 cob ,

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where co = co (suP~~[~,TJ II Au(t)ll,c,T) and h E @ho], h 0 sufficiently small, in L2 and H’, respec- tively. Finally we want to get boundedness in H2. From the moment inequality and the assumptions

II us- %I( I ch ut uniformly bounded in DA,

we get the following estimate for the error w0 at time t = 0

(4.1)

1 II Vwc.II 5 Clhl,

where cl = E ci I] Aw,ll) 5 i; c* (11 Au,Iji+ 11 Au!Jlli) . T o e g th er with (4.1) this estimate gives the

H’- error of w0 at t = 0. Now with the help of the H’- error of w0 at time t = 0 and the H’- error of wk at time t D 0, by the same method we used to prove L2- and H’- convergence, we get the following estimate

11 A’&l12 < c2.

Further we have the well known inequality [5]

11 “hvz’ 5 M (11 4l+ ll ~11) , ‘21 E DA.

Now a8 a direct consequence we conclude

11 WkllH2 5 c3 , where g = M (6 + Ah) .

Together with the first order L2- convergence from this by means of the moment inequality we get the following estimate

11 A’wkll 5 c4 9 where q = ~7r@. &l-ph’-p,

that is convergence of the Rothe approximations in H (fl) for /3 < 1. Additionally by the inequality

11 u~]L, 5 MO II A~114 II A~uII~ + Mz 11 A~uII~ II uII~, u E DA.

from Masuda [5] we even get the following explicit convergence rate in Loo:

11 wkll,p 5 cghf , where c5 = MO (c~c,)~ + Mi (co); hi.

5. RESULTS

THEOREM 5.1. a) Let u E C” ([O,T], DA)(f or some T > 0) denote a Navier- Stokes solution of problem (1.1). Then to initial values uk = u0 E DA, the Rothe approximations uk E DA exist, are uniformly bounded in H2(Cn) and converge of first order in L2(Cn) to u(kh) with K + cc uniformly in k = 1,. . . , I< for h= 5. b) From a) and the moment inequality we get convergence of the Rothe’ s approximations in H(“) for o < 1, and together with the first order L2- convergence we get explicit convergence rates e.g. in H’(Cn) and Loo.

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REFERENCES

1. TEMAM R., Behaviour at time t = 0 of the solutions of semi- linear evolution equations, MRC Technical Summary Report 2162, Madison: University of Wisconsin (1980).

2. MIYAKAWA T., On nonstationary solutions of the Navies- Stokes equations in an exterior domain, Hiroshima Math. .I. 12, 115- 140 (1982).

3. RAUTMANN R., A remark on the convergence of Rothe’s scheme to the Navier- Stokes equations, Stability and Applied Analysis of Continuous Media 3 229- 246 (1993).

4. RAUTMANN R., He- convergence of Rothe’s scheme to the Navier- Stokes equations, Journal of Nonlinear Analysis 24 1081- 1102 (1995).

5. MASUDA K., On the stability of incompressible viscous fluid motions past objects, J. Math. Sot. Japan 27, 294- 327 (1975).

6. TANABE H., Equations of evolutions, Pitman, London (1979). 7. VARNHORN W., Time stepping procedures for the nonstationary Stokes equation, preprint 1353, Technische

Hochschule Darmstadt (1991). 8. HEYWOOD J., On the Existence, Regularity and Decay of Solutions, Indiana University Math. Journal, 29, 639-

681 (1980).

Supported by Deutsche Forschuugsgemeinschaft.