DECAY ESTIMATES FOR LINEARIZED UNSTEADY INCOMPRESSIBLE VISCOUS FLOWS AROUND ROTATING AND TRANSLATING BODIES

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1 JEPE Vol, 205, p DECAY ESIMAES FOR LINEARIZED UNSEADY INCOMPRESSIBLE VISCOUS FLOWS AROUND ROAING AND RANSLAING BODIES PAUL DEURING, SANISLAV KRAČMAR, ŠARKA NEČASOVÁ, AND PEER WIWER Abstract. We consider the time-dependent Oseen sytem with rotational terms. his system is a linearized model for the flow of a viscous incompressible fluid around a rigid body moving at a constant velocity and rotating with constant angular velocity. We present results on temporal and spatial decay of solutions to this system in the whole space. he spatial asymptotics we establish exhibit a wake.. Introduction Consider the motion of a viscous incompressible fluid around a rigid body translating with constant velocity and rotating at constant angular velocity. Suppose the fluid flow is described with respect to a coordinate system in which the body is at rest. hen the flow in question is usually represented by a modified Navier-Stokes system which reads like this:.) t v ν z v + v z )v V + ω z) z v + ω v + z q = F, div z v = 0 in R 3 \D) 0, ). Here D R 3 is a bounded domain representing the rigid body. he function v denotes the velocity field of the fluid, and the function q its pressure field. he vector V describes the constant translation of the body, and the vector ω its constant angular velocity. We suppose that V and ω are parallel. he function F stands for an exterior force exerted on the fluid, and the parameter ν 0, ) characterizes the viscosity of the fluid. By a suitable normalization and some changes of variables see [6]), system.) may be rewritten in the form.2) t u x u + τ x u + τu x )u ϱ e x) x u + ϱ e u + x σ = f, div x u = 0, where τ 0, ) is the Reynolds number and ϱ R\{0} the aylor number. In recent years, many articles dealt with flows around a rotating body. As examples we mention [0] [4], [7] [8], [20] [2]. In the present context, an article by Chen and Miyakava [] is relevant. hese authors proved existence of a global weak solution to.) in the whole space R n with n = 2 and n = 3, and derived algebraic decay rates as t ) for the kinetic energy associated with this solution. hey assumed F = 0, ν = but considered 200 Mathematics Subject Classification. 35Q30, 76D05. Key words and phrases. Whole space, viscous incompressible flow, rotating body, fundamental solution, Navier-Stokes system. Received 3/2/206, accepted 29/02/206. he research of Š. N. and S. K. was supported by Grant Agency of Czech Republic P S. Moreover research of Š.N. was supported by RVO

2 326 PAUL DEURING, SANISLAV KRAČMAR, ŠARKA NEČASOVÁ, AND PEER WIWER nonzero initial data and admitted the case that V and ω are functions depending on time, and need not be parallel. We will show results related to those in [], but pertaining to the Oseen system with rotational terms, that is, to the following system obtained by dropping the nonlinearity in.2),.3) t u x u + τ x u ϱ e x) x u + ϱ e u + x σ = f, div x u = 0. Under the assumption that f does not depend on time and decays in an appropriate way, we will study the asymptotics of Ux) ux, t) and x Ux) ux, t) ) with respect to both the space variable x and the time variable t, where u is the velocity part of a solution to.3) with initial data zero, and U the velocity part of a solution to the stationary variant of.3), that is,.4) U + τ U ϱ e x) x U + ϱ e U + Π = f, div U = 0. he decay bounds we obtain for Ux) ux, t) exhibit a wake. In addition, they imply optimal rates of spatial decay for ux, t) when x. hese rates are uniform with respect to t. Our estimates of Ux) ux, t) further yield that u, t) converges to U with respect to a weighted W, -norm, which we will denote by,,w,ɛ. he rate of this convergence is t ɛ, where ɛ may be arbitrarily chosen in 0, /2) but enters into the definition of,,w,ɛ. his convergence result means in particular that U is unconditionally asymptotically stable with respect to the norm,,w,ɛ. For more details on our results we refer to heorem 2, Corollary and the comments in Section Notations, definitions and auxiliary results If A R 3, we write A c for the complement R 3 \A of A. he symbol denotes the Euclidean norm of R 3 and also the length of a multiindex from N 3 0, that is, α := α +α 2 +α 3 for α N 3 0. he open ball centered at x R 3 and with radius r > 0 is denoted by B r x). If x = 0, we will write B r instead of B r 0). Put e :=, 0, 0). Let x y denote the usual vector product of x, y R 3. he parameters τ 0, ) and ϱ R\{0} will be kept fixed throughout. Put s τ x) := + τ x x ) for x R 3. Define the matrix Ω R 3 3 by Ω := ϱ , 0 0 so that ϱ e x = Ω x for x R 3. By the symbol C, we denote constants only depending on τ or ω. We write Cγ,..., γ n ) for constants that additionally depend on parameters γ,..., γ n R, for some n N. For p [, ) and for open sets A R 3, we write W,p A) for the usual Sobolev space of order and exponent p. If B R 3 is open, define W,p loc B) as the set of all functions g : B R such that g U W,p U) for any open bounded set U R 3 with U B. If V is a normed space whose norm is denoted by V, and if n N, we equip the product space V n with a norm n) V defined by v n) n ) /2 V := j= v j 2 V for v V n. But for simplicity, we will write V instead of n) V. Let K denote the usual fundamental solution to the heat equation, 2.) Kx, t) = 4πt) 3/2 e x 2 /4t) for x R 3, t 0, ).

3 DECAY ESIMAES FOR LINEARIZED UNSEADY INCOMPRESSIBLE VISCOUS FLOWS 327 Recall that the Kummer function F,, ) is given by F, c, u) := Γc)/Γu + c))u n for all u R, c 0, ), n=0 where Γ denotes the usual Gamma function. We put H jk x) := x j x k x 2 for x R 3 \ {0}, Λ jk x, t) := Kx, t)δ jk H jk x) F, 5/2, x 2 /4t)))δ jk /3 H jk x)) for x R 3 \ {0}, t 0, ), j, k {, 2, 3}. In what follows, the letter Γ will always stand for a matrix-valued function defined by Γ jk y, z, t)) j,k 3 := Λ rs y τte e tω z, t)) r,s 3 e tω for y, z R 3, t 0, ) with y τte e tω z 0. his function is the velocity part of the fundamental solution to.3) introduced by Guenther, homann [22]. Our following lemma restates [3, Corollary 3.]. Lemma. he function Γ jk may be extended continuously to a function belonging to C R 3 R 3 0, ) ). We will use the ensuing technical lemmas: Lemma 2. see [4, Lemma 2.9]) Let x R 3, t R. hen e tω x = x, e tω x) = x, e tω e = e. Lemma 3. see [2, Lemma 4.8]) For x, y R 3 we have s τ x y) C + y )s τ x). Lemma 4. see [9, Lemma 4.3]) Let β, ). hen B r s τ x) β do x Cβ)r for r 0, ). Lemma 5. see [4], Lemma 2.4) Let S 0, ), x B c S. hen Lemma 6. see [3, Lemma 3.2]) x CS)s τ x). 2.2) β y Γ jk y, z, t) + β z Γ jk y, z, t) C y τte e tω z 2 + t) 3/2 β /2 for y, z R 3, t 0, ), β N 3 0 with β. Lemma 7. see [3, heorem 3.]) Let k {0, }, R 0, ), y, z B R with y z. hen 2.3) 0 y τte e tω z 2 + t) 3/2 k/2 dt CR) y z k. Due to the preceding lemma and by 2.2), we may define Z jk y, z, ) := Γ jk y, z, t)dt for [0, ), y, z R 3 with y z, j, k 3. he function Z,, 0) is the velocity part of the fundamental solution of.3) proposed by Guenther, homann [22].

4 328 PAUL DEURING, SANISLAV KRAČMAR, ŠARKA NEČASOVÁ, AND PEER WIWER Lemma 8. Let j, k {, 2, 3}, [0, ). hen Z jk,, ) C R 3 R 3 ) \ {x, x) : x R 3 } ), and 2.4) yn Z jk y, z, ) = yn Γ jk y, z, t)dt, zn Z jk y, z, ) = z n Γ jk y, z, t)dt for y, z R 3 with y z, n {, 2, 3}. If R 0, ), y, z B R with y z, α N 3 0 with α, we have 2.5) α y Z jk y, z, ) + α z Z jk y, z, ) CR) y z α. Proof. he proof of [4, Lemma 2.5] carries over to the present situation [0, ) instead of = 0). Note that 2.5) follows from 2.4) and 2.3). heorem. Let S, δ 0, ), ν, ), 0, ) and 0 ɛ < ν, or = 0 and ɛ = 0. hen 2.6) for y B c +δ)s, z B S, and 2.7) y τe e tω z 2 + t) ν dt CS, δ, ɛ, ν) ɛ y s τ y)) ν+ɛ+/2 y τe e tω z 2 + t) ν dt CS, δ, ɛ, ν) ɛ z s τ z)) ν+ɛ+/2 for z B c +δ)s, y B S. Moreover, 2.8) α y Z jk y, z, ) + α z Z jk y, z, ) Cs, δ, ɛ) ɛ y s τ y)) α /2+ɛ for j, k {, 2, 3}, α N 3 0 with α, y B c +δ)s, z B S, 2.9) α y Z jk y, z, ) + α z Z jk y, z, ) Cs, δ, ɛ) ɛ z s τ z)) α /2+ɛ for y B S, z B c +δ)s and j, k, α as above. Proof. In the case = 0, ɛ = 0, heorem restates [4, heorem 2.9]. Now suppose that > 0 and 0 ɛ < ν. hen inequalities 2.6) and 2.7) may be reduced to the preceding reference. In fact, take y, z as in 2.6) or 2.7). hen y τe e tω z 2 + t) ν dt y τe e tω z 2 + t) ν+ɛ t ɛ dt ɛ y τe e tω z 2 + t) ν+ɛ dt ɛ y τe e tω z 2 + t) ν+ɛ dt. Since ν + ɛ <, we may now use [4, heorem 2.9] with ν replaced by ν ɛ, obtaining 2.6) and 2.7). he estimates in 2.8) and 2.9) follow from 2.2), 2.4), 2.6) and 2.7). 3. Volume potentials We will study volume potentials involving the kernel Z, ). Lemma 9. Let p, ), q, 2), f L p loc R3 ) 3 with f BS c Lq BS c )3 for some S 0, ). hen, for j, k {, 2, 3}, α N 3 0 with α, we have 3.) y α Γy, z, t) dt f k z) dz < for a.e. y R 3. R 3 0 0

5 DECAY ESIMAES FOR LINEARIZED UNSEADY INCOMPRESSIBLE VISCOUS FLOWS 329 Let [0, ). We define Rf), ) : R 3 R 3 by putting 3 R j f)y, ) := Γ jk y, z, t)dt f k z)dz = R 3 k= R 3 k= 3 Z jk y, z, ) f k z)dz for y R 3 such that 3.) holds; otherwise we set R j t)y, ) := 0 j 3). hen Rf), ) W, loc R3 ) 3 and 3 3.2) l R j f)y, ) = yl Z jk y, z, ) f k z)dz R 3 k= for j, l {, 2, 3} and for a.e. y R 3. Moreover, if f L R 3 ) 3, > 0, we have 3.3) α Rf)y, ) C 2 α 2 f for α N 3 0 with α, y R 3. Proof. In view of 2.8) and 2.9) with ɛ = 0, and due to Lemma 8, all the statements of Lemma 9 except 3.3) may be proved in exactly the same way, without any modification, as analogous statements in [4, Lemma 3.]. As for 3.3), we use 2.2) to obtain for y R 3, j, k 3 that y α Γ jk y, z, t) dt fz) dz R 3 C y τte e tω z 2 + t) 3/2 α /2 dt fz) dz R 3 C t 3/2 α /2 dt fz) dz C /2 α /2 f. R 3 Inequality 3.3) now follows with 3.2) and 2.4). heorem 2. Let 0, ), S, S, γ 0, ) with S < S, p, ), A [2, ), B R, 0 ɛ < /2 + α /2, f : R 3 R 3 measurable with f B S L p B S ) 3, fz) γ z A s τ z) B for z B c S, A + min{, B} 3. Let i, j {, 2, 3}, y B c S. hen 3.4) R j f)y, ) CS, S, A, B, ɛ) ɛ f B S + γ) y s τ y) ) +ɛ la,b y), 3.5) where yi R j f)y, ) CS, S, A, B, ɛ) ɛ f B S + γ) y s τ y) ) 3/2+ɛ la,b y) { l A,B y) = if A + min{, B} > 3 max, ln y ) if A + min{, B} = 3 s τ y) max0, 7/2 A B 2ɛ), Proof. We modify the proof of [4, heorem 3.]. Since A 2, we have f BS c L q BS c ) 3 for any q 3/2, ). But f B S L p B S ) 3, so we get, say, f L min{p,2} loc R 3 ) 3. herefore f satisfies the assumptions of Lemma 9, hence Rf), ) is well defined, belongs to W, loc R3 ) 3 and verifies 3.2).

6 330 PAUL DEURING, SANISLAV KRAČMAR, ŠARKA NEČASOVÁ, AND PEER WIWER By 2.8), we find for k {, 2, 3}, α N 3 0 with α that y α Z jk y, z, ) fz) dz CS, S, ɛ) ɛ y s τ y) ) α /2+ɛ 3.6) f BS. B S We further get with 2.4), 2.2), a change of variables and Lemma 2 that 3.7) A α := y α Z jk y, z, ) fz) dz BS c Cγ y τ te e t Ω z 2 + t) 3/2 α /2 z A s τ z) B dz dt BS c = Cγ y τ te x 2 + t) 3/2 α /2 x A s τ e t Ω x) B dx dt BS c = Cγ ɛ y τ te x 2 + t) 3/2 α /2+ɛ dt x A s τ x) B dx. B c S he preceding integral over BS c is split into a sum of integrals over BS c B S/2 y) and BS c \B S/2 y), respectively. In order to estimate the integral over BS c B S/2 y), we observe that for x R 3, the term y τ te x 2 +t) 3/2 α /2+ɛ is bounded by y τ te x 2 +t) 2 if y τ te x 2 + t. Else it may be bounded by min{, t 3/2 α /2+ɛ }. hus we get by 2.3) with y z in the place of y and with z = 0 that y τ te x 2 + t) 3/2 α /2+ɛ dt x A s τ x) B dx B c S B S/2 y) B c S B S/2 y) CS) CS, ɛ) B c S B S/2 y) B c S B S/2 y) y τ te x 2 + t) 2 + min{, t 3/2 α /2+ɛ } ) dt y x x A s τ x) B dx min{, t 3/2 α /2+ɛ } ) ) dt x A s τ x) B dx y x 2 + ) x A s τ x) B dx, where we used the assumption ɛ < /2 + α /2 in the last inequality. On the other hand, we apply 2.6) with y, ν replaced by y x, 3/2 α /2+ɛ, respectively, and with z = 0, ɛ = 0, to obtain y τ te x 2 + t) 3/2 α /2+ɛ dt CS, ɛ) y x s τ y x) ) α /2+ɛ for x B c S \B S/2 y). Here the assumption ɛ < /2 + α /2 is again relevant. Now we may deduce from 3.7), 3.8) A α CS, ɛ)γ ɛ y x 2 + ) x A s τ x) B dx BS c B S/2 y) + y x sτ y x) ) α /2+ɛ x A s τ x) dx) B. BS c \B S/2 y)

7 DECAY ESIMAES FOR LINEARIZED UNSEADY INCOMPRESSIBLE VISCOUS FLOWS 33 Next we observe that for x B S/2 y), we have x y y x y S/2 y /2, and by Lemma 3, For x B S/2 y) c, we find s τ x) C + y x )s τ y) CS)s τ y). y x = y x /2 + y x /2 S/4 + y x /2 min{s/4, /2} + y x ), and for x BS c we get x CS ) + x ). herefore from 3.8), 3.9) A α CS, S, A, B, ɛ) ɛ γ y A s τ y) B y x 2 + ) dx B S/2 y) + + y x )sτ y x) ) α /2+ɛ + x ) A s τ x) dx) B BS c \B S/2 y) CS, S, A, B, ɛ)γ ɛ y A s τ y) B + + y x )sτ y x) ) ) α /2+ɛ + x ) A s τ x) B dx. R 3 By Lemma 5 and because y BS c, A 3/2 > 0, A + B A + min{, B} 3, we further observe that 3.0) y A s τ y) B CS, A) y 3/2 s τ y) A+3/2 B CS, A) y 3/2 s τ y) 3/2. Moreover, by the proof of [9, heorem 3.] we get R 3 + y x )sτ y x) ) +ɛ + x ) A s τ x) B dx Cɛ, A, B) y s τ y) ) +ɛ la,b y). Similarly, the proof of [9, heorem 3.2] yields R 3 + y x )sτ y x) ) 3/2+ɛ + x ) A s τ x) B dx Cɛ, A, B) y s τ y) ) 3/2+ɛ la,b y)s τ y) max0, 7/2 A B 2ɛ). he two preceding estimates together with 3.6), 3.9) and 3.0) imply 3.4) and 3.5). Corollary. Consider the situation of heorem 2. As additional assumptions, suppose that A + min{, B} > 3, > 0 and ɛ < /2. hen f L R 3 ) 3. For v W, loc R3 ) 3, define hen v,,w,ɛ := sup{ vx) [ + x ) s τ x)] ɛ : x R 3 } + sup{ vx) [ + x ) s τ x)] 3/2 ɛ s τ x) max0, 7/2 A B 2ɛ) : x R 3 }. Rf), ),,w,ɛ CS, S, A, B, ɛ) f + γ) max{ ɛ, }. Proof. Put B := min{, B}, δ := min{a 2)/2, A+B 3)/2}. he assumption A+B > 3 implies A > 2, so δ > 0 and A + 2 δ < 0. hus we get with Lemma 5 that 3.) x A s τ x) B = x 2 δ x A+2+δ s τ x) B CS, A) x 2 δ s τ x) A B+2+δ

8 332 PAUL DEURING, SANISLAV KRAČMAR, ŠARKA NEČASOVÁ, AND PEER WIWER for x B c S. We further observe that A B +2+δ A B +2+δ <, where the last inequality follows from the choice of δ and the assumption A + B > 3. Now Lemma 4 and 3.) yield B c S f dx <, so we may conclude f L R 3 ) 3 in view of the assumption f B S L p B S ) 3. At this point 3.3) implies 3.2) α y Rf)y, ) CS ) /2 α /2 f for y B S, α N 0 with α. Obviously CS ) + y ) s τ y) for y B S and y CS ) + y ) for y BS c. hus Corollary follows from heorem 2 in the case y ) and inequality 3.2) in the case y ). 4. Comments Let f C0 R 3 ) 3. It is implicit in the proof of [3, heorem 4.2] that the function U := Rf), 0) is the velocity part of a classical solution to the stationary problem.4) in the whole space R 3. On the other hand, according to [22, heorem.2], the velocity part u of a solution to.3) in R 3 0, ) with initial data zero is given by ux, t) := t Γx, z, t 0 R 3 s) fz) dz ds. But U u, ) = Rf), ) for > 0, so heorem 2 yields a decay estimate of Ux) ux, t) with respect to the space variable x and the time variable t. In addition, the function Rf), 0) is known to satisfy all the statements of heorem 2 with ɛ = 0 [4, heorem 3.]). herefore these statements with ɛ = 0 carry over to u, t), yielding pointwise spatial decay estimates of u, t) which are uniform with respect to t 0, ). hese estimates are optimal in the sense that the fundamental solution of the stationary Oseen system without rotational terms) decays with those same rates [9]). he powers of s τ appearing in the estimates stated in heorem 2 should be considered as a mathematical manifestation of the wake extending behind a body which moves in a viscous incompressible fluid. Corollary means that U u, t) converges to zero for t with respect to the weighted W, -norm,,w,ɛ. As already mentioned in Section, this convergence result means in particular that U is unconditionally asymptotically stable with respect to this norm. he notion of stability which we refer to here is the one introduced in [5, Definition 5.2] in a Hilbert space setting. Obviously it may also be used in the context of Banach spaces. References [] Z. Chen,. Miyakawa, Decay properties of weak solutions to a perturbed Navier-Stokes system in R n, Adv. Math. Sci. Appl ), [2] P. Deuring, S. Kračmar, Exterior stationary Navier-Stokes flows in 3D with non-zero velocity at infinity: approximation by flows in bounded domains, Math. Nachr ), [3] P. Deuring, S. Kračmar, Š. Nečasová, A representation formula for linearized stationary incompressible viscous flows around rotating and translating bodies, Discrete Contin. Dyn. Syst. Ser. S 3 200), [4] P. Deuring, S. Kračmar, Š. Nečasová, On pointwise decay of linearized stationary incompressible viscous flow around rotating and translating bodies, SIAM J. Math. Anal ), [5] P. Deuring, S. Kračmar, Š. Nečasová, Linearized stationary incompressible flow around rotating and translating bodies: asymptotic profile of the velocity gradient and decay estimate of the second derivatives of the velocity, J. Differential Equations ),

9 DECAY ESIMAES FOR LINEARIZED UNSEADY INCOMPRESSIBLE VISCOUS FLOWS 333 [6] P. Deuring, S. Kračmar, Š. Nečasová, A linearized system describing stationary incompressible viscous flow around rotating and translating bodies: improved decay estimates of the velocity and its gradient, In: Dynamical Systems, Differential Equations and Applications, Vol. I, Ed. by W. Feng, Z. Feng, M. Grasselli, A. Ibragimov, X. Lu, S. Siegmund and J. Voigt. Discrete Contin. Dyn. Syst. Supplement 20, 8th AIMS Conference, Dresden, Germany) 20), [7] P. Deuring, S. Kračmar, Š. Nečasová, Pointwise decay of stationary rotational viscous incompressible flows with nonzero velocity at infinity, J. Differential Equations ), [8] P. Deuring, S. Kračmar, Š. Nečasová, Linearized stationary incompressible flow around rotating and translating bodies - Leray solutions, Discrete Contin. Dyn. Syst. Ser. S 7 204), [9] R. Farwig, he stationary exterior 3D-problem of Oseen and Navier-Stokes equations in anisotropically weighted Sobolev spaces, Math. Z ), [0] R. Farwig, An L q -analysis of viscous fluid flow past a rotating obstacle, ôhoku Math. J ), [] R. Farwig,. Hishida, D. Müller, L q -heory of a singular winding integral operator arising from fluid dynamics, Pacific J. Math ), [2] R. Farwig, M. Krbec, Š. Nečasová, A weighted Lq - approach to Oseen flow around a rotating body, Math. Methods Appl. Sci ), [3] G.P. Galdi, On the motion of a rigid body in a viscous liquid: a mathematical analysis with applications, Handbook of Mathematical Fluid Dynamics, Vol., Ed. by S. Friedlander, D. Serre, Elsevier, [4] G.P. Galdi, M. Kyed, A simple proof of L q -estimates for the steady-state Oseen and Stokes equations in a rotating frame. Part I: Strong solutions, Proc. Amer. Math. Soc ), [5] G.P. Galdi, S. Rionero, Weighted Energy Methods in Fluid Dynamics and Elasticity, Lecture Notes in Mathematics 34, Springer, Berlin e.a., 985. [6] G.P. Galdi, A.L. Silvestre, he steady motion of a Navier-Stokes liquid around a rigid body, Arch. Rat. Mech. Anal ), [7]. Hishida, L q estimates of weak solutions to the stationary Stokes equations around a rotating body, J. Math. Soc. Japan ), [8] S. Kračmar, Š. Nečasová, P. Penel, Lq - approach of weak solutions of Oseen flow around a rotating body, Quart. of Appl. Math ), [9] S. Kračmar, A. Novotný, M. Pokorný, Estimates of Oseen kernels in weighted L p spaces, J. Math. Soc. Japan ), 59. [20] M. Kyed, On a mapping property of the Oseen operator with rotation, Discrete Contin. Dyn. Syst. Ser. S 6 203), [2] Š. Nečasová, K. Schumacher, Strong solution of the Stokes flow around a rotating body in weighted L q -spaces, Math. Nachr ), [22] E.A. homann, R.B. Guenther, he fundamental solution of the linearized Navier-Stokes equations for spinning bodies in three spatial dimensions time dependent case, J. Math. Fluid Mech ), Univ. Littoral Côte d Opale, Laboratoire de mathématiques pures et appliquées Joseph Liouville, F Calais, France address: Paul.Deuring@lmpa.univ-littoral.fr Department of echnical Mathematics, Czech echnical University, Karlovo nám. 3, 2 35 Praha 2, Czech Republic, and Mathematical Institute of the Academy of Sciences of the Czech Republic Žitná 25, 5 67 Praha, Czech Republic address: Stanislav.Kracmar@fs.cvut.cz Mathematical Institute of the Academy of Sciences of the Czech Republic, Žitná 25, 5 67 Praha, Czech Republic address: matus@math.cas.cz University of Geneva 24, Quai Ernest Ansermet 205 Geneva - Switzerland address: Peter.Wittwer@unige.ch