COUPLING THE STOKES AND NAVIER STOKES EQUATIONS WITH TWO SCALAR NONLINEAR PARABOLIC EQUATIONS

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1 Mathematical Modelling and Numerical Analysis M2AN, Vol. 33, N o 1, 1999, p Modélisation Mathématique et Analyse Numérique COUPLING THE STOKES AND NAVIER STOKES EUATIONS WITH TWO SCALAR NONLINEAR PARABOLIC EUATIONS Macarena Gómez Mármol 1 and Francisco Ortegón Gallego 2 Abstract. This work deals with a system of nonlinear parabolic equations arising in turbulence modelling. The unknowns are the N components of the velocity field u coupled with two scalar quantities θ and ϕ. The system presents nonlinear turbulent viscosity Aθ, ϕ and nonlinear source terms of the form θ 2 u 2 and θϕ u 2 lying in L 1. Some existence results are shown in this paper, including L -estimates and positivity for both θ and ϕ. Résumé. Nousétudions un système non-linéaire d équations du type parabolique provenant de la modélisation de la turbulence. Les inconnues sont les N composantes du champ des vitesses u couplées avec deux grandeurs scalaires θ et ϕ. Cesystème présente un terme de diffusion non-linéaire sous forme matricielle Aθ, ϕ etlestermessourcesnon-linéaires θ 2 u 2 et θϕ u 2 appartenant à L 1. On démontre alors quelques résultats d existence de solutions, ainsi que des estimations dans L et positivité pourθet ϕ. AMS Subject Classification. 35K45, 35K6, Received: December 3, Revised: November 18, S 1. Introduction In this paper, we state some existence results of a weak solution to the nonlinear parabolic system u Aθ, ϕ u+ pf, u, in t θ t Aθ, ϕ θ 1 θ2 u 2, in ϕ Aθ, ϕ ϕ ϕ θ u 2 + 1, in t θ + r ux, u x, θx, θ x, ϕx, ϕ x, in Ω ux, t, θx, t a, ϕx, t b, on Ω, T where Ω,T, Ω R N is a bounded domain with Lipschitz boundary Ω, N 2, T>, a and b are non-negative constants and r> is a small parameter. Positivity of both θ and ϕ is also shown in this work. System S derives from the so-called k-ɛ turbulence model see [8]. Here, u u 1,...,u N stands for the mean velocity field the symbol here means vector transposition, p is the mean pressure and f f 1,...,f N Key words and phrases. Nonlinear parabolic systems, Stokes and Navier-Stokes equations. Research partially supported by DGICYT grant PB Departamento de Ecuaciones Diferenciales y Análisis Numérico Universidad de Sevilla, Spain. ortegon@numer.us.es 2 Departamento de Matemáticas, Universidad de Cádiz, Spain. francisco.ortegon@uca.es c EDP Sciences, SMAI 1999

2 158 M. GÓMEZ MÁRMOL AND F. ORTEGÓN GALLEGO is a given function describing a distributed force field over. The magnitudes θ and ϕ are obtained from both, the mean turbulent kinetic energy, k, and the mean turbulent viscous dissipation, ɛ. Indeed, θ kɛ 1, ϕ ɛ 2 k 3 and the θ and ϕ equations are both deduced from those of k and ɛ see [6 8] for details. The resulting viscosity is of the form Aθ, ϕ ν + 1 θϕ I, ν > being a constant value. In order to avoid a zero denominator we must change this expression in some way; for example, we may take a perturbation of the form Aθ, ϕ ν + 1 θϕ +r I. Here we shall consider a general matrix expression for Aθ, ϕ. Also, we point outthattheterm ϕ θ appearing in the original modelling of the equation for ϕ has been changed to ϕ θ +r for the same reason as above. System S lacks transport terms; we have first considered these equations without transport terms in order to establish some existence results, independently on the space dimension. The resolution of the full model C is discussed in Section 6 below. As we are just considering system S from the mathematical standpoint, we may assume that the involved physical quantities are dimensionless and that the physical constants which actually appear are taken to be equal to one. The interest in introducing the θ-ϕ approach in turbulence modelling is that S may be considered as a stabilization of the k-ɛ model in the sense that we can state some results concerning the existence, regularity, positivity and L -estimates of certain solutions of S. As one can readily see, system S presents some mathematical difficulties, namely 1. nonlinear source terms and nonlinear viscosity; 2. the whole system is coupled through these nonlinear terms; 3. the regularity result for the Stokes or even the Navier-Stokes equations yields a velocity field u L 2 [,T]; HΩ 1 N if, for example, f L 2 [,T]; H 1 Ω N. So both the θ and ϕ equations contain nonlinear source terms, namely, θ 2 u 2 and θϕ u 2, which lie in L 1 ifθand ϕ belong to, say, L. Some partial results concerning the existence and positivity of solutions θ, ϕ may be found in [6,7]. In these papers, it is assumed that u L [,T]; W 1, Ω is a given data and verifies the rather restrictive condition ess inf u >. The goal of this work is to state the existence of a weak solution to the whole system S, such that θ, ϕ L when the initial data lie in L Ω, together with θ andϕ almost everywhere in Theorems 1 and 2. The proofs are based in some different standard techniques, including truncation [2] and a priori estimates. But from the physical and numerical standpoint, the results about the L regularity and positivity of both, θ and ϕ, are very important since, in general, there is an enormous lack of this kind of results in turbulence modelling e.g. in the k-ɛ model. The resolution of a system like S, or C in section 6, may be regarded, from the mathematical point of view, as a pioneering work in turbulence modelling, since it is the first time that an existence result is shown for a two equations turbulence model. 2. Functional spaces and weak formulation The following notation and functional spaces will be adopted throughout this paper: Ω R N is a bounded domain with Lispchitz boundary Ω; N 2 is the space dimension, and is the cylinder Ω,T, with T> the final time. For an integer m and 1 p + we define DΩ def space of C functions with compact support in Ω. W m,p Ω def v L p Ω / α v x α1 1 xαn N L p Ω,

3 COUPLING THE STOCKES AND NAVIER STOKES EUATIONS } α α 1,...,α N Z N +, α α 1 + +α N m. Also, for s>, and 1 p + one can introduce the space W s,p Ω by interpolation see [1]. W s,p Ω def closure of DΩ with the standard norm of W s,p Ω, W s,p Ω def dual space of W s,p Ω, 1 p + 1 p 1, 1 p< H 1 Ω def W 1,2 Ω, H 1 def Ω W 1,2 Ω, H 1 Ω def W 1,2 Ω, v def v 1 x v N x N, divergence of v v 1,...,v N, V def v H 1 ΩN / v }, V def dual space of V. Let n nx be the outward unitary normal vector to Ω inx Ω, then we define H def v L 2 Ω N / vinω, v n on Ω }, For a Banach space X and 1 p +, wedenotebyl p XthespaceL p [,T]; X, that is, the set of equivalence class of functions f :[,T] X measurables and such that t [,T] ft X is in L p,t. Forafunctionf L p X we put f L p X def 1/p T ft p X, 1 p<+, f L X def ess sup t [,T ] ft X. It is well-known that L p X, L p X is a Banach space. Notice that by Fubini s theorem we can identify the space L p L p Ω with L p the reader is refer to [4] for more properties about these spaces. W 1 def v L 2 V / dv } L2 V, W q 2 def v L 2 H 1 Ω / dv } L2 W 1,q Ω, 1 q<+. W q 3 def v L q W 1,q Ω / dv } L1 W 1,q Ω, 1 q<+. In these definitions, all derivatives are assumed to be taken in the sens of distributions. It is well known that all these spaces are Banach spaces provided with their standard norms. Moreover, V, H and W 2 2 are in fact

4 16 M. GÓMEZ MÁRMOL AND F. ORTEGÓN GALLEGO Hilbert spaces. Also, the following imbeddings hold algebraically and topologically W 1 CH; W q 2 CL 2 Ω, q 2N N 2 < ; W q 3 CW 1,q Ω, q < ; 1 where we are denoting CX C[,T]; X, X a Banach space, the space of continuous functions v :[,T] X, provided with the norm v CX max [,T ] vt X. We will make use of the following compactness lemma see [9]: Lemma 1. Let X, B and Y be three Banach spaces such that X B Y, every imbedding being continuous and the inclusion X B compact. For 1 p, q < +, let W be the Banach space defined as W v L p X / dv Lq Y }. Then, the inclusion W L p B holds and is compact. Finally, we will use the abbreviation a.e. meaning almost everywhere. Now, we may introduce the weak formulation of system S. Let the initial data u, θ and ϕ in Ω, the boundary constants a and b, and the forcing term f in be given. Then, we search for u u 1,...,u N, θ and ϕ, in certain suitable spaces, such that: T du T,v + Aθ, ϕ u v f,v, v L 2 V, 2 T dθ, Ψ + Aθ, ϕ θ Ψ 1 θ 2 u 2 Ψ, Ψ D, 3 T dϕ, Ψ + Aθ, ϕ ϕ Ψ ϕ θ u Ψ, Ψ D, 4 θ + r ux, u, θx, θ x, ϕx, ϕ x, in Ω, 5 θx, t a, ϕx, t b, on Ω, T. 6 Remarks 1. In 2 4, the symbol, stands for certain duality products specified below. 2. As usual, the pressure p does not appear in the weak formulation 2 6 and it can be retrieved by the classical de Rham s argument see [1]. 3. We will show below that, when θ,ϕ L Ω resp. θ,ϕ L 1 Ω then, θ a and ϕ b belong to some space W q 2 resp. W q 3 forsomeq 1, 2. In particular, thanks to the imbeddings given in 1, the initial conditions 5 will make sense at least in L 2 Ω N for u, andinl 2 Ω resp. W 1,q Ω for θ and ϕ. We will consider the following hypotheses: H1 f L 2 H 1 Ω N, u H; 3. The main results

5 COUPLING THE STOCKES AND NAVIER STOKES EUATIONS H2 a, b are real constants; H3 θ, ϕ ; H4 A: R R R N N is continuous and there exists a constant α> such that As 1,s 2 ξξ α ξ 2, for all s 1, s 2 R, ξ R N. The main results now follow. The difference between them is the assumed regularity for the initial data θ and ϕ. Theorem 1. L initial data Under hypotheses H1 H4, ifθ,ϕ L Ω, then there exists u, θ, ϕ solution to 2-6, such that [ u W 1, θ a, ϕ b W q 2 L N, q 1,, 7 N 1 θx, t max θ L Ω,a}+t, a.e. in, 8 ϕx, t max ϕ L Ω,b}, a.e. in. 9 Theorem 2. L 1 initial data Let assume hypotheses H1 H4 and also H5 there exists β>such that As 1,s 2 β for all s 1, s 2 R, being some matrix norm. If θ,ϕ L 1 Ω, then there exists u, θ, ϕ solution to 2-6, such that [ u W 1, θ a, ϕ b W q 3 L L 1 Ω, q 1, N +2, 1 N+1 θ, ϕ, a.e. in, 11 θ 2 u 2 L 1, ϕθ u 2 L Remarks 1. In both theorems, due to 1, the initial conditions given in 5 make sens at least in L 2 Ω N for u and in W 1,q Ω for θ and ϕ. 2. In Section 6 we give two existence results concerning the full system in 2D with transport terms. 3. We may assume a more general version of the diffusion matrix A. Indeed, Theorem 1 also holds if H4 is changed to: A: R R R N N is a Caratheodory matrix function and there exist a constant α> and a non-decreasing function d: R + R + such that H4 α ξ 2 Ax, t, s 1,s 2 ξξ d s 1 + s 2 ξ 2, a.e. in, forall s 1,s 2 R. Also, instead of just A, we may think of three different viscosities A u, A θ and A ϕ for the respective equations of u, θ and ϕ. If these three matrix functions verify H4 then Theorem 1 still holds, and if d is constant, then Theorem 2 also holds.

6 162 M. GÓMEZ MÁRMOL AND F. ORTEGÓN GALLEGO The next sections are devoted to develop the proofs of Theorems 1 and 2. The basic ideas of the proof are summarized here: 1. To divide the two main difficulties, namely i the presence of the terms θ 2 u 2 and θϕ u 2,whichonly belong to L 1 ; and ii the coupling of the N + 2 equations 2 4 through nonlinearities. 2. To avoid L 1 -terms, we use a regularization method based on truncations. This will lead to approximated systems. 3. Finally, the same arguments due to Boccardo and Gallouët [2] may be applied in our context, and this yields the necessary estimates for the approximated solutions to pass to the limit. 4. Proof of Theorem Setting of the approximated problem P M For every M>, we define the truncation function at height M, T M,as s if s M, T M s Msign s if s >M, sign s s/ s if s, if s. 13 Then, we consider the approximated problem P M consisting in finding u M, θ M and ϕ M such that P M u M W 1, θ M a, ϕ M b W 2 2 L and, T dum,v + Aθ M,ϕ M u M v f,v, v L 2 V, dθm, Ψ + Aθ M,ϕ M θ M Ψ 1 θm θ M T M um 2 Ψ, Ψ L 2 H 1 Ω, T T T dϕm, Ψ + Aθ M,ϕ M ϕ M Ψ ϕ M θ M T M um 2 + u M u, θ M θ, ϕ M ϕ. The existence of solution to P M is guaranteed by the next 1 θ M + r Ψ, Ψ L 2 H 1 Ω, Lemma 2. Under hypotheses H1 H4, there exists a solution u M,θ M,ϕ M of problem P M such that θ M x, t maxa, θ L Ω} + t, a.e. in Ω, t [,T] 14 ϕ M x, t maxb, ϕ L Ω}, a.e. in Ω, t [,T]. 15 The existence of u M,θ M,ϕ M solutiontop M and verifying the L estimates can be found in [5]. Basically, it is obtained by an application of Schauder s fix point theorem Estimates for u M, θ M,ϕ M Estimates for u M The classical estimates for the Stokes or Navier-Stokes equations follows inmediately by taking v u M as a test function in the equation verified by u M in P M. This leads to the existence of a constant

7 C 1 C 1 α, f L 2 H 1 Ω,T, u L2 Ω such that Estimates for θ M and ϕ M COUPLING THE STOCKES AND NAVIER STOKES EUATIONS u M L L 2 Ω N C 1, u M L 2 V C First of all, Lemma 2 gives uniform L bounds for θ M andϕ M. Now, taking φ θ M a and φ ϕ M b in the respective equations for θ M and ϕ M, yield θ M L 2 H 1 Ω C 2, ϕ M L 2 H 1 Ω C 3 17 where C 2 C 2 C 1,a, θ L Ω andc 3 C 3 C 2,b, ϕ L Ω. Time derivatives estimates From the estimates derived in the preceeding paragraphs, we may deduce that du M is bounded in L 2 H 1 Ω N, whereas dθ M and dϕm are bounded in L 2 H 1 Ω + L 2 L 1 Ω. On the other hand, we know by Sobolev s imbedding see [3] that L 1 Ω W 1,q Ω whenever q < N N 1, which implies the inclusion L 2 L 1 Ω L 2 W 1,q Ω. Finally, L 2 H 1 Ω + L 2 L 1 Ω L 2 W 1,q Ω, q < N N 1 and we may conclude that dθm, dϕm are bounded in L 2 W 1,q Ω, q < N N 1 Now, the inclusions X H 1 Ω B L 2 Ω Y W 1,q Ω verify the hypotheses of Lemma 1, and consequently the imbedding W W q 2 L 2 is compact. In particular, this implies that θ M andϕ M are relative compact in L Passing to the limit in P M From the estimates obtained in the last section, we deduce that from u M, θ M andϕ M we may extract subsequences, denoted in the same way, such that in L 2 H 1 Ω N -weakly, in L 2 N -strongly, u M u W 1 a.e. in, in L L 2 Ω N -weak-, θ M θ, ϕ M ϕ, θ a, ϕ b W q 2, q < N N 1, in L 2 H 1 Ω-weakly, in L 2 -strongly, a.e. in, in L -weak-, du M du, in L2 H 1 Ω-weakly

8 164 M. GÓMEZ MÁRMOL AND F. ORTEGÓN GALLEGO dθ M dθ, dϕ M dϕ, in L2 W 1,q Ω-weakly, q < N N 1 These convergences are enough to pass to the limit in the velocity equation of P M, obtaining then 2. But passing to the limit in the θ M and ϕ M equations cannot be done directly unless the convergence of u M to uis in L 2 H 1ΩN -strongly. Fortunately, this is the case as it can be shown by taking v u M in P M and passing to the limit. Consequently, u M 2 u 2 in L 1 -strongly and, without loss of generality, we may assume that the convergence also holds almost eveywhere in. Now, we may pass to the limit in the θ M and ϕ M equations and deduce 3 and 4, respectively. On the other hand, using 1, we readily obtain 5. This ends the proof of Theorem Proof of Theorem 2 Now, we assume hypothesis H5 and that θ,ϕ L 1 Ω. For M>, we consider the new approximated problem M M u M W 1, θ M a, ϕ M b W 2 2 L andsuchthat T dum,v + Aθ M,ϕ M u M v f,v, v L 2 V, dθm, Ψ + Aθ M,ϕ M θ M Ψ 1 θ 2 M T M u M 2 Ψ, Ψ L 2 H 1 Ω, T T T dϕm, Ψ + Aθ M,ϕ M ϕ M Ψ ϕ M θ M T M u M θ M +r u M u, θ M T M θ, ϕ M T M ϕ. Ψ, Ψ L 2 H 1 Ω, Since T M θ,t M ϕ L Ω we can apply Lemma 2 and deduce that M admits a solution u M,θ M,ϕ M. It is straightforward that the estimates for u M given in 16 still hold. For θ M andϕ M wehavethe Lemma 3. Let θ M θ M a and ϕ M ϕ M b. Then, there exists a constant C > depending on f L 2 H 1,α,a,b,T, θ L1 Ω, ϕ L1 Ω such that for all M> Ω θ M t C, t [,T], θ 2 MT M u M 2 C; 18 Ω ϕ M t C, t [,T], ϕ M θ M T M u M 2 C; 19 T j θ M L2 H 1 Ω C, T j ϕ M L2 H 1 Ω C, j >; 2

9 COUPLING THE STOCKES AND NAVIER STOKES EUATIONS B j M Aθ M,ϕ M θ M θ M C, C j M Aθ M,ϕ M ϕ M ϕ M C, j >, 21 where B j M x, t /j θ M <j+1} and C j M x, t /j ϕ M <j+1}; lim θmt 2 M u M 2 lim ϕ M θ M T M u M j θ M >j} j ϕ M >j} Proof. All the estimates are obtained by using in M suitable test functions. Indeed, 18 and 19 are deduced taking φ 1 ε T ε θ M andφ 1 ε T ε ϕ M intheθ M and ϕ M equations respectively, and then passing to the limit in ε. Estimates in 2 are straightforward by putting φ T j θ M resp.φt j ϕ M. Finally, to obtain 21 and 22 we just take φ g j θ M resp.φg j ϕ M where g j is given by if s <j g j s sign s if s j+1 s jsign s if j s <j+1. Now, we may apply a result due to Boccardo and Gallouët [2] and deduce, from 2, 21 that θ M andϕ M are bounded in L q W 1,q Ω, for all q< N+2 N+1. Going back to M, we see that du M is bounded in L 2 H 1 Ω N, whereas dθ M and dϕm are bounded in L 1 +L q W 1,q Ω, for all q< N+2 N+1 here, we have explicitly used hypothesis H5. But now, we have L 1 L 1 W 1,r Ω whenever r< N 2, we also have L 1 +L q W 1,q Ω L 1 W 1,q Ω, q < N+2 N+1 N N 1, and since N+2 N+1 < N N 1, This means that θ M and ϕ M lie in a bounded set of the space W q N+2 3, q < N+1, which is compactly imbedded in L 1 L q Ω thanks to Lemma Passing to the limit in M From the estimates obtained above, we deduce that from u M, θ M andϕ M we may extract subsequences, denoted in the same way, such that in L 2 H 1ΩN -weakly, in L 2 N -strongly, u M u W 1 a.e. in, in L L 2 Ω N -weak-, θ M θ, ϕ M ϕ, in L q W 1,q Ω-weakly, θ a, ϕ b W q 3, q <N+2 N+1, in L 1 L q Ω-strongly, a.e. in,

10 166 M. GÓMEZ MÁRMOL AND F. ORTEGÓN GALLEGO du M du, in L2 H 1 Ω-weakly dθ M dθ, dϕ M dϕ, in D. Remark Notice that from the respectively convergences of dθ M and dφm, we just obtain dθ, dφ D. The conclusion dθ, dφ L 1 W 1,q Ω is derived a posteriori, that is, from the equations verified by θ and ϕ, respectively. As in P M, we can show that the convergence u M u still holds L 2 H 1 Ω-strongly and we may assume that u M 2 u 2, a.e. in. 23 By 18, 19, 23 and Fatou s lemma, we derive 12. Then, using this fact and 22 and 23, we obtain θ 2 M T M u M 2 θ 2 u 2, ϕ M θ M T M u M 2 ϕθ u 2, in L 1. Consequently, all terms in M pass to the limit. It remains to prove that θ, ϕ L L 1 Ω; but this is a straightforward consequence of Fatou s lemma. This ends the proof of Theorem Concluding remarks Notice that Theorems 1 and 2 hold for all N 2, the space dimension, though it intervenes in the regularity of the solutions θ and ϕ. In order to study the full system, including transport terms, namely C u +u u Aθ, ϕ u+ pf, t u, in θ t + u θ Aθ, ϕ θ 1 θ2 u 2, in ϕ + u ϕ Aθ, ϕ ϕ ϕ θ u 2 + 1, in t θ + r ux, u x, θx, θ x, ϕx, ϕ x, in Ω ux, t, θx, t a, ϕx, t b, on Ω, T we ought to restrict ourselves to study the case N 2, since the process described in this work fails because it is not known if the energy identity is still verified i.e. in N 3, the convergence u M 2 u 2 in L 1 -strongly is not guaranted.

11 COUPLING THE STOCKES AND NAVIER STOKES EUATIONS By a weak solution of system C we mean a triplet u, θ, ϕ, such that remember that the pressure may be retrieved by the usual de Rham s argument D u W 1, T du,v + T dθ, Ψ Ψ D, T θ a, ϕ b W q 3, θ2 u 2,ϕθ u 2 L 1, dϕ, Ψ ϕ u uv + Aθ, ϕ u v u Ψθ + Aθ, ϕ θ Ψ u Ψϕ + θ u θ + r Aθ, ϕ ϕ Ψ u u, θ θ, ϕ ϕ, in Ω. T Ψ, Ψ D, f,v, v L 2 V, 1 θ 2 u 2 Ψ, Then, the following theorem holds Theorem 3. Let N 2and assume hypotheses H1 H4. i If θ,ϕ L Ω, then there exists a solution to D such that 7 9 are verified. ii If θ,ϕ L 1 Ω and H5 is assumed, then there exists a solution to D such that 1 12 are verified. Proof. We apply the same technique described above and we just need to show that all transport terms pass to the limit. To this end, we know that, when N 2,u M L 4 andu M ustrongly in this space; now, in the first case, we have u M θ M uθ, u M ϕ M uϕ in L 4 -strongly, and then u M θ M uθ, u M ϕ M uϕ inl 4 W 1,4 Ω-strongly. In the second case, the interpolation between L q W 1,q q<4/3 and L L 1 Ω yields θ M andϕ M bounded in L r, for all r<2, so that u M θ M uθ, u M ϕ M uϕ in L r -weakly. Hence, u M θ M uθ, u M ϕ M uϕ inl q W 1,q Ω-weakly, for all q<4/3. There is no uniqueness result for this kind of problems up till now. The authors wish to thank Dr. E. Fernández Cara, Dr. R. Lewandowski and Dr. F. Murat for fruitful discussions and useful comments and suggestions. References [1] R. Adams, Sobolev Spaces. Academic Press [2] L. Boccardo and T. Gallouët, Non-linear Elliptic and Parabolic Equations Involving Meausure Data. J. Funct. Anal [3] H. Brezis, Analyse Fonctionnelle. Théorie et Applications. Masson, Paris [4] T. Cazenave and A. Haraux, Introduction aux problèmes d évolution semi-linéaires. Série Mathématiques et Applications, Ellipses 199. [5] M. Gómez Mármol, Ph.D. thesis. Universidad de Sevilla to appear. [6] R. Lewandowski, Modèles de turbulence et équations paraboliques. C. R. Acad. Sci. Paris [7] R. Lewandowski and B. Mohammadi, Existence and positivity results for the φ-θ model and a modified k-ɛ turbulence model. Math. Model and Methods in Applied Sciences [8] B. Mohammadi and O. Pironneau, Analysis of the k-ɛ turbulence model. Research in Applied Mathematics. Wiley-Masson, Paris [9] J. Simon, Compact sets in the space L p [,T]; B. Annali di Matemàtica Pura et Applicata [1] Temam, R. Navier Stokes equations. Theory and numerical analysis. North Holland Publishing Company, Amsterdam 1979.