Capriz, G.

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A model for phenomena of instability

Institute of Mathematics AS CR, v. v. i.

EQUADIFF 3Gianfranco CaprizA model for phenomena of instabilityIn: Miloš Ráb and Jaromír Vosmanský (eds.): Proceedings of Equadiff III, 3rd CzechoslovakConference on Differential Equations and Their Applications. Brno, Czechoslovakia, August 28 -September 1, 1972. Univ. J. E. Purkyně - Přírodovědecká fakulta, Brno, 1973. Folia FacultatisScientiarum Naturalium Universitatis Purkynianae Brunensis. Seria Monographia, Tomus I.pp. 217--224.Persistent URL: http://dml.cz/dmlcz/700083Terms of use:© Masaryk University, 1973Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czA MODEL FOR PHENOMENA OF INSTABILITY byG. CAPRIZ 1. I N T R O D U C T I O N The analysis of some phenomena of instability in hydrodynamics (in particular of the simpler forms of unstable behaviour for the Couette flow of a viscous fluid between two rotating coaxial cylinders) is well developed (see, e.g., [1]). For instance, the important contributions of VELTE particularly [2], [3], and then of Kirchgassner [4] and others have strengthened the significance of earlier results of linear analyses. Extensive numerical experiments have also been carried out [5], [6]. The peculiar difficulty of the case treated by VELTE in [3] is in part connected with the fact that the differential operators involved are not gradient operators, as happens in the case of the analytical schemes for phenomena of instability in elastic systems, for instance. Again, partially as a consequence of this fact, there are still many open questions even within the relatively narrow field of study of branching phenomena. Analysis and numerical work cannot go much beyond indications of behaviour of solutions in the neighbourhood of branching points and actually it is usual to consider in detail only one of the branching points. Therefore it was thought worthwhile to study a special differential problem which, though relatively simple, nevertheless maintains essential features occurring in some of the much more difficult problems quoted above. In itself the problem belongs to a class which has direct relevance to the description of certain natural phenomena. In fact, the best way to introduce the problem is to recall, even if rather superficially, certain developments contained in a recent paper devoted to the study of a biophysical question [7]. Let * and if be two sets of functions u(P)and v(P), respectively; assume that the domain of both u and v is a certain set DofE2. Let F be a mapping defined over % and with values in if\ then, for each choice of ve'V, let us look for functions w e * such that F(u) = v. (1.1) Naturally, to decide upon the significance of the problem thus set, regularity properties of * , if and F are needed; we refer to [7] for a statement of these properties. We recall here only the mathematical-physical hypothesis which is essential for the developments which follow: we assume that the relationship (1.1) is invariant to orthogonal changes of coordinates in E2. Then we proceed to examine the following question of approximation. Let {va} be a class of functions of if, dependent upon the values of a real parameter a in a neighbourhood of zero; and such that v tends, in appropriate sense, to a constant function when a -> 0. Then, again in an 217 appropriate sense, (and we refer again to [7] for details), an approximate solution of the problem F(u) = va, for a ~ 0 must be sought among the functions w such that a Aw 4- b | grad w |2 4- c = va, w e °U, (1.2) where a, b, c are certain functions of w. This result is not surprising: the orthogonal invariants which can be formed starting from a function w (defined over a set of E2) and its first and second derivatives are (w apart) the modulus of the gradient and the invariants of the hessian matrix; the "simplest" of the latter is the laplacian. Our problem involves a special case of eqn (1.2). Other special cases have been already amply studied; apart from the case b = 0, we can quote the case a = 0, which is the fundamental equation in geometrical optics (see, for instance, [8], p. 88 and ff., p. 369 and ff.). It is of interest to remark here that, when b is not identically zero, the left-hand side of eqn (1.2) is not a gradient operator. 2. STATEMENT OF THE PROBLEM We must now specify the domain D, the functions a, b, c, v, and the class %. In this task we are guided by some general requirements: (i) Our equation must be of the second order and without singularities; hence a # 0. (ii) w = 0 must be a solution of the problem; hence c(0) — va = 0, and the boundary conditions must be homogeneous. These conditions focus all the attention on cases of lack of unicity as a consequence of the existence of "non trivial" solutions, (iii) The interest lies exclusively in real solutions. (iv) For the associated ""linearized" problem eigenvalues and eigenfunctions can be given explicitly. Obviously these requirements do not lead to a unique problem; reasons of simplicity and of analogy with cases quoted in the Introduction suggest the choice made below. Other relevant cases will be dealt with in later papers. The special case of (1.2) studied here is the following one Aw — w | grad w |2 4- Xw = 0, (2.1) (X, a real parameter). °li is the set of real continuous functions w(x, y) defined in the strip S S: 0 = x = T, -oo < v < 4-oo, even in y and periodic in y of period 2; w has partial derivatives up to order two, square integrable over the rectangle R R: 0 = x g 1, - 1 = y = 1; 218 Л - *( j + Яs = °' xe(°» П- ^2-4) w vanishes along the boundary of the strip. A strong solution of our problem, which we will call henceforth problem A, is a function of °U which satisfies eqn (2A). We consider only functions even in y to avoid certain disturbing ambiguities in our analysis. It is now appropriate to remark that the associated linear problem, which we will call problem At, admits in <% the set of eigenfunctions wh,k(x> y) — sin hnx cos k7cy, he N, k e N0, (2.2) which correspond to eigenvalues X = (h2 + k2) 7r2, heN,keN0. (2.3) In the analysis which follows we must consider two other auxiliary problems, which we specify now. The first of these, problem B, involves an ordinary differential equation d2s / <*-\2 dx2 s(0) = 0, s(l) = 0. (2.5) The linear problem Bt associated with problem B is elementary; the eigenvalues are simple and form that part of the sequence (2.3) which corresponds to the choice k = 0. In the third problem, problem C(h, k), (h, k e N) we seek a function z(x, y) defined over the rectangle R(h,k);0£xg±, 0^y^±, having in R(h, k) first and second derivatives, the latter square integrable in R(h, k), and satisfying eqn (2.1) and the boundary conditions z(0,>0 = 0, *(*>-^) = °» (£) - • (") -"• \SxJx=1/2h \oy Jy=0 The linearized problem Cj(h, k) associated with problem C(h, k) admits the functions sin hn(2m + l)x cos k7c(2n + \)y, m,neN0, as eigenfunctions; the corresponding eigenvalues are [h2(2m + l)2 + k2(2n + l)2] 7c2, m,neN0. For us it is important to remark here that the lowest eigenvalue (h2 + k2) 7c2 is simple for problem Q(h, k); notice that the same value is also (perhaps multiple) eigenvalue for problem A/. 219 3. EXISTENCE OF SOLUTIONS OF PROBLEM A We remark first of all that a strong non trivial solution w of problem A might exist only if A _ n2. In fact from eqn (2.1) we get by partial integration over the rectangle R J | grad w |2 di* J w2 | grad w |2 dR A = ^ — - — + * — . (3.1) f w2 dR J w2 dR R R On the other hand, for any non null function in fy the first term in the right-hand side of (3.1) is not less than the lowest eigenvalue of problem At (see for instance [8]); hence the property. It follows that, instead of looking for solutions of problem A, we could seek that function / of °il that satisfies the equation A/ - A/| grad/l2 + A/ = 0, A > 0, (3.2) and then take w = yjlf There is the following advantage of eqn (3.2) over eqn (2.1): let g(x, y) be a function defined over 5, even in y and of period 2; consider the problem A / = g , fe% (3.3) and the relevant Green's function G(P, Q) such that f(P)^\G(P,Q)g(Q)dR; (3.4) R then eqn (3.2) can be substituted by the integral equation 1 f(P) = J G(P, Q) {f(Q) [| grad / |2 - 1]} dR, (3.5) R which we will write briefly, using a standard notation, rf = Tf \i = -j. We turn now to questions of existence of solutions of our problem. Consider again the set of functions tfl, introduce the norm H w || = M w 2 d i M + M | g r a d w | 2 d i n + R and let °U be the closure of °U with respect to this norm. Notice that % is a Sobolev space and also a Hilbert space. 220 If g(P) is a function L2 in R, then eqn (3.3) admits a solution in °U; i.e. there exists a Green's function such that formula (3.4) defines a functionf(P) of $11. Further-more if f(P) belongs to °U, the expression within curly brackets in the right-hand side of (3.5) is in L2. Moreover the mapping y \f -+ f Vf = Tf is a compact mapping of °ll into itself. A non-trivial fixed point of 3~, if it exists, is a solution in a generalized sense of our problem; actually, known regularization theorems assure then the existence also of a strong solution. Similarly one can consider the mapping 3TX \f -*f where Mf'= Tf ff'BW. Its fixed points are solutions of problem At. Now the mapping ^ is the Frechet differential of the mapping ZT at the null element of ^ ; one can use then the theorem of Leray — Schauder on bifurcation of solutions of functional equations to reach the proof of the existence of non trivial fixed point for the mapping ZT for X in the neighbourhood of all eigenvalues of At which have odd multiplicity, see for instance [9] and references quoted there, in particular [10]. Many eigenvalues of problem Az (such as n2, 2n2, An2, etc.) are in fact simple, but many others are double (such as 5n2, I0n2, etc.). Even when triple eigenvalues are considered (such as 25n2), there is an interest in deciding if they correspond to a multiple branching. These questions may be settled through an analysis of the auxiliary problems B and C. 4. E X I S T E N C E AND P R O P E R T I E S OF THE SOLUTIONS OF PROBLEM B For problem B detailed properties of solutions can be found. We do not enter here into details of proofs; we only quote results. Consider the function q 0 for q ^ 0; F(q) has the following properties lim F(q) = %-; F(q) > q\ q-*0 Z F'(q)>0 for q>0, limF,(q) = 0; 4 - 0 l i m Ä = i. 3-+ + ao q 221 For each positive integer h, consider the real function #/,(A) defined implicitly by the relation qh(X) has values only for X — h2n2, vanishes for X — h27i2, is always less than yJXjlh and is such that lim qh(X) — ~ — = 0. A-oo L 2h J Then, the problem B has a sequence of non trivial solutions sh(x, X), he N. sh(x, X) exists only for X = h2n2 and has h — 1 zeros for x e (0, 1), its values over the whole interval (0, 1) can be easily defined when its values over the interval ( 0, —- J are known, in exactly the same way as the values of sin h7ix in the whole interval (0, 1) can be defined by the values in I 0, — - J. In I 0, ——- 1, sh(x, X) is defined by the inverse relation v^= [ SҺ(X, Я) áa [1 _ ^-чЫDrү 0 sh(x, X) reaches its maximum for x — -—- and In ,(-^Г,ЯJ = íł(A). For large values of X, sh(x, X) is equal approximately to sh(x, X) __ jX x in f 0, -y—i, so that sh(x, X) approaches, as X -> oo, a continuous function with — 2m + 1 jumps in the first derivative amounting in absolute value to 2yJX at all points — — — (m = 0, ...h - 1). We conclude then, in particular, that our problem A has at least a solution branch­ing away from the trivial one at the values of X belonging to the sequence h27i2, irrespective of the fact if these values are or not eigenvalues of odd multiplicity for A/. 5. E X I S T E N C E OF S O L U T I O N S OF P R O B L E M C(h, k). The existence of branching points for solutions of each problem C(h, k) can be ascertained with the same techniques which have been used in dealing with problem A. We are interested in particular in the following facts: 222 (i) The lowest eigenvalue of Q(h, k) is simple; hence it corresponds to a branching point for solutions of C(h, k) (which we will call zhk(x, y)). (ii) Only the trivial solution exists for C(h, k) when X < (h2 -f- k2) n2. (iii) A solution of problem A branching away from the trivial one at the eigenvalue X = (h2 + k2) n2 can be found once zhk(x9 y) is known. This statement is based on the following circumstance: suppose a function z(x9 y) of class C2 is given over R(h9 k)9 so that it satisfies eqn (2.1) and the boundary conditions (2.6); then a function w(x9 y) of class C2 can be defined over the whole rectangle R so that it coincides with z over R(h9 k) and has over the rest of R the symmetry properties of the function sin hnx cos kny. The doubts expressed at the end of Sect. 3 are thus completely settled. A number of open questions remains; in particular the following conjectures require proof: (i) zhik(x9 y) is defined for all X > (h2 + k2) n2 ; (ii) the maximum of zhfk(x9 y) increases with increasing X9 with the order 1/2; (iii) as X tends to infinity zhik(x9 y) tends to a "roof function" rhk(x9 y): r(x, y) = y/Xx for y rg -^-- — x 2k 1 2k for y ^ — г - x. REFERENCES [1] J. T. STUART: Hydrodynamicstability, in L. Rosenhead, ed., Laminar boundary layers. Clarendon Press, Oxford (1963). [2] W. VELTE: Stabilitätsverhalten und Verzweigung stationärer Lösungen der Navier—Stokesschen Gleichungen. Arch. Rat. Mech. Anal., 16 (1964), 97-125. [3] W. VELTE: Stabilität und Verzweigung stationärer Lösungen der Navier—Stokesschen Glei-chungen beim Taylorproblem. Arch. Rat. Mech. Anal., 22 (1966), 1—14. [4] K. КIRCHGÄSSNER, P. SORGER: Branching analysis for the Taylorproblem. Quart. J. Mech. Appl. Math., 22 (1969), 183-209. [5] G. CAPRIZ: he numericaì approach to hydrodynamic problems, in J. L. Lions, ed., Numerical analysis of partial differential equations, Cremonese, Roma (1968). [6] R. C DA PRШA, E. H. ROGERS: Computing problems in nonlinear hydrodynamic stability Physics of Fluids, (1969), Suppl. II, 155-165. [7] B. COLEMAN: A mathematical theory of lateral sensory inЫЫťюn. Arch. Rat. Mech. Anal., 43 (1971), 79-100. [S] R. COURANT, D. HILBERT: Methods of mathematical physics, vol. II, Interscience Publ., New Yorк (1962). 223 [9] G. PRODI: ProЫemi di diramazione per eąuazioni funzionaii. Atti VIII Congresso U.M.I.,Trieste (1967). [10] J. SCHWARZ: Nonlinear functional analysis. New York Univ. (1963). Лuthoŕs address: Gianfranco Capriz ístituto di Elaborazione della Jnformazione C.N.R., Via S. Maria, 46 56100 PISA, Italy 224 

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A model for phenomena of instability

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Institute of Mathematics AS CR, v. v. i.