N

v(z) ----- limt;n(,z) = inf vn(z) z G M+ , in C,20'ca(I+). In addition,

0 <v(z)<v(z) z GM+, for any arbitrary bounded positive solution v to (21) satisfying (22). Since v(z) is also a solution to (21), this means that v(z) is in fact the maximal positive solution to (21) in such class. Observe now that for every vector r G

vT(z) =v(«+(r,0)), is again as positive solution to (21) bounded by 1. In particular

This immediately implies that v(z' + r, zN) =v(z',zN), for all z G K+ and r G Hence, v depends only on z^.

It can be shown in the same way that problem (21) has a minimal solution v in the class of positive solutions bounded by 1, which only depends onz^r. In fact, consider the problem,

(-Av = -a0vp zeDR,h \ v = C zedDR)h, where DR]h is as before while ( e C2'a(dDRth), 0 < C < 1> satisfying in addition £(z) = 1 for zn = 0, \z'\ < R — 2e, e > 0 small enough, and £ = 0 outside {\z'\ < R — e, z^ == 0} in the whole of Problem (25)

and the corresponding ones in Dn admit a unique positive classical solution wn(z) such that the relation

0 < wn(z) < wn+k(z) < v(z), holds for any n, k € N, z € Dn and an arbitrary nonnegative solution v to (21) satisfying (22). Then, the limit, v(z) = limw„(z) = supu;ri(z)

n _jy exists in Clo'°(R+) and defines the minimal solution to (21). Therefore, v only depends on zm-

Finally, observe that the one-dimensional problem,

(v" = aovp t> 0 \«(0) = 1, has a unique positive solution v = v\ (t) with the property of being defined in the whole interval [0, oo). Such solution is explicitely given by

with

In particular, the maximal and minimal solutions satisfy:

v(z) = v(z) = vi (zN) z e , and problem (21) has v\(zn) as its unique positive bounded solution. □ Remark 5.1.

a) In many cases it can be shown that supu(-, w0) = «o, (27)

n for uo large. In fact, notice that since u{x, uq) becomes finite in every x e ft as uo ► oo (Theorem 4.1), then dist(x„0,ft) —> 0 as uq —i► oo where xUo is any point in ft where u(-,uo) achieves its maximum. On the other hand X(x) — a(x)u(x, uo)p~1 > 0 at x = xUo if xUa G ft.

If, for instance, a > 0 on 9ft then it follows that xUo G 9ft (and so (27) holds) if uo is large. Otherwise, with d(x) = dist(x, 9ft) and 5 some sufficiently small positive number. That is not possible since sup u(-, uo) —> oo as uq —► oo.

b) It should be stressed that regarding the asymptotic rate (13), a(x) is allowed to freely vary on 9ft provided it keeps positive there. On the other hand, the limit (23) not only provides asymptotic information on u(-,uo) at xq but also in nearby points of the form x = xo + a-1 y, y fixed, with a = 4P_1)/2 and uo —> oo.

We are now studying the asymptotic behavior of the solution u(-,uo) to (12) near a point xq G 9ft where the coefficient a vanishes. To this proposal we are allowing a(x) to decay as a power of the distance to the boundary, with a variable rate, possibly depending on the location of the reference point xo G 9ft. Since it will be assumed that ft is C2,a no generality is lost if it is assumed as before that xq — 0 while the outward unit normal to 9ft atx = Oisf = — ejv- Let us introduce the decaying restriction on the coefficient a to be used in the forthcoming analysis. The notation of the proof of Theorem 5.1 describing 9ft near x = 0 as x^ = f(x') is kept. Supposing a(0) = 0 it will be assumed the existence of functions oo, ai and g defined on a neighborhood of xo — 0 and a positive constant 7 such that, a(x) = a0(x') +a1(x')yji + g(x',yN)yJf, yN = xN - f(x'), (28)

where ao>°i are nonnegative, ao(x') = o(Jx'|7) as \x'\ 0, aj(0) / 0 and g(x',yN) —> 0 uniformly in x' (|x'| < 5) as j/^ —> 0. The meaning of the condition is clarified by observing that yN has the status of the "vertical distance" to the boundary 9ft, ao gives the restriction ci|9n, i.e., ao(a;') = a(x', f(x')), while a\ measures the rate of deviation of a(x) regarding its value in the boundary as the distance yjv tends to 0.

Remark 5.2. Condition (28) is commonly used in the literature under the more restrictive form, a(x) = A(P+o((P), 0, d(x) = dist(x, dCl) with a coefficient A either constant or variable but positive (see [10], [14], [27], and references quoted there). This corresponds to set ao(x') = 0 in (28). On the other hand it is remarked that a\(x') is allowed to be variable in (28).

We can already state the following improved version of Theorem 5.1.

Theorem 5.2. Let a S Ca(Q) be nonnegative such that its null set {a(x) = 0} = Qo, with fio C Qo C fi is a C2,a subdomain ofQ, fulfilling the conditions of Theorem 5.1.

If a(xo) =0 at xo € dQ. and a(x) satisfies the decaying condition (28) at Xq then the normal derivative of the solution u(-,uo) to problem (12) exhibits the exact asymptotic behavior, du(-,u0) .

Of |x=xo where k is a universal constant depending only on p> 1.

Proof. Recall we are taking xq = 0 with {xm =0} the tangent hyperplane to dil at x = 0. The course of the proof of Theorem 5.1 can be followed, firstly rectifying dfl near x = 0 by means of the change (15) (new coordinates y) and then performing the alternative blow-up scaling, z = ay, a = u^, u{y) = u0v(z). Problem (12) is thus transformed near zero into

V(z',0) = l, \z'\ < (TSOt with \z'\ < creo, 0 < z^ < oe, where z> zJ z a{z) = a7a0(—) + ai(—)z1N + g(-)zjf.

Since the family v(z, a) is uniformly bounded in aVi and aV\ —> R^ as a —> oo, by arguing as in the proof of Theorem 5.1 a subfamily v(z, a') is found such that

n _ft in Clo'"(R+). Moreover, v(z) solves the problem, f—Av = —aizJfVp z€Rf

(31) v(z',0) = 1, with ai = ai(0). The same reasoning used in Lemma 5.1 ensures that (31) has a minimal solution v(z) and a maximal solution v(z), both positive and exclusively depending on zn- In addition, the following lemma (whose proof is delayed to the end of the current one) holds.

Lemma 5.2. The initial value problem,

(v» = tW> t> 0 1«(0) = 1, has a unique solution v(t) with the property of being defined in the whole interval [0, oo). In addition, v(t) is positive, convex, decreasing and v(t) = At~e + 0(t-("+'>) as t —> oo,

2 + 7 i with 6 = —-f, A = (6(6 + 1))^ and every /x satisfying p- 1

Qoca(®+) With V(Zn) = vi(<h^2+^zn)i vi(t) being the unique solution to (32) whose existence is ensured by Lemma 5.2. Set, in particular,

Then we finally get du<■ un) p+!+-y dvl• <j) -i- P+1+T £±i±2

UU\ ' "0> — _„ 2+T ( ' ' _ Kn 2 + T „ , ( 2+-, X

OV \x=xo OZN |z=o as uq —■► oo. Thus, the proof of Theorem 5.2 is concluded. □

Proof of Lemma 5.2. Consider the initial value problem, 1 v(0) = 1, t/(0) = a,

(T regarded as a parameter and set v(t,a) its unique non continuable solution. We claim that a solution to (33) defined for all t > 0 must have v'(t) < 0 in 0 < t < oo. As a consequence of this fact and arguing by convexity one finds that limu = limn' = 0 as t —> oo and so 0 < v(t) < 1 for t > 0.

To show the claim suppose u'(to) > 0 at some to. If v(to) > 0 (of course, the case v = v' = 0 at t = to is discarded by uniqueness) then v' > 0 together with v"v' > t^vpv' for t > to. This implies v'2 > ~p[(vP+1 ~ vo+1)>

and v must blow-up at a finite t\ > to- If v(to) < 0, v' is positive at the right of to and by convexity v becomes positive with positive derivative at finite time. Thus, blow-up occurs again.

Next, let us show the uniqueness of a solution defined in [0,oo). It can be checked that v(t,ai) < v(t,(t2) if <7i < a2 in any interval (0,6) where both solutions are positive. In particular, this must be true if b = oo and they are two possible different solutions in the conditions of the statement. By integrating in (33) using their behavior at t — oo one gets

-<7! = / Fvfrviy < / tJv(t,<J2)P = -<72, Jo Jo which contradicts the assumption on the a^s.

To construct the solution define imin = tmin(a), a < 0, as that t > 0 where infu(-,<r) is achieved. It can be proved that tmin increases when a decreases provided that v(-,a) keeps positive. Set a* = inf{a < 0 : infu(-,(j) > 0}.

Then —oo < a* < 0 while the solution v(t,a*) is defined in all t > 0 and provides the desired solution. In fact, infw(-,cr) > 0 for a < 0 small, by smooth perturbation, and so a* < 0. On the other hand, for 0 < t < tm[n(a) and <7 > a* one gets

Since such inequality cannot be true if cr —y — oo with t fixed then it follows that cr* > — oo. Finally we achieve t* := lim imin(c) = sup tmin(<7) = OO.

Otherwise, v(t,a*) = infcr>cr* v(t,a) would vanish together with its derivative at t = t*, which is impossible.

To finish the proof we study the asymptotic behavior of v(t,a*) (v(t) for short) as t —> oo. By performing the scaling (see [15], [4])

v{t) = At~ez(t), with the exponent and coefficient 6, A introduced in the statement, the normalized solution z(t) observed in t > 0 satisfies tV -26tz' = 6{6+l)(zp -z).

After Euler's change r = logi, z — z(t) solves the autonomous equation, z"-(20 + l)z' + 0(l + 0)(z-z") =0, ' = (34)

in the whole — oo < r < oo. Let us study the phase space of (34) to show that, lim z(t) = 1,

which yields the desired asymptotic behavior. To achieve this observe that (z, zi) = (0,0) is an unstable node while (z, zi) = (1,0) defines a saddle for the associated equation, z = z\

Also, the function E(z, zi) = -f- + 0(1 + 9) ( —---- ) strictly increases

on solutions.

Now, orbits T entering the quadrant 2 > 0, z\ > 0 from (0,0) at t = —oo can only exhibit the following behaviors (see Figure 2):

i) The orbit T keeps in the region z\ > —~{{z - zv) h(z) for

0 < z < 1. Since E(z,zi) increases on the orbit this implies that the parameterizing solution z(r) is increasing and blows-up at finite time.

ii) The orbit F reaches z\ = h(z) at some 0 < z < 1, enters the region 0<z<l,0<zi< h(z) remaining there for all future times r.

iii) Orbit T behaves as in ii) but enters the region z\ < 0 at some 0 < z < 1. In this case z(r) vanishes at some finite r.

Observe now that the solution z(t) obtained from our global solution v(t) = v(t,a*) satisfies z(t) = A-1eeTv(eT) ~ A~1effT as t-* -oo, z'(r) = A~1eeT(0v + eTv'(eT)) ~ 0A~1eeT as r -> -oo. Thus, its orbit enters z > 0, z\ >0 but does not vanish, or blows-up at finite time. Therefore such orbit fulfills ii), which means that it necessarily defines a branch of the stable manifold corresponding to (1,0). This implies that wfj) = l + 0(e~Mr) as r oo,

Figure 2. Phase space configuration for equation (35).

for every /x G (0, -/x-) with /x_ the negative eigenvalue of the linearization of (34) at the saddle (1,0). This concludes the proof of the lemma. □

Remark 5.3.

a) As an application of the results in this section we are performing an alternative construction of the finite supersolution u* appearing in Sections 3 and 4. Assume that fi0 C f2o C ft consists of m connected pieces {fto.i}™i fulfilling condition (10). For 5 > 0 small introduce Qi = (fto.i)5 and D = ft \ U£Li Qii aH them being C2'a domains. The problem i -Au = A(x)u - a(x)up x G D \ u — uo x G dD, has a unique positive solution ud(-,uo) while each problem j -Av - X(x)v = 0 x G Qi 1^ = 1 x G dQi,

has also a unique positive solution Vi(x) € C2'Q(£Ji), the existence being ensured by the conditions A^ (-A - A) > 0, 1 < i < m. Define

Then u* defines a supersolution to — Au = Au — aup both in D and in each dxt ( y \

Qi. Thanks to Theorem 5.1, the normal derivative —D\ ' at x G dQi av dv'(x)

grows faster than — un—as uo -> oo. According to [1] this implies that ov u* defines a positive supersolution for Uq large.

b) A positive supersolution u* to —Aw = Xu — aup in fi having = oo can be constructed exactly in the same way by replacing ud(x,uo) in (38) by the solution ud(x,uo) to,

-An = A(a;)tt — a{x)up x € D u = u0 x€dD\d n (39)

With an appropriate handling, Theorems 5.1 and 5.2 can be adapted to this new scenario to show that ^-— satisfies the same growth estimates ov regarding uo, as uq —> oo. The use of this supersolution permits to obtain the existence assertion in Theorem 4.1 under an alternative approach.

Acknowledgments

This work has been supported by DGES and FEDER under grant BFM2001-3894.

References

1. H. Berestycki, P. L. Lions, Some applications of the method of super and subsolutions, in Bifurcation and nonlinear eigenvalue problems, Proc., Session, Univ. Paris XIII, Villetaneuse, 1978, pp. 16-41, Lecture Notes in Math., 782, Springer, Berlin, 1980.

2. L. Bieberbach, Au = e" und die Automorphen Funktionen, Math. Ann. 77 (1916), 173-212.

3. H. Brezis, L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal. 10 (1986), no. 1, 55-64.

4. M. Chuaqui, C. Cortázar, M. Elgueta, C. Flores, J. GarcíarMelián, R. Lete-lier, On an elliptic problem with boundary blow-up and a singular weight: the radial case, Proc. Royal Soc. Edinburgh 133 (6) (2003), 1283-1297.

5. M. Chuaqui, C. Cortázar, M. Elgueta, J. GarcíarMelián, Uniqueness and boundary behavior of large solutions to elliptic problems with singular weights, Comm. Pure Appl. Anal. 3 (4) (2004), 653-662.

6. N. Dancer, Y. Du, Effects of certain degeneracies in the predator-prey model, SIAM J. Math. Anal. 34 (2002), 292-314.

7. M. Del Pino, R. Letelier, The influence of domain geometry in boundary blow-up elliptic problems, Nonlinear Anal. 48 (2002), no. 6, Ser. A: Theory Methods, 897-904.

8. Y. Du, Effects of a degeneracy in the competition model. Part I: classical and generalized steady-state solutions, J. Diff. Eqns. 181 (2002), 92-132.

9. Y. Du, Effects of a degeneracy in the competition model. Part II: perturbation and dynamical behaviour, J. Diff. Eqns. 181 (2002), 133-164.

10. Y. Du, Q. Huang Blow-up solutions for a class of semilinear elliptic and parabolic equations, SIAM J. Math. Anal. 31 (6) (1999), 1-18.

11. J. Fraile, P. Koch-Medina, J. Lopez-Gomez, S. Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation, J. Differential Equations 127 (1996), no. 1, 295-319.

12. J. Garcia-Melian, A remark on the existence of positive large solutions via sub and supersolutions, Electronic J. of Differential Equations 2003 (2003), n. 110, 1-4.

13. J. Garcia-Melian, R. Gomez-Renasco, J. Lopez-Gomez, J. Sabina de Lis, Point-wise growth and uniqueness of positive solutions of sublinear elliptic problems where bifurcation from infinity occurs, Arch. Rational Mech. Anal. 145 (1998), 261-289.

14. J. Garcia-Melian, R. Letelier-Albornoz, J. Sabina de Lis, Uniqueness and asymptotic behaviour for solutions of semilinear problems with boundary blowup, Proc. Amer. Math. Soc. 129 (2001), no. 12, 3593-3602.

15. J. Garcia-Melian, R. Letelier-Albornoz, J. Sabina de Lis, The solvability of an elliptic system under a singular boundary condition, Proc. Royal Soc. Edinburgh, to appear.

16. J. Garcia-Melian, J. Rossi, Boundary blow-up solutions to elliptic systems of competitive type, J. Differential Equations 206 (2004), 156-181.

17. J. Garcia-Melian, J. Sabina de Lis, Elliptic problems under a singular boundary condition, Proceedings of the XVII Conference on Differential Equations and Applications, CD-Rom ISBN 84-699-6144-6, Salamanca (Spain), 2001.

18. J. Garcia-Melian, A. Suarez, Existence and uniqueness of positive large solutions to some cooperative elliptic systems, Advanced Nonlinear Studies 3 (2003), 193-206.

19. B. Gidas, J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 (1981), no. 8, 883-901.

20. D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, 1983.

21. J. B. Keller, Electrohydrodynamics I. The equilibrium of a charged gass in a container, J. Rational Mech. Anal. 5 (1956), 715-724.

22. J. B. Keller, On solution of Au = f(u), Comm. Pure Appl. Math. 10 (1957), 503-510.

23. C. Loewner, L. Nirenberg, Partial differential equations invariant under con-formal or projective transformations, Contributions to Analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, pp. 245-272.

24. J. Lopez-Gomez, Permanence under strong competition. Dynamical systems and applications, 473-488, World Sci. Ser. Appl. Anal., 4, World Sci. Publishing, River Edge, NJ, 1995.

25. J. Lopez-Gomez, The maximum principle and the existence of principal eigen values for some linear weighted boundary value problems, J. Differential Equations 127 (1996), no. 1, 263-294.

26. J. Lopez-Gomez, Existence and metacoexistence states in competing species models, Houston J. Math. 29 (2003), 483-536.

27. J. Löpez-Gömez, The boundary blow-up rate of large solutions, J. Differential Equations 195 (2003), no. 1, 25-45.

28. R. Osserman, On the inequality Au > f(u), Pacific J. Math. 7 (1957), 16411647.

29. S. I. Pohozaev, The Dirichlet problem for Au = u2, Dokl. Akad. Nauk. SSSR 134 (1960), 769-772.

30. H. Rademacher, Einige besondere probleme partieller Differentialgleichungen in Die Differential und Integralgleichungen der Mechanik und Physik (by P. Prank and R. von Mises), vol. I, Rosenberg, New York, 1943, 838-845.

This page intentionally left blank

WELL POSEDNESS AND ASYMPTOTIC BEHAVIOUR OF A CLOSED LOOP THERMOSYPHON*

A. JIMÉNEZ-CASAS Departamento de Matemática Aplicada y Computación Grupo de Dinámica No lineal, Universidad Pontificia Comillas de Madrid, 28015-Madrid, SPAIN E-mail: [email protected]

We analyze the motion of a fluid containing a soluble substance in the interior of a closed loop under the effects of natural convection and a given external heat flux. This motion is governing by a coupled differential system. First, we show the well posedness of this system in a framework generalizing the existence results of [5], Finally, we prove some result about the asymptotic behaviour for solutions of above system generalizing the results of [12].

1. Introduction

In the engineering literature a thermosyphon is a device composed of a closed loop containing a fluid where some soluble substance has been dissolved. The motion of the fluid is driven by several actions such as gravity and natural convection. In particular, we will consider the convective movements caused by inner solute fluctuations generated by a temperature gradient; this fact is known as the Soret effect, and it has been studied experimentally by Hart [1] between others. We study the evolution of the velocity of the fluid v, of the temperature T of the fluid and of the solute concentration S.

We assume that the section of the loop is constant and small compared with the dimensions of the physical device, so that the arc length coordinate along the loop (x) gives the position in the circuit. The velocity v of the fluid is assumed to be independent of the position in the circuit, i.e., it is

♦Work supported by projects BFM2003-03810 and BFM2003-07749-C05-05/FISI-CICYT, Spain.

assumed to be a scalar quantity depending only on time, v = v(t). The other relevant quantities, namely temperature, T(t,x), and concentration of the solute, S(t,x), are assumed to depend on time and position along the loop.

We assume that the average circulation is generated by the net buoyancy torque exerted by both solute concentration and temperature, and is retarded by viscous drag at the wall, i.e., by friction forces. In addition we assume that the variations of temperature are independent on the solute concentration. We consider the distribution equation of solute into the loop as in [1] and [8], where has been used the conservation of mass for the solute and has been assumed that the fluid also transports the solute, and be generated by Soret diffusion and reduced by molecular diffusion.

The evolution of the above quantities is given by the following ODE/PDE system (cf. [4],[5],[6],[14] for further details)

It is important to note that all functions considered are 1-periodic respect to the spatial variable. The function / describes the geometry of the loop and the distribution of gravitational forces, so note that § f = 0 where § = f0 dx denotes integration along the closed path of the circuit. We consider the general geometries as in [14]. In the sequel we assume that G and h, are given continuous functions, such that G(v) > Go > 0, h(v) > ho > 0 and Ta is the (given) ambient temperature. The functions G(v), which specifies the friction law at the inner wall of the loop ([8],[9],[11],[14]), and h which prescribes the heat flux at the wall of the loop (cf. [9]), are given by different forms. The diffusion coefficients b, c are positive constants and we note that b is proportional to the Soret coefficient, therefore if we assume it to be zero, i.e., if we neglect the Soret effect, and we start with an homogeneous initial concentration of solute, then S remains constant in time and space in Eq. (1) and, since / / = 0, then Eq. (1) reduces exactly to the model in [11],[14] and [12].

In the present work two main results are presented:

- First, in Section 2, we prove that Theorem 3.1 in [5] is also true when we consider initial data in other more general space.

- Finally, in Section 3, under suitable conditions we prove that any solution either converges to the rest state or the oscillations of velocity around v = 0 must be large enough. This result generalizes the one proposed in [12] for a thermosyphon model including a two-component fluid.

2. Well posedness, existence and uniqueness of solutions

To prove the existence and uniqueness of solutions we will consider the suitable phase space.

2.1. Well posedness. Discussion about the suitable framework

Using the notation in [5], we define:

Definition 2.1. We denote by admissible space any Banach space X of 1-periodic functions, verifying the following properties

0 ||s(- + k)\\x = Hflllx, for every k eR and g € X. ii) Given g G X, then the function R —► X, defined by k i—> g(-+k) G X is continuous.

Remark 2.1. Examples of admissible spaces are W™jP( 0,1), Cper(0,1) or Cp£(0,1) among others.

In this section, we will show the reasons to consider the phase space used in the next theorem, as suitable framework.

Indeed, we assume that To,Ta G X, where X is an admissible space in the sense of Definition 2.1. In order to prove the existence of solution of system Eq. (1) with initial data (uo, To, So) G y, we develop a more general framework for So and /. Let consider the third equation of system Eq. (1)

It is known that it is possible solve this working in the fractional power spaces, Za, associated to the sectorial operator + J) (cf. [2]). We note also Za+1 is an admissible space in the sense of Definition 2.1 for T.

We know that the map t >—► T(t) is continuous in X but we also need that the map t -£?T(t) is well defined in Za that is t ^ T(t) G Za+1. To get this we consider a suitable relationship between X and Za, that is, X Za+1.

On the other hand we need that the operator ^ : Za+¿ —» Za will be well defined, for this S{t) G Za+¿, and hence it suffices to consider So G Za+i.

Therefore, we can consider the initial data (vq,Tq,So) e y — R x X x with X <—► Za+1, and choosing a such that the map F(t,v,S) — ~ + CS, has good properties that allow us to use a fixed point argument. Arguing as in [5], first to prove the existence of solutions of system Eq. (1) considering an initial data in we note that the condition iii) of Lemma 3.1 in [5], is now given by this rj 2

sup II —{T^{r)-T^(r)) \\z. < Mr\\Vl - v2\U (2)

Now, taking into account that

we have that Eq. (2) is true if sup \\Tv*(r)-Tv*(r)\\Za+1 < Mr\\vx - t^»«, (3)

is satisfied.

Next, we know that Eq. (3) is true if the functions To, Ta have Lipschitz translations in Za+1. Also it is suffices to suppose that To,Ta G Za+i to get Eq. (3) (cf. [4],[5] and [10]). Thus, from the above arguing we get the following result:

Lemma 2.1. Let r > 0 and v G C[0,t] and we consider To,Ta G X being X an admissible space, then the function Tv defined by

Tv(t,x)=T0(x- /%)e-•»(«)+ f[h{v{r))e-S>^Ta{x- f v)]dr (4) Jo Jo Jr is the integral solution of dTv dTv

and verifies the following properties i) For every t G [0, r] and for every X admissible space, we have that imi)||x<maoc{||T0|U,||T0||x}. (6)

ii) Suppose that the translations of Ta and To are Lipschitz in Za+1, i.e., there exists a positive constant Cd > 0 with d = a and d = 0 such that

IITd(- + k) - Td(-)\\Zc+i < cd\k\ for every k G R. Then, the function Tv, defined by Eq. (4) is Lipschitz in t with values in Za+1.

Hi) We consider the particular case X = H2er = and we denote by H~el = (H^ery the dual space of Hlper = Thus, if T0,Ta G

Hper, then Eq. (4) satisfies Eq. (5) pointwise, and Tv £ C([0,r], Hper) n CJ((0,T),H^er). Moreover, if h £ C1 then Tv £ C2((0, r), L2er) and given Vi £ C[0,t] with i £ {1,2}, we have that

Finally, we have to choose the suitable space for the function / such that f(T - S)f will be well defined. Taking into account that T(t) £ X, and S(t) £ Za+ 3, it is suffices to take / G X' fl (Za+i)'. In this way, if x za+ f Za+3 then we can consider / € (Za+%)'.

Using the sectorial operator theory from [2], we have that — + I is sectorial in Lper = Z, 1 < p < oo with domain = Z1 and Za C W2™r,p, a > 0, is the domain of (—+ I)a. In the same way we have that Z1/2 = and Z"1/2 is the dual space of with 1 + 1 = 1.

Under the above assumptions, taking different values of a we get the following particular cases with ^Rxlx Za+3.

i) If a — -5, then the phase space is y = R x X x L£er with X an admissible space such that X <-► Ta £ W£p and / £ Lqper.

It should be pointed out that this choice in the particular case when p = 2, corresponds to the hypothesis of Theorem 3.1 in [5].

ii) If we consider now a = — 1 instead of a = — then the phase space is}' = RxXx Z~ 3 with Z~ 3 = (W^Y and X an admissible space such that X ^ Z3 = with Ta G and f £ In this situation, if we consider p = 2, we get Ta, / G and (vq, Jo, 5o) G M x h*er x iy~eJ..

2.2. Existence and uniqueness of solutions

In this section we will show that under the above conditions we have the existence and uniqueness of solutions of system Eq. (1).

Theorem 2.1. Suppose that H(r) = rG(r) is locally Lipschitz and Ta G X where X is an admissible space such that one of the following assertions is satisfied:

i) X c W2fr and f G Lper, where 1 + 1 = 1,1 < p < oo. In this case we consider )* = RxIx Lper-

ii) X C and f G W^. In this case we consider y = R x X x

Then, in both situations, given (vo,To,So) G y, there exists a unique solutions (v,T,S) G C([0,oo),}'). Moreover the map:

defines a C° semigroup in

Proof. Taking into account the above section together Lemma 2.1, and arguing as in [5], we conclude the result. To get this we cover several steps. First, we prove there exists a unique local solution (v,T,S) G C([0,t],^) of Eq. (1). To prove this we use a fixed point argument. We consider the space from i) (analogous working in ii )), W = {(v,S) G C([0,r];R x LPer), v(0) = wo, S(0) = So such that |v(t) -< 71, ||S(t) - < 72

for every t G [0, r]}, with r,7¿,i G {1,2}, some fixed positive constants. Then (W, || ■ ||oo) is a Banach space with the norm ||(v,S)||w =

SUPt€[0,r] ll(«(i).'S(i))||RxLf.P = IMloo + IISIIL°°(L?eT.)-

Let U = (y, S) G W and J(U(t)) = (w(t), R(t, x)) be the solution of the system

Jo where A is the operator defined by A = i ^ ^ J! + J ' nonlinearity

Tv is defined in Eq. (4). Then we have J{U(t)) = ( j^^jj) with w(t) = Ji(U(t)) = vo - T G(v)vdr + f[l(Tv{r) - S(r))f}dr (9) Jo Jo J

+ J* e-c(-&+I)(t-r)[cS(r) - v(r)^ - b^Tv(r)}dr. (10)

Since Ta,T0 £ X C from Lemma 2.1, Tv verifies T° £ C([0,r],X)

and £ C([0,r], Thus, from the continuity of G, we have that Ji(U{t)) e C([0,t];R). Moreover e~c(-^+/)tSo € C([0,r];L£er) and cS(t)-v{t)^-b^Tv(t) £ L°°(0,t, (W^)"1), so applying the regularity results from [10], we obtain j\-<-^+m-r)[cS{r)_v{r)^_b^_TV{r)]dr e c([0;r]ijLPer);

Next, we prove that there exist r > 0,71 and 72 small enough such that J is a contraction in (W, || • (|oo)-

In second step, we prove the local solution is defined for every t > 0. For this, we suppose the solution has been extended to a maximal interval of time [0,r), with (v,T,S) £ C([0,r);^) and we prove the norm of the solution in y remains bounded in finite time intervals. □

If we consider the case p = 2, we get the next result.

Corollary 2.1. Suppose that H(r) = rG(r) is locally Lipschitz and the functions Ta and f £ Hper. Then given (v0,T0, SQ) e^ = Rx Hper x there exists a unique solution (v,T,S) £ C([0,00), [V). Moreover we have that v £ C1(0,oo),T £ G(0,00; Hlper) n C1 (0,00; L2per),

S £ C((0,00); Hlper), St £ C((0,00); ff-J.) for every e £ (0,1).

Proof. We note that the nonlinearity F2(t,S) = b^T - i>ff is locally Lipschitz in S and Lipschitz in t, when we consider the map F2 : R x H^). —> H~2r, where H~e2r is the dual space of H2er. Now applying the sectorial theory from [2], we get S £ C((0, r)-,L2er) and St £ C((0,r); Hp*r), for every e e (0,1).

Next, we note that S verifies the equation — = F2(t, S) — , by this way taking into account the above regularity of S and T we obtain that F2(t,S) - G and by elliptic regularity we get that

From above section, we have also the following result:

Corollary 2.2. We suppose that H(r) = rG(r) is locally Lipschitz, h G C\Ta G Hjer and f G L2er. Then, given {v0,T0,S0) G y = R x H%er x Lper, i/iere exists a unique solution of system Eq. (1) globally defined, i.e., (v,T,S)eC([0,oo),y).

Moreover, the function S*(t)(v0,T0,S0) = (v(t),T(t),S(t)),t > 0 defines a C° semigroup in y.

We will see a result about the existence and behavior of constant solutions with respect to the spatial variable, i.e., depending only on time.

Proposition 2.1. Under the hypothesis of Theorem 2.1 if we suppose that Ta is a constant function, i.e., Ta G M, then the unique solutions of system Eq. (1) depending only on time, i.e., the solutions such that T = T(t) and S = S(t), are given by the solute concentration constant in (x,t), i.e., S(t,x) = So G M. Moreover these solutions (v(t),T(t), So) converge to (0,T0,So) exponentially when the time goes to infinity.

Proof. If ^ = = 0 then taking into account that § f = 0 we have that §{T — S)f = 0. Therefore from the system for the constant solutions in x, we get that S(t) = S0 G M, v = v0e~ -ft G^dr and T(t) = Ta + (T0 -Ta)e~J~oh(^dr, and hence v(t) -> 0 and T(t) Ta exponentially, when t—KX>. □

3. Asymptotic behaviour for solutions under orthogonality condition

In previous works, like [4] and [5], the asymptotic behaviour of the system Eq. (1) for large enough time is studied.

In this sense the existence of a inertial manifold associated to the functions / (loop-geometry) and Ta (ambient temperature) have been proved. The abstract operators theory ([2],[13] and [10]) has been used for this purpose.

In this section we prove in Proposition 3.1 the results which rise an important consequence: for large time the velocity reaches the equilibrium —null velocity—, or takes a value to make its integral diverge, which means that either it remains with a constant value without changing its sign or it will alternate an infinite number of times so the oscillations around zero become large enough to make the integral diverge.

3.1. Previous results and notations

First, we note that in this section we consider the case in which all periodic functions in Eq. (1) have zero average, i.e., we work in y — R x H2er x L2er, where H2er = {T e H%er with §T = 0} and L2per = {S e L2per with /S = 0}.

In effect, we note that integrating the third equation of Eq. (1) with respect to x, since T and S are periodic functions we have § ^ = / §§ — Therefore, [/ Sdx] = 0 and f S is constant respect to t, i.e., § S = § So = mo-

Moreover, integrating with respect to x the second equation of Eq. (1) and taking into account again the periodicity of T, we have that =

h{v){§ Ta-§ T). Therefore if we consider now r = T-§T and a = S-§ S0, then from the second and third equation of system Eq. (1), we obtain that r and a verify the equations

Qt d0 f

da d2a da l82t /x „In dt-CW = -Vte~bd*> ^)=°o = So-fSo (12)

Finally, since § f = 0, we have that §(T - S)f = f(r - a)f, and the equation for v reads

Thus, from Eqs. (11), (12) and (13) we have (v, r, a) verifies system Eq. (1) with rQ,r0, ao replacing Ta,To, So respectively and now §r = §a — §ra = $ ao = § To — 0. Note that if Ta is constant, then ra = 0. Observe moreover that from the equation above for the average of T and taking into account that h > ho > 0, we have that § T converges to § Ta exponentially as i —> oo. Also note that to obtain the original dynamics we put v,T = r+f T,S = a + mo, which shows that the dynamics is essentially independent of mo. Thus, using again variables v, T and S instead of v, r and a we consider the system Eq. (1) with § T0 = 0, § So = 0, § Ta = 0 and §T{t) = § S{t) = 0 for every t > 0. We then consider the semigroup S*(t) generated by Eq.

(1) and given by Corollary 2.2, which is defined by S*(t)(v0,T0, S0) = (v(t),T(t,-),S(t,-)).

In this section we assume also that G*(r) = rG(r) is locally Lipschitz, h € C1, Ta € Hper and / € L2er are given by following Fourier expansions

Ta(x) = he2*kix', f(x) = Y, c^kiX: where Z* = Z \ {0} . fcez* fee z*

T0(x) = £ akoe2*kix,SQ(x) = £ 40e27rfcM fcez* fcez*

Finally assume that T(i, x) G #2ej. and 5(t, x) € ¿2er are given by

T(t, x) = ak{t)e2*kix and S(i, x) = £ dk(t)e2*kix Z* = Z \ {0}. feez* fcez*

We note that a^ = —ak (dk = — dk) since all functions consider are real and also ao = do = 0 since they have zero average.

Now we observe the dynamics of each Fourier mode and from Eq. (1), we get the following system for the new unknowns, v and the coefficients ak(t) and dk(t):

f + G(v)v = £fcez.(afe(i) - dk{t))c.k dk{t) + \2nkiv(t) + h(v(t))] ak(t) = h(v{t))bk (16)

dfc(t) + [2irkiv(t) + 4c7r2fc2] dk(t) = -4bir2k2ak(t).

We consider the functions Ta and / given by following Fourier expansions

where

J = {k 6 Z'/cfc ^ 0}, if = {fc e Z*/6fc ^ 0} with Z* = Z \ {0}.

First, from the equations Eq. (16) we can observe the velocity of the fluid is independent of the coefficients for temperature ak(t) and the salinity dk{t) for every k £ Z* \ (K n J). That is, the relevant coefficients for the evolution of the velocity are only ak(t) and dk(t) with k belonging to the set K fl J. This important result about the asymptotic behaviour has been proved in [4] and [5] using the inertial manifold theory (cf. [2],[13],[10]).

The aim is to prove the Proposition 3.1 which generalize the result of thermosyphon model without solute of [12]. To do so we examine which are these steady-state solutions, also called equilibrium points.

We have to make the difference between equilibrium points (constants respect to the time) null velocity, called rest equilibrium, and equilibrium points with non-vanishing constant velocity.

Equilibrium conditions.

i) The system Eq. (16) presents the rest equilibrium v = 0, ak = bk and dk = \bk Vfc € K fi J under the assumption of the following orthogonality condition:

keKnj

This equilibrium corresponds to v = 0, T = Ta and S = So = with the orthogonality condition $ Taf = 0.

ii) Any other equilibrium position will have a non-vanishing velocity and the equilibrium is given by:

{G(v)v = Efcexnj h0(v)+Lkivbkc-k ~ Y,keKnjdkC-k ak = Hvnllkiv^ (19)

, _ -AbK2k2h(v) , ak ~ {h(v)+2Trkiv)(2nkiv+4cn2k2)°k •

3.2. Asymptotic behaviour

Lemma 3.1. If we assume that a solution of Eq. (16) satisfies Io° ltl(s)M's < oo, then for every rj > 0 there exists to such that rt

/ h(r)e~~ h(e~ £ 2*ikv - 1 )dr < r) with t > t0 (20)

Jo where h(r) = h(v(r)). Moreover ¡■t limsup | / {ak{r)e-X2*kiv-b^e-^^^dr|<

< limsup |a^(i) — bk| + rj\bk\ with t > to- (21)

Proof. If J0°° |u(s)jds < oo, then for all 5 there exists io > 0 such that for every io < r < i we have | < 5. Then, for any 77 > 0 we can take to large enough such that

\e~ & 2nikv -l\<r} for all to < r < t. (22)

/ h(r)e~ f' /' a*«™ _ i )dr < t/(1 - £h) < v with t > t0. J to

Taking into account that rj —> 0 for t —► oo and h is strictly bounded away from zero, we get Eq. (20). To prove Eq. (21), we write

Jto f\ak{r)-bk)e-£2*ikve-4ar'k^t-r)dr+ f bk(e~ -l)e-^2k2^t-r)dr

Jto Jto and taking modulus in this expression the first term in the right member remains rt

Was this article helpful?

0 0

Post a comment