GLOBAL SOLVABILITY FOR DOUBLE-DIFFUSIVE CONVECTION SYSTEM BASED ON BRINKMAN FORCHHEIMER EQUATION IN GENERAL DOMAINS

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1 Ôtani, M. and Uchida, S. Osaka J. Math. 53 (216), GLOBAL SOLVABILITY FOR DOUBLE-DIFFUSIVE CONVECTION SYSTEM BASED ON BRINKMAN FORCHHEIMER EQUATION IN GENERAL DOMAINS MITSUHARU ÔTANI and SHUN UCHIDA (Received April 7, 215, revised August 28, 215) Abstract In this paper, we are concerned with the solvability of the initial boundary value problem of a system which describes double-diffusive convection phenomena in some porous medium under general domains, especially unbounded domains. In previous works where the boundedness of the space domain is imposed, some global solvability results have been already derived. However, when we consider our problem in general domains, some compactness theorems are not available. Hence it becomes difficult to follow the same strategies as before. Nevertheless, we can assure the global existence of a unique solution via the contraction method. Moreover, it is revealed that the global solvability holds for higher space dimension and larger class of the initial data than those assumed in previous works. 1. Introduction We consider the following double-diffusive convection system based upon Brinkman Forchheimer equation. (DCBF) t u ½u au Ö p gt hc f 1, t T u ÖT ½T f 2, t C u ÖC ½C ½T f 3, Ö u, T u, n, C n, u(, ) u ( ), T (, ) T ( ), C(, ) C ( ), (x, t) ¾ [, S], (x, t) ¾ [, S], (x, t) ¾ [, S], (x, t) ¾ [, S], (x, t) ¾ [, S], where Ê N is a general domain with some suitable conditions (provided later in Section 2.2) and is the boundary of. The unit outward normal vector on is denoted by n and T n Ï ÖT n. Unknown functions of this system are u(x, t) 21 Mathematics Subject Classification. Primary 35K45; Secondary 35Q35, 76D3. The first author was partly supported by the Grant-in-Aid for Scientific Research, The Ministry of Education, Culture, Sports, Science and Technology, Japan, grant no The second author was supported by the Grant-in-Aid for JSPS Fellows, The Ministry of Education, Culture, Sports, Science and Technology, Japan, grant no

2 856 M. ÔTANI AND S. UCHIDA (u 1 (x, t), u 2 (x, t),, u N (x, t)) t, T (x, t), C(x, t) and p(x, t) which represent the fluid velocity, the temperature of fluid, the concentration of solute and the pressure of fluid respectively. Positive constants, and a are called the viscosity coefficient, Soret s coefficient and Darcy s coefficient respectively. Constant vectors g (g 1, g 2,, g N ) t and h (h 1, h 2,, h N ) t are derived from the gravity. The terms f 1 (x, t) ( f1 1 (x, t), f1 2 N (x, t),, f1 (x, t))t, f 2 (x, t) and f 3 (x, t) are given external forces. Double-diffusive convection is a model of convection in the fluid reflecting some interactions between the temperature and the concentration of solute. When the distribution of the temperature is far from homogeneous, the behavior of the concentration of solute becomes more complicated than the simplified diffusion model. Double-diffusive convection phenomena can be described by the second equation and the third equation of (DCBF) which originate from results of the irreversible thermodynamics. The term ½T describes one of interactions between the temperature of fluid and the concentration of solute, the so-called Soret s effect. It is well known that double-diffusive convection is mainly characterized by this Soret s effect. Strictly speaking, the second equation also contains an interaction term ¼ ½C, which is called Dufour s effect. However, Dufour s effect is generally much smaller than Soret s effect, especially for the case where the fluid is a liquid. Therefore we here consider only Soret s effect. A great number of researches in double-diffusive convection phenomena have been carried out (see, e.g., Brandt Fernando [2] and Nield Bejan [5]). Among them, the study of double-diffusive convection in porous media is one of subjects which attracted a lot of researchers, since the model of double-diffusive convection in porous media has a large area of application, for example, the behavior of polluted water in the soil. When we deal with these models, the void space of porous medium is assumed to be relatively large. In order to describe the behavior of the fluid velocity under these situations, it is appropriate to apply the so-called Brinkman Forchheimer equation. The first equation of (DCBF) is based on Brinkman Forchheimer equation. Originally, Brinkman Forchheimer equation has a convection term and another nonlinear term, and in each term of the equation, there appears another space-dependent function which stands for the rate of void space in the porous medium (which is called the porosity). However, we use a linearized Brinkman Forchheimer equation in our system under some physical conditions, for example, homogeneity of the porous medium. Here gt and hc are the effects from the gravity. For the case where the space domain is bounded, there exist some results for global solvability. In [13], for instance, the initial boundary value problem of (DCBF) with homogeneous Dirichlet boundary condition is considered. They showed the existence of a unique global solution for the case where the initial data belongs to H 1 () and the space dimension N 3. Moreover, the time periodic problem of (DCBF) with homogeneous Dirichlet boundary condition is examined in [8], where the existence of time periodic solutions is showed for N 3. In the case where homogeneous Neumann boundary condition is imposed, the solvability of the initial boundary value problem

3 DOUBLE-DIFFUSIVE CONVECTION SYSTEM IN GENERAL DOMAINS 857 and the time periodic problem are solved in [9] with N 3. In this way, some results for global solvability can be assured even for N 3, in spite of the existence of convection terms u ÖT, u ÖC which possess the nonlinearity quite similar to that appearing in the Navier Stokes equations. In previous works referred above, (DCBF) is reduced to some abstract problem in an appropriate Hilbert space and some abstract theories are applied, which are developed in [6] and [7], results for Cauchy problem and periodic problem of an abstract equation governed by subdifferential operators with non-monotone perturbations. When we apply these abstract results, Rellich Kondrachov s compactness theorem plays an essential role. Therefore, it is difficult to follow the same strategy as before for unbounded domains. The main purpose of this paper is to show that the global solvability result still holds true for (DCBF) with unbounded space domains. In order to carry out this, we rely on Banach s fixed point theorem. In Section 2, our main results are stated and some notations and function spaces are fixed. We give an outline of the proof for our main result in Section 3. In Section 4, we show the well-definedness of some mappings, to which the fixed point theorem will be applied. In Section 5, we assure that the composition of these mappings becomes a contraction mapping in some appropriate function space. Finally, we shall show that the time-local solution constructed in Sections 4 and 5 can be globally extended in Section Main result 2.1. Notation. In order to formulate our result, we use the following notation: ½ () Ï {u (u1, u 2,, u N ) t Á u j ¾ C ½ Ä 2 () Ï (L 2 ()) N, À k () Ï (H k ()) N (W k,2 ()) N, Ä 2 ()Ï the closure of ½() under the Ä2 ()-norm, À 1 ()Ï the closure of ½() under the À1 ()-norm, P Ï the orthogonal projection from Ä 2 () onto Ä 2 (), (), j 1, 2,, N, Ö u }, A Ï P ½Ï the Stokes operator with domain D(A) À 2 () À 1 (). Moreover, we define the following function spaces where solutions and external forces should belong. W S Ï C([, S]Á À 1 ()) L2 (, SÁ À 2 ()), X S Ï { f ¾ L 1 (, SÁ L 2 ())Á Ô t f ¾ L 2 (, SÁ L 2 ())}, Y S Ï {U ¾ C([, S]Á L 2 ()) L 2 (, SÁ H 1 ())Á Ô t½u, Ô t t U ¾ L 2 (, SÁ L 2 ())},

4 858 M. ÔTANI AND S. UCHIDA Z S Ï ¼ u T C ½ ¾ C([, S]Á Ä 2 () L2 () L 2 ())Á u ¾ W S, T, C ¾ Y S, t u ¾ L 2 (, SÁ Ä 2 ()). Here norms for W S, X S and Y S are defined as follows respectively: u WS Ï sup u(t) À 1 ts () f X S Ï f L 1 (,SÁL 2 ()) U YS S S u(t) 2 À 2 () dt 12, t f (t) 2 L 2 () dt 12, Ï sup U(t) L2 () ÖU(t) L2 (,SÁL 2 ()) ts S t½u(t) 2 L 2 () dt 12 S t t U(t) 2 L 2 () dt Main results. Our main result can be stated as follows: Theorem 2.1. Let N 4 and let the space domain satisfy the following condition #. #: The space domain falls under any of the following: 1. Whole space Ê N. 2. A domain belonging to red uniformly C 2 (resp. C 3 )-regular class for N 2, 3 (resp. N 4) (see, e.g., Amann [1], Browder [4] and Sohr [1]). Moreover, assume that the initial data satisfy u ¾ À 1 (), T, C ¾ L 2 () and the external forces satisfy f 1 ¾ L 2 (, SÁ Ä 2 ()), f 2, f 3 ¾ X S. Then, for each S, (DCBF) admits a unique solution (u, T, C) t ¾ Z S. REMARK 1. (1) The following cases are sufficient to assure condition 2 of #. A bounded domain with C 2 (or C 3 )-class boundary. A exterior of some bounded domain with C 2 (or C 3 )-class boundary. (2) The assumption that is uniform C 2 -domain implies that we can apply Sobolev s embedding theorem and the elliptic estimates for the operators ½ and A. Therefore, throughout this paper, one can guarantee to use the following facts under condition #, i.e., there exist some constants s, el, es such that the following inequalities hold. U L p () s U W 1, p (), U H 2 () el (U L2 ()½U L2 ()), u À 2 () es (u Ä 2 ()Au Ä 2 ()), U ¾ W 1, p () (1 p N, 1p 1p 1N), U ¾ D( ½){U ¾ H 2 ()Á Un }, u¾ D(A), where el, es depends only on, N and s is a embedding constant depending on, N and p. Moreover, the assumption that is uniform C 3 -domain for N 4 implies

5 DOUBLE-DIFFUSIVE CONVECTION SYSTEM IN GENERAL DOMAINS 859 that the embedding D(A 2 ) À 3 () is valid (see Theorem 1.5.1, Section III in Sohr [1]), where A 2 AÆA and D(A 2 ) stands for its domain. Namely, under the condition #, we obtain the embedding C([, S]Á D(A 2 )) L ½ (, SÁ Ä ½ ()) for N 4, which will be used in our proof of Lemma 4.1 later on. (3) Even if the homogeneous Neumann boundary condition is replaced by the homogeneous Dirichlet boundary condition, we can derive almost the same result as Theorem 2.1. (4) If T, C ¾ H 1 (), we can show the existence of a unique solution with the same regularity as that derived in [13]. Indeed, we can carry out the same arguments as in proofs of Lemmas 4.1 and 4.2 given later without the weight Ô t and with T,C ¾ L 2 () and f 2, f 3 ¾ X S replaced by T,C ¾ H 1 () and f 2, f 3 ¾ L 2 (, SÁ L 2 ()) respectively, in order to construct a unique global solution belonging to the space Y ¼ S defined by with the norm U Y ¼ S Y ¼ S Ï {U ¾ C([, S]Á H 1 ())Á ½U, t U ¾ L 2 (, SÁ L 2 ())}, Ï sup U(t) H 1 () ½U L 2 (,SÁL 2 ()) t U L 2 (,SÁL 2 ()). ts Hence, we can obtain the following result. Theorem 2.2. Let N 4 and let f 1 ¾ L 2 (, SÁ Ä 2 ()), f 2, f 3 ¾ L 2 (, SÁ L 2 ()). Moreover, let the initial data satisfy u ¾ À 1 (), T,C ¾ H 1 (). Then, for each S, (DCBF) admits a unique solution (u, T, C) t satisfying the following regularities: u ¾ C([, S]Á À 1 ()), Au, t u ¾ L 2 (, SÁ Ä 2 ()), T, C ¾ C([, S]Á H 1 ()), ½T, ½C, t T, t C ¾ L 2 (, SÁ L 2 ()). 3. Outline of the proof for Theorem 2.1 Operating the projection P to the first equation of (DCBF), we have the following equation: (1) t u Au au P gt P hc P f 1, t T ½T u ÖT f 2, t C ½C u ÖC ½T f 3. It is well known that the system of above equations is equivalent to our system (DCBF) (see, e.g., Temam [12]). Therefore, here and after, we consider system (1).

6 86 M. ÔTANI AND S. UCHIDA (2) In this section, we give an outline of our proof. Our proof consists of four steps: STEP 1: For each u fixed in W S, we consider the following problem in Y S Y S : t T ½T u ÖT f 2, t C ½C u ÖC ½T f 3, T C n, n, T (, ) T ( ), C(, ) C ( ), where u in the second and third equations of (1) are replaced by given u. Then, we define a mapping T,C Ï W S Y S Y S by T,C (u) (T, C) t, where (T, C) t denotes a unique global solution of (2). STEP 2: By substituting T, C by the unique solution T, C in the first equation of (1), we consider the following problem: (3) t Æu A Æu a Æu P gt P hc P f 1, Æu, Æu(, ) u ( ), and we show that (3) admits a unique global solution Æu in W S. Moreover, we define u Ï Y S Y S W S by u ((T, C) t ) Æu. STEP 3: We show that the mapping u Æ T,C becomes a contraction mapping in W S for a sufficiently small S ¾ (, S]. Hence we can show that (1) has a unique local solution in Z S. STEP 4: Establishing appropriate a priori estimates, we assure that time-local solutions can be extended globally onto [, S] (the whole of the prescribed interval). 4. Construction of solutions for Steps 1 and 2 In what follows, we carry out the program given in the previous section. To begin with, we discuss the procedures Steps 1 and 2. Namely, we ensure the solvability of problems (2) and (3) in this section Well-definedness of T,C. First we show the unique solvability of (2). Lemma 4.1. Let N 4 and assume that T ¾ L 2 (), u ¾ W S and f 2 ¾ X S. Then the following problem (4) has a unique global solution T in Y S. (4) t T ½T u ÖT f 2 in [, S], T n, T (, ) T ( ). Proof. We first consider the case where u possesses higher regularity. To this end, we recall that the following heat equation has a unique global solution T ¾ Y S

7 DOUBLE-DIFFUSIVE CONVECTION SYSTEM IN GENERAL DOMAINS 861 for any T ¾ L 2 () and f ¾ X S (see, e.g., Brézis [3]). (5) t T ½T f in [, S], T n, T (, ) T ( ). Therefore, since Û ÖU ¾ L 2 (, SÁ L 2 ()) X S for any U ¾ L 2 (, SÁ H 1 ()) and Û ¾ C([, S]Á D(A 2 )) (where A 2 Ï A ÆA), the following equation also has a unique global solution T ¾ Y S. (6) t T ½T Û ÖU f 2 in [, S], T n, T (, ) T ( ). Then we can define a mapping Û T Ï L 2 (, SÁ H 1 ()) Y S L 2 (, SÁ H 1 ()) by the correspondence Û T (U) T. Let Û T (U i ) T i (i 1, 2) and ÆU U 1 U 2, ÆT T 1 T 2. Obviously, ÆU and ÆT satisfy the following problem: (7) t ÆT ½ÆT Û ÖÆU in [, S], ÆT n Multiplying (7) by ÆT, we have, ÆT (, ). (8) 2 dt ÆT 2 L 2 ÖÆT 2 L 2 Û ÖÆU L 2ÆT L 2 MÖÆU L 2ÆT L 2, where M is the constant given by M Ï esssup (x,t)¾ [,S] Û(x, t) (we here remark that C([, S]Á D(A 2 )) L ½ (, SÁ Ä ½ ()) for N 4). Therefore, integrating (8) over [, t], we have (9) 1 2 ÆT (t)2 L 2 M t M S 12 ÖÆU L 2ÆT L 2 ds sup ÆT (t) L 2ÖÆU L 2 (,SÁL 2 ()). ts Hence we obtain (1) sup ÆT (t) L 2 ÖÆT L 2 (,SÁL 2 ()) (2 Ô 2)M S 12 ÖÆU L 2 (,SÁL 2 ()). ts Thus, from (1), we can assure that Û T becomes a contraction mapping with a sufficiently small S ¾ (, S], i.e., we can show that the following problem has a unique

8 862 M. ÔTANI AND S. UCHIDA local solution T ¾ Y S for any smooth function Û ¾ C([, S]Á D(A 2 )). (11) t T ½T Û ÖT f 2 in [, S], T n, T (, ) T ( ). Moreover, it is easy to see that this local solution can be extended globally. Indeed, since (12) T Û ÖT dx 1 2 Û ÖT 2 dx 1 2 T 2 Ö Û dx, multiplying (11) by T, we can obtain a priori bound for T (t) L 2. We here remark that there exists a sequence {u n } n¾æ satisfying u n ¾ C([,S]Á D(A 2 )) and u n u in C([,S]ÁÀ 1 ())L2 (,SÁÀ 2 ()) for each u ¾ W S (for a typical example, see Remark 2 given later). Then, for each n ¾ Æ, the following problem has a unique global solution T n ¾ Y S. (13) t T n ½T n u n ÖT n f 2 in [, S], T n n, T n (, ) T ( ) ¾ L 2 (). In order to show the existence of a global solution of (4) for u ¾ W S, we here discuss the convergence of approximate solutions T n. To this end, we begin with establishing some a priori estimates for T n. Multiplying (13) by T n, we have (14) 2 dt T n 2 L ÖT 2 n 2 L f 2 2 L 2T n L 2, since Ê T n u n ÖT n dx. Hence, we obtain (15) sup T n (t) L 2 ÖT n L 2 (,SÁL 2 ()) 1, ts where enotes a general constant independent of n. Next, multiplying (13) by t½t n, we have (16) 2 dt töt n 2 L t 2 4 ½T n 2 L tu 2 n ÖT n 2 L t 2 2 f 2 2 L ÖT n 2 L. 2 Here, applying Hölder s inequality, Sobolev s inequality, Riesz Thorin s interpolation

9 DOUBLE-DIFFUSIVE CONVECTION SYSTEM IN GENERAL DOMAINS 863 theorem and the elliptic estimate, we can derive the following estimate: u n ÖT n 2 L 2 u n 2 Ä 8 ÖT n 2 L 83 1 u n 2 Ï 1,83 ÖT n 2 L 83 (17) 1 u n À 1u n Ï 1,4ÖT n L 2ÖT n L 4 1 u n À 1u n À 2ÖT n L 2T n H 2 1 u n À 1u n À 2ÖT n L 2(½T n L 2 T n L 2). Then, by substituting (17) into (16), we have (18) 2 dt töt n 2 L t 2 8 ½T n 2 L 2 1 tu n 2 À ÖT 2 n 2 L t 2 2 f 2 2 L t 2 2 T n 2 L ÖT n 2 L. 2 Moreover, applying Gronwall s inequality to (18), we obtain S S (19) töt n (t) 2 L (s f L st 2 n 2 L ÖT 2 n 2 L ) ds exp u n 2 À ds. 2 Therefore, we obtain sup ts töt n (t) 2 L 2 1 with some constant 1 independent of n and Ê S t½t n 2 L 2 dt 1 by integrating (18) over [, S]. Similarly, multiplying (13) by t t T n, we have t 2 t T n 2 L d t 2 dt 2 ÖT n 2 L 2 1 tu n 2 À ÖT 2 n 2 L t f L 2 t(½t n L 2 T n L 2) ÖT n 2 L 2, whence follows Ê S t t T n 2 L 2 dt 1. By using these estimates, we show that {T n } becomes a Cauchy sequence in Y S. Let Æu u m u n, ÆT T m T n. Then Æu and ÆT satisfy the following equation: (2) t ÆT ½ÆT Æu ÖT m u n ÖÆT in [, S], ÆT n, ÆT (, ). Multiplying (2) by ÆT, we have (21) 2 dt ÆT 2 L ÖÆT 2 2 L 2 ÆT Æu ÖT m dx T m Æu ÖÆT dx 1 2 ÖÆT 2 L T mæu 2 Ä ÖÆT 2 L 2 1 T m 2 H 1 Æu 2 À 1.

10 864 M. ÔTANI AND S. UCHIDA Therefore, integrating (21) and using (15), we can derive the following estimate of ÆT : (22) sup ÆT (t) 2 L 2 ts S Next, multiplying (2) by t½æt, we have ÖÆT 2 L 2 ds 1 sup Æu 2 À. 1 ts (23) 2 dt töæt 2 L t 2 8 ½ÆT 2 L 2 1 u n 2 À töæt 2 2 L t 2 2 ÆT 2 L ÖÆT 2 L 2 1 Æu À 1Æu À 2Ô tötm L 2Ô t(½tm L 2 T m L 2), where we used the following estimates of convection terms derived by procedures similar to that in (17): (24) t 2 Æu ÖT Ô Ô m 2 L 2 1 Æu À 1Æu À 2 tötm L 2 t(½tm L 2 T m L 2), tu n ÖÆT 2 L 2 1 tu n À 1u n À 2ÖÆT L 2(½ÆT L 2 ÆT L 2). Since sup ts u n À 1, Ê S u n 2 À 2 ds and sup ts Ô tötm L 2 are uniformly bounded, we can derive the following inequality by applying Gronwall s inequality to (23): (25) töæt (t) 2 L 2 1 S {Æu À 1Æu À 2Ô s(½tm L 2 T m L 2) sæt 2 L 2 ÖÆT 2 L 2 } ds. Moreover, by using the uniform boundedness of Ê S t½t m 2 L 2 dt and (22), we can obtain (26) sup töæt (t) 2 L 2 ts Hence again by (23), we have (27) S 1 t½æt 2 L 2 dt 1 sup Æu(t) 2 À Æu 2 1 L 2 (,SÁÀ 2 ()) ts ts. sup Æu(t) 2 À Æu 2 1 L 2 (,SÁÀ 2 ()). Similarly, multiplying (2) by t t ÆT, we can derive S (28) t t ÆT 2 L dt 2 1 sup Æu(t) 2 À Æu 2 1 L 2 (,SÁÀ 2 ()). ts Therefore, from (22), (27) and (28), we can assure that {T n } n¾æ in Y S, hence (4) has a unique global solution in Y S. is a Cauchy sequence

11 DOUBLE-DIFFUSIVE CONVECTION SYSTEM IN GENERAL DOMAINS 865 REMARK 2. We can construct the approximation sequence {u n } of u ¾ W S used in (13) by the following procedure. Let J n be the resolvent of A, i.e., J n Ï (I (1n)A) 1 (we note that J n Û ¾ D(A 2 ) is valid for any Û ¾ D(A) due to the definition of J n and due to the elliptic regularity for the Stokes operator A). Moreover, let ÉÚ be the extension of Ú ¾ C([, S]Á Ä 2 ()) defined by ÉÚ(t) Ú(t), t ¾ [, S], Ú( t), t ¾ [ S, ], Ú(2S t), t ¾ [S, 2S],, t ¾ Ê Ò [ S, 2S], (we can adopt other extensions of Ú ¾ C([, S]Á Ä 2 ()) instead of ÉÚ if their extensions also assure the arguments given below). Then it can be shown that u n Ï ( 1n J n u) [,S] satisfies u n ¾ C([, S]Á D(A 2 )) and u n u in W S as n ½, where 1n is the Friedrichs mollifier. Indeed, we get u() u n () u() J n u() J n u() u n () u() J n u() u() J n u() 1n 1n 1n 1n 1n (s)(j n u() J n u( s)) ds 1n (s)(j n u() J n u(s)) ds 1n (s)(j n u() J n u( s)) ds. Since J n is a contraction mapping on Ä 2 () and u belongs to C([, S]Á Ä2 ()), we can see that u n () u() in Ä 2 () as n ½. Likewise, we can obtain the convergence u n (t) u(t) for any t ¾ [, S]. Moreover, from the uniform continuity of u on [, S], we can assure that u n converges to u in C([, S]Á Ä 2 ()) (uniform convergence). Let A 12 denote the fractional power of the Stokes operator A of order 12. Then A 12 satisfies D(A 12 ) À 1 () and A12 Û Ä 2 ÖÛ Ä 2 for any Û ¾ D(A 12 ) (see, e.g., Sohr [1] and Tanabe [11]). By the linearity and closedness of A 12 and by the fact that u ¾ C([, S]Á À 1 ()), we have A 12 u() A 12 u n () A 12 u() J n A 12 u() 1n 1n 1n (s)(j n A 12 u() J n A 12 u(s)) ds 1n (s)(j n A 12 u() J n A 12 u( s)) ds. Therefore, we obtain A 12 u n () A 12 u() in Ä 2 () since u ¾ C([, S]Á À1 ()). By the same argument as above, we can assure that A 12 u n A 12 u in C([, S]Á Ä 2 ()).

12 866 M. ÔTANI AND S. UCHIDA Hence u n converges to u in C([, S]Á À 1 ()). Ä 2 Since Au ¾ L 2 (Ê 1 Á Ä 2 ()), it is easy to see that 1n Au[,S] Au in L 2 (, SÁ ()) as n ½. We can also derive that 1n Au(t) 1n J n Au(t) Ä 2 1n Au J n Au Ä 2 (t). Therefore, by using Young s inequality, we have 1n Au 1n J n Au L2 (Ê 1 ÁÄ 2 ()) 1n L1 (Ê 1 ) Au J n Au L2 (Ê 1 ÁÄ 2 ()) 3Au J n Au L 2 (,SÁÄ 2 ()). Since the right hand side converges to zero by virtue of Lebesgue s dominated convergence theorem, we can assure that 1n J n Au strongly converges to Au in L 2 (, SÁ Ä 2 ()). Next, we consider the third equation. Lemma 4.2. Let N 4. Moreover, assume that C ¾ L 2 (), u ¾ W S, T ¾ Y S and f 3 ¾ X S. Then the following problem (29) has a unique global solution C ¾ Y S. (29) t C ½C u ÖC ½T f 3 in [, S], C n, C(, ) C ( ). This problem is quite similar to the previous problem (4). However, we can not use our argument in the proof of Lemma 4.irectly, since it is not known whether ½T ¾ X S. Therefore, we need some additional argument. Proof. Let Ï [, S] Ê be the cut-off function defined by (3) (t) if t, 1 if t S. Since T ¾ Y S implies that ½T ¾ X S, we can show that the following problems have a unique global solution C ¾ Y S for each by applying Lemma 4.1. (31) t C ½C u ÖC ½T f 3 in [, S], C n, C (, ) C ( ). Here by showing that {C } becomes a Cauchy sequence in Y S, we can assure the existence of a unique global solution. Indeed, let 2 1 and ÆC C 1 C 2.

13 DOUBLE-DIFFUSIVE CONVECTION SYSTEM IN GENERAL DOMAINS 867 Moreover, let É 1 2, i.e., É (t) 1, 1 t 2,, otherwise. Then, ÆC satisfies the following problem in Y S : (32) t ÆC ½ÆC u ÖÆC É ½T in [, S], ÆC n, ÆC(, ). Multiplying (32) by ÆC, we have (33) 2 dt ÆC2 L ÖÆC 2 2 L 2 É ÖÆC L 2ÖT L ÖÆC2 L É ÖT 2 L 2. Therefore, integrating (33) over [, t] and [, S], we obtain (34) sup ts ÆC(t) 2 L 2 ÖÆC 2 L 2 (,SÁL 2 ()) 2 2 Next, multiplying (32) by t½æc and using (24), we have 1 ÖT 2 L 2 ds. (35) 2 dt töæc2 L t 2 8 ½ÆC2 L 2 2 u 2 À 2 töæc 2 L 2 t2 2 É 2 ½T 2 L 2 t 2 ÆC2 L ÖÆC2 L 2, where 2 is a general constant independent of 1, 2. Applying Gronwall s inequality to (35), we obtain (36) töæc(t) 2 L 2 S {s 2 2 É ½T 2 L sæc 2 2 L ÖÆC 2 2 L } ds 2 S exp 2 2 u 2 À ds. 2 Therefore, owing to (34), we derive sup ts töæc(t) 2 L 2 2 Ê 2 1 {s½t 2 L 2 ÖT 2 } ds from (36). Moreover, integrating (35) over [, S], we can get L 2 (37) S t½æc 2 L 2 dt {s½t 2 L 2 ÖT 2 L 2 } ds.

14 868 M. ÔTANI AND S. UCHIDA Similarly, multiplying (32) by t t ÆC, we can obtain (38) S t t ÆC 2 L 2 dt {s½t 2 L 2 ÖT 2 L 2 } ds. Thus, we can assure that {C } forms a Cauchy sequence in Y S since T ¾ Y S. Hence, the problem (29) has a unique global solution. Hence it follows that we can obtain a unique global solution T, C of (2) and the well-definedness of T,C Well-definedness of u. It is easy to see that (3) has a unique global solution. Indeed, by using the standard arguments of evolution equations governed by maximal monotone operators, we can derive the following (see, e.g., Brézis [3]): Lemma 4.3. Let N ¾ Æ and assume that u ¾ À 1 () and F ¾ L2 (, SÁ Ä 2 ()). Then the following problem (39) has a unique global solution u ¾ W S. (39) t u Au au F in [, S], u, u(, ) u ( ). Substituting F by P gt P hc P f 1, we can assure the existence of a unique global solution Æu of (3) and the well-definedness of u. 5. Application of contraction mapping principle In this section, we assure the local existence of a unique solution of (DCBF) by using Banach s contraction mapping principle. Let u i ¾ W S (i 1, 2), (T i, C i ) t Ï T,C (u i ) and Æu i Ï u ((T i, C i ) t ). Moreover, let Æu u 1 u 2, ÆT T 1 T 2, ÆC C 1 C 2 and Æ Æu Æu 1 Æu 2. Then Æu, ÆT, ÆC and Æ Æu satisfy the following equations: (4) t Æ Æu AÆ Æu aæ Æu P gæt P hæc, t ÆT ½ÆT u 1 ÖÆT Æu ÖT 2, t ÆC ½ÆC u 1 ÖÆC Æu ÖC 2 ½ÆT, ÆT ÆC Æ Æu, n, n, Æ Æu(, ), ÆT (, ), ÆC(, ). Multiplying the first equation of (4) by Æ Æu and AÆ Æu, we obtain (41) 2 dt Æ Æu2 Ä ÖÆ Æu 2 2 Ä aæ Æu 2 2 Ä gæt 2 L 2Æ Æu Ä 2 hæc L 2Æ Æu Ä 2, 2 dt ÖÆ Æu2 Ä 2 4 AÆ Æu2 Ä aöæ Æu 2 2 Ä g2 2 2 ÆT 2 L h2 2 ÆC2 L. 2

15 DOUBLE-DIFFUSIVE CONVECTION SYSTEM IN GENERAL DOMAINS 869 Integrating (41) over [, t] and [, S], we can derive (42) S S sup Æ Æu(t) Ä 2 g ÆT L 2 ds h ÆC L 2 ds ts gs sup ÆT (t) L 2 hs sup ÆC(t) L 2, ts ts and (43) sup tsöæ Æu(t) 2 Ä 2 2 g2 S S AÆ Æu 2 Ä 2 ds sup ÆT (t) 2 L 2h2 S 2 ts sup ÆC(t) 2 L. 2 ts Next, from the facts that Ê ÆT Æu ÖT 2 dx Ê T 2Æu ÖÆT dx and Ê ÆT u 1 ÖÆT dx, multiplying the second equation of (4) by ÆT, we have (44) d dt ÆT 2 L 2 ÖÆT 2 L 2 T 2 Æu 2 Ä 2 T 2 2 L 4 Æu 2 Ä 4 T 2 2 H 1 Æu 2 À 1, where is a constant depending only on Sobolev s embedding constant. multiplying the third equation of (4) by ÆC, we have (45) Therefore, we obtain d dt ÆC2 L ÖÆC2 L 2 2 ÖÆT 2 L 2 2C 2 2 H 1 Æu 2 À 1. Similarly, (46) sup ÆT (t) 2 L 2 ts S sup ÆC(t) 2 L sup 2 ts ts S ÖÆT 2 L dt sup Æu(t) 2 2 À 1 ts Æu(t) 2 À T H ds S T 2 2 H s, S C 2 2 H s. Hence, combining (42) and (43) with (46), we can derive (47) sup Æ Æu(t) 2 À 1 ts S AÆ Æu 2 Ä 2 ds S 3 S(1 S) sup Æu(t) 2 À T H ds 1 ts S C 2 2 H ds, 1 Ê where Ê S 3 is a constant depending only on, g, h, and. Since T 2(s) 2 ds H 1 S and C 2(s) 2 ds are bounded by some constant depending only on the initial data H 1 and the external forces (see next section), we can assure that u Æ T,C becomes a contraction mapping in W S with sufficiently small S ¾ (, S]. Namely, we can assure the existence of a unique local solution of (DCBF).

16 87 M. ÔTANI AND S. UCHIDA 6. Global existence In this section, we shall show that the unique time-local solution constructed in the previous section can be extended up to S by establishing some a priori estimates. Multiplying the second equation of (1) by T, we can obtain the following a priori bounds of T for any S ¾ (, S] (by using the same procedures as those for (14) and (15)): (48) sup ts T (t) 2 L 2 ÖT 2 L 2 (,S ÁL 2 ()) 4, where 4 is a suitable constant which depends only on T L 2 Multiplying the third equation of (1) by C, we have and f 2 L 1 (,SÁL 2 ()). i.e., 2 dt C2 L ÖC2 L C(t)2 L ÖT 2 L 2 f 3 L 2C L 2, t f 3 (s) L 2C(s) L 2 ds, where 5 denotes some general constant which depends only on, C L 2 and 4. Here we recall the following variant of Gronwall s inequality (see, e.g., Lemma A.5 of Brézis [3]): if (t) 1 t 2 b2 R(s)(s) ds is satisfied by ¾ C([,S]ÁÊ 1 ), a non-negative constant b and a non-negative L 1 (,SÁÊ 1 )- function R, then t (t) b R(s) ds is valid for any t ¾ [, S]. Hence we easily get (49) sup ts C(t) L 2 6, with some suitable general constant 6 which depends only on f 3 L 1 (,SÁL 2 ()) and 5. By the same way as for (41), (42) and (43), multiplying the first equation of (1) by u and Au, we obtain (5) sup ts u(t) 2 Ä 2 sup ts Ö u(t) 2 Ä 2 7, where 7 is a constant depending on, g, h, u À 1, f 1 L 2 (,SÁÄ 2 ()) and 6. Thus

17 DOUBLE-DIFFUSIVE CONVECTION SYSTEM IN GENERAL DOMAINS 871 we find that there exists a priori bound independent of S such that sup ts {T (t) L 2 C(t) L 2 u(t) À 1 }, which implies that the local solution constructed in previous sections can be extended onto the whole of the prescribed interval [, S], whence follows our result. REMARK 3. Throughout our argument in this paper, the positivity of a is not used. Therefore, the existence of a unique solution of (DCBF) still holds even if a. We note that for this case, the operator u Auau may lose its coercivity in Ä 2 (). ACKNOWLEDGEMENT. The authors would like to thank the referees for carefully reading the manuscript and for giving constructive comments which substantially helped improving the quality of this paper. References [1] H. Amann: Existence and regularity for semilinear parabolic evolution equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11 (1984), [2] A. Brandt and H.J.S. Fernando: Double-Diffusive Convection, Amer. Geophysical Union, Washington, [3] H. Brézis: Opérateurs Maximaux Monotones et Semigroupes de Contractions dans un Espace de Hilbert, North Holland, Amsterdam, [4] F.E. Browder: On the spectral theory of elliptic differential operators, I, Math. Ann. 142 (196), [5] D.A. Nield and A. Bejan: Convection in Porous Medium, third edition, Springer, New York, 26. [6] M. Ôtani: Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, Cauchy problems, J. Differential Equations 46 (1982), [7] M. Ôtani: Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, periodic problems, J. Differential Equations 54 (1984), [8] M. Ôtani and S. Uchida: The existence of periodic solutions of some double-diffusive convection system based on Brinkman Forchheimer equations, Adv. Math. Sci. Appl. 23 (213), [9] M. Ôtani and S. Uchida: Global solvability of some double-diffusive convection system coupled with Brinkman Forchheimer equations, Lib. Math. (N.S.) 33 (213), [1] H. Sohr: The Navier Stokes Equations, Birkhäuser/Springer Basel AG, Basel, 21. [11] H. Tanabe: Equations of Evolution, Monographs and Studies in Mathematics 6, Pitman, Boston, MA, [12] R. Temam: Navier Stokes Equations, third edition, North-Holland, Amsterdam, [13] K. Terasawa and M. Ôtani: Global solvability of double-diffusive convection systems based upon Brinkman Forchheimer equations; in Current Advances in Nonlinear Analysis and Related Topics, GAKUTO Internat. Ser. Math. Sci. Appl. 32, Gakkōtosho, Tokyo,

18 872 M. ÔTANI AND S. UCHIDA Mitsuharu Ôtani Department of Applied Physics School of Science and Engineering Waseda University 3-4-1, Okubo Shinjuku-ku, Tokyo, Japan Shun Uchida Department of Applied Physics School of Advanced Science and Engineering Waseda University 3-4-1, Okubo Shinjuku-ku, Tokyo, Japan