On the Navier-Stokes Equations with Variable Viscousity in a Noncylindrical Domains

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1 On the Navier-Stokes Equations with Variable Viscousity in a Noncylindrical Domains G. M. de ARAÚJO1, M. MILLA MIRANDAand L. A. MEDEIROS Departamento de Matemática 1, UFPA, Rua Augusto Corrêa s/n CEP:66, Belém - Pa - Brasil Instituto de Matemática, UFRJ, C.P.6853 CEP , Rio de Janeiro - RJ - Brasil Abstract. In this paper we study the existence of weak solutions when n 4 to the sistem (NC 1 ) defined in a noncylindrical domain Q, where Q is the image of a cylinder Q of IR n+1. Uniqueness of solutions for n 3 is also studied. Key words and phrases: Navier-stokes equation with variable viscousity, noncylindrical domain, weak solutions. Subject Classification 1 Introduction The mathematical model for description of the motion of a viscous incompressible fluid is given by following system of partial differencial equation in Euler coordinates u t ν u + n u u i = f grad p x i (P 1 ) div u = In (P 1 ) u = (u 1, u 2,.., u n ) is a vector with u i = u i (x, t), for x = (x 1, x 2,..., x n ) IR n and t is a real number. Note that u is the velocity of fluid, f is the density of forces acting on it, p = p(x, t) it s pressure at point (x, t). This article is a resume of thesis 1 formulated for obtain the doctor science degree with defense predict for december 23 1

2 By ν we represent the viscousity of the fluid and when ν = (P 1 ) reduces the Euler system. We soppose ν >. The mathematical analysis of (P 1 ) was done, first time, by Jean Laray em 193, cf.[2]. After that it was systematicaly investigated by O. A. Ladyzhenskaya[1], 1963; Jacques-Louis Lions[3], 1969; Roger Temam[8], 1979; Luc Tartar[7], 1999 and many others investigators. The problem we investigate in this article is proposed by Ladyzheskaya[1] which consists in suppose that in (P 1 ) the viscousity ν is a function of u of a certain form. This problem was investigated by Lions[3] in a cylindrical domain Q of IR n+1. In the Lions [3] he investigate the model (P 1 ) when the viscosity ν is of the form ν = ν + ν 1 ( u(t) 2 ), for ν >, ν 1 > real numbers. More precisely, he investigate the mixed problem u ( t ν + ν 1 u(t) 2) u + div u = em Q u = em Σ u(x, ) = u (x) em. u i u x i = f grad p em Q (P 2 ) Note that in (P 2 ) Q is cylinder of IR n+1. He proves the existence of weak solution for n 4 and uniqueness for n 3. For the case ν 1 = as we know we have up to now, uniqueness for n < 3, cf. Lions-Prodi[4]. The problem we investigate in the present article is the mixed problem (P 2 ) in one noncylindrical domain Q which is diffeomorphe to a cylinder Q. Let T > be a real number and t, t T, a family of bounded open sets of IR n with regular boundary Γ t. Consider the noncylindrical domain of IR n+1 Q = t {t}, whose lateral boundary Σ = <t<t <t<t Γ t {t}, supposed regular. Consider the following Navier-Stokes Sistem with variable viscousity u ( ) t ν + ν 1 u(t) 2 u V ( t) u + u i = f grad p em x Q i div u = em Q (NC 1 ) u = sobre Σ u(x, ) = u (x), x, 2

3 where u(x, t) is a vectorial function u(x, t) = (u 1 (x, t), u 2 (x, t),..., u n (x, t)), ( (x, t) Q, u = ( u ) 1, u 2,..., u n ), is the gradient operator,,...,, ν, ν 1 are positives constants and x 1 x 2 x n u(t) 2 V ( t) = n t ( ) 2 ui (x, t) dx. x j Let K(t) be a function, such that for t [, T ] to correspond a n n, matrix, that is, K : [, T ] IR n2. Let be a bounded open sets of IR n with regular boundary Γ. Consider the sets t = {x = K(t)y, y }, (1) whose boundary we represent by Γ t. We study the existence of weak sulution for the problem (NC 1 ) with n 4 and t defined by (1), also we study the uniqueness of solutions when n = 2 or n = 3. For that, by a suitable change of variable, we transform the noncylindrical problem (NC 1 ) in a problem defined in cylinder Q = ], T [. In Q we follow the ideas of J.L.Lions [3]. 2 Notation and Main Resultats We fixe the following hypotesis on K(t) (H 1 ) K(t) = k(t)m where k : [, T ] IR, k C 1 ([, T ]), k(t) k > and M is an invetible n n matrix whose entries are real constants. Consider a notation K(t) = (α ij (t)) and K 1 (t) = (β ij (t)). (2.1) By, we will represent the duality pairing between V and V, V being the topological dual of the space V. In order to transform the noncylindrical problem (NC 1 ) into a problem defined in the cylinder Q, we introduce the functions u(x, t) = v(k 1 (t)x, t), f(x, t) = g(k 1 (t)x, t) p(x, t) = q(k 1 (t)x, t), u (x) = v (K 1 ()x) (2.2) 3

4 Then obtain from (NC 1 ), the following problem defined in the cylinder Q v ( t n ν + ν 1 det K(t) β lj (t) v ) 2 i dy a lr (t) 2 v + y l=1 l y l,r=1 l y r v + β li (t)v i + β v y lr(t)α rj (t)y j = g ( q)k 1 (t) em Q (C i,l=1 l y 1 ) j,l,r=1 l div(m 1 v T ) = em Q v = sobre Σ v(y, ) = v (y) em where a lr (t) = β lj (t)β rj (t) and v T is the transposed of the row vector j=1 v = (v 1,..., v n ). The equivalence of problems (NC 1 ) and (C 1 ) is given in Teorem 5. We define the following spaces V t = {ϕ (D( t )) n ; div ϕ = }, V ( t ) = V (H1 (t))n t, with inner product and norm denoted, respectively in ((u, z)) V (t) = t u i x j (x) z i x j (x) dx, u 2 V ( t) = n t ( ) 2 ui (x) dx, x j and H( t ) = V (L2 ( t)) n t, with inner product and norm denoted, respectively in (u, v) H(t) = u i (x)v i (x) dx, u 2 H( = n t) u i (x) 2 dx t t Through analogy,we define V = {ψ (D()) n ; div(m 1 ψ T ) = }, V = V (H1 ())n, with inner product and norm denoted, respectively in ((v, w)) = u i y j (y) w i y j (y) dy, v 2 = ( ) 2 vi (y) dy, and y j H = V (L2 ()) n, with inner product and norm denoted, respectively in (v, w) = v i (y)w i (y) dy, v 2 = v i (y) 2 dy. 4

5 In order to state the variational formulation of problems (NC 1 ) and (C 1 ), we introduce some bilinear and trilinear forms. Concerning to the noncylindrical problem we introduce the notations u i â(t; u, z) = (x) z i (x) dx = ((u, z)) x j x V (t), j b(t; u, z, ξ) = t and to the cylindrical problem a(t; v, w) = b(t; v, w, ψ) = c(t; v, w) = t u i (x) z j x i (x)ξ j (x) dx. i,l,r=1 i,j,l=1 i,j,l,r=1 a lr (t) v i y r (y) w i y l (y) dy, (2.3) β li (t)v i (y) w j y l (y)ψ j (y) dy, (2.4) β lr(t)α rj (t)y j v i y l (y)w i (y) dy. (2.5) The spaces L p (, T ; V ( t )), L p (, T ; H( t )) and L p (, T ; V ( t )) (1 p ) are defined of same way the cylindrical case. Definition 2.1 A function u L (, T ; H( t )) L 4 (, T ; V ( t )) is called a weak solution of problem (NC 1 ) when it verifies T T (u(t), ξ (t)) H(t) dt + ν â(t; u(t), ξ(t)) dt+ T T + Â(t)u(t), ξ(t) V ( t)v ( t) dt + b(t; u(t), u(t), ξ(t)) dt = (NC 2 ) T = f(t), ξ(t) V ( t)v ( t) dt, ξ L 4 (, T ; V ( t )), ξ L 1 (, T ; H( t )), ξ() = ξ(t ) = u() = u. Definition 2.2 A function v L (, T ; H) L 4 (, T ; V ) is called a weak solution of problem(c 1 ) when it verifies T T T (v(t), ψ (t)) dt + ν a(t; v(t), ψ(t)) dt + A(t)v(t), ψ(t) dt+ T T T + b(t; v(t), v(t), ψ(t)) dt + c(t; v(t), ψ(t)) dt = g(t), ψ(t) dt, ψ L 4 (, T ; V ), ψ L 1 (, T ; H), ψ() = ψ(t ) = v() = v. 5

6 We represent by Â(t)u(t) = ν 1 u(t) 2 V ( t) u(t), and ( A(t)v(t) = ν n 1 det K(t) β lj (t) v ) 2 i (y, t) dy y l=1 l a lr (t) 2 v (t) y l y r. l,r=1 Next we shall state the main resultats of this paper Theorem 2.1 (Weak Solutions).Assume tha hypotesis (H 1 ) is satisfied. If f L 4/3 (, T ; V ( t )) and u H( ), then there exists u : Q IR n, solution to problem (NC 2 ) Theorem 2.2 If g L 4/3 (, T ; V ()) and v H, then there exists v : Q IR n solution to problem (C 2 ). Theorem 2.3 Supose n = 2, 3 and that (H 1 ) is verified. If f L 4/3 (, T ; V ( t )) and u H( ), then there exists a unique u : Q IR n in the class u L (, T ; H) L 4 (, T ; V ) such that u + ν Âu + Âu + Bu + Ĉu = f in L4/3 (, T ; V ( t )) u() = u. Theorem 2.4 Supose n = 2, 3, g L 4/3 (, T ; V ) and v H, then there exists a unique v : Q IR n in the class v L (, T ; H) L 4 (, T ; V ) such that v + ν Av + Av + Bv + Cv = g in L 4/3 (, T ; V ) v() = v. Theorem 2.5 The problems (C 2 ) and (NC 2 ) are equivalents 3 Proofs of Results We begin with two lemmas Lemma 3.1 Concerning the bilinear form a(t; v, w) defined by (2.3) and the operator A(t) = a l,r (t) 2 v, we have y l y r l,r i) A(t)v, w = a(t; v, w), v, w V. ii) a(t; v, v) a v 2, v V (a positive constant). iii) a(t; v, w) a 1 v w, v V (a 1 positive constant). 6

7 Lemma 3.2 Let b(t; v, w, ψ), c(t; v, w) and n 4 be the trilinear and bilinear forms defined, respectively, by (2.3) and (2.4). Then i) b(t; v, w, ψ) c v w ψ, v, w, ψ V. ii) b(t; v, v, w) = b(t; v, w, v), v, w V. iii) for all v V, the linear form w b(t; v, v, w) is continous on V and b(t; v, v, w) = B(t)v, w, where B(t)v V and B(t)v V c v 2, v V. iv) c(t; v, w) c v w, v V and w H. v) for all v V, the linear form w c(t; v, w) is continous on H, c(t; v, w) = (C(t)v, w), where C(t)v H and C(t)v c v. Lemmas 3.1 except ii)(see M. Milla Miranda and J. L. Ferrel[6] and 3.2, with slight modifications, by the proofs presented by J. L. Lions [3] for similar results. Proof of Theorem 2.2. we employ the Faedo-Galerkin method. Since V comp H V, we have the following espectral problem ((w, v)) = λ(w, v), v V. (3.1) Let (w ν ), (λ ν ) be solutions of (3.1). Consider V m = [w 1, w 2,..., w m ] the subspace generate by the m first vectors of Hilbert s base (w ν ) and m v m (t) = h jm (t)w j solution of aproximate problem j=1 (v m, w j ) + ν a(t; v m, w j ) + A(t)v m, w j + b(t; v m, v m, w j )+ +c(t; v m, w j ) = g(t), w j, j = 1, 2,...m v m() = v m, v m v em H, v m V m. where A(t)v m, w j = ν 1 det K(t) ( n l=1 β lj (t) v m i y l (y) ) 2 dy (3.2) a(t; v m, w j )}. The sistem (3.2) has a solution v m (t) in [, t m [, t m >, and the estimates we obtain permit extend the solution to interval [, T ] and pass to the limits ESTIMATIVAS PRIORI Estimativa I. We obtain for usual way the following convergence v m v fraco em L (, T ; H) (3.3) v m v fraco em L 2 (, T ; V ) (3.4) v m v fraco em L 4 (, T ; V ) (3.5) 7

8 Estimativa II. By the special choise of (w µ ), using ortogonal projection and Aubin-Lions Theorem s we can take limit in nonlinear term to obtain v + ν Av + χ + Bv + Cv = g em L 4/3 (, T ; V ). It is necessary show to prove that χ = Av. This is done by monotony of the operator A as in J. L. Lions [3]. The proof of Theorem 2.4 follows, with slight modifications, by the proof presented by J. L. Lions[3] and the proof of Theorem 2.5 see M. Milla Miranda an J. L. Ferrel[6]. Referências [1] Ladyzhenskaya, O.A., The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach-London(Second Edition), [2] Leray, J., Essai sur les Mouvements Plans d un Liquide Visqueux que limitent des Parois, J. Math Pures Appl. t. XIII(1934), [3] Lions, J.L., Quelques Méthodes de Resolution Des Problmes Aux Limites Non Linéaires, Dunod, paris, [4] Lions, J.L. et Prodi, G., Um Thorme d Existence et Unicité dans les Equations de Navier-Stokes en Dimension 2, C. R. Acad. Sci. Paris, 248(1959, pp ), Ouvres Choisies de Jacques-Louis Lions, Vol.1- EDP-sciences, Paris, 23, pp. 117 [5] Medeiros, L.A., Lições de Equações Diferenciais Parciais, Instituto de Matemática-UFRJ, 22 [6] Milla Miranda, M., Ferrel, J.L., The Navier-Stokes Equation in Noncylindrical Domain, Comp. Appl. Math., V. 16, no. 3, pp , [7] Tartar, L., Partial Differential Equations Models in Oceanography, Carnegie Mellon University [8] Temam, R., Navier- Stokes Equations, Theory and Numerical Analysis, North-Holland Publishing Company,