Hidden Regularity for a Nonlinear Hyperbolic Equation with a Resistance Term

Transcription

1 International Mathematical Forum, 4, 2009, no. 11, Hidden Regularity for a Nonlinear Hyperbolic Equation with a Resistance Term G. O. Antunes Instituto de Matemática e Estatística Universidade do Estado do Rio de Janeiro Rio de Janeiro - Brasil gladsonantunes@hotmail.com R. S. Busse Instituto de Matemática e Estatística Universidade do Estado do Rio de Janeiro Rio de Janeiro - Brasil ronaldobusse@yahoo.com.br H. R. Crippa Instituto de Matemática Universidade Federal do Rio de Janeiro Rio de Janeiro - Brasil crippa@im.ufrj.br Abstract In this paper we obtain the hidden regularity for the weak solution of the following nonlinear system u Δu + F (u) = p in, div u =0 in, u = 0 on Σ, u (0) = u 0, u (0) = u 1 in Ω. The linear model is originated from a problem of elasticity with a resistance of the material and was proposed in the first time in a work of J. L. Lions. Mathematics Subject Classification: 35L10, 49N60 Keywords: Hidden Regulariry, Hyperbolic Equation

2 512 G. O. Antunes, R. S. Busse and H. R. Crippa 1 Introduction Let Ω be a bounded domain of R n with smooth boundary Γ. We represent by the cylinder Ω (0,T) with lateral boundary Σ = Γ (0,T),T an arbitrary positive real number. In this article we consider the system u Δu + F (u) = p in, div u =0 in, (1) u = 0 on Σ, u (0) = u 0, u (0) = u 1 in Ω, that appear in the theory of elasticity of incompressible material. We observe that u : R n, u =(u 1,u 2,..., u n ), where u i : R for i =1,..., n, div u = n u i, F : R n R n is given by x i i=1 F (u) = f f f n where we assume that each f i : R R, i =1,..., n is a continuous function that satisfies f L f i (s) (R), lim = α i,i=1,..., n. (2) s s This model, without the nonlinear term, was proposed in the first time in a work of J. L. Lions [6]. The aim of this paper it shows that, under the condition (2) and a geometrical condition on Ω, the weak solution of the problem (1) has the following regularity on the lateral boundary Σ u ν [ L 2 (Σ) ]n, (3) where ν is the exterior normal vector to Γ. The property (3) was proved by J. L. Lions [6], motivated by problems of optimal control for the equation u Δu =0in. In [7], J. L. Lions proves that the weak solution u of the mixed problem for the equation u Δu + u ρ u =0in, also satisfies (3) and called this property of Hidden Regularity, because that one does not follow from the equation or from the spaces where the solution u 1 u 2. u n,

3 Hidden regularity for a hyperbolic equation 513 lies. This result was generalized by M. Milla Miranda and L. A. Medeiros [8] for the equation u Δu + F (u) =0in, where F : R R satisfies the hypothesis of W. A. Strauss [10], i.e., F is continuous and sf (s) 0, s R. In [11], M. C. de Campos Vieira obtain the Hidden Regularity for the following Coupled System of Nonlinear Hyperbolic Equations u Δu + α v ρ+2 u ρ u = g 1 in v Δv + α u ρ+2 v ρ v = g 2 in. The Hidden Regularity for the linear system associated to (1) was proved in [5] by Lions. For a physical interpretation of this model see Rocha dos Santos [9]. See about similar questions in Kapitonov [3], [4]. 2 Existence and Uniqueness of Solution In this section we study the existence and uniqueness of solution for the problem (1) that will be written in the following form: u i Δu i + f i (u i )= p in, x i n u i =0 in, j=1 x (4) j u i = 0 on Σ, u i (0) = u 0i, u i (0) = u 1i in Ω, for i =1,..., n, where the prime means the derivative with respect to t and all the derivatives are in the sense of the theory of distributions. Before beginning the study of (4), the notation will be established. We consider the Hilbert spaces V = { u ( H 1 0 (Ω) ) n ; div u =0 } and H = { u ( L 2 (Ω) ) n ; div u = 0 and u.ν = 0 on Γ }, equipped with the inner product and norm given respectively by ((u, v)) = n ((u i,v i )) H 1 0 (Ω), u 2 = i=1 n u i 2 H0 1(Ω) i=1

4 514 G. O. Antunes, R. S. Busse and H. R. Crippa and (u, v) = n (u i,v i ) L 2 (Ω), u 2 = i=1 n u i 2 L 2 (Ω). Remark 1 Note that it is necessary to show that for each w H 1 0 (Ω), f i (w) L 2 (Ω). In fact, from the hypothesis (2) follows that given ε =1there exists N i = N i (ε) such that this is, i=1 f i (w) α i w < w, w >N i, f i (w) 2 < 2 ( α i w 2 + w 2), w >N i. (5) From (5) we obtain f i (w) 2 dω < f i (w) 2 dω+ f i (w) 2 dω < Ω w N i w >N i <C i +2 ( α i 2 +1 ) w 2 dω, Ω because f i is continuous and Ω is bounded. Theorem 2.1 Given u 0 V, u 1 H and f i satisfying (2), there exists a unique function u such that u L (0,T; V ), (6) u L (0,T; H), (7) u Δu + F (u) = p in D ( 0,T;(D (Ω)) n), (8) u (0) = u 0, u (0) = u 1 in Ω. (9) Before beginning the proof of the Theorem 2.1, let us consider the following lemma: Lemma 2.1 Considering u 0, u 1 and f i in the conditions of the Theorem 2.1. Then there exists a unique function u satisfying u L (0,T; V ), (10) u L (0,T; H), (11) d dt (u (t),v)+((u (t),v))+(f (u),v)=0, v V in D (0,T), (12) u (0) = u 0, u (0) = u 1 in Ω. (13)

5 Hidden regularity for a hyperbolic equation 515 Proof. We employ the Faedo-Galerkin s method. Let {w ν } ν N a Hilbertian basis of V. We consider V m =[w 1,..., w m ] the subspace of V generated by the m first vectors of {w ν } ν N. The approximate problem of (12) (13) consists in to find u m (t) V m satisfying for each i =1,..., n d dt (u im (t),v)+((u im (t),v))+(f i (u im (t)),v)=0, v V m, u im (0) = u 0im u 0i strong in H0 1 (Ω), (14) u im (0) = u 1im u 1i strong in L 2 (Ω). By Carathèodory s theorem, (14) has solution on an interval [0,t m [, t m <T. This solutions extend to the whole interval [0,T] as a consequence of the estimates that shall be obtained in the next step. Estimates. Taking v = u im (t) in (14) 1, we have after some calculations that and (u im ) is bounded in L (0,T; V ) (15) (u im ) is bounded in L (0,T; H), i =1,..., n. (16) To obtain the convergence of the nonlinear term we observe that from Theorem of Lions-Aubin there exists a subsequence (u im ) such that: u im u i strong in L 2 ( 0,T; L 2 (Ω) ). (17) From (17) and the continuity of the f follows that f i (u im (x, t)) f i (u i (x, t)) a.e. in. (18) We also have, from Remark 1, that f i (u im ) L 2 (), for i =1, 2,..., n. Therefore, by Lemma of Lions we get f i (u im ) f i (u i )inl 2 ( 0,T; L 2 (Ω) ) weak as m +, for 1 i n. (19) From (15), (16) and the convergence in (19), it is permissible to pass to the limit in (14) 1 and obtain, by standard procedure, a solution u satisfaying (10) (13). Proof of Theorem 2.1. To prove the Theorem 2.1, it is sufficient to prove that u satisfies (10) (13) if, and only if, u satisfies (6) (9). In fact, we consider the following functional: L (v) = u Δu + F (u),v (H 1 (Ω)) n (H 1 0 (Ω))n,

6 516 G. O. Antunes, R. S. Busse and H. R. Crippa where u is the solution given by the Lemma 2.1. Observe that L (H 1 (Ω)) n. From (12) we have that L (v) = 0 for all v V,inD (0,T). By the De Rham s lemma [2], there exists p L 2 (Ω) R such that L (v) = pdiv vdx, v V. Ω In particular, for all ϕ V= {ϕ (D (Ω)) n ; div ϕ =0}. So, we obtain this is, u Δu + F (u),ϕ = p, ϕ, u Δu + F (u) = p in D ( 0,T;(D (Ω)) n). Reciprocally, if u satisfies (6) (9), then multiplying (8) by ϕ V and integrating in Ω, we conclude that (u (t),ϕ)+((u (t),ϕ)) + (F (u),ϕ)= p, ϕ, in D (0,T), ϕ V, thus the theorem is proved. 2.1 Energy Conservation From the weak formulation of the (4), taking v = u we obtain 1 d [ u i 2 dt (t) 2 + u i (t) 2] +(f i (u i (t)),u i (t)) = 0. (20) If we represent by G i (s) the function then G i (s) = s 0 f i (r) dr (f i (u i (t)),u i (t)) = d G i (u i (t)) dx. (21) dt Ω Substituing (21) in (20) we obtain where E (t) = 1 2 E (0) = E 0. d E (t) =0, dt [ u i (t) 2 + u i (t) 2] + G i (u i (t)) dx. Therefore E (t) = Ω

7 Hidden regularity for a hyperbolic equation Main Result In what follows we shall prove (3) under the assumption m (x) ν γ>0, m(x) =x, for x Γ. (22) Theorem 3.1 We assume that (2) and (22) holds true. Then, if u denotes the solution of (1) with initial data {u 0,u 1 } V H, one has u ν ( L 2 (Σ) )n. (23) Proof: In the follows, let us to prove (23) for u 0, u 1 smooth, this is, u 0, u 1 U where U = {ϕ (D (Ω)) n such that div ϕ =0}. If we denote by u μ the solution associated to u 0μ, u 1μ U such that, than we have u 0μ u 0 in V u 1μ u 1 in H, u μ L (0,T; H 2 (Ω) V ) u μ L (0,T; V ) u μ L (0,T; H), (24) and so the integrations by parts to follow are valid. Multiply both sides of (4) 1 by m (x) u μ = n on, we obtain: ( u u μi u ki μim k Δu μi m k + f i (u μi ) m k k=1 m k ) dxdt =, and integrate Remark 2 If we integrate by parts the last term of the (25) we obtain p m k dxdt =0. x i Now we will analyze each term of the first member of the (25). ( ) u μi m k dxdt = u μi x,m T k + n ( ) u 2 k 0 2 μi dxdt. p m k dxdt x i (25)

8 518 G. O. Antunes, R. S. Busse and H. R. Crippa u ki Δu μi m k dxdt = 1 m k ν k 2 Σ ν 2 dσ+ 1 n 2 2 dxdt. Remark 3 For the analysis of the nonlinear term we use the Strauss approximation cf. [10] and [8] to prove that G i is C 1 with bounded derivative and G i (0) = 0, this is, G i (u i ) L 2 (0,T; H0 1 (Ω)), for i =1, 2,..., n. Therefore we have, f i (u μi ) m k dxdt = n G i (u μi ) dxdt. Substituing the estimates above in (25) and from (22) we obtain, after some calculations that 2 γ 2 Σ ν dσ nt E 0 +2n G i (u μi (x, t)) dxdt + ( ) + u (t) T (26) μi (t),m k. 0 From (26) follows that 2 Σ ν dσ C (n, T, Ω) E 0, (27) because G i (u μi (x, t)) dxdt C (Ω). Therefore, from (27) we have ν χ in ( L2 0,T; L 2 (Γ) ). (28) Now observe that by (24) and Theorem of Lions-Aubin there exists a susbsequence (u μi ) such that u μi u i strong in L 2 ( 0,T; H 1 (Ω) ). (29) By the continuity of the trace we have T 0 T γ 1 (u μi (t)) γ 1 (u i (t)) 2 H 1/2 (Γ) dt C u μi (t) u i (t) 2 H 1 (Ω) dt, 0 (30) for each i =1,..., n.

9 Hidden regularity for a hyperbolic equation 519 From (29) and (30) we conclude that ν u i ν in L2 ( 0,T; H 1/2 (Γ) ) as μ +, i =1,..., n. (31) From (28) and (31) follows that χ = u i ν L2 (Σ), for each i =1,..., n, this is, u ν ( L 2 (Σ) ) n. ACKNOWLEDGEMENTS. We acknowledge to Professor Luis Adauto Medeiros who drew our atention to this question. We also appreciate the constructive conversation with Professor Maria Darci G. da Silva about the subject of the paper. References [1] Brezis, H., Analyse Fonctionelle, Théorie et Applications, Dunod, Paris, (1999 ). [2] De Rham, G., Variétés Différentiables, Hermann, Paris (1960 ). [3] Kapitonov, B.V., Uniform stabilization and exact controllability of a class of coupled hyperbolic systems, Computational and Applied Mathematics, v. 15, n. 3, p , [4] Kapitonov, B.V., Uniform stabilization and simultaneous exact control for a couple of hyperbolic systems, Sibirskij Matematiceskij Zurnal, n. 4, [5] Lions, J. L., On some Hyperbolic Equations with a Pressure Term, Proceedings of the conference dedicated to Louis Nirenberg held in Trento, Italy, september 3-8, Harlow: Longman Scientific and Technical, Pitman Res. Notes Math. Ser 269, p (1992 ). [6] Lions, J. L., Exact Controllability, stabilization and pertubations for distributed systems, J. Von Neuman Lecture, Boston, Siam Review (1986 ). [7] Lions, J. L., Hidden regularity in some nonlinear Hyperbolic Equations, Math. Aplic. and Computational, 6:n. 1, p (1987 ). [8] Milla Miranda, M and Medeiros, L. A., Hidden Regularity for Semilinear Hyperbolic Partial Differential Equations, Ann. Fac. Sci., Toulouse, 9:n 1, p (1988 ).

10 520 G. O. Antunes, R. S. Busse and H. R. Crippa [9] Rocha dos Santos, A., Exact Controllability in Dynamic Incompressible Materials, Ph. Dissertation. Instituto de Matemática - UFRJ, Rio de Janeiro -RJ- Brasil (1996 ). [10] Strauss, W. A., On weak solutions of semilinear hyperbolic equations, An. Acad. Bras. Ciências 42 (4) (1970 ), p [11] Vieira, M. C. de Campos, Hidden Regularity for a Coupled System of Nonlinear Hyperbolic Equations, An. Acad. Bras. Ciências 63 (2) p (1991 ). Received: September 23, 2008