Regularity of solutions of a phase field model

Transcription

1 Regularity of solutions of a phase field model K.-H. Hoffmann, T. G. Amler, N. D. Botkin, K. Ruf Center for Mathematics, M6, Technische Universität München, Boltzmannstr. 3, Garching/Munich, Germany 1 Introduction Nowadays, phase field techniques for modeling of solidification and freezing processes become very popular (see e.g. 1], ], and 3]). They are based on the consideration of the Gibbs free energy which depends on an order parameter that assumes values from -1 (solid) to 1 (liquid) and changes sharply but smoothly over the solidification front so that the sharp liquid/solid interface becomes smoothed. The rate of smoothing is controlled by a small parameter, which enables to reach arbitrary approximation of the sharp interface. Phase field models are also appropriate for the description of phase transitions when modeling CO sequestration. The supercritical carbon dioxide, CO that has been pressurized to a phase between gas and liquid, may be injected into a saline aquifer where it may either dissolve in the brine, react with the dissolved minerals or the surrounding rock, or become trapped in the pore space of the aquifer. In this paper, we consider a phase field model proposed by G. Caginalp in 1]. The aim of the investigation is to prove the existence and uniqueness of solutions to this model for very general initial data (comp. with 4], 5], and 6]). Moreover, it will be proved that the solutions are continuous in time, and their values at each time instant lie in the same space as the initial data. Thus, the model can be considered as a dynamical system. The Model The phase field model derived by G. Caginalp (see 1]) is given by (1). The temperature u and the phase-function φ are defined on a bounded domain R N, N {,3}. The evolution of This publication is based on work supported in part by Award No. UK-C, made by King Abdullah University of Science and Technology (KAUST). This publication is based on work supported in part by Stiftung der Deutschen Wirtschaft (sdw). This work is supported in part by the German Research Society (Deutsche Forschungsgemeinschaft), SPP

2 these functions is given by the initial boundary value problem u t l φ t k u = in (,T), τ φ t u 1 ( φ 3 φ ) ξ φ = in (,T), k ν u = λ(u g), ν φ = on (,T), u(x,) = u (x), φ(x) = φ (x) for t = on. (1) The constants k and l appearing in (1) denote the heat conductivity and the latent heat, respectively. The boundary temperature regime is defined by a given function g. The region is assumed to be a bounded domain in R N with the Lipschitz boundary, i.e. is of class C,1. The evolution of u and φ is considered on the time interval,t] where the finial time instant T is an arbitrary positive and finite real number. The paper is structured as follows. The precise formulation of the considered problem and the results on existence, uniqueness, and regularity of solutions are given in Section 3. Approximate solutions are constructed in Section 4, and the existence of weak solutions is shown in Section 5. The uniqueness and stability of solutions is proved in Section 6. 3 Statement of the problem and main result Instead of considering the unknown temperature u with an initial value u, introduce the functions v := u l φ, v (x) := u (x) l φ (x). () If the functions u and φ satisfy system (1), the unknowns (v,φ) solve the following initialboundary value problem: v t k v kl φ = in, ( τφ t v l 1 ) φ 1 φ3 ξ φ = in, ν φ =, k ν v = λ (v l ) φ g on, (3) v(x) = v (x), φ(x) = φ (x) for t = on.

3 If not stated differently, the following notation is used throughout the paper: T := (,T), T := (,T), HT(,T;L 1 ()) := { f H 1 (,T;L ()) : f(t) = }, CT 1 (,T]) := { f C 1 (,T]) : f(t) = }, Y := L (,T;H 1 ()) L (,T;L ()), y Y = y L (,T;H 1 ()) y L (,T;L ()), X := L (,T;H 1 ()) L 4 ( T ), x X = x L (,T;H 1 ()) x L 4 ( T ), (4) X := L (,T;(H 1 ()) ) L 4/3 ( T ), { } f X = inf max f 1 L (,T;(H 1 ()) ), f L 4/3 ( T ), f 1 L (,T;(H 1 ()) ) f L 4/3 ( T ) f 1 f =f W := { u X : u t X }, u W := u X u t X. Indeed, (X, X ) is the normed dual of the separable and reflexive Banach space (X, X ), see 7, Chap. IV]. The next definition states the sense of weak solutions. Definition 3.1 (Weak solution of problem (3)). A pair of functions (v, φ) with v L (,T;H 1 ()) and φ X is called weak solution of problem (3), if the following equation = T T v ψ τφ η ] T dx v ψ t τφη t ] dxdt k v kl ] T φ ψ dxdt λ v l ] φ g ψ dsdt { ( v l 1 )φ 1 } ] φ3 η ξ φ η dxdt, (5) holds true for all ψ, η H 1 T (,T;L ()) L (,T;H 1 ()) with η L 4 ( T ). The main result on problem (3) is stated in the following theorem. Theorem 3.. Let v, φ L (), g L ( T ) be arbitrary functions. Then system (3) has a unique weak solution (v,φ) in the sense of Definition 3.1. The components v and φ of 3

4 the solution have the regularity: v, φ C (,T];L () ). Moreover, if vi, φ i L (), and g i L ( T ), i = 1,, then v Y, φ Y C ( 1 T e T) v L () ] φ L () ḡ L ( T ), where ḡ = g 1 g, v = v 1 v, φ = φ 1 φ, v = v1 v, and φ = φ 1 φ. The constant C is independent of v i, φ i, vi, φ i, and g i, i = 1,. 4 Construction of Approximations Let {ω i } be an orthogonal basis in H1 () and L () simultaneously (such a basis really exists). Consider Galerkin-Approximations of the form v m (x,t) = a m i (t)ω i(x), φ m (x,t) = i= b m i (t)ω i(x), (6) where a m i (t) and bm i (t) are unknown functions. To determine the coefficients a m i and b m i, we derive a system of ordinary differential equations. Extend g by zero for t > T, and (similar to 8, Chap. II, 4]) let g m be the Steklov average of g g m (x,t) = m t1/m t i= g(x,r)dr. Then g m C(R;L ( )) L ( R ), and g m g strongly in L ( T ) as m. Substitute the approximations (6) into (5), cancel the integration over the time, and replace the couples of test functions, (ψ, η), first by (ω j, ) and second by (, ω k ), j, k = 1,...,m. This yields the ordinary differential equations = λ = {ȧ mi (t) ω i ω j k a m i a m i (t) l bm i (t) τ ḃm i (t) ω i ω k ] { a mi (t) ( l 1 ] } kl (t) bm i (t) ω i ω j ω i ω j ds λ g m (t)ω j ds ξ b m i (t) (j = 1,...,m), ω i ω k 1 ( m ) 3 b m i (t)ω i ω k ] )b }ω mi (t) i ω k (k = 1,...,m). Assuming that {ω i } is also orthonormal in L (), equation (7) can be written as a system of ordinary differential equations determining the coefficients a m i (t) and bm i (t) as follows: ȧ m j (t) Am j (t, am 1,..., am m, bm 1,..., bm m ) = } τ ḃm j (t) Bj m (t, a m 1,..., a m m, b m 1,..., b m for j = 1,...,m, m) = (7) 4

5 where the functions A m j and Bj m depend analytically on their variables. This system can be rewritten as ]ȧm ] ] Im A m τ I m ḃ m B m =, (8) where I m denotes the m m identity-matrix. The initial conditions for (8) are given by a m j () = ω j v and b m j () = ω j φ, for j = 1,...,m. (9) Since the functions A m and B m are smooth with respect to all their variables, the theory of ordinary differential equations shows that, for each fixed m, there exist a non-empty time interval,t m ] on which (8) is solvable. Next, we show that T m can be chosen independently of m. To this end, note that if a m i and b m i are the solution of (8), they are continuously differentiable. Multiply the first equations in (7) by a m j (t) and sum up over j = 1,...,m. Multiply the second equations in (7) by α bm k (t) (the constant α will be specified later) and sum up over k = 1,..., m. By the ansatz (6), this yields = v m vt m k v m k l ] λ v m l ] φm v m g m v m ds α τ φ m φ m t ( l 1 φm v m ) φ m 1 ] φm 4 ξ φ m. Integrating over a time interval (,t), t (,T m ], using the product rule for the time derivatives, and applying Young s inequality yield (with ǫ > ) 1 λ v m (t) α τ φ m (t) ] t 1 λ t t v m ds v α τ φ ] t ǫ v m gm ǫ k v m α φm 4 αξ φ m ] l 8ǫ φm ( α l 1 ) φ m k vm l k ] ds Choose ǫ = 1 / and use the embedding H 1 () L ( ) to obtain l λ 4 φ m C ] φm (1) φ m φ m ], (11) where C is a constant that is independent of φ m. Now, choose α such that β := α ξ l k C >. (1) 5

6 Reinserting (11) and (1) into (1) yields 1 v m (t) ατ φ m (t) ] t t λ 1 v m ds v ατ φ ] t λ By Gronwall s inequality, we obtain that k vm α ] φm 4 β ξ φ m g m ds t v m bounded in L (,T m ;L ()) L (,T m ;H 1 ()) ( C l 1 ) φ m. φ m bounded in L (,T m ;L ()) L (,T m ;H 1 ()) L 4 (,T m ;L 4 ()), (13) (14) and moreover the bounds are independent of m and t. Due to the choice of {ω j } j N, it holds m v m (t) L () = m a m i (t) and φ m (t) L () = b m i (t). Therefore (14) shows that a m i and b m i (i = 1,...,m) are bounded on,t m ] and can be continued beyond T m. Consequently, there exists no maximal T m, and a m i and b m i can be defined on, ) for each m. 5 Existence of weak solutions The next lemma establishes the existence of solutions to problem (3) and a regularity result for the time derivatives of the solution components. This regularity is needed in Section 6 to show the uniqueness of the solution. Lemma 5.1. Problem (3) has at least one weak solution (v, φ) in the sense of Definition 3.1. Each of the weak solutions satisfies: v t L (,T;(H 1 ()) ) and φ t L (,T;(H 1 ()) ) L 4/3 (,T;L 4/3 ()). Proof. Due to (14), there exist functions v, φ, and ζ such that (up to subsequences) v m v in Y, φ m φ in Y X, (φ m ) 3 ζ in L 4/3 (,T;L 4/3 ()), (15) see definitions (4). By the construction of the approximate solutions in (6), (7), and (9), the 6

7 functions v m and φ m satisfy the equation = v,m ψ φ,m η ] T ( v m ψ t k v m kl ) ] φm ψ T λ (v m l ) φm g m ψ ds T τφ m η t { ( v m l 1 )φ m 1 } ] (φm ) 3 η ξ φ m η (16) for all ψ and η which are linear combinations of functions c j (t)ω j (x), c j (t) CT 1(,T]), j = 1,...,m. Consider the limit as m. Due to (9), the initial functions v,m and φ,m can be replaced by v and φ in the first integral on the right-hand side. By the properties of Steklov averages, g m can be replaced by g and, by (15), (v m,φ m,(φ m ) 3 ) can be replaced by the limiting functions (v, φ, ζ) in (16). Note that linear combinations of functions c j (t)ω j (x),c j CT 1,T] lie dense in the spaces HT 1(,T;L ()) L (,T;H 1 ()) and HT 1(,T;L ()) L (,T;H 1 ()) L 4 (,T;L 4 ()). Therefore, v, φ, and ζ satisfy the equation = v ψ φ η ] for all λ T T T ( v ψ t k v kl (v l φg ) ψ ds τφη t ) φ ] ψ { ( v l 1 )φ 1 } ] ζ η ξ φ η ψ H 1 T(,T;L ()) L (,T;H 1 ()), η H 1 T (,T;L ()) L (,T;H 1 ()) L 4 (,T;L 4 ()). It remains to show that ζ = φ 3. To this end, show first that φ m φ in L (,T;H 1 ǫ ()) strongly (ǫ > ) by applying 9, Section 8, Corollary 4]. To apply this corollary, φ t has to be estimated. Choose ψ = in (17), use the boundedness of T and the embedding H 1 () L 4 (), and apply Hölder s inequality to derive the estimate T φ t ;η = C φη t = 1 T { ( v τ v L 3/4 ( T ) φ L 3/4 ( T ) ζ L 3/4 ( T ) C φ L (,T;H 1 ()) η L (,T;H 1 ()) l 1 )φ 1 } ζ η ξ φ η] ] η L 4 ( T ) (17) (18) 7

8 for all η of the form η(x,t) = c(t)ω(x), c D(,T) and ω H 1 (). Inequality (18) and the bounds (14) show that φ t is a continuous functional on L (,T;H 1 ()) L 4 (,T;L 4 ()). Therefore, (see also the proof of Lemma 6.1) it holds φ t ( L (,T;H 1 ()) L 4 (,T;L 4 ()) ) = L (,T;(H 1 ()) ) L 4/3 (,T;L 4/3 ()) L 4/3 (,T;(H 1 ()) ) L 1(,T;(H 1 ()) ), because T is finite. Applying 9, Section 8, Corollary 4] with X = H 1 (), B = H 1 ǫ (), Y = ( H 1 () ), and p = we obtain φ m φ strongly in L (,T;H 1 ǫ ()) for a subsequence still denoted by {φ m }. This implies the convergence φ m (x,t) 3 φ(x,t) 3 almost everywhere in T. Now ζ = φ 3 follows from (φ m ) 3 ζ in L 4/3 ( T ) (see (15)) and 1, Chap. 1, Lemma 1.3]. 6 Uniqueness and stability In order to show the uniqueness of the solution (v, φ) obtained in Sections 4 and 5, we need a certain regularity of the phase function φ. Using methods presented in book 7], the following lemma that provides a formula for the integration by parts can be proved. Lemma 6.1. Let H be a Hilbert space, and V 1 and V be reflexive and separable Banach spaces which are continuously embedded in H. Assume that V := V 1 V is dense in H. For given 1 < p 1, p, denote by p 1 and p the Lebesgue conjugate exponents. Define the following normed spaces X := L p 1 (,T;V 1 ) L p (,T;V ), x X := x L p 1(,T;V1 ) x L p (,T;V ), W := { u X : u t X }, u W := u X u t X. Then W C(, T]; H) with the continuous embedding (because T is finite). The formula u(t);v(t) H H u(s);v(s) H H = t s ut (τ);v(τ) V V v ] t(τ);u(τ) V V, (the integration by parts) holds for arbitrary u, v W and s, t, T]. Proof. Denote the norms in V i by i, i = 1,. The spaces V 1 V and V 1 V are Banach spaces when they are endowed with the norms v V1 V := v V1 v V, v V 1 V. z V1 V := inf max { } v 1 V1, v V. v 1 V 1, v V, v 1 v =z. 8

9 Since V i H and H is locally convex, 7, Chap. I, Th. 5.13] yields the relations V 1 V = (V 1 V ), (V 1 V ) = V 1 V. (19) Set V := V 1 V, then (19) implies that V = V 1 V. The normed dual X of X (defined in the lemma) is given by X := L p 1 (,T;V 1 ) Lp (,T;V ), f } f X := inf max{ L 1 p 1(,T;V1 ), f L p, (,T;V ) f 1 L p 1(,T;V 1 ), f L p (,T;V ), f 1 f =f. see 7, Chap. I, Th and Chap. IV, Th. 1.14] which also imply that X is reflexive. The space W defined in the lemma is a Banach space, see 7, Chap. IV, Th. 1.16]. Using the method of the proof of 7, Chap. IV, Lemma 1.11], we obtain the continuity of the embedding W C(,T];V 1 V ) because T is assumed to be finite. Adopting the proof of 7, Chap. IV, Lemma 1.1], we deduce that C 1 (,T];V ) W is dense in W. Moreover, the proof of 7, Chap. IV, Th. 1.17] shows that the embedding W C(,T];H) is continuous because T is finite. It remains to show the formula for integration by parts. Let u, v W and u n, v n C 1 (,T];V ) W, n N, be such that Then u n and v n satisfy the formula u u n W, v v n W 1/n. u n (t);v n (t) H H u n (s);v n (s) H H t = u n t (τ);v n (τ) V V vn t (τ);u n ] () (τ) V V dτ, s for arbitrary s, t,t] and for all n N. Consider the limit as n in the first summand on the left-hand side of (). The embedding W C(,T];H) implies that u n (t);v n (t) H H u(t);v(t) H H u n (t);v n (t) v(t) H H u n (t) u n (t);v(t) H H C ] W CW n n u C(,T];H) v C(,T];H), where C W is the norm of the identity map from W C(,T];H). A similar argument with t replaced by s shows that the left-hand side of () satisfies u n (t);v n (t) H H u n (s);v n (s) H H u(t);v(t) H H u(s);v(s) H H (1) 9

10 as n. Consider the limit as n of the first summand on the right-hand side of () to obtain: t u n t (τ);v n (τ) V V u ] t(τ);v(τ) V V s t u n t (τ);v n (τ) v(τ) V V ] un t (τ) u t (τ);v(τ) V V s 1 ] 1 n n u t X v X. Interchanging the roles of u and v shows that the right-hand side of () satisfies t s u n t (τ);v n (τ) V V ] vn t (τ);un (τ) V V t s ut (τ);v(τ) V V v ] t(τ);u(τ) V V as n. Substituting (1) and () into () yields the claimed formula. This completes the proof of the lemma. Lemma 6.1 is applied with V 1 = H 1 (), p 1 =, V = L 4 (), p = 4 and H = L (). Using the notation introduced in (4), we obtain from (18) that φ t X. Thus, Lemma 6.1 yields φ W C(,T];L ()). The following lemma establishes the uniqueness of weak solutions to problem (3) and a stability result in the space Y. Lemma 6.. Let vi, φ i L (), and g i L ( T ), i = 1, be given functions. Let (v i, φ i ) be weak solutions of problem (3) with v i L (,T;H 1 ()) and φ i X, i = 1,. Then v Y, φ Y C ( 1 T e T) v L () ] φ L () ḡ L ( T ), () where ḡ = g 1 g, v = v 1 v, and φ = φ 1 φ. The constant C is independent of v i, φ i, v i, φ i and g i, i = 1,. Proof. Use the notations from (4). Let ḡ, v, and φ be the functions defined in the lemma. Set ζ = φ 1 φ 1 φ φ. Then v and φ satisfy the equation T ( = v t ;ψ k v kl τ φt ;η T X X ) φ { v T ψ dxdt λ ( v l ) φ g ψ dsdt (l 1 1 ) } ] ζ φ η ξ φ η dxdt, where ; denotes the duality product between L (,T,(H 1 ()) ) ans L (,T;H 1 ()). The right-hand side of (3) is a continuous linear functional defined on functions ψ L (,T;H 1 ()) and η X. Therefore, (ψ,η) = ( v, φ) is an admissible choice, and φ W. (3) 1

11 Denote by χ (,t) the characteristic function of the interval (,t) for a fixed t (,T]. Let the constants α and β satisfy (1), and set (ψ,η) = (χ (,t) v,αχ (,t) φ) in (3) to obtain 1 v(t) ατ φ(t) ] t k v α ξ φ α ζ φ ] λ 1 v ατ φ ] t λ v l4 φ 1 ] ḡ t ( α v α l 1 ) φ l k 8 φ k ] v. Note that Lemma 6.1 is used here to evaluate φ t ; φ X X. Due to (1) and the imbedding H 1 () L ( ), inequality (4) yields the estimate v(t) φ(t) ] T { C v φ ] T v φ ] t T ḡ v φ ]}, with an independent of v and φ constant C. By Lemma 6.1, it holds v, φ C(,T];L ()) so that Gronwall s inequality yields v Y, φ Y C ( 1 T e T) v L () ] φ L () ḡ L ( T ). In particular, for v = φ = ḡ, we obtain the uniqueness of the solution. This proves the lemma. v (4) References 1] G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rat. Mech. Anal., vol. 9, pp. 5 45, ] M. Frémond, Non-smooth thermomechanics. Springer-Verlag, Berlin,. 3] P. Colli and K.-H. Hoffmann, A nonlinear evolution problem describing multicomponent phase changes with dissipation, Numer. Funct. Anal. and Optimiz., vol. 14, 3 & 4, pp , ] J. L. Lions, Control of Singular Distributed Systems. Gauthier Villars, ] K.-H. Hoffmann and Jiang Lishang, Optimal control of a phase field model for solidification, Numer. Funct. Anal. Optimiz., vol. 13, 1 &, pp. 11 7, ] C. Eck, A Two-Scale Phase Field Model for Liquid-Solid Phase Transitions of Binary Mixtures with Dendritic Microstructure. Habilitation, Universität Erlangen, 4. 11

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