A class of time discrete schemes for a phase

In this paper, a phase eld system of Penrose Fife type with non conserved order parameter is considered. A class of time discrete schemes for an initial boundary value problem for this phase eld system is presented. In three space dimensions, convergence is proved and an error estimate is derived. For one scheme, this error estimate is linear with respect to the time step size. 1 Introduction In [PF90], Penrose and Fife derived a phase eld system modeling the dynamics of di usive phase transitions. In the case of a non conserved order parameter, their approach leads to the following system: c0 t + 0( ) t + 1 = g; (1.1) t " + ( ) 0( ) 3 0( ) : (1.2) This system determines the evolution of the order parameter and the absolute temperature . Here, c0 and denote the physical data speci c heat and thermal conductivity, which are supposed to be positive constants. The datum g represents heat sources or sinks, and stands for a positive space dependent relaxation coe cient. Choosing this coe cient in a particular way, an anisotropic growth can be simulated. " is a positive relaxation coe cient and denotes the subdi erential of the convex but non smooth part of a potential on R, while corresponds to the non convex but di erentiable part of the potential. The latent heat of the phase transition is represented by 0( ). In the context of solid liquid phase transitions, one typically has a quadratic or linear function and (s) = (s) C + s2; 8 s 2 R; (1.3) where C denotes some critical temperature and some positive constant. For (s) = 2 s3, we see that (s) 0(s)+ 1 C 0(s) is the derivative of the double well potential 2(s 1)2(s+1)2. If is the subdi erential of the indicator function I[ 1;1] of the interval [ 1; 1], we see that (s) 0(s) + 1 C 0(s) corresponds to the derivative of the double obstacle potential I[ 1;1](s) + (1 s2), which has been introduced for the standard phase eld system by Blowey and Elliott (see [BE94]). In the mean eld theory of the Ising ferromagnet as in [PF90, Sec. 4], one has quadratic functions and , D( ) = (0; 1), and (s) = @ @s s ln s+ (1 s) ln(1 s) ln 1 2 = ln s 1 s ; 8 s 2 D( ); where is some positive constant. 1 The results in this work cover all these situations. Its main novelty is a time discrete scheme for an initial boundary value problem for the phase eld system (1.1) (1.2) such that in three space dimensions an error estimate linear with respect to the time step size h can be derived. Moreover, a general class of time discrete schemes is investigated, including some which are explicit in the approximation of 0( ) or 0( ). For these schemes an error estimate is derived, which is linear with respect to h in two space dimensions and still nearly linear in three space dimensions. In [Hor93], Horn considers a time discrete scheme in one space dimension for the Penrose Fife system for quadratic and . He derives an error estimate of order ph. In previous works [Kle97a, Kle97b] of the author a time discrete scheme for a simpli ed Penrose Fife system with linear and linear or quadratic has been considered and an error estimate of order ph has been shown. Using the time discrete scheme, the existence of a unique solution to the Penrose Fife system is proved. This result is a minor novelty of this paper, because of the weakened regularity assumption used for and . These functions are supposed to be C1 functions on R with 0 and 0 locally Lipschitz continuous such that the Lipschitz constants ful ll some growth conditions. Until now, in papers concerning existence, uniqueness, and regularity of similar Penrose Fife systems these functions are supposed to be at least C2 functions with 00 bounded (see, e.g. [HLS96, HSZ96, Lau93, Lau95, SZ93] or C1 functions with 0 global Lipschitz (see [KN94]) resp. convex (see [DK96]). The same holds for papers like [CL98, CS98, CLS, Lau98], where more general heat ux laws are considered. The layout of this paper is as follows: In Section 2, a precise formulation of the considered phase eld system is given, the class of time discrete schemes is introduced, and the existence and approximation results are presented. The remaining sections are devoted to the proof of these results. In Section 3, estimates concerning the approximation of the data are derived and the existence of a solution to the scheme is shown under the additional assumption that the domain D( ) is bounded. Uniform estimates for a solution to the scheme are derived in Section 4. Based on these results, the existence of a unique solution to the scheme is proved in Section 5. This is done by considering the time discrete scheme with replaced by + @I[ C;C], where I[ C;C] denotes the indicator function of the interval [ C;C] for some su ciently big C > 0. In Section 6, the error estimates are derived, and the existence of a unique solution to the Penrose Fife system is proved. 2 The Penrose Fife system and the time discrete schemes In this section, a precise formulation of the considered phase eld system of Penrose Fife type is given. Moreover, existence results and approximation results for a class of time discrete schemes are presented. 2 2.1 The phase eld system In the sequel, RN with N 2 f2; 3g denotes a bounded, open domain with smooth boundary and T > 0 stands for a nal time. Let T := (0; T ) and T := (0; T ). We consider the following Penrose Fife system: (PF): Find a quadruple ( ; u; ; ) ful lling 2 H1(0; T ;L2( )); u 2 L2(0; T ;H2( )) \ L1(0; T ;H1( )); (2.1a) 2 H1(0; T ;L2( )) \ L1(0; T ;H2( )); (2.1b) 2 L1(0; T ;L2( )); (2.1c) > 0; u = 1 ; 2 D( ); 2 ( ) a.e. in T ; (2.1d) c0 t + 0( ) t + u = g a.e. in T ; (2.1e) t " + 0( ) = 0( )u a.e. in T ; (2.1f) @u @n + u = ; @ @n = 0 a.e. in T ; (2.1g) ( ; 0) = 0; ( ; 0) = 0 a.e. in : (2.1h) For dealing with this system, the following assumptions will be used: (A1): Let be a maximal monotone graph on R and : R ! [0;1] a convex, lower semicontinuous function : R! [0;1] satisfying = @ ; 0 2 D( ); 0 2 (0); int D( ) 6= ;: (A2): There are positive constants C 1 ; p; q such that 2 C1(R); 2 C1(R); p < 1; q < 4; (s) C 1 ( (s) + 1); ( 0(s))2 C 1 ( (s) + 1); 8 s 2 D( ); j 0(s) 0(r)j js rjC 1 (jsjp + jrjp + 1) ; 8 s; r 2 D( ); j 0(s) 0(r)j js rjC 1 (jsjq + jrjq + 1) ; 8 s; r 2 D( ): (A3): We have positive constants c ; c , and c such that g 2 H1(0; T ;L1( )); 2 L1( ); c a.e. in ; 2 L1(0; T ;C1( )); t 2 L1( T ); c a.e. in T ; 2 H1(0; T ;L2( )) \ L1( T ) \ L1(0; T ;H 1 2 ( )); c a.e. in T : (A4): We consider initial data 0; 0; u0; 0 such that 0; u0 2 H1( ) \ L1( ); 0 2 H2( ); 0 2 L2( ); ( 0) 2 L1( ); 0 > 0; u0 = 1 0 ; 0 2 D( ); 0 2 ( 0) a.e. in ; @ 0 @n = 0 a.e. in : 3 2.2 The class of time discrete schemes To allow for variable time steps, we consider decompositions of (0; T ) that do not need to be uniform, but satisfy the following assumption. (A5): The decomposition Z = ft0; t1; : : : ; tKg with 0 = t0 < t1 < < tK = T ful lls 0:01(tm tm 1) tm+1 tm 2(tm tm 1); 8 1 m < K: (2.2) Remark 2.1. In the estimate (2.2), the rst constant could be replaced by any positive constant smaller than one and the second by any constant bigger than one. We de ne the width jZj of the decomposition by jZj := max 1 m K(tm tm 1), and, for 1 m K, hm := tm tm 1; gm(x) := 1 hm tm Z tm 1 g(x; t) dt ; 8x 2 ; (2.3a) m( ) := 1 hm tm Z tm 1 ( ; t) dt ; m( ) := 1 hm tm Z tm 1 ( ; t) dt ; 8 2 : (2.3b) Now, the following time discrete scheme (DZ) for the Penrose Fife system is considered (DZ): For 1 m K, nd m 2 L2( ); um; m 2 H2( ); m 2 L2( ) (2.4a) such that 0 < um; m = 1 um ; m 2 D( ); m 2 ( m) a.e. in ; (2.4b) c0 m m 1 hm + 0d( m; m 1) m m 1 hm + um = gm a.e. in ; (2.4c) m m 1 hm " m + m 0 d( m; m 1) = 0d ( m; m 1) um a.e. in ; (2.4d) @um @n = mum m; @ m @n = 0 a.e. in ; (2.4e) with 0 := 0; u0 := u0; 0 := 0; 0 := 0: (2.4f) Here, approximations 0d and 0 d for 0 and 0 are used such that the following assumption is satis ed: 4 (A6): Let 0d; 0 d : R R! R be continuous functions, and let C 2 ; p; q be positive constants with p < 1, q < 4 such that, for all r; s; r0; s0 2 D( ), 0d(s; s) = 0(s); 0 d(s; s) = 0(s); ( 0 d(r; s))2 C 2 ( (r) + (s) + 1); j 0d(r; r0) 0d(s; s0)j C 2 (jr sj+ jr0 s0j) jrjp + jr0jp + jsjp + js0jp + 1 ; j 0 d(r; r0) 0 d(s; s0)j C 2 (jr sj+ jr0 s0j) jrjq + jr0jq + jsjq + js0jq + 1 ; 0d(r; s)(r s) (r) + (s) + C 2(r s)2: (2.5) Remark 2.2. The time discrete scheme (DZ) is an Euler scheme in time for the Penrose Fife system (PF), which is fully implicit, except for the treatment of the nonlinearities 0 and 0. By introducing the general approximations 0d( m; m 1) and 0 d( m; m 1) in (DZ), the same formulation can be used to investigate a bunch of di erent time discrete schemes. A full implicit scheme corresponds to the choices 0d(r; s) = 0(r) and 0 d(r; s) = 0(r). A fully explicitly treatment of nonlinearities 0 and 0 corresponds to 0d(r; s) = 0(s) and 0 d(r; s) = 0(s). The following choices for 0 d and 0d ful ll (A6), if (A2) is satis ed (see Lemma 3.1). a) Any convex combination of 0( m) and 0( m 1) can be used for 0 d( m; m 1). b) One particular choice for 0d is the following approximation for a derivative, which has been used by Niezgódka and Sprekels in [NS91, (2.3)]: 0 (r; s) := ( (r) (s) r s ; if r 6= s; 0(r); if r = s: (2.6) If one chooses this function as 0d , the approximation for 0( ) t used in the discrete energy balance (2.4c) will coincide with the discrete di erential quotient arising in the approximation of ( ( ))t . c) Assume that 2 C2(R). If we have a uniform upper and a uniform lower bound for 00 on D( ), we can use every convex combination of 0( m) and 0( m 1) for 0d( m; m 1). If we have a uniform upper bound for 00 onD( ), we can use the explicit approximation 0d( m; m 1) = 0( m 1). If we have a uniform lower bound for 00 on D( ), we can use the implicit approximation 0d( m; m 1) = 0( m). For the time discrete scheme there holds: Theorem 2.1. Assume (A1) (A6). Then, the scheme has a unique solution, if jZj is su ciently small. Remark 2.3. We use the solution to (DZ) to construct an approximate solution b Z; b uZ; b Z; Z in (L1(0; T ;L2( )))4 to the Penrose Fife system (PF). The function 5 b Z is de ned to be linear in time on [tm 1; tm] for m = 1; : : : ;K such that b Z(tk) = k holds for k = 0; : : : ;K. The functions b uZ and b Z are de ned analogously. We de ne Z piecewise constant in time by Z(t) = k for t 2 (tk 1; tk] and k = 1; : : : ;K. The following corollary allows to check, if for a given decomposition Z the scheme has a unique solution. Corollary 2.1. Assume that (A1) (A6) hold. There exists a solution to (DZ), if jZj h , where h and C 5 are positive constants with h 2 ( 0 d(r; s))2 C 5( (s) + 1) c (r); 8 r; s 2 D( ): (2.7) The solution to the scheme is unique, if, in addition, 0d(r; s) = 0(s); 2 jZj j 0 d(r; s) 0 d(r0; s)j c jr r0j ; 8 r; r0; s 2 D( ): (2.8) Remark 2.4. Assume that (A1) (A6) hold. If D( ) is bounded, Corollary 2.1 yields that the scheme has a solution. If D( ) is unbounded, the upper bound h can be calculated from (2.7) for given and 0 d. Thanks to (A6), we can always nd positive h and C 5 , such that (2.7) is satis ed. If 0 is approximated explicitly and 0 d is globally Lipschitz continuous in the rst variable on D( ) D( ), the conditions (2.8) and (2.7) lead to an computable upper bound for the time step size to ensure the existence of a unique solution. For 0 d explicit, i.e. 0 d(r; s) = 0(s), we do not get any restriction for the time step size from (2.7) or (2.8). 2.3 Existence and approximation results Theorem 2.2. Assume that (A1) (A4) and (A6) hold. Then there is a unique solution ( ; u; ; ) to the Penrose Fife system (PF). For this solution it holds that 2 L1(0; T ;H1( )) \ L1 ( T ) \ W 1;1(0; T ;H1( ) ); (2.9) u 2 H1(0; T ;L2( )) \ L1 ( T ) ; (2.10) 2 W 1;1(0; T ;L2( )) \ H1(0; T ;H1( )) \ L1 ( T ) : (2.11) As, for decompositions Z with (A5), jZj tends to 0, we have, b Z ! weakly in H1(0; T ;L2( )); (2.12) weakly star in L1(0; T ;H1( )) \ L1 ( T ) ; (2.13) weakly star in W 1;1(0; T ;H1( ) ); (2.14) b uZ ! u weakly in H1(0; T ;L2( )); (2.15) weakly star in L1(0; T ;H1( )) \ L1 ( T ) ; (2.16) weakly in L2(t ; T ;H2( )); 8 0 < t < T; (2.17) 6 b Z ! weakly in H1(0; T ;H1( )); (2.18) weakly star in W 1;1(0; T ;L2( )) \ L1(0; T ;H2( )); (2.19) Z ! weakly star in L1(0; T ;L2( )): (2.20) The following error estimate is the main result of this work. Theorem 2.3. Assume that (A1) (A6) hold and that jZj is su ciently small. Let ( ; u; ; ) be the solution to the Penrose Fife system (PF). a) If 0d = 0 (cf. (2.6)), then we have a positive constant C, independent of Z, such that b Z L2(0;T ;L2( ))\C([0;T ];H1( ) ) + b uZ u L2(0;T ;L2( )) + b Z C([0;T ];L2( ))\L2(0;T ;H1( )) C jZj : (2.21) b) If 0d 6= 0 and R2, then (2.21) still holds. c) If 0d 6= 0 and R3, then (2.21) holds with jZj replaced by jZj 20 23 . Remark 2.5 (Numerical implementation). In a lot of physically relevant situations, see [PF90], the considered functions and are quadratic and has a quadratic lower bound, i.e. we have positive constants C 3 ; C 4 with (s) + C 3 C 4s2; 8 s 2 D( ): (2.22) In this situation, the scheme with 0 d(r; s) := 0(r); 0d(r; s) := 0(s); 8 r; s 2 R is the most promising one to perform numerical computations, because of the following properties of this scheme: The coupling between the two equations (2.4c) and (2.4d) is a linear one, since 0d( m; m 1) does not depend on m. Moreover, 0 d( m; m 1) is linear in m. Thus, a nite element discretization and a nonlinear Gauss Seidel scheme similar to the one in [Kle97a, Sec. 10] can be used to nd approximative solutions to (DZ). Corollary 2.1 allows us to calculate an upper bound for the time step size to ensure the existence of a unique solution. In two space dimensions, Theorem 2.3 yields a convergence linear with respect to the time step size, and in three dimensions the convergence in still nearly linear. Remark 2.6. If the regularity assumption for g in (A3) is weakened to g 2 L1 ( T ), all results of this work still holds, except the error estimates in Theorem 2.3. 3 Some properties of the approximation of the data and a special existence result To prepare the proof of the theorems and the corollary in the last section, some notations will be xed and some properties for the approximation of the data will be proved. Moreover, 7 the existence of a unique solution will be shown, under the additional condition that D( ) is bounded. In the sequel, we use the notation k kp for the Lp( ) norm, for all p 2 [1;1]. Moreover, k k2 will also be used for the (L2( ))2 resp. (L2( ))3 norm. 3.1 Properties of the data and their approximations In the following lemma it is shown that those approximations discussed in Remark 2.2 ful ll the condition (A6). Lemma 3.1. Assume that (A2) holds. Let ! 2 [0; 1] be given and de ne 0 d : R R! R by 0 d(r; s) = ! 0(r) + (1 !) 0(s); 8 r; s 2 R: (3.1) a) If 0d = 0 (cf. (2.6)), we have (A6) and 0 (r; s)(r s) = (r) (s); 8 r; s 2 R: (3.2) b) Assume, in addition, 2 C2(R) and 0d(r; s) = ! 0(r) + (1 ! ) 0(s); 8 r; s 2 R; (3.3) with some ! 2 [0; 1]. If we have positive constants C1; C2 such that C1 00(s) C2 for all s 2 D( ), the assumption (A6) holds. If ! = 0 and we have a positive constant C3 with 00(s) C3 for all s 2 D( ), the assumption (A6) is satis ed. If ! = 1 and we have a positive constant C4 with C4 00(s) for all s 2 D( ), the assumption (A6) holds. Proof. First, we consider part (a) of the lemma. Thanks to (2.6), we have (3.2) and 0 (r; s) = 1 R0 0(s+ (r s)) d : Hence, for 0d = 0 , we can use (3.1), Schwarz's inequality, and (A2), to show that (A6) is satis ed. This yields part (a) of the Lemma. To prove part (b) of the lemma, we need only to show that the last estimate in (A6), i.e. (2.5), is satis ed, since the remaining assumptions in (A6) follow by an argumentation similar to the one above. For r; s 2 D( ), applying Taylor's formula and (3.3) gives ; 2 D( ) such that 0d(r; s)(r s) + (r) (s) = 1 2 ( ! 00( ) + (1 ! ) 00( )) (r s)2: Now, we see immediately that (2.5) holds under the considered assumptions. 8 Lemma 3.2. Assume that (A3) holds. Then there exist positive constants C1; C2; : : : ; C6, such that, for all decompositions Z with (A5), the functions gm, m, and m de ned in (2.3) ful ll, for 1 m K, C1 kvk2H1( ) krvk22 + Z mv2 d C2 kvk2H1( ) ; 8 v 2 H1( ); mv 2 H 1 2 ( ); k mvkH 1 2 ( ) C3 kvkH1( ) ; 8 v 2 H1( ); c m a.e. in ; Z mv d + Z gmv dx C4 kvkH1( ) ; 8 v 2 H1( ); kgmk1 + k mkC1( ) + k mkL1( ) + k mkH 1 2 ( ) C5; and max 1 m K 1 m+1 m hm L1( ) + K 1 X m=1 hm m+1 m hm 2L2( ) C6; where the positive constants c ; c are speci ed in (A3). Proof. This lemma follows from (A3), (A5), the trace mapping from H1( ) to H 12 ( ), and the interpolation of H 12 ( ) by H1( ) and L2( ). 3.2 The existence proof for D( ) bounded Lemma 3.3. Assume that (A1) (A6) hold and that D( ) is bounded. Then there exists a solution to (DZ). Proof. From (2.4f), we get 0; u0; 0; 0. Now, we assume that m 1 2 L2( ); m 1 2 H2( ) for some m 2 f1; : : : ;Kg are given. To show that there exists a solution to the system in (DZ), i.e. to (2.4a) (2.4e), we will rst consider the discrete energy balance equation and the discrete equation for the order parameter separately. Afterwards, we will rewrite the system as a xed point problem and apply Schauder's xed point theorem. Lemma 3.4. For every 2 L1( ), there is a unique ~ u 2 H2( ) such that 0 < ~ u a.e. in ; 1~ u 2 L2( ); @~ u @n = m~ u m a.e. in ; (3.4) c0 ~ u hm ~ u = c0 m 1 hmgm + 0d( ; m 1) ( m 1) a.e. in : (3.5) Proof. Let 2 L1( ) be given. Thanks to (A6) and m 1 2 C( ), we have 0d( ; m 1) ( m 1) 2 L2( ): By translating the proof of [Bré71, Corollary 13], we see that the operator corresponding to (3.4) and the left hand side of (3.5) is maximal monotone. By showing that this operator is also coercive, we obtain that the operator is also surjective. The injectivity follows by estimating the di erence between two given solutions. Details can be found in [Kle97a, Lemma 5.1]. 9 Lemma 3.5. For every 2 L1( ), ~ u 2 L2( ) there exists a unique ~ such that ~ 2 H2( ); ~ 2 D( ) a.e. in ; @ ~ @n = 0 a.e. in ; (3.6) ~ m 1 hm " ~ + (~ ) 3 0 d( ; m 1) 0d ( ; m 1) ~ u a.e. in ; (3.7) ~ m 1 hm + " ~ + 0 d( ; m 1) 0d ( ; m 1) ~ u 2 L2( ): (3.8) Proof. By (A1) and (A3), we can rewrite (3.6) (3.8) as c hm ~ +B ~ 3 0 d( ; m 1) 0d ( ; m 1) ~ u+ hm m 1; (3.9) where B : L2( ) ! fW L2( )g is a nonlinear operator. Using [Bré71, Corollary 13], we see that this operator is maximal monotone. Details can be found in [Kle97a, (5.7) (5.8) and Lemma 5.5]. Because of (A6), (A3), 2 L1( ), m 1 2 H2( ) C( ), we see that the right hand side of (3.9) is in L2( ). Hence, [Bré71, Theorem 2] yields that there is a unique solution ~ to (3.6) (3.8). In this proof, Ci, for i 2 N, will always denote generic positive constants, independent of 2 M with M := n 2 L2( ) : 2 D( ) a.e. in o : (3.10) This is a closed and convex set. We have Lemma 3.6. The functions 0 d( ; m 1) and 0d( ; m 1) are Lipschitz continuous on D( ) and there is a positive constant C1 such that, for all 2 M, k 0d( ; m 1)k1 + k 0 d( ; m 1)k1 + k k1 + k m 1k1 C1: (3.11) Proof. Since D( ) is bounded and m 1 2 H2( ) C( ), (A6) yields that the assertions of this lemma hold. Combining Lemma 3.4 and Lemma 3.5, we see that for every 2 M there is a unique ~ u 2 H2( ) and a unique ( ) := ~ 2 H2( ) such that (3.4) (3.5) and (3.6) (3.8) hold. This de nes a mapping : M ! M and any xed point of leads to a solution to the system in (DZ), i.e. to (2.4a) (2.4e). Therefore, it is su cient to prove that has a xed point. We test (3.5) by hm~ u, apply Green's formula, Lemma 3.2, Hölder's inequality, (3.4), (3.11), and Young's inequality to conclude that C2 k~ uk2H1( ) c0 j j+ hm Z m~ ud + Z ( c0 m 1 hmgm + 0d( ; m 1) ( m 1)) ~ udx C3 + C2 2 k~ uk2H1( ) : (3.12) 10 Owing to (A1), we have ws 0 for all s 2 D( ), w 2 (s). Therefore, by testing (3.7) by ~ and applying (A3), Green's formula, (3.6), (3.11), Hölder's inequality, (3.12), and Young's inequality, we get C4 k~ k2H1( ) m 1 hm + 0 d( ; m 1) 0( ; m 1)~ u 2 k~ k2 C5 + C4 2 k~ k22 : Hence, we see that ~ 2 M1 with M1 :=  2 M : k k2H1( ) 2C5 C4 : Therefore, we observe that M1 is a nonempty, convex, compact set in L2( ) and, by construction, that mapsM1 into itself. Thanks to Lemma 3.7, is on M1 continuous. Now, Schauder's xed point theorem yields the existence of a xed point of in M1. Lemma 3.7. :M!M is L2( ) continuous. Proof. Let 1; 2 in M be arbitrary, and ~ 1 := ( 1); ~ 2 := ( 2); := 1 2; ~ := ~ 1 ~ 2: Combining (3.4) (3.5), (3.6) (3.8), and the de nition of , we nd ~ u1; ~ u2 2 H2( ), ~ 1; ~ 2 2 L2( ) such that ~ u1 > 0; ~ u2 > 0; ~ 1 2 (~ 1); ~ 2 2 (~ 2) a.e. in ; (3.13) c0 1 ~ u1 1 ~ u2 hm (~ u1 ~ u2) = 0d( 1; m 1)( 1 m 1) 0d( 2; m 1)( 2 m 1) a.e. in ; (3.14) ~ hm " ~ + ~ 1 ~ 2 = 0d( 1; m 1)~ u1 + 0d( 2; m 1)~ u2 + 0 d( 1; m 1) 0 d( 2; m 1) a.e. in ; (3.15) @ (~ u1 ~ u2) @n = m (~ u1 ~ u2) ; @ ~ @n = 0 a.e. in : (3.16) Testing (3.14) by ~ u := ~ u1 ~ u2, integrating by parts, and using (3.16), (3.13), Lemma 3.2, Hölder's inequality, Lemma 3.6, and Young's inequality, we deduce c0 ~ u p~ u1~ u2 22 + C6 k~ uk2H1( ) Z 0d( 1; m 1) + ( 0d( 1; m 1) 0d( 2; m 1)) ( 2 m 1) ~ udx C7 k k2 k~ uk2 C8 k k22 + C6 2 k~ uk22 : (3.17) 11 We test (3.15) by ~ and use (3.13), the monotonicity of , (A3), (3.16), and the generalized Hölder's inequality (see Lemma AP.2) to derive C9 k~ k2H1( ) k 0d( 1; m 1)k32 k~ uk6 k~ k6 + k 0d( 1; m 1) 0d( 2; m 1)k 3 2 k~ u2k6 k~ k6 + k 0 d( 1; m 1) 0 d( 2; m 1)k2 k~ k2 : Because of Lemma 3.6, (AP.1), (3.17), and (3.12), we see C9 k~ k2H1( ) C10 k k2 k~ kH1( ) : Hence, thanks to Young's inequality, we have shown that is L2( ) continuous. 4 Uniform estimates In this section, uniform estimates for the solutions to the time discrete scheme are derived. Assume that (A1) (A6) hold and that jZj h , where h and C 5 are positive constants such that (2.7) is satis ed. Let := @ and : R! [0;1] be either or the function de ned by (s) = ( (s); if jsj B; 1; otherwise; (4.1) for some B > k 0k1. In the light of (A1), we see that is a convex, lower semicontinuous function with 0 on R; 0 2 D( ); intD( ) 6= ;; 0 2 (0); jD( ) = jD( ) : (4.2) Now, a modi ed version of the time discrete scheme is considered, where in (DZ), i.e. in (2.4b), is replaced by . Let any solution to this scheme be given. In the sequel, Ci, for i 2 N, will always denote positive generic constants, independent of the decomposition Z, the considered choice of , and the solution itself. Remark 4.1. Recalling (2.4a), (2.4b), (2.4e), (2.4f), (A4), and the de nition of , we see that 0 < um = 1 m ; m 2 D( ) D( ); m 2 ( m) = @ ( m) a.e. in ; m 2 H2( ); @ m @n = 0 a.e. in ; 8 0 m K: (4.3) Applying (2.4c), Green's formula, and (2.4e), we deduce that Z c0 m m 1 hm + m m 1 hm v dx Z rum rv dx Z mumv d = Z gmv dx Z mv d ; 8 v 2 H1( ); 1 m K; (4.4) with 0 := ( 0); m := m 1 + 0d( m; m 1)( m m 1) a.e. in ; 8 1 m K: (4.5) 12 The following Lemmas use ideas from [HSZ96, SZ93, CS97, Hor93, HS, Lau93, Lau94, Kle97a] Lemma 4.1. a) There is a positive constant C1 such that k ( 0)k1 + k 0k2 + k ( 0)k1 + k 0k6 + k 0d( 0; 0)k2 + k 0 d( 0; 0)k2 + k 0kH2( ) + k 0kH1( )\L1( ) + ku0kH1( )\L1( ) + kln( 0)k1 C1: (4.6) b) Let 1 2 L2( ) be de ned by 0 1 h0 " 0 + 0 + 0 d( 0; 0) = 0d( 0; 0)u0 a.e. in ; (4.7) with h0 := jZj. We have a positive constant C2 such that p 0 1 h0 22 C2: (4.8) Proof. If = , we use the initial condition (2.4f), (A2), (A4), Sobolev's embedding Theorem, (A6), and (4.5) to show that (4.6) is satis ed. If 6= , in addition, (4.1) and B > k 0k1 are applied. Combining (4.7), (4.6), and (A3) leads to (4.8). Lemma 4.2. There are two positive constants C3; C4 such that max 0 m K k mk1 + kln( m)k1 + k mk2H1( ) + k ( m)k1 + K X m=1hm kumk2H1( ) + K X m=1hm m m 1 hm 22 + K X m=1 k m m 1k2H1( ) C3; (4.9) max 1 m K k 0 d( m; m 1)k2 C4: (4.10) Proof. Testing (2.4d) by ( m m 1), taking the sum from m = 1 to m = k, and using (A3), Green's formula, (4.3), (AP.5), (4.6), (4.2), (4.5), Schwarz's inequality, and Young's inequality, we deduce c 2 k X m=1 hm m m 1 hm 22 + " 2 kr kk22 + " 2 k X m=1 kr m r m 1k22 + k ( k)k1 C5 k X m=1 Z ( m m 1)um dx + 1 2c k X m=1 hm k 0 d( m; m 1)k22 : (4.11) Let := min 1 2C 1 ; c 6C 2T , with C 1 ; C 2 as in (A2) and (A6). For 1 m K, we insert v = hm hmum in (4.4), use (4.3) and that 1 s is the derivative of the convex function ln(s), take the sum from m = 1 to m = k, and apply Lemma 3.2, (4.6), and Young's inequality, to show that c0 Z ( ln( k)) dx + c0 k kk1 + C6 k X m=1hm kumk2H1( ) C7 + k X m=1Z ( m m 1)(um ) dx : (4.12) 13 Because of (4.5), (A6), (A2), (4.6), Young's inequality, and the de nition of , we have k X m=1Z ( m m 1) dx C8 + 1 2 Z ( k) dx + c 6 k X m=1 hm m m 1 hm 22 : Hence, by using Lemma AP.8 and adding (4.12) to (4.11), we deduce C9 k kk1 + c0 kln( k)k1 + C6 k X m=1 hm kumk2H1( ) + c 3 k X m=1hm m m 1 hm 22 + " 2 kr kk22 + " 2 k X m=1 kr ( m m 1)k22 + 12 k ( k)k1 C10 + 1 2c k X m=1 hm k 0 d( m; m 1)k22 : (4.13) Since (A6), (2.7), and jZj h yield k 0 d( m; m 1)k22 C 2 0@k ( m)k1 + k ( m 1)k1 + Z 1 dx1A ; 8 1 m K; (4.14) hk 2c k 0 d( k; k 1)k22 1 4 k ( k)k1 + C11 (hk k k 1k1 + 1) ; we obtain from (4.13), (A5), the discrete version of Gronwall's lemma, and (4.6) that (4.9) is satis ed. Therefore, (4.10) holds because of (4.14). Lemma 4.3. There exists a constant C12 such that max 0 m K m m 1 hm 22 + kumk2H1( )!+ K X m=1 hm m m 1 hm 2H1( ) + K X m=1 ( m m 1) m m 1 hm 1 + K X m=1 m m 1 hm m 1 m 2 hm 1 22 + K X m=1 hm um um 1 hmpumum 1 22 + K X m=1 kum um 1k2H1( ) C12; (4.15) with 1, h0 as in Lemma 4.1. Proof. Inserting v = (um um 1) in (4.4), taking the sum from m = 1 to m = k, and applying (4.3), (AP.5), (AP.4), Lemma 3.2, (4.9), (4.6), the generalized Hölder's inequality, hm 2hm 1, and Young's inequality, we deduce that co 2 k X m=1hm um um 1 hmpumum 1 22 + C13 kukk2H1( ) + C13 k X m=1 kum um 1k2H1( ) C14 + k X m=1Z m m 1 hm (um um 1) dx : (4.16) 14 For 2 m K, we test the di erence of (2.4d) for m and m 1 by m m 1 hm . By applying (A3), Green's formula, (4.3), the monotonicity of , (4.5), and (AP.5), we obtain that 1 2 p m m 1 hm 22 12 p m 1 m 2 hm 1 22 + c 2 m m 1 hm m 1 m 2 hm 1 22 + "hm r m m 1 hm 22 + ( m m 1) m m 1 hm 1 Z m m 1 hm um 0d;m 1 m m 1 hm um 1 dx + Z 0 d( m; m 1) 0 d;m 1 m m 1 hm dx ; (4.17) with 0d;m 1 := 0d( m 1; m 2); 0 d;m 1 := 0 d( m 1; m 2) a.e. in : (4.18) Testing the di erence of (2.4d) for m = 1 and (4.7) by 1 0 h1 and using the same argumentation as above, we deduce that (4.17) holds also for m = 1 with 0d;0 := 0d( 0; 0); 0 d;0 := 0 d( 0; 0) a.e. in : (4.19) Summing up (4.17) from m = 1 to m = k, adding the resulting estimate to (4.16), and using (A3), (4.9), and (4.8), we conclude that c 2 k k 1 hk 22 + c 2 k X m=1 m m 1 hm m 1 m 2 hm 1 22 + C15 k X m=1 hm m m 1 hm 2H1( ) + k X m=1 ( m m 1) m m 1 hm 1 + co 2 k X m=1 hm um um 1 hmpumum 1 22 + C13 kukk2H1( ) + C13 k X m=1 kum um 1k2H1( ) C16 + I1;k + I2;k; (4.20) with I1;k := k X m=1 Z 0d;m 1 m m 1 hm m m 1 hm um 1 dx ; (4.21) I2;k := k X m=1 Z 0 d( m; m 1) 0 d;m 1 m m 1 hm dx : (4.22) Using (4.5), the generalized Hölder's inequality, and Schwarz's inequality, we deduce that I1;k max 1 m k m m 1 hm 2 pI3;kvuut k X m=1 hm 1 kum 1k26; with I3;k := k X m=1 1 hm 1 0d;m 1 0d( m; m 1) 23 : (4.23) 15 Now, owing to (AP.1), (4.9), (4.6), and Young's inequality, we observe that I1;k = c 4 max 1 m k m m 1 hm 22 + C17I3;k: (4.24) Since 1 3 = 1 p1 + p6 holds for p1 := 6 2 p , we obtain, by (4.18), (4.19), (A6), the generalized Hölder's inequality, hm 2hm 1, (AP.1), and (4.9), that I3;k C18 k X m=2 h2m hm 1 m m 1 hm 2p1 + hm 1 m 1 m 2 hm 1 2p1! k pmk 6 p + pm 1 6p + pm 2 6 p + 1 2 + C19 h21 jZj 1 0 h1 2p1 k p1k 6 p + k p0k 6p + 1 2 C20 k X m=1hm m m 1 hm 2 6 2 p : Because of p < 1, we can use the Gagliardo Nirenberg inequality (see Lemma AP.5) and Young's inequality to deduce C17I3;k C15 4 k X m=1 hm m m 1 hm 2H1( ) + C21 k X m=1 hm m m 1 hm 22 : (4.25) De ning q1 := 12 6 q , we have 1 = 1 q1 + q6 + 1 q1 . It follows from (4.22), (4.18), (4.19), (A6), and the generalized Hölder's inequality that I2;k C22 k X m=2 hm m m 1 hm q1 + hm 1 m 1 m 2 hm 1 q1! k qmk6 q + qm 1 6 q + qm 2 6q + 1 m m 1 hm q1 + C23h1 1 0 h1 q1 k q1k 6 q + k q0k 6 q + 1 1 0 h1 q1 : Using (AP.1), (4.9), Young's inequality, (A5), the Gagliardo Nirenberg inequality, and q < 4, we obtain that I2;k C15 4 k X m=1 hm m m 1 hm 2H1( ) + C24 k X m=1 hm m m 1 hm 22 : (4.26) 16 Combining (4.20), (4.24) (4.26), and (4.9), we conclude that c 2 k k 1 hk 22 + c 2 k X m=1 m m 1 hm m 1 m 2 hm 1 22 + C15 2 k X m=1hm m m 1 hm 2H1( ) + k X m=1 ( m m 1) m m 1 hm 1 + co 2 k X m=1hm um um 1 hmpumum 1 22 + C13 kukk2H1( ) + C13 k X m=1 kum um 1k2H1( ) C25 + c 4 max 1 m k m m 1 hm 22 : (4.27) By taking the maximum from m = 1 to m = K, we see that (4.15) holds, because of (4.6). Lemma 4.4. There exists a positive constant C26 such that max 1 m K k mk2 + max 0 m K k mkH2( ) C26: (4.28) Proof. To test (2.4d) by m, we use the Yosida approximation 1 n of , which is, see [Bré71, p. 104], a nondecreasing, Lipschitz continuous function on R. The construction of the Yosida approximation and 0 2 (0) yield that 0 = 1 n (0), for all n 2 N. Since 1 n is the derivative of a convex function on R, we can apply [Bré71, Corollary 13] to show that for every n 2 N there exists a unique m;n 2 H2( ) and a unique m;n 2 L2( ) such that m;n " m;n + m;n = fm a.e. in ; (4.29) m;n = 1 n ( m;n) a.e. in ; @ m;n @n = 0 a.e. in ; (4.30) with fm 2 L2( ) de ned by fm := 0d( m; m 1)um + 0 d( m; m 1) + m hm ( m m 1) : (4.31) Since 1 n is globally Lipschitz continuous on R and, by Sobolev's embedding Theorem, m;n 2 H1;6( ), we obtain, by [MM79, Theorem 1], that 1 n ( m;n) = m;n 2 H1( ) and, by [MM72, Lemma 2.1 and Remark 2.1], that for this function the generalized chain rules holds. Therefore, since 1 n is nondecreasing on R, we see that Z r( m;n) r m;n dx = Z 1 n 0 ( m;n) (r m;n)2 dx 0: (4.32) 17 We test (4.29) by m;n, and use Green's formula, (4.32), (4.30), and Young's inequality, to derive k m;nk22 Z (fm m;n) m;n dx 1 2 kfm m;nk22 + 1 2 k m;nk22 : (4.33) Testing (4.29) by m;n and using Green's formula, (4.30), 0 2 1 n (0), the monotonicity of 1 n , and Young's inequality, we observe that the sequence ( m;n)n2N is bounded in H1( ). Hence, the sequence ( m;n)n2N is bounded in L2( ), because of (4.33). Comparing the terms in (4.29), using (4.30) and Lemma AP.4, we see that ( m;n)n2N is also bounded in H2( ). Thus, there is a  2 H2( ) and a  2 L2( ) such that, for some subsequences, m;ni !  weakly in H2( ); strongly in H1( ); (4.34) m;ni !  weakly in L2( ): (4.35) Now, a passage to the limit in (4.29) (4.30) and using [Bar76, Cha. II Prob. 1.1(iv)] lead to 2 D( );  2 ( );  " +  = fm a.e. in ; @ @n = 0 a.e. in : Since (4.31), (4.3), and (2.4d), yield that ( m; m) is also a solution to this system, which has, by [Bré71, Corollary 13], a unique solution, we see that m =  and m = . Now, (4.33) (4.35), (4.31), (4.10), (A3), and (4.15) lead to 1 2 k mk22 12 kfm mk22 C27 + C28 k 0d( m; m 1)umk22 : (4.36) Applying (A6), the generalized Hölder's inequality, p < 1, (AP.1), (4.9), and (4.15), we obtain k 0d( m; m 1)umk2 j 0d(0; 0)j kumk2 + C29 (k m 0k6 + k m 1 0k6) k pmk6 + pm 1 6 + 1 kumk6 C30: (4.37) Comparing the terms in (2.4d), and using (A3), (4.15), (4.10), (4.36), and (4.37), we see that k" mk2 = m m 1 hm + m 0 d( m; m 1) + 0d ( m; m 1) um 2 C31: Now, using Lemma AP.4, (4.9), and (4.3), we conclude k mkH2( ) C32: Combining this with (4.36), (4.37), and (4.6), we see that (4.28) is satis ed. Lemma 4.5. There exists a positive constant C33 such that max 1 m K k 0d( m; m 1)k1 + m m 1 hm 2 + m m 1 hm H1( ) ! + K X m=1 hm m m 1 hm 2H1( ) + max 0 m K k mkH1( ) C33: (4.38) 18 Proof. By looking at the terms in (4.4) and using (4.15) and Lemma 3.2, we see that max 1 m K c0 m m 1 hm + m m 1 hm H1( ) C34: (4.39) Thanks to (4.28), Sobolev's embedding Theorem, and (A6), we have max 0 m K k mkH1;6( ) + max 1 m K k 0d( m; m 1)k1 C35: (4.40) Combining this with (A6), and [MM79, Theorem 1], we see that 0d( m; m 1) 2 H1;6( ) and max 1 m K kr 0d( m; m 1)k6 C36: Therefore, owing to (4.5), Young's inequality, the generalized Hölder's inequality, (4.40), (4.15), and Sobolev's embedding Theorem, we have max 1 m K m m 1 hm 22 + K X m=1hm r m m 1 hm 22 max 1 m K k 0d( m; m 1)k1 m m 1 hm 2 2 + 2 K X m=1 hm k 0d( m; m 1)k21 r m m 1 hm 22 + 2 K X m=1 hm kr 0d( m; m 1)k26 m m 1 hm 23 C37: Combining this with (4.39) and (4.6), we see that (4.38) is satis ed. Lemma 4.6. We have m 2 H1( ) for 0 m K. Proof. We have 0 2 H1( ) by (2.4f) and (A4). For 1 m K with m 1 2 H1( ), we de ne the approximation m;n 2 H1( ) \ L1( ) for m by m;n := um + 1 n 1 a.e. in ; 8n 2 N: The Lebesgue dominated convergence theorem and m 2 L2( ) yield that m;n ! n!1 m strongly in L2( ): (4.41) By applying (4.4) with v = 3 m;n and using (4.3), Hölder's inequality, Lemma 3.2, (4.38), (AP.1), and Young's inequality, we see that this sequence is bounded in H1( ). Combining this with (4.41), we conclude that m 2 H1( ). 19 Lemma 4.7. There exists a constant C38 such that max 0 m K k mk2 C38: (4.42) Proof. We multiply (2.4c) by hm and use (4.5). Summing up the resulting equation for m = 1 to m = i, we nd c0 i + i + i X m=1 hm um = c0 0 + 0 + i X m=1 hmgm a.e. in : (4.43) We test (4.43) by hi ui, take the sum from i = 1 to i = k, and apply Green's formula, (2.4e), (4.3), m 2 H1( ), (AP.3), (AP.2), Lemma 3.2, and Schwarz's inequality, to derive c0 k Xi=1 hi rui ui 22 + 2 k Xi=1 hi ui 22 + 2 k Xi=1 h2i k uik22 + c0c k Xi=1 hi k ikL1( ) C39 + Z c0 0 + 0 + k Xi=1 higi! k Xi=1 hi ui! dx k 1 Xi=1 hi+1 Z gi+1 i X m=1 hm um dx + k Xi=1 hi Z r i rui dx + k Xi=1 hi 1 Z i ( iui i) d : Now, by utilizing Young's inequality, (4.6), Lemma 3.2, (4.15), (4.38), and hm 2hm 1, we derive c0 k X m=1 hm rum um 22 + 4 k X m=1 hm um 22 + 2 k X m=1h2m k umk22 C40 + C41 k 1 X m=1 hm m Xi=1 hi ui 22 : (4.44) By applying the discrete version of Gronwall's lemma, we get a uniform upper bound for the left hand side of (4.44). Looking at the terms in (4.43) and applying (4.38), (4.6), and Lemma 3.2, we see that (4.42) holds. Lemma 4.8. There are two positive constant C42; C43 such that max 0 m K kumkC( ) + k mkC( )\H1( ) + K X m=1 hm um um 1 hm 22 + m m 1 hm 22! C42; (4.45) K X m=1 hm kumk2H2( ) C43: (4.46) 20 Proof. We deduce, by Lemma 3.2, (4.38), and (AP.1), that k X m=1hm gm m m 1 hm 26 C44: Thanks to (4.3) (4.6), (4.15), (4.38), (4.42), and Lemma 3.2, we can apply Moser's technique as in [Kle97a, Lemma 6.11 and 6.12, for " > 0 xed], and derive, by using (4.6), that max 0 m K kumkL1( ) + k mkL1( ) C45: Combining this with um 2 H2( ) C( ), (4.3), (4.15), and Hölder's inequality, we see that (4.45) holds. Now, by looking at the terms in (2.4c), and using (4.5), (4.38), and Lemma 3.2, we see that K X m=1 hm k umk22 C46: Now, Lemma AP.4 yields that (4.46) is satis ed, because of (2.4e), Lemma 3.2, and (4.15). Lemma 4.9. We have k ( k) kk 5 3 C47 jZj ; 8 1 k K: (4.47) If at least one of the assumptions R2 or 0d = 0 is satis ed, we have k ( k) kk2 C48 jZj ; 8 1 k K: (4.48) Proof. Applying (4.5), (A2), the mean value theorem, (A6), (4.28), and Sobolev's embedding Theorem, we deduce j ( k) kj C49 k X m=1 h2m m m 1 hm 2 a.e. in : (4.49) Hence, recalling Hölder's inequality, the Gagliardo Nirenberg inequality, and (4.15), we conclude k ( k) kk 5 353 C50 jZj 5 3 k X m=1hm m m 1 hm 10 3 10 3 C51 jZj 5 3 k X m=1hm m m 1 hm 2H1( ) m m 1 hm 4 3 2 C52 jZj 53 : Thus, we have shown (4.47). We use (4.49) and Hölder's inequality to show that k ( k) kk22 C53 jZj2 k X m=1hm m m 1 hm 44 : Therefore, if R2, recalling the Gagliardo Nirenberg inequality and (4.15) leads to (4.48). If 0d = 0 , then (3.2) and (4.5) yield that ( k) = k. Hence, (4.48) is satis ed. 21 5 Proof of Theorem 2.1 and Corollary 2.1 We assume that (A1) (A6) hold. In the framework of Theorem 2.1, we obtain from (A6) that we have positive constants h and C 5 such that (2.7) is satis ed. We assume that jZj h . In the framework of Corollary 2.1, it is part of the assumptions that jZj h where h and C 5 are positive constants ful lling (2.7). Because of (A4) and Sobolev's embedding Theorem, we see that k 0k1 is nite. For any B > k 0k1, we can consider as in (4.1), , and the corresponding modi ed version of the time discrete scheme as in the last section. Lemma 3.3 yields that there exists a solution B m; uBm; Bm; B m Km=0 to this modi ed version of the scheme. Since the assumptions used in the last section are satis ed, the estimates derived therein hold for this solution. Now, because of (4.28) and Sobolev's embedding Theorem, there is some positive constant C 0, independent of B, such that max 0 m K Bm C( ) C 0: (5.1) Now, we consider B := C 0 + k 0k1 + 2. Thanks to (4.1), = @ , and = @ , we have j[ C0 1;C0+1] = j[ C0 1;C0+1] : This yields, by (4.3) and (5.1), that the solution to the modi ed version of scheme is also a solution to the unmodi ed version of the scheme (DZ). It remains to show the uniqueness of the solution. Assume that we have two solutions (1) m ; u(1) m ; (1) m ; (1) m Km=0 and (2) m ; u(2) m ; (2) m ; (2) m Km=0 to the scheme (DZ). Hence, the estimates in the last section are valid for both solutions. In the sequel, Ci, for i 2 N, will always denote positive generic constants, independent of the decomposition Z and the considered solutions. Thanks to (2.4f), we have (1) 0 = (2) 0 ; u(1) 0 = u(2) 0 ; (1) 0 = (2) 0 ; (1) 0 = (2) 0 a.e. on . To prove by induction that the two solutions coincide, we now assume that 1 m K is given such that (1) m 1 = (2) m 1; u(1) m 1 = u(2) m 1; (1) m 1 = (2) m 1 =: a.e. in : (5.2) Now, let um := u(1) m u(2) m and m := (1) m (2) m . Using (2.4b), (2.4c), (2.4e), Green's formula, and (5.2), we deduce (1) m (2) m = um u(1) m u(2) m a.e. in ; (5.3) 1 hm Z c0 um u(1) m u(2) m + 0d( (1) m ; ) (1) m 0d( (2) m ; ) (2) m v dx Z rum rv dx Z mumv dx = 0; 8 v 2 H1( ): 22 This yields for v = hmum, by Lemma 3.2, c0 um qu(1) m u(2) m 22 + hmC1 kumk2H1( ) Z 0d( (1) m ; ) mum dx + Z 0d( (1) m ; ) 0d( (2) m ; ) ( (2) m )um dx : (5.4) Recalling (2.4d) and (5.2), we have m hm " m + (1) m (2) m 0 d( (1) m ; ) + 0 d( (2) m ; ) = 0d (1) m ; um 0d (1) m ; 0d (2) m ; u(2) m a.e. in : (5.5) Testing this equation by m and using (A3), Green's formula, (2.4e), (2.4b), and the monotonicity of , and adding the resulting estimate to (5.4), we obtain, by (4.45), C2 kumk22 + hmC1 kumk2H1( ) + c hm k mk22 + " kr mk22 I1 + I2; (5.6) with I1 := Z 0d( (1) m ; ) 0d( (2) m ; ) ( (2) m )um u(2) m m dx ; (5.7) I2 := Z 0 d( (1) m ; ) 0 d( (2) m ; ) m dx : (5.8) Now, we consider the framework of Corollary 2.1 and Theorem 2.1 separately. If we are in the framework of Corollary 2.1, the uniqueness needs only to be shown under the additional assumption that (2.8) holds. Therefore, we have I1 = 0 and I2 c 2 jZj Z ( m)2 dx c 2hm k mk22 : Hence, (5.6), (5.3), and (5.5) yield that um = m = 0; (1) m = (2) m ; (1) m = (2) m a.e. in : (5.9) This nishes the proof of Corollary 2.1. Now, we consider the framework of Theorem 2.1. (A6), (4.28), and Sobolev's embedding Theorem yield that 0d( (1) m ; ) 0d( (2) m ; ) + 0 d( (1) m ; ) 0 d( (2) m ; ) C3 j mj a.e. in : Hence, by applying the generalized Hölder's inequality, (4.28), (4.45), and Young's inequality, we deduce I1 + I2 C3 k mk2 (2) m m 1 1 kumk2 + u(2) m 1 k mk2 + C3 k mk22 C2 2 kumk22 + C4 k mk22 : 23 Therefore, if we assume that jZj c 2C4 , we obtain I1+I2 C2 2 kumk2+ c 2hm k mk2 : Combining this with (5.6), (5.3), and (5.5), we see that (5.9) is satis ed. Since we have shown that the scheme has a unique solution, if jZj is su ciently small, Theorem 2.1 is proved. 6 Proof of Theorem 2.2 and Theorem 2.3 We assume that (A1) (A4) and (A6) hold. Thanks to (A6), we have positive constants h and C 5 such that (2.7) is satis ed. 6.1 Properties of the approximations In this section, we only consider decompositions Z with (A5) and jZj su ciently small. Hence, Theorem 2.1 yields that there exists a unique solution to the time discrete scheme (DZ). Let b Z; b uZ; b Z; Z be the corresponding approximations derived from the solution to (DZ) as in Remark 2.3. For ( m)Km=0 as in (4.5), we de ne the piecewise linear function b Z analogously to b Z. The piecewise constant functions Z, uZ, Z , Z, Z, gZ , Z are de ned analogously to Z, and Z 2 L1(0; T ;H2( )) is de ned by Z(t) = m 1; 8 t 2 (tm 1; tm); 1 m K: (6.1) Then, by the de nition of the approximations, (2.4a) (2.4f), and (4.5), we have b Z; b uZ;b Z 2 H1(0; T ;H1( )); uZ 2 L2(0; T ;H2( )); b uZ 2 L2(jZj ; T ;H2( )); (6.2a) b Z 2 H1(0; T ;H2( )); Z ; Z 2 L1(0; T ;H2( )); (6.2b) Z 2 L1(0; T ;L2( )); (6.2c) 0 < b uZ; 0 < uZ; Z = 1 uZ ; Z; b Z; Z 2 D( ); Z 2 Z a.e. in T ; (6.2d) c0b Z t + b Zt + uZ = gZ a.e. in T ; (6.2e) b Zt " Z + Z 0 d( Z; Z) = 0d( Z ; Z)uZ a.e. in T ; (6.2f) @uZ @n = ZuZ Z; @b Z @n = 0; @ Z @n = 0 a.e. in T ; (6.2g) b Z( ; 0) = 0; b uZ( ; 0) = u0; b Z( ; 0) = 0; b Z( ; 0) = ( 0) a.e. in : (6.2h) In the sequel, Ci, for i 2 N, will always denote positive generic constants, independent of the decomposition Z. We nd, from (4.15), (4.28), (4.38), (4.45), and (4.46): b Z W 1;1(0;T ;H1( ) )\H1(0;T ;L2( ))\C( T )\L1(0;T ;H1( )) + Z L1( T )\L1(0;T ;H1( )) + b uZ C([0;T ];H1( ))\H1(0;T ;L2( ))\C( T ) + b uZ L2(jZj;T ;H2( )) + uZ L1(0;T ;H1( ))\L1( T )\L2(0;T ;H2( )) C1; (6.3) 24 b Z W 1;1(0;T ;L2( ))\H1(0;T ;H1( ))\C([0;T ];H2( )) + Z L1(0;T ;H2( )) + Z L1(0;T ;H2( )) + Z L1(0;T ;L2( )) + b Z W 1;1(0;T ;L2( ))\H1(0;T ;H1( )) C2: (6.4) The di erence between the piecewise linear and the piecewise constant approximations can be estimated, by using (4.15), (A2), (4.28), Sobolev's embedding Theorem, (4.38), (4.45), and (4.47): b Z Z L2(0;T ;L2( ))\L1(0;T ;H1( ) ) + b uZ uZ L2(0;T ;L2( )) C3 jZj ; (6.5) b Z Z L1(0;T ;L2( ))\L2(0;T ;H1( )) + b Z Z L1(0;T ;L2( ))\L2(0;T ;H1( )) + (b Z) ( Z) L1(0;T ;L2( )) C4 jZj ; (6.6) b Z Z L1(0;T ;L2( ))\L2(0;T ;H1( )) + ( Z) Z L1(0;T ;L 5 3 ) C5 jZj ; (6.7) b uZ uZ L2(0;T ;H1( )) C6pjZj: (6.8) For the approximation of the data, we have, by (A3): Lemma 6.1. The functions gZ; Z ; Z ful ll gZ L1( T ) + Z L1(0;T ;C1( )) + Z L1( T )\L1(0;T ;H 12 ( )) C7; (6.9) g gZ L2(0;T ;L1( )) + Z L1( T ) + Z L2(0;T ;L2( )) C8 jZj : (6.10) Now, estimates similar to [NSV] are used to prove the following lemma, which is important to improve the order of the error estimate from pjZj to jZj. Lemma 6.2. We have a positive constant C9 such that s Z0 Z Z b Z dx dt C9 jZj2 ; 8 s 2 [0; T ]; (6.11) for all ; 2 L2(0; T ;L2( )) with 2 D( ); 2 ( ) a.e. in T : (6.12) Proof. From (6.12), (6.2d), and = @ , we get A1 := s Z0 Z Z b Z dx dt s Z0 Z ( Z) + Z Z b Z + (b Z) dx dt : For lZ : (0; T ]! [0; 1] de ned by lZ(t) = tm t hm ; 8 t 2 (tm 1; tm]; 1 m K; (6.13) holds b Z = 1 lZ Z + lZ Z = Z + lZ Z Z a.e. in T : 25 We apply the convexity of , to show that A1 s Z0 lZ Z ( Z) + ( Z) + Z Z Z dx dt : Since (6.2d) and = @ yield that the integrand is a.e. non negative, we see, by (6.13), (2.4b), (2.4f), (A4), and = @ , that A1 K X m=1 tm Z tm 1 tm t hm dt Z ( ( m) + ( m 1) + m ( m m 1)) dx 1 2 jZj2 K X m=1 ( m m 1) m m 1 hm 1 : Hence, (6.11) holds because of (4.15). 6.2 Error estimates Now, we estimate the di erence between the approximation and one exact solution. Here, ideas from [CS97, Col96, Kle97a, NSV] are used. Lemma 6.3. For every solution ( ; u; ; ) to the Penrose Fife system (PF) there are positive constants C10; C11 such that max 0 s T s Z0 u( ) uZ( ) d 2H1( ) + max 0 s T s Z0 ( ) u( ) uZ( ) d 2 + u uZ puuZ 2L2(0;T ;L2( )) + u uZ 2L2(0;T ;L 3 2 ( )) + Z 2L2(0;T ;L1( )) + r Z 2L2(0;T ;L2( )) + b Z 2L2(0;T ;H1( ))\L1(0;T ;L2( )) C10AZ + C11 jZj2 + jZj u uZ L2(0;T ;L2( )) (6.14) ! 0; as jZj ! 0; (6.15) with AZ := T Z0 (b Z) b Z u uZ 1 dt (6.16) C12 jZj u uZ 17 20 L2(0;T ;L2( )) : (6.17) Proof. The generic constants may depend on the solution to the Penrose Fife system. Thanks to (2.1a), (2.1b), Sobolev's embedding Theorem, and (A2), we have k kL1(0;T ;L2( )) + kukL1(0;T ;H1( )) + kukL2(0;T ;H2( )) + k kL1( T ) + k 0( )kL1( T ) C13: (6.18) 26 First, we work on the equation for and u. Integrating the di erence of (2.1e) and (6.2e) in time, and testing the corresponding equation by v, and using (2.1g), (2.1h), (6.2g), and (6.2h), we obtain for all v 2 H1( ), Z c0( (t) b Z(t)) + ( (t)) b Z(t) v dx t Z0 Z r u( ) uZ( ) rv dx d = Z t Z0 g( ) gZ( ) d v dx + t Z0 Z ( ) u( ) uZ( ) v d d + t Z0 Z ( ) Z( ) uZ( ) ( ) Z( ) v d d ; 8 t 2 (0; T ): (6.19) For a.e. t 2 (0; T ), this yields, with v = u(t) uZ(t) , by (2.1d) and (6.2d), Z c0 u uZ 2 uuZ Z b Z u uZ ! dx Z ( ) b Z u uZ dx = Z t Z0 g( ) gZ( ) d u uZ dx Z 0@ t Z0 ( ) u( ) uZ( ) d 1A u uZ d Z t Z0 ( ) Z( ) uZ( ) ( ) Z( ) d u uZ d Z t Z0 r u( ) uZ( ) d r u uZ dx =: A2 +A3 +A4 +A5: (6.20) Owing to (6.2d), (2.1d), the generalized Hölder's inequality, (AP.1), (6.3), and (6.18), we see that s Z0 u uZ 232 + Z 21 dt C14 s Z0 u uZ puuZ 22 dt : (6.21) We have, by Hölder's inequality, (6.10), and Young's inequality, A2 C15 g gZ L2(0;T ;L1( )) u uZ 32 C16 jZj2 + c0 4C14 u uZ 232 ; (6.22) 27 A3 = 12@t 1 p (t) t Z0 ( ) u( ) uZ( ) d 2L2( ) Z t(t) 2( (t))2 0@ t Z0 ( ) u( ) uZ( ) d 1A2 d ; (6.23) A5 = 2@t t Z0 r u( ) uZ( ) d 22 : (6.24) By integrating (6.20) from 0 to s and using (6.16), (6.21) (6.23), we obtain c0 2C14 s Z0 12 u uZ 232 + Z 21 dt + c0 2 s Z0 u uZ puuZ 22 dt + 2 r s Z0 u( ) uZ( ) d 22 + 12 1 p (s) s Z0 ( ) u( ) uZ( ) d 2L2( ) s Z0 Z ( ) (b Z) u uZ dx dt +AZ + c0 s Z0 Z Z b Z u uZ dx dt + s Z0 Z t Z0 ( ) Z( ) ( ) Z( ) uZ( ) d u uZ d dt + TC16 jZj2 s Z0 Z t(t) 2( (t))2 0@ t Z0 ( ) u( ) uZ( ) d 1A2 d dt =: A6 +AZ +A7 +A8 + TC16 jZj2 +A9: (6.25) Applying Poincaré's inequality and Hölder's inequality, we conclude that C17 s Z0 u( ) uZ( ) d 2H1( ) 2 r s Z0 u( ) uZ( ) d 22 + c0 8C14 s Z0 u( ) uZ( ) 232 d : (6.26) Using Hölder's inequality, (A2), (6.18), (6.4), Sobolev's embedding Theorem, and (6.5), we deriveA6 +A7 C18 s Z0 b Z u uZ 1 dt + C19 jZj u uZ L2(0;T ;L2( )) : (6.27) 28 Partial integration with respect to time and Hölder's inequality results in A8 0B@ s Z0 Z dt L2( ) + s Z0 Z L1( ) dt uZ L1(0;T ;L2( ))1CA s Z0 u( ) uZ( ) d L2( ) + s Z0 Z L2( ) + Z L1( ) uZ L2( ) t Z0 u( ) uZ( ) d L2( ) dt : Because of the trace theorem, (6.3), (6.10), and Young's inequality, we observe A8 C17 2 s Z0 u( ) uZ( ) d 2H1( ) + 12 s Z0 t Z0 u( ) uZ( ) d 2H1( ) dt + C20 jZj2 : (6.28) In the light of Hölder's inequality and (A3), we see A9 C21 s Z0 1 p (t) t Z0 ( ) u( ) uZ( ) d 2L2( ) dt : (6.29) Hence, we get, by using Hölder's inequality, (6.25) (6.29), and Young's inequality, C17 2 s Z0 u( ) uZ( ) d 2H1( ) + c0 2C14 s Z0 1 4 u uZ 232 + Z 21 dt + c0 2 s Z0 u uZ puuZ 22 dt + 12 1 p (s) s Z0 ( ) u( ) uZ( ) d 2L2( ) C18 s Z0 b Z u uZ 1 dt +AZ + C19 jZj u uZ L2(0;T ;L2( )) + 12 s Z0 t Z0 u( ) uZ( ) d 2H1( ) dt + (C20 + TC16) jZj2 + C21 s Z0 1 p (t) t Z0 ( ) u( ) uZ( ) d 2L2( ) dt : (6.30) 29 Now, estimates for will be derived. Subtracting (6.2f) from (2.1f), we obtain that ( t b Zt ) " Z + Z 0( ) + 0 d( Z ; Z) = 0( )u+ 0d( Z ; Z)uZ a.e. in T : (6.31) Testing this with b Z and recalling (A3), (2.1g), and (6.2g), we end up with 1 2@t p b Z 22 + "Z r Z r b Z dx + Z Z b Z dx Z 0( ) 0 d( Z; Z) b Z dx Z 0( )u 0d( Z; Z)uZ b Z dx =: A10 +A11: (6.32) We have "Z r Z r b Z dx =" 2 r Z 22 + " 2 r b Z 22 " 2 r Z b Z 22 : (6.33) Using (6.32), (A6), (A2), (6.18), (6.4), Sobolev's embedding Theorem, Hölder's inequality, (6.6), and Young's inequality, we conclude A10 =Z 0( ) 0(b Z) b Z dx + Z 0 d(b Z; b Z) 0 d( Z ; Z) b Z dx C22 b Z 22 + C23 jZj2 a.e. in (0; T ): (6.34) In the light of (6.32), (A6), the generalized Hölder's inequality, (A2), (6.18), (6.3), (6.4), Sobolev's embedding Theorem, (6.6), and Young's inequality, we see that A11 = Z 0( ) u uZ + 0( ) 0(b Z) uZ b Z dx Z 0d(b Z; b Z) 0d( Z; Z) uZ b Z dx C24 u uZ b Z 1 + C25 b Z 22 + C26 jZj2 : (6.35) 30 Combining (6.32) (6.35), integrating in time, using (A3), (2.1h), (6.2h), (6.11), (6.6), and adding the resulting estimate to (6.30), we get C17 2 s Z0 u( ) uZ( ) d 2H1( ) + C27 s Z0 u uZ 232 + Z 2L1( ) dt + c0 2 s Z0 u uZ puuZ 22 dt + 1 2 1 p (s) s Z0 ( ) u( ) uZ( ) d 2L2( ) + c 2 (s) b Z(s) 22 + " 2 s Z0 r Z 22 dt + " 2 s Z0 b Z 2H1( ) dt A12 +AZ + C19 jZj u uZ L2(0;T ;L2( )) + C28 jZj2 + 12 s Z0 t Z0 u( ) uZ( ) d 2H1( ) dt + " 2 + C22 + C25 s Z0 b Z 22 dt + C21 s Z0 1 p (t) t Z0 ( ) u( ) uZ( ) d 2L2( ) dt (6.36) with A12 := (C18 + C24) s Z0 b Z u uZ 1 dt : Using Hölder's inequality, Young's inequality, and the Gagliardo Nirenberg inequality, we obtain A12 C27 2 s Z0 u uZ 232 dt + " 4 s Z0 b Z 2H1( ) dt + C29 s Z0 b Z 22 dt : Hence, (6.36), Gronwall's lemma, and (A3) yield that (6.14) is satis ed. By applying (6.16), Hölder's inequality, (6.6), (6.7), and the Gagliardo Nirenberg inequality, we get AZ T Z0 (b Z) b Z 5 3 u uZ 52 dt C30 jZj T Z0 u uZ 3 20 H2( ) u uZ 17 20 2 dt : Hence, using Hölder's inequality, (6.18), and (6.3), we deduce that (6.17) and (6.15) are satis ed. 6.3 Proof of Theorem 2.2 Proof. Thanks to the estimates (6.3), (6.4), Sobolev's embedding Theorem, and compactness (see, e.g., [Zei90, Prop. 23.7, 23.19, Prob. 23.12]), we get ( ; u; ; ; ) ful lling (2.1b) (2.1c), 31 (2.9) (2.11), and 2 H1(0; T ;L2( )); u 2 L1(0; T ;H1( )); 2 W 1;1(0; T ;L2( )): such that we have, for some subsequence with jZj ! 0, the convergences (2.12) (2.20), and b Z ! weakly star in W 1;1(0; T ;L2( )): (6.37) We obtain the convergences (2.12) (2.20) for the whole sequence, if we can show that ( ; u; ; ) is the unique solution to the Penrose Fife system (PF). Hence, we need only to prove this, to nish the proof of Theorem 2.2. Thanks to the convergences for b Z in (2.18), (6.4), the Aubin compactness lemma (see, e.g., [Lio69, p. 58]), and (6.6), we also get b Z ! ; Z ! ; Z ! strongly in L2(0; T ;L2( )): (6.38) Hence, after possibly extracting a further subsequence, we have Z ! ; Z ! a.e. in T : This yields, thanks to (A2), (A6), (6.4), and the Lebesgue dominated convergence theorem, that ( Z) ! ( ); 0d( Z ; Z) ! 0( ); 0 d( Z; Z) ! 0( ) strongly in L2( T ): (6.39) Thus, (6.37), (6.6), and (6.7) yield that = ( ) a.e. on T . Hence, using (2.12) (2.20), (6.37) (6.39), and (6.3) (6.10), we can pass to the limit in (6.2a) (6.2h) and obtain that ( ; u; ; ) is a solution to the Penrose Fife system (PF). Details can be found in [Kle97a, Sec. 8]. It remains to show that this solution is unique. Let ( ; u ; ; ) be any solution to the Penrose Fife system (PF). Since we can apply Lemma 6.3 for this solution, using (6.15) and the convergences (2.12) (2.19) yields that = ; u = u; = a.e. in T : Comparing the terms in (2.1f), we see that the two solutions coincide. 6.4 Proof of Theorem 2.3 Proof. Thanks to (2.1d), (6.2d), Hölder's inequality, (2.9), (2.10), and (6.3), we have u uZ 2L2(0;T ;L2( )) + Z 2L2(0;T ;L2( )) C31 u uZ puuZ L2(0;T ;L2( )) : Moreover, we have b Z 2 C([0; T ];L2( )), because of (6.2b) and (2.1b). Hence, we obtain from (6.14) and Young's inequality that A13 := max 0 s T s Z0 u( ) uZ( ) d 2H1( ) + max 0 s T s Z0 ( ) u( ) uZ( ) d 2 + 1 2C31 u uZ 2L2(0;T ;L2( )) + 1 C31 Z 2L2(0;T ;L2( )) + b Z 2L2(0;T ;H1( ))\C([0;T ];L2( )) C10AZ + C32 jZj2 : (6.40) 32 Therefore, by comparing the terms in (6.19), and using (6.10) and (6.3), we get c0 b Z + ( ) b Z 2L1(0;T ;H1( ) ) C33 A13 + jZj2 : (6.41) Now, (A2), (6.18), (6.4), (6.6), (6.7), L 53 ( ) H1( ) , (6.2a), and (2.1a) yield that b Z 2C([0;T ];H1( ) ) C34 A13 + jZj2 : (6.42) Thanks to (6.17) and Young's inequality, we deduce C10AZ 1 4C31 u uZ 2L2(0;T ;L2( )) + C35 jZj 40 23 : Hence, (6.40), (6.42), and (6.5) yield that (2.21) holds with jZj replaced by jZj 20 23 . If we assume that at least one of the assumptions R2 or 0d = 0 is satis ed, applying (6.16), Schwarz's inequality, (4.48), (6.7), (6.6), and Young's inequality leads to C10AZ C36 jZj u uZ L2(0;T ;L2( )) C37 jZj2 + 1 4C31 u uZ L2(0;T ;L2( )) : Combining this with (6.40), (6.42), and (6.5), we see that (2.21) is satis ed. A Appendix For convenience, we list some inequalities and equalities used throughout this paper. Lemma AP.1 (Young's inequality). For a; b 2 R, > 0, p > 1, q := p p 1 , it holds ab 1p jajp + 1 q jbjq ; ab 1p (p 1) jajp + 1 q jbjq ; jajps jbjp(1 s) s 1 s s 1 s jajp + jbjp ; 8 0 < s < 1: Lemma AP.2 (Generalized Hölder's inequality). For a bounded, open domain RN with N 2 N, p; p1; p2; p3 2 [1;1], f1 2 Lp1( ), f2 2 Lp2( ), and f3 2 Lp3( ) such that 1 p1 + 1 p2 + 1 p3 = 1 p ; we have f1 f2 f3 2 Lp( ) and kf1 f2 f3kLp( ) kf1kLp1( ) kf2kLp2( ) kf3kLp3 ( ) : Thanks to Sobolev's embedding Theorem, we have Lemma AP.3. For a bounded, open domain RN with N 2 f2; 3g and Lipschitz boundary, there is a positive constant C such that kvpkL 6 p ( ) = kvkpL6( ) Cp kvkpH1( ) ; 8 v 2 H1( ); p 2 (0; 6]: (AP.1) 33 The following classical elliptic estimate can be found in [Ama93, Remark 9.3 d].Lemma AP.4. For a bounded, open domain with @ smooth there is a positive constantC such thatkvk2H2( ) C kvk2L2( ) + @v@n2H 12 ( ) +kvk2L2( )! ; 8 v 2 H2( ):In particular, for all v 2 H2( ) with @v@n = 0 a.e. on ,kvk2H2( ) C kvk2L2( ) +kvk2L2( ) :The following version of the Gagliardo Nirenberg inequality is a special case of those con-sidered in [Zhe95, Th. 1.1.4ii]Lemma AP.5. LetRN with N 2 f2; 3g be a bounded domain with @ smooth. Let2 < p < 6 be given and a := 32 3p ; Then there is a positive constant C such thatkukLp( ) CkukaH1( ) kuk1 aL2( ) ; kukLp( ) C kuka2H2( ) kuk1 a2L2( ) :If R2, then the rst estimate is also satis ed for a = 12p .Elementary calculations lead toLemma AP.6. For n 2 N, a0; a1; : : : ; an, b0; b1; : : : ; bn 2 R, we havenXi=1 ai iXj=1 bj =nXi=1 ai! nXi=1 bi! n 1Xj=1 bj+1 jXi=1 ai;(AP.2)nXi=1 ai iXj=1 aj =12 nXi=1 ai!2 + 12 nXi=1 a2i ;(AP.3)nXi=1 ai(bi bi 1) = anbn a1b0 n 1Xi=1 (ai+1 ai) bi:(AP.4)Lemma AP.7. Let H be a Hilbert space with scalar-product h ; iH and norm k kH. Thenwe haveha; a biH = 12 kak2H 12 kbk2H +12 kabk2H ; 8 a; b 2 H:(AP.5)The next lemma follows form elementary analysis.Lemma AP.8. Let a; b > 0 be given. Then there exists a constant C > 0, such thata2s+ b jln sj as b ln s+ C; 8 s > 0:34 AcknowledgementsI wish to thank the European Science Foundation programme on Mathematical Treatment ofFree Boundary Problems for supporting this research with a fellowship held at the Universityof Pavia. My thanks are due to Prof. Pierluigi Colli and Dr. Giuseppe Savaré for variousfruitful discussions.References[Ama93] H. Amann. Nonhomogeneous linear and quasilinear elliptic and parabolic boundaryvalue problems. In H.-J. 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