convergence to equilibrium of solutions of the backward euler scheme ...

CONVERGENCE TO EQUILIBRIUM OF SOLUTIONS OF THE BACKWARD EULER SCHEME FOR ASYMPTOTICALLY AUTONOMOUS SECOND-ORDER GRADIENT-LIKE SYSTEMS MAURIZIO GRASSELLI AND MORGAN PIERRE Dedicated to Michel Pierre Abstract. Following a result of Chill and Jendoubi in the continuous case, we study the asymptotic behavior of sequences (U n )n in Rd which satisfy the following backward Euler scheme: ¡ n+1 ¢ U − 2U n + U n−1 U n+1 − U n ε + + ∇F (U n+1 ) = Gn+1 , n ≥ 0, 2 ∆t ∆t where ∆t > 0 is the time step, ε ≥ 0, (Gn+1 )n is a sequence in Rd which con1,1 verges to 0 in a suitable way, and F ∈ Cloc (Rd , R) is a function which satisfies a L Ã ojasiewicz inequality. We prove that the above scheme is Lyapunov stable and that any bounded sequence (U n )n which complies with it converges to a critical point of F as n tends to ∞. We also obtain convergence rates. We assume that F is semiconvex for some constant cF ≥ 0 and that 1/∆t < cF /2; in the case ε = 0, these last two assumptions can be dropped off if the scheme is defined by a minimization algorithm. Applications to space and time discretizations of the damped wave equation and of the modified Swift-Hohenberg equation are given.

Keywords: L Ã ojasiewicz inequality, gradient-like systems, backward Euler scheme, proximal method. AMS Classification 65L20, 65L12, 65M12, 65M60. 1. Introduction In this paper, we are concerned with the asymptotic behavior, as time goes to infinity, of the solution of the backward Euler scheme applied to asymptotically autonomous second-order gradient-like systems with analytic nonlinearities. Our model, which also includes the first order gradient case, is the following: εU 00 (t) + U 0 (t) + ∇F (U (t)) = G(t) t ≥ 0,

(1.1)

1,1 (Rd , R), and where ε ≥ 0, G ∈ L1loc (R+ , Rd ), U = (u1 , . . . , ud )t , F ∈ Cloc µ ¶ ∂F ∂F ∂F t ∇F = , ,..., . ∂u1 ∂u2 ∂ud

If F is real analytic and G satisfies Z 1+δ sup t t∈R+

∞

kG(s)k2 ds < ∞,

(1.2)

t

for some positive constant δ, Chill and Jendoubi [10] showed that any bounded solution of (1.1) converges to a critical point of F as t tends to ∞. Notice that assumption (1.2), roughly speaking, says that G tends to 0 fast enough as t → ∞, and it is obviously satisfied in the autonomous case G ≡ 0. If d ≥ 2 and F is only 1

2

MAURIZIO GRASSELLI AND MORGAN PIERRE

assumed C ∞ , convergence to equilibrium may fail: counterexamples can be found in [1, 22] (see also references therein) in the case of the gradient system ε = 0, G ≡ 0; counterexamples of the same type are expected in the case ε > 0 (if d = 1 and G ≡ 0, convergence to equilibrium holds true, even if F is not analytic; cf., e.g., [18]). The result of Chill and Jendoubi [10] also includes infinite dimensional cases, such as the asymptotically autonomous semilinear damped wave equation or the asymptotically autonomous Cahn-Hilliard equation. Rates of convergence to equilibrium for second-order evolution equations and their optimality were discussed in [5, 19]. Other related asymptotically autonomous systems were analyzed in [16, 17]. The results in [10] extend previous ones on the autonomous case for second-order gradientlike systems [20, 21, 25] and results on nonautonomous gradient flows [23]; they are based on the celebrated L Ã ojasiewicz-Simon gradient inequality for analytic functions [24, 26, 27, 31]. In fact, convergence to equilibrium is a general feature of systems or partial differential equations which possess a Lyapunov functional satisfying an angle condition and a L Ã ojasiewicz-Simon inequality (see [4, 11, 22] and references therein for an overview). The aim of this paper is to adapt the result of Chill and Jendoubi to the following time discrete version of (1.1): ¡ n+1 ¢ U − 2U n + U n−1 U n+1 − U n + ∇F (U n+1 ) = Gn+1 , n ≥ 0, (1.3) + ε ∆t2 ∆t where ∆t > 0 is the (fixed) time step, (Gn+1 )n∈N is a given sequence in Rd which converges to 0 as n tends to ∞, and (U n )n∈N is a sequence in Rd . The scheme (1.3) can be obtained by applying the backward Euler scheme to the first order system equivalent to (1.1), and by letting Z tn+1 1 n+1 G := G(t)dt with tj = j∆t, j = 0, 1, 2, . . . (1.4) ∆t tn Although the scheme (1.3) has order O(∆t) only, it has very interesting properties with respect to asymptotic behavior. Consider for instance the well-known case where ε = 0 and G ≡ 0, in which case the scheme (1.3) reduces to U n+1 − U n + ∇F (U n+1 ) = 0, ∀n ≥ 0. ∆t This is the backward Euler scheme for the gradient flow U 0 (t) + ∇F (U (t)) = 0 t ≥ 0.

(1.5)

(1.6)

In this case, the backward Euler scheme can more efficiently be written as an optimization algorithm (known as proximal algorithm [7]) as follows: let U 0 ∈ Rd and for n = 0, 1, 2, . . . , let ¾ ½ kW − U n k2 d n+1 + F (W ) : W ∈ R n ≥ 0, (1.7) U ∈ argmin 2∆t where k·k denotes the Euclidean norm on Rd . Existence of a sequence (not necessarily uniquely) defined by (1.7) is guaranteed for instance if inf Rd F > −∞. Notice that any sequence (U n )n≥0 which is defined by (1.7) satisfies (1.5) and the following Lyapunov stability estimate: kU n+1 − U n k2 + F (U n+1 ) ≤ F (U n ), 2∆t

∀n ≥ 0.

(1.8)

BACKWARD EULER SCHEME FOR SECOND-ORDER GRADIENT-LIKE SYSTEMS

3

Such stability properties of the backward Euler scheme for gradient flows are widely used in the approximation of partial differential equations (cf., for instance, [6, 8]). As a consequence of Lyapunov stability, it is possible to prove that if F is real analytic (or more generally, if F satisfies a L Ã ojasiewicz inequality), then any bounded sequence defined by (1.7) converges to a critical point of F : this has been proved in a finite dimensional setting in [2] under quite general assumptions (see also [29]), and in [7] in an infinite dimensional setting; convergence rates were also obtained. There has recently been a growing interest on similar applications of the L Ã ojasiewicz-Simon inequality in an optimization context [1, 3, 14]. It turns out that the backward Euler scheme (1.3) also has Lyapunov stability, although this is no longer an easy consequence of the definition. This is known when F is convex or quasiconvex (cf., e.g., [28]), and we prove it in Theorem 2.3 under the assumption that F is semiconvex and that ∆t is small. Recall that F is semiconvex if the functional W 7→ F (W ) + cF kW k2 /2 (1.9) is convex for some (best) constant cF ≥ 0. The main result of our paper, Theorem 3.4, shows that if F : Rd → R is a semiconvex real analytic function, if ∆t < 2/cF (see (1.9)) and if (Gn+1 )n∈N satisfies a discrete version of (1.2), then any bounded sequence (U n )n∈N which complies with (1.3) converges to a critical point of F . We also obtain convergence rates. Our proof is based on the L Ã ojasiewicz inequality for real analytic functions and on the Lyapunov stability result. The semiconvexity assumption seems natural for the stability and convergence of discrete schemes in an infinite dimensional setting [7, 14], or even in finite dimension for modified backward Euler schemes such as the θ-scheme [29, 33]. However, semiconvexity can be removed in the first order case ε = 0, as well as the smallness assumption on ∆t, if the sequence (U n )n∈N is defined by a minimization algorithm similar to (1.7): we prove this in Theorem 4.4 (see also [2, 29] for the autonomous case). Notice that the boundedness assumption on (U n )n , which could seem hard to check, is in fact an easy consequence of the Lyapunov stability when F is coercive, i.e., limkW k→∞ F (W ) = ∞; notice also that the semiconvexity of F includes situations where F has a continuum of critical points, so that convergence to equilibrium is not obvious. The paper is organized as follows: in Section 2, we prove existence, uniqueness and stability results for the scheme (1.3), when ε ≥ 0; in Section 3, we prove our main convergence result in the case ε > 0. Section 4 is devoted to the case ε = 0. In the last section, we show how these results apply to space and time discretizations of the damped wave equation and of the modified Swift-Hohenberg equation. 2. Existence, uniqueness and Lyapunov stability For the time discretization of (1.1), we first rewrite it as the equivalent first order system ( U 0 (t) = V (t) t ≥ 0. (2.1) εV 0 (t) = −V (t) − ∇F (U (t)) + G(t) We choose a fixed time step ∆t > 0. The backward Euler scheme applied to (2.1) reads: let (U 0 , V 0 ) ∈ Rd × Rd and for n = 0, 1, 2, . . . , let (U n+1 , V n+1 ) ∈ Rd × Rd

4


solve

 n+1 U − Un   = V n+1   ∆t (2.2) ¡ n+1 ¢  n  V − V   ε = −V n+1 − ∇F (U n+1 ) + Gn+1 ∆t where (Gn+1 )n∈N is a given sequence in Rd . Throughout the paper we assume that F ∈ C 1 (Rd , R) at least. Notice that we recover (1.3) from (2.2), by eliminating V n and V n+1 . In Sections 2 and 3 we assume, and this is our main assumption for the proof of stability, that F is semiconvex, i.e., h∇F (U ) − ∇F (W ), U − W i ≥ −cF kU − W k2 ,

∀U, W ∈ Rd ,

(2.3)

for some constant cF ≥ 0. Here and in the sequel, h·, ·i denotes the Euclidean scalar product on Rd . Assumption (2.3), which can be understood as a one-sided Lipschitz condition, is equivalent to the assumption that the map cF kW k2 W 7→ F (W ) + 2 is convex. As a consequence of (2.3), F satisfies cF kU − W k2 , ∀U, W ∈ Rd . (2.4) 2 Notice that if F satisfies (2.3) for some cF < 0, then F is strictly convex and has at most one critical point. In this case, convergence to equilibrium is well known (see, e.g., [8]). The convex case cF = 0 has also been extensively studied (see for instance [28] and references therein). The novelty here is the case cF > 0. The sequence (Gn+1 )n satisfies either ∞ X kGn+1 k2 < ∞, (2.5) F (U ) ≥ F (W ) + h∇F (W ), U − W i −

n=0

or

Ã 1+δ

sup n n∈N

∞ X

! kG

k+1 2

k

< ∞,

(2.6)

k=n

for some δ > 0. Notice that condition (2.6) implies (2.5), and that both conditions imply that Gn+1 → 0 as n → ∞, so that the scheme (2.2) is asymptotically autonomous. Also, if (Gn+1 )n is defined by (1.4) and G satisfies (1.2), then by the Cauchy-Schwarz inequality, (Gn+1 )n satisfies (2.6). Conversely, if (Gn+1 )n satisfies (2.6), then the function G defined by G ≡ Gn+1 on [tn , tn+1 ) satisfies (1.2), so that condition (2.6) is the discrete counterpart of (1.2). We have: Proposition 2.1 (Existence). If F satisfies (2.3) and ε/∆t2 + 1/∆t > cF , or if inf Rd F > −∞ (without any assumption on ∆t > 0), then for all (U 0 , V 0 ) ∈ Rd ×Rd , there exists a least one sequence (U n , V n )n∈N which complies with (2.2). Proof. Let (U n , V n ) ∈ Rd × Rd . Eliminating V n+1 from (2.2), we see that any solution U n+1 satisfies ¶ µ 1 ε n ε + (U n+1 − U n ) − V + ∇F (U n+1 ) = Gn+1 , (2.7) 2 ∆t ∆t ∆t


5

and conversely, if U n+1 satisfies (2.7), by setting V n+1 = (U n+1 − U n )/∆t, we see that (U n+1 , V n+1 ) satisfies (2.2). Now, define µ ¶ ε 1 ε n G (W ) := + kW − U n k2 − h V n + Gn+1 , W − U n i + F (W ). 2 2∆t 2∆t ∆t Notice that U n+1 satisfies (2.7) if and only if ∇Gn (U n+1 ) = 0. If inf Rd F > −∞, then Gn (W ) → ∞ as kW k → ∞, and since Gn is continuous on Rd , we can choose U n+1 in n o argmin Gn (W ) : W ∈ Rd , (2.8) which is nonempty. Similarly, if F satisfies (2.3), then by applying (2.4) with W = U n and U = W , we see that µ ¶ ε 1 cF ε n + G (W ) ≥ − kW − U n k2 − h V n + Gn+1 , W − U n i + F (U n ). 2 2∆t 2∆t 2 ∆t Thus, if ε/∆t2 + 1/∆t − cF > 0, Gn (W ) → ∞ as kW k → ∞. As previously, we can choose U n+1 as a minimizer of Gn on Rd . ¤ The semiconvexity of F also provides uniqueness, provided that ∆t is small enough: Proposition 2.2 (Uniqueness). If F satisfies (2.3) and ε/∆t2 +1/∆t > cF , then for every (U n , V n ) ∈ Rd ×Rd , there exists at most one (U n+1 , V n+1 ) which satisfies (2.2). Proof. We know that (U n+1 , V n+1 ) satisfies (2.2) if and only if U n+1 satisfies (2.7) ¯ n+1 both satisfy (2.7), their and V n+1 = (U n+1 − U n )/∆t. Now if U n+1 and U n+1 n+1 ¯ difference δU := U −U satisfies µ ¶ ε 1 ¯ n+1 ) = 0. δU + ∇F (U n+1 ) − ∇F (U + 2 ∆t ∆t On multiplying by δU and using (2.3), we obtain µ ¶ ε 1 kδU k2 ≤ cF kδU k2 . + ∆t2 ∆t Using the smallness assumption on ∆t, we find δU = 0.

¤

The energy of the system is defined (as in the continuous case) by ε E(U, V ) := kV k2 + F (U ). 2 Theorem 2.3 (Lyapunov stability). Assume that F satisfies (2.3) and 1/∆t > cF /2. If (U n , V n )n∈N is a sequence which satisfies (2.2), then for µ > 0 small enough, ¶ µ ∆t n+1 2 cF ∆t n+1 n+1 − µ ∆tkV n+1 k2 ≤ E(U n , V n ) + kG k , (2.9) E(U ,V )+ 1− 2 4µ for all n ≥ 0. Proof. Taking the scalar product of the second equation of (2.2) with U n+1 − U n = ∆tV n+1 , we obtain, for all n ≥ 0, ¡ ¢ ε kV n+1 k2 − hV n , V n+1 i = −∆tkV n+1 k2 − h∇F (U n+1 ), U n+1 − U n i +∆thGn+1 , V n+1 i.

6


We now use inequality (2.4) with U = U n and W = U n+1 , and the Cauchy-Schwarz inequality. This yields ¢ ε ¡ n+1 2 cF kV k − kV n k2 ≤ −∆tkV n+1 k2 + F (U n ) − F (U n+1 ) + kU n+1 − U n k2 2 2 1 ∆tkGn+1 k2 , +µ∆tkV n+1 k2 + 4µ which is the required inequality. ¤ Remark 2.4. Notice that if ε ≥ 0 is small, we may choose ∆t such that the stability condition 1/∆t > cF /2 is satisfied and the uniqueness condition ε/∆t2 + 1/∆t > cF is not satisfied. Define the ω-limit set of a sequence (U n )n as n o ω ((U n )n ) := U ? ∈ Rd : ∃nk → ∞ such that U nk → U ? .

(2.10)

Let also

S := {U ? ∈ Rd : ∇F (U ? ) = 0}. (2.11) For a subset A of Rd and a point U ∈ Rd , d(U, A) denotes the distance from U to A, defined by d(U, A) := inf kU − W k. W ∈A

Lyapunov stability has the following well-known consequences. Corollary 2.5. Assume that F satisfies (2.3), that (Gn+1 )n satisfies (2.5) and that 1/∆t > cF /2. Let (U n , V n )n∈N be a sequence which satisfies (2.2). If (U n )n∈N is bounded, then V n → 0 as n → ∞, ω ((U n )n ) is a nonempty compact connected subset of S, and ³ ´ d U k , ω ((U n )n ) → 0 as k → ∞. (2.12) Proof. By induction, from (2.9) we deduce that µ ¶ n−1 n−1 X ∆t X n+1 2 cF ∆t n n ∆tkV k+1 k2 ≤ E(U 0 , V 0 ) + kG k . E(U , V ) + 1 − −µ 2 4µ k=0 k=0 (2.13) By assumption, (U n )n is bounded and (Gn+1 )n satisfies (2.5), so that ∞ X kV k+1 k2 < ∞. k=0 n

In particular, V → 0 as n → ∞, as stated. The boundedness of (U n )n and the fact that U n+1 − U n → 0 also implies, as a general result on sequences [2], that ω ((U n )n∈N ) is a nonempty compact connected set and that (2.12) holds. Let U ? ∈ ω ((U n )n∈N ) and nk → ∞ such that U nk +1 → U ? . By passing to the limit in the second equation of (2.2), we see that ∇F (U ? ) = 0, i.e. U ? ∈ S, as stated. The proof is complete. ¤ Remark 2.6. In many applications, F is coercive, i.e. F (W ) → ∞ as kW k → ∞.

(2.14)

This is a general assumption which ensures that Lyapunov stable sequences are bounded. Under the assumptions of Corollary 2.5, if, in addition F satisfies (2.14),


7

then any sequence (U n )n∈N which satisfies (2.2) is automatically bounded, by the stability estimate (2.13). Existence of such a sequence for every (U 0 , V 0 ) ∈ Rd × Rd is also guaranteed by Proposition 2.1, since (2.14) implies inf Rd F > −∞. As a consequence, we have:

P n+1 k < ∞, under the assumptions of CorolCorollary 2.7. If d = 1 and ∞ n=0 kG lary 2.5, there exists U ? ∈ S such that U n → U ? . Proof. We only need to show that ω ((U n )n ) reduces to a singleton. We argue as in the continuous case [18]. Assume by contradiction that there exist two values a, b ∈ ω ((U n )n ), with a < b, and let c := (a + b)/2 ∈ [a, b]. Using that ω ((U n )n∈N ) is connected and Corollary 2.5, we can choose n0 large enough such that kU n0 −ck ≤ β, ∞ X ∆t kGn+1 k ≤ β and kU n+1 − U n k ≤ β, εkV n k ≤ β, (2.15) n=n0

for all n ≥ n0 , where β < (b − a)/8. Assume by contradiction that the set © ª n ≥ n0 : U n+1 6∈ [a, b]

(2.16)

is nonempty, and let n1 denote the infimum of this set (notice that n1 > n0 + 2 by (2.15) and by the triangle inequality). For all n ∈ {n0 , . . . , n1 − 1}, U n+1 ∈ [a, b] ⊂ S, so that ∇F (U n+1 ) = 0. By (2.2), U n+1 − U n = −ε(V n+1 − V n ) + ∆tGn+1 ,

∀n ∈ {n0 , . . . , n1 − 1}.

(2.17)

Summing from n = n0 to n = n1 − 1, we obtain U

n1

−U

n0

= −ε(V

kU n1

n1

−V

n0

) + ∆t

nX 1 −1

Gn+1 .

n=n0 that kU n1

U n0 k

In particular, by (2.15), − ≤ 3β, so − ck ≤ 4β < (b − a)/2 and this contradicts U n1 6∈ [a, b]. The set defined by (2.16) is therefore empty and we can set n1 = ∞. Equation (2.17) holds for all n ≥ n0 , and summing from n0 to n − 1 ≥ n0 , we find as previously that kU n − U n0 k ≤ 3β,

∀n > n0 .

By the triangle inequality, kU n − ck ≤ 4β,

∀n > n0 .

U nk

By choosing a subsequence such that → b, we obtain that |b−c| ≤ 4β < (b−a)/2, a contradiction. Thus, ω ((U n )n∈N ) is reduced to a single point U ? , and U n → U ? as n → ∞. ¤ P∞ Remark 2.8. Notice that assumption n=0 kGn+1 k < ∞ is stronger than condition (2.5), but weaker than (2.6). Indeed, if (Gn+1 )n satisfies (2.6), then    1/2 ∞ 2k+1 ∞ 2k+1 ∞ X X−1 X X−1 X  kGn+1 k = kGn+1 k ≤ 2k/2  kGn+1 k2  n=1

k=0

k=0

n=2k

≤

∞ X

n=2k

2k/2 C 1/2 (2k )−(1+δ)/2

k=0 1/2

= C

(1 − 2−δ/2 )−1 ,

8


for some constant C < ∞. 3. Convergence to equilibrium The main tool of the proof is the L Ã ojasiewicz inequality, which is defined as follows. Definition 3.1. We say that F ∈ C 1 (Rd , R) satisfies the L Ã ojasiewicz inequality near ? d some point U ∈ R if there exist constants θ ∈ (0, 1/2], γ ≥ 0 and σ > 0 such that ∀U ∈ Rd ,

kU − U ? k < σ ⇒ |F (U ) − F (U ? )|1−θ ≤ γk∇F (U )k.

(3.1)

The number θ which appears in this definition is called a L Ã ojasiewicz exponent of U ? [2]. If F satisfies (3.1) for some exponent θ ∈ (0, 1/2], then, by changing the constants σ and γ if necessary, it is easy to see that F also satisfies the L Ã ojasiewicz inequality for every exponent θ0 ∈ (0, θ]. The result of L Ã ojasiewicz [27] shows that if F : Rd → R is real analytic near ? some point U ∈ Rd , then F satisfies the L Ã ojasiewicz inequality near U ? (for some exponent which is not known explicitely, in general). For the reader’s convenience, we first state and proof the result of convergence in the continuous case. In this case, the ω-limit set of U ∈ C 0 (R+ , Rd ) is defined as n o ω(U ) = U ? ∈ Rd : ∃tn → ∞ such that U (tn ) → U ? . 2,1 (R+ , Rd ) be a solution of (1.1) with ε ≥ 0, and assume Theorem 3.2. Let U ∈ Wloc that (1) F ∈ C 2 (Rd , R), (2) the set {U (t) : t ≥ 0} is bounded in Rd , (3) there exists U ? ∈ ω(U ) such that F satisfies the L Ã ojasiewicz inequality near U ? in the sense of Definition 3.1, with a L Ã ojasiewicz exponent θ, (4) G ∈ L1loc (R+ , Rd ) satisfies (1.2) for some δ > 0. Then limt→∞ U (t) = U ? . Moreover, there exists a constant C > 0 such that for all t ≥ 0, we have ½ ¾ θ δ ? −α kU (t) − U k ≤ C(1 + t) , where α = min , . 1 − 2θ 2 Proof. We prove here the result of convergence when ε > 0, by adapting the proof in [10]. The convergence rate, which is optimal in general, is derived in [5]. We let V = U 0 ∈ L1loc (R+ , Rd ), so that (U, V ) satisfies (2.1). First define Z ∞ ε Φ0 (t) = kV (t)k2 + F (U (t)) + Cµ kG(s)k2 ds, 2 t where µ ∈ (0, 1) and Cµ > 1/(4µ). On computing, we find

−Φ00 (t) = kV (t)k2 − hG(t), V (t)i + Cµ kG(t)k2 ¶ µ 1 2 kG(t)k2 . ≥ (1 − µ)kV (t)k + Cµ − 4µ

(3.2)

In particular, Φ0 is nonincreasing, and since F (U ) is bounded, the limit limt→∞ Φ0 (t) exists and V is bounded. Moreover, by (3.2), the function h(t) = kV (t)k2 is integrable; h is also uniformly continuous on R+ : indeed, h0 (t) = 2hV (t), V 0 (t)i, V is bounded and, by (2.1) and assumption (1.2), V 0 ∈ L∞ (R+ , Rd ) + L2 (R+ , Rd ),


9

As a consequence, h(t) → 0 as t → ∞. A standard argument (see [10] for details) implies that F is constant on ω(U ) and that ω(U ) is a nonempty compact connected subset of S. Up to now, we have obtained a continuous version of Corollary 2.5. Define now Z ∞ ε 2 Φ(t) = kV (t)k + F (U (t)) + βh∇F (U (t)), V (t)i + Cµ kG(s)k2 ds, (3.3) 2 t where β > 0 is a small constant which will be specified below. Using (3.2) and (2.1), we find that −Φ0 (t) = −Φ00 (t) − βh∇2 F (U (t))V (t), V (t)i − βh∇F (U (t)), V 0 (t)i ¶ µ 1 2 kG(t)k2 ≥ (1 − µ − βC(R))kV (t)k + Cµ − 4µ ¢ β¡ + h∇F (U (t)), V (t)i + k∇F (U (t))k2 − h∇F (U (t)), G(t)i , ε where R = supt≥0 kU (t)k < ∞ and C(R) = supkW k≤R k∇2 F (W )k < ∞. By the Cauchy-Schwarz inequality, µ ¶ β β 0 −Φ (t) ≥ 1 − µ − βC(R) − kV (t)k2 + k∇F (U (t))k2 ε 2ε µ ¶ 1 β + Cµ − − kG(t)k2 . 4µ ε Thus, for β > 0 small enough, β (kV (t)k + k∇F (U (t))k)2 . 4ε In particular, Φ is nonincreasing. As a consequence, the function −Φ0 (t) ≥

(3.4)

Φ? (t) = Φ(t) − F (U ? ) tends to 0 as t → ∞, and Φ? (t) ≥ 0 for all t ≥ 0. By changing the constants θ, σ and γ in (3.1) if necessary, we can choose a L Ã ojasiewicz exponent θ ∈ (0, 1/2) of F near U ? so that δ≥

2θ = 2α. 1 − 2θ

(3.5)

Next, we estimate Φ1−θ ? (t). We have ³ ε ´1−θ Φ1−θ (t) ≤ kV (t)k2(1−θ) + |F (U (t)) − F (U ? )|1−θ ? 2 +β 1−θ k∇F (U (t))k1−θ kV (t)k1−θ µ Z ∞ ¶1−θ 2 + Cµ kG(s)k ds , t

where we used the Cauchy-Schwarz inequality and the inequality (a + b)1−θ ≤ a1−θ + b1−θ ,

∀a, b ≥ 0.

By Young’s inequality, k∇F (U (t))k1−θ kV (t)k1−θ ≤ k∇F (U (t))k + kV (t)k(1−θ)/θ .

(3.6)

10


Let t ≥ 0 such that kU (t) − U ? k < σ. Using the L Ã ojasiewicz inequality (3.1), we obtain µ Z ∞ ¶1−θ 1−θ 2 Φ? (t) ≤ C0 (kV (t)k + k∇F (U (t))k) + Cµ kG(s)k ds , (3.7) t

for some constant C0 < ∞. If kV (t)k + k∇F (U (t))k > γ

−1

µ Z Cµ

∞

¶1−θ kG(s)k ds , 2

(3.8)

t

then −

d θ Φ (t) = −θΦ0? (t)Φθ−1 ? (t) dt ? θβ ≥ (kV (t)k + k∇F (U (t))k) , 4ε(C0 + γ)

by (3.4) and (3.7). On the contrary, if (3.8) does not hold, then kV (t)k + k∇F (U (t))k ≤ Ct−(1+δ)(1−θ) , for some constant C, by (1.2). In any case, for all t such that kU (t) − U ? k < σ, we have 4ε(C0 + γ) 0 kU 0 (t)k = kV (t)k ≤ − Φ? (t) + Ct−(1+δ)(1−θ) . (3.9) θβ Since (1 + δ)(1 − θ) ≥ (1 − θ)/(1 − 2θ) > 1 by (3.5), the right hand side of this estimate is an integrable function. Now, let t¯ be large enough so that Z ∞ 4ε(C0 + γ) ? ¯ ¯ kU (t) − U k < σ/2, and Φ? (t) + Ct−(1+δ)(1−θ) dt < σ/2. θβ t¯ Define t+ = sup{t ≥ t¯ : kU (s) − U ? k < σ ∀s ∈ [t¯, t)}, and assume by contradiction that t+ < ∞, so that kU (t+ ) − U ? k = σ. For all t ∈ [t¯, t+ ), estimate (3.9) holds, and by the choice of t¯, we obtain Z t kU (t) − U (t¯)k ≤ kU 0 (s)kds ≤ σ/2. (3.10) t¯

By the triangle inequality, kU (t+ ) − U ? k ≤ kU (t+ ) − U (t¯)k + kU (t¯) − U ? k < σ, and we obtain a contradiction. Thus, t+ = ∞, estimate (3.10) is valid for all t ≥ t¯, and in particular, U (t) has a limit as t → ∞. This concludes the proof. The convergence rate can also be derived from the previous estimates (see [16, 17]). ¤ Now, we turn back to the discrete dynamical system (2.2). We first define, for all n ≥ 0 (compare with (3.3)), ∞

X ε kGk+1 k2 , Φn := kV n k2 + F (U n ) + βh∇F (U n ), V n i + Cµ ∆t 2 k=n

where Cµ > 1/(4µ) is fixed (µ > 0 is small enough so that 1 − cF ∆t/2 − µ > 0) and β > 0 is a small constant which will be specified later on. We have the following strong Lyapunov stability, which is a discrete version of (3.4):


11

1,1 Lemma 3.3. Assume that F ∈ Cloc (Rd , R) satisfies (2.3), G satisfies (2.5), ε > 0 n n and 1/∆t > cF /2. Let (U , V )n∈N be a sequence which satisfies (2.2). If (U n )n∈N is bounded, then for β > 0 small enough, ¢2 β∆t ¡ n+1 Φn − Φn+1 ≥ kV k + k∇F (U n+1 )k , ∀n ≥ 0. (3.11) 8ε Proof. Let R > 0 be such that kU n k ≤ R for all n ≥ 0, and let L(R) be the Lipschitz constant of ∇F on the closed ball {W ∈ Rd : kW k ≤ R}. Let also µ > 0 be small enough such that 1 − cF ∆t/2 − µ > 0. By Theorem 2.3, we have, for all n ≥ 0, ¶ µ ¶ µ cF ∆t 1 n n+1 n+1 2 ∆tkGn+1 k2 Φ −Φ ≥ 1− − µ ∆tkV k + Cµ − 2 4µ

+βh∇F (U n ) − ∇F (U n+1 ), V n i + βh∇F (U n+1 ), V n − V n+1 i µ ¶ µ ¶ cF ∆t 1 n+1 2 ≥ 1− − µ ∆tkV k + Cµ − ∆tkGn+1 k2 2 4µ ∆t −βL(R)∆tkV n+1 kkV n k + β h∇F (U n+1 ), V n+1 i ε ∆t ∆t n+1 2 +β k∇F (U )k − β h∇F (U n+1 ), Gn+1 i, (3.12) ε ε where, in the second line, we used (2.2), and in particular ¢ ∆t ¡ n+1 V + ∇F (U n+1 ) − Gn+1 , ∀n ≥ 0. V n − V n+1 = ε This equality implies in particular that ¶ µ ∆t ∆t n+1 ∆t n kV n+1 k + k∇F (U n+1 )k + kG k, kV k ≤ 1 + ε ε ε for all n ≥ 0. Using this in (3.12), together with the Cauchy-Schwarz inequality |ha, bi| ≤ kak2 /4 + kbk2 , we find µ µ ¶¶ cF ∆t ∆t Φn − Φn+1 ≥ 1− − µ − βL(R) 1 + kV n+1 k2 ∆t 2 ε β β − (1 + 2L(R)2 ∆t2 )kV n+1 k2 + k∇F (U n+1 )k2 ε 4ε µ ¶ 1 5β + Cµ − − kGn+1 k2 . 4µ 4ε For β > 0 small enough, we have ¢ Φn − Φn+1 β ¡ n+1 2 ≥ kV k + k∇F (U n+1 )k2 , ∆t 4ε and from this estimate we immediately deduce (3.11).

¤

The discrete version of Theorem 3.2 reads: Theorem 3.4. Let (U n , V n )n∈N be a sequence which satisfies (2.2) with ε ≥ 0, and assume that 1,1 (1) F ∈ Cloc (Rd , R) satisfies (2.3), (2) 1/∆t > cF /2, (3) (U n )n∈N is bounded,

12


(4) there exists U ? ∈ ω((U n )n ) such that F satisfies the L Ã ojasiewicz inequality near U ? in the sense of Definition 3.1, with a L Ã ojasiewicz exponent θ, (5) there exists a constant δ > 0 such that (Gn+1 )n∈N satisfies (2.6). Then limn→∞ U n = U ? . Moreover, there exists a constant C such that for all n > 0, we have ½ ¾ θ δ n ? −α kU − U k ≤ Cn , where α = min , . (3.13) 1 − 2θ 2 Proof. We prove here the case ε > 0; the proof in the (easier) case ε = 0 is postponed until the end of the next section. By changing the constants θ, σ and γ in the L Ã ojasiewicz inequality (Definition 3.1) if necessary, we can assume that θ ∈ (0, 1/2) satisfies 2θ δ≥ = 2α. (3.14) 1 − 2θ Define Φn? := Φn − F (U ? ). By choosing a sequence of integers nk → ∞ such that U nk → U ? , and by Corollary 2.5, we see that Φn? k → 0 as k → ∞. And since (Φn? )n is nonincreasing by Lemma 3.11, we have Φn? ≥ 0 for all n ≥ 0. We first estimate (Φn+1 )1−θ . We have ? ³ ε ´1−θ n+1 1−θ kV n+1 k2(1−θ) + |F (U n+1 ) − F (U ? )|1−θ (Φ? ) ≤ 2 +β 1−θ k∇F (U n+1 )k1−θ kV n+1 k1−θ !1−θ Ã ∞ X kGk+1 k2 + Cµ ∆t , (3.15) k=n+1

where we used the Cauchy-Schwarz inequality and the inequality (3.6). By Young’s inequality, k∇F (U n+1 )k1−θ kV n+1 k1−θ ≤ k∇F (U n+1 )k + kV n+1 k(1−θ)/θ .

(3.16)

If kU n+1 − U ? k < σ, we can use the L Ã ojasiewicz inequality (3.1) and we deduce from (3.15) that Ã !1−θ ∞ X ¡ n+1 ¢ n+1 k+1 2 n+1 1−θ (Φ? ) ≤ C1 kV k + k∇F (U )k + Cµ ∆t kG k , (3.17) k=n+1

where

½

C1 = max sup

n∈N

µ³ ´ ε 1−θ 2

¶ kV

n+1 1−2θ

k

+β

1−θ

kV

n+1 1/θ−2

k

¾ ,γ + β

1−θ

,

and C1 < ∞ since (V n+1 )n is bounded and θ ∈ (0, 1/2). Note that here and in the sequel, the notation Ci is used to denote various constants which do not depend on n (but which can depend on other parameters). Now, let n ∈ N? such that kU n+1 − U ? k < σ and Φn? ≤ 1. If Ã !1−θ ∞ X kV n+1 k + k∇F (U n+1 )k ≤ γ −1 Cµ ∆t kGk+1 k2 , (3.18) k=n+1


13

then ∆tkV n+1 k ≤ C2 n−(1+δ)(1−θ) ,

(3.19)

for some constant C2 = C2 (∆t, Cµ , γ, θ) > 0, by (2.6). On the contrary, if Ã !1−θ ∞ X kV n+1 k + k∇F (U n+1 )k > γ −1 Cµ ∆t kGk+1 k2 ,

(3.20)

k=n+1

then either Φn+1 ≤ Φn? /2, in which case, by Lemma 3.3, ? µ ¶ ¤1/2 8ε∆t 1/2 £ n n+1 ∆tkV k ≤ Φ? − Φn+1 ? β µ ¶ 8ε∆t 1/2 n θ ≤ [Φ? ] β h i ≤ C3 (Φn? )θ − (Φn+1 )θ , ? where

µ C3 :=

8ε∆t β

¶1/2 ³

1 − 2−θ

´−1

(3.21)

,

or Φn+1 > Φn? /2. In the latter case, we write ? Z Φn? n θ n+1 θ (Φ? ) − (Φ? ) = θsθ−1 ds ≥ ≥

Φn+1 ? θ(Φn? )θ−1 [Φn? − Φn+1 ] ? θ−1 n+1 θ−1 n θ2 (Φ? ) [Φ? −

Φn+1 ]. ?

(3.22)

By (3.17) and (3.20), ¡ ¢ (Φn+1 )1−θ ≤ (C1 + γ) kV n+1 k + k∇F (U n+1 )k , ?

(3.23)

Using this last inequality and (3.22), together with Lemma 3.3, we see that (Φn? )θ − (Φn+1 )θ ≥ ?

¢ θ2θ−1 β∆t ¡ n+1 kV k + k∇F (U n+1 )k . (C1 + γ) 8ε

(3.24)

Adding together (3.19), (3.21) and (3.24), for all n ≥ 0 such that kU n+1 − U ? k < σ and Φn? ≤ 1, we have ³ ´ θ ∆tkV n+1 k ≤ C2 n−(1+δ)(1−θ) + C4 (Φn? )θ − (Φn+1 ) , (3.25) ? where C4 = C3 + 8ε(C1 + γ)/(θ2θ−1 β). Let us now choose n0 ∈ N? large enough such that Φn? 0 ≤ 1, kU n0 − U ? k < σ/3, and C2 Notice that

P∞

n=1 n

∞ X

n−(1+δ)(1−θ) + C4 (Φn? 0 )θ < σ/3.

n=n0 −(1+δ)(1−θ)

< ∞ since (1 + δ)(1 − θ) > 1 by (3.14).

(3.26)

14


Let N ≥ n0 be the largest integer such that kU n − U ? k < 2σ/3 for all n0 ≤ n ≤ N (we set N = ∞ if kU n − U ? k < 2σ/3 for all n ≥ n0 ). Assume by contradiction that N < ∞. By Lemma 3.3, we have kU N +1 − U ? k ≤ kU N +1 − U N k + kU N − U ? k 1/2 ≤ C3 (ΦN + kU N − U ? k ? ) < σ,

where we used that ∆tkV N +1 k = kU N +1 − U N k (recall (2.2)), that (Φn? )n is nonincreasing and (3.26). Thus, we can apply (3.25) to every n ∈ {n0 , . . . , N } and summing from n = n0 to n = N . We find N X

kU

n+1

n

− U k ≤ C2

n=n0

∞ X

n−(1+δ)(1−θ) + C4 (Φn? 0 )θ

(3.27)

n=n0

< σ/3. kU N +1 − U ? k

σ/3 + kU n0

Thus, ≤ − U ? k < 2σ/3, and this contradicts the definition of N . So N = ∞, estimate (3.27) is still true, and so the whole sequence (U n )n converges to U ? . Next, we prove the convergence rate (3.13). If Φn? = 0 for some n0 ∈ N, then n Φ? = 0 for n ≥ n0 and V n+1 = 0 for n ≥ n0 by Lemma 3.3, so that U n = U ? for n large enough and (3.13) is obviously true. Now, we assume that Φn? > 0 for all n ≥ 0 and we choose n0 ∈ N? large enough so that 0 ≤ Φn? 0 ≤ 1

and

kU n+1 − U ? k < σ

for all n ≥ n0 .

We define J1 = {n ≥ n0 : (3.18) holds}

and

J2 = {n ≥ n0 : (3.20) holds} ,

so that J1 and J2 are disjoint and J1 ∪ J2 = {n ≥ n0 }. Let n ∈ J2 . Then, either Φn+1 ≤ Φn? /2, in which case it follows that ? ³ ´ (Φn+1 )−1+2θ − (Φn? )−1+2θ ≥ (Φn? )−1+2θ 21−2θ − 1 ≥ 21−2θ − 1, ? or Φn+1 > Φn? /2. In the latter case, we have ? (Φn+1 )−1+2θ − (Φn? )−1+2θ = (1 − 2θ) ?

Z

Φn ?

Φn+1 ?

s−2+2θ ds

£ ¤ ≥ (1 − 2θ)(Φn? )−2+2θ Φn? − Φn+1 ? £ ¤ ≥ (1 − 2θ)2−2+2θ (Φn+1 )−2+2θ Φn? − Φn+1 . ? ? Using Lemma 3.3 and estimate (3.23), we find that (Φn+1 )−1+2θ − (Φn? )−1+2θ ≥ (1 − 2θ)2−2+2θ ?

β∆t . 8ε(C1 + γ)2

Thus, in both cases, for all n ∈ J2 , we obtain (Φn+1 )−1+2θ − (Φn? )−1+2θ ≥ C5 , ?

(3.28)

for some constant C5 > 0. By (3.21) and (3.24), we also have, for all n ∈ J2 , ³ ´ ∆tkV n+1 k ≤ C4 (Φn? )θ − (Φn+1 )θ . (3.29) ? On the other hand, for all n ∈ J1 , estimate (3.19) holds.


15

We now consider three cases. If J1 contains {n ≥ n1 } for some n1 ≥ n0 , then n

?

kU − U k ≤

∞ X

∆tkV

k+1

k≤

k=n

∞ X

C2 k −(1+δ)(1−θ) ≤ C6 n−δ(1−θ)+θ ,

k=n

for all n ≥ n1 . We therefore obtain (3.13) since δ(1 − θ) − θ ≥ θ/(1 − 2θ) by (3.14). If J2 contains {n ≥ n1 } for some n1 ≥ n0 , then by summing (3.28) from n1 to n − 1, we obtain ³ ´−1/(1−2θ) Φn? ≤ C5 (n − n1 ) + (Φn? 1 )−1+2θ ≤ C7 n−1/(1−2θ) , for all n ≥ n1 . By summing (3.29) from n ≥ n1 to ∞, we obtain also ∞ X

kU n − U ? k ≤

∆tkV k+1 k ≤ C4 (Φn? )θ ≤ C4 C7θ n−θ/(1−2θ) ,

k=n

for all n ≥ n1 . We thus obtain (3.13) again. If neither of the previous two cases holds, then we can find two sequences of positive integers, (np )p∈N? and (mp )p∈N? , such that J2 ∩ {n ≥ n1 } = ∪p∈N {np , . . . , mp }, with n1 > n0 , np ≤ mp and np − 1 6∈ J2 , for all p ≥ 1. Let n ∈ {np , . . . , mp }. By summing (3.28) from np to n − 1, we find that ³ ´−1/(1−2θ) n Φn? ≤ C5 (n − np ) + (Φ? p )−1+2θ . (3.30) By using (3.17) and (3.18) for n = np − 1, we find ³ ´1−θ n . (Φ? p )1−θ ≤ (C1 + γ)C2 (∆t)−1 (np − 1)−(1+δ)(1−θ) ≤ C8 n−(1+δ) p Plugging this into (3.30), we get ³ ´−1/(1−2θ) . Φn? ≤ C5 (n − np ) + C8−1+2θ n(1+δ)(1−2θ) p Changing C5 into C50 = min{C5 , C8−1+2θ }, and noticing that (1 + δ)(1 − 2θ) ≥ 1 by (3.14), we see that Φn? ≤ (C50 n)−1/(1−2θ)

∀n ∈ {np , . . . , mp }.

For all n ≥ n1 , we have, by (3.19) and (3.29), n

?

kU − U k ≤

∞ X

∆tkV k+1 k

k=n

≤

X

C2 k −(1+δ)(1−θ) +

k≥n,k∈J1

≤ C6 n

−δ(1−θ)+θ

X

³ ´ θ C4 (Φk? )θ − (Φk+1 ? )

k≥n,k∈J2

+

C4 (C50 n)−θ/(1−2θ) .

Using (3.14), we find the convergence rate (3.13) again, and the proof is complete. ¤ Remark 3.5. The convergence rate O(n−α ) with α = min{θ/(1−2θ), δ/2} is optimal in general, even in dimension d = 1, as shown by the following examples. However, in the specific case where θ = 1/2 and Gn+1 = 0 for all n, it is possible to obtain a geometric convergence (see [2, 29] for the case ε = 0).

16


(1) If ε ≥ 0, U n = n−λ (λ > 0) and F = 0, then F satisfies the L Ã ojasiewicz inequality for any exponent θ ∈ (0, 1/2]. On the other hand, V n+1 ∼ −λn−λ−1 /∆t so that

∞ X

and

Gn+1 ∼ V n+1 ,

kGk+1 k2 ∼ Cn−1−2λ ,

k=n

for some constant C > 0. Thus, δ = 2λ is the best constant in (2.6), and kU n k = n−λ . Therefore the exponent α = δ/2 = λ is optimal. (2) If F (U ) = |U |p with p > 2 and Gn+1 = 0 for all n, then θ = 1/p and the solution (U n )n of (1.3) in the case ε = 0 satisfies kU n k ≥ Cn−1/(p−2) for some constant C > 0 (see [29] for details). This shows that the exponent α = θ/(1 − 2θ) = 1/(p − 2) cannot be improved here. 4. The first order case The aim of this section is twofold: we want to give the proof of Theorem 3.4 in the case ε = 0, and we want to point out that, in this case, a similar convergence result can be obtained with no restriction on the time step, if the scheme is defined by a minimization algorithm. We begin by the second point. When ε = 0, the scheme (2.2) reduces to U n+1 − U n + ∇F (U n+1 ) = Gn+1 , ∀n ≥ 0, (4.1) ∆t where (Gn+1 )n∈N is a sequence in Rd . We assume that F is C 1 (Rd , R), but we no longer have to assume that F is semiconvex. The sequence (Gn+1 )n∈N in Rd satisfies either (2.5) or (2.6). Existence of a solution for (4.1) can be obtained by choosing ½ ¾ kW − U n k2 n+1 n+1 n d U ∈ argmin + F (W ) − hG ,W − U i : W ∈ R . (4.2) 2∆t The above set is nonempty for instance if inf Rd F > −∞. As in the autonomous case, we will use (4.2) as a definition of the backward Euler scheme for the asymptotically autonomous gradient system. Concerning uniqueness, we notice that the result and the proof of Proposition 2.2 are still valid in the case ε = 0. The stability result reads: Proposition 4.1 (Lyapunov stability). If (U n )n satisfies (4.2) and µ ∈ (0, 1), then F (U n+1 ) +

(1 − µ) n+1 ∆t n+1 2 kU − U n k2 ≤ F (U n ) + kG k , 2∆t 2µ

(4.3)

for all n ≥ 0. Proof. Let n ∈ N. By definition (4.2), kU n+1 − U n k2 + F (U n+1 ) ≤ F (U n ) + hGn+1 , U n+1 − U n i. 2∆t By the Cauchy-Schwarz inequality, ∆t n+1 2 µ hGn+1 , U n+1 − U n i ≤ kG k + kU n+1 − U n k2 . 2µ 2∆t

(4.4)


Putting together these two inequalities, we find (4.3).

17

¤

Lyapunov stability has the following consequence. Corollary 4.2. Assume that (Gn+1 )n satisfies (2.5). If (U n )n is a bounded sequence which satisfies (4.2), then ω((U n )n ) is a nonempty compact connected subset of S, and ¡ ¢ d U k , ω((U n )n ) → 0 as k → ∞. Recall that (as in the second-order case) the set ω((U n )n ) is defined by (2.10) and the set S by (2.11). Proof. Let µ ∈ (0, 1). From (4.3), we deduce by induction that n−1

n−1

k=0

k=0

(1 − µ) X k+1 ∆t X k+1 2 F (U ) + kU − U k k2 ≤ F (U 0 ) + kG k , 2∆t 2µ n

(4.5)

for all n ≥ 0.P From this estimate, from (2.5) and the fact that (U n )n is bounded, we n+1 − U n k2 < ∞, so that U n+1 − U n → 0 as n → ∞. The rest deduce that ∞ n=0 kU of the proof is similar to that of Corollary 2.5. ¤ Arguing as in the proof of Corollary 2.7, we also obtain: P n+1 k < ∞, under the assumptions of CorolCorollary 4.3. If d = 1 and ∞ n=0 kG ? lary 4.2, there exists U ∈ S such that U n → U ? . For the first-order case, convergence to equilibrium can be stated as follows. Theorem 4.4. Let (U n )n∈N be a sequence in Rd which satisfies (4.2), and assume that (1) F ∈ C 1 (Rd , R), (2) (U n )n∈N is bounded, (3) there exists U ? ∈ ω((U n )n ) such that F satisfies the L Ã ojasiewicz inequality near U ? (Definition 3.1), with a L Ã ojasiewicz exponent θ, (4) there exists a constant δ > 0 such that (Gn+1 )n∈N satisfies (2.6). Then limn→∞ U n = U ? . Moreover, there exists a constant C such that for all n > 0, ½ n

?

−α

kU − U k ≤ Cn

,

where

α = min

θ δ , 1 − 2θ 2

¾ .

Proof. We define, for all n ≥ 0, Φn := F (U n ) − F (U ? ) + C1/2 ∆t

∞ X

kGk+1 k2 ,

k=n

where C1/2 > 1. Let n ∈ N. By choosing µ = 1/2 in the the stability estimate (4.3), we find Φn − Φn+1 ≥

1 kU n+1 − U n k2 + (C1/2 − 1)∆tkGn+1 k2 . 4∆t

(4.6)

18


Replacing Gn+1 by its expression given in equation (4.1), we see that µ ¶ 1 kU n+1 − U n k2 n n+1 Φ −Φ ≥ + (C1/2 − 1) 4 ∆t +(C1/2 − 1)∆tk∇F (U n+1 )k2 −2(C1/2 − 1)hU n+1 − U n , ∇F (U n+1 )i. By the Cauchy-Schwarz inequality, we have µ ¶ 1 kU n+1 − U n k2 n n+1 Φ −Φ ≥ + (C1/2 − 1)(1 − β) 4 ∆t µ µ ¶¶ 1 + (C1/2 − 1) 1 − ∆tk∇F (U n+1 )k2 , β for all β > 1. For β > 1 close enough to 1, we find that µ n+1 ¶2 Φn − Φn+1 − U nk ? kU n+1 ≥β + k∇F (U )k , ∆t ∆t

(4.7)

for some small constant β ? > 0 which depends only on C1/2 . In particular, (Φn )n is nonincreasing; by choosing a sequence of integers nk → ∞ such that U nk → U ? , we see that Φnk → 0, and so Φn ≥ 0 for all n ≥ 0. By changing the constants θ, σ and γ in the L Ã ojasiewicz inequality (Definition 3.1) if necessary, we can assume without loss of generality that δ≥

2θ = 2α. 1 − 2θ

Let now n ∈ N such that kU n+1 − U ? k < σ and Φn ≤ 1. If !1−θ Ã ∞ X kU n+1 − U n k k+1 2 n+1 −1 kG k , + k∇F (U )k ≤ γ C1/2 ∆t ∆t

(4.8)

k=n+1

then kU n+1 − U n k ≤ C9 n−(1+δ)(1−θ) . for some constant C9 , by (2.6). If, on the contrary, we have Ã !1−θ ∞ X kU n+1 − U n k n+1 −1 k+1 2 + k∇F (U )k > γ C1/2 ∆t kG k , ∆t

(4.9)

(4.10)

k=n+1

then either Φn+1 ≤ Φn /2, in which case, by (4.7), µ ¶1/2 £ n ¤1/2 ∆t kU n+1 − U n k ≤ Φ − Φn+1 ? β h i ≤ C10 (Φn )θ − (Φn+1 )θ ,

(4.11)

for some constant C10 > 0, or Φn+1 > Φn /2. In the latter case, we write (as in (3.22)) £ ¤ (Φn )θ − (Φn+1 )θ ≥ θ2θ−1 (Φn+1 )θ−1 Φn − Φn+1 . (4.12)


19

On the other hand, by the L Ã ojasiewicz inequality and (4.10), we obtain Ã !1−θ ∞ X (Φn+1 )1−θ ≤ |F (U n+1 ) − F (U ? )|1−θ + C1/2 ∆t kGk+1 k2 µ ≤ 2γ

k=n+1

kU n+1

− ∆t

U nk

¶ n+1 + k∇F (U )k .

(4.13)

Thus, putting together (4.7), (4.12) and (4.13), we find that ³ ´ kU n+1 − U n k (4.14) + k∇F (U n+1 )k ≤ C11 (Φn )θ − (Φn+1 )θ . ∆t Combining (4.9), (4.11) and (4.14), for all n ≥ 0 such that kU n+1 − U ? k < σ, we have ³ ´ (4.15) kU n+1 − U n k ≤ C9 n−(1+δ)(1−θ) + C12 (Φn )θ − (Φn+1 )θ . From this estimate, arguing as in the proof of Theorem 3.4, we conclude that for n0 P n+1 − U n k < ∞. Thus (U n ) converges to U ? . large enough, ∞ n n=n0 kU In order to obtain the convergence rate (4.6), we choose n0 ∈ N? large enough so that 0 ≤ Φn0 ≤ 1 and kU n+1 − U ? k < σ for all n ≥ n0 , and we define J1 = {n ≥ n0 : (4.8) holds}

and

J2 = {n ≥ n0 : (4.10) holds} .

Notice that J1 and J2 are disjoint and J1 ∪ J2 = {n ≥ n0 }. For all n ∈ J1 , estimate (4.9) holds. By arguing as in the proof of Theorem 3.4, we also obtain that, for all n ∈ J2 , (Φn+1 )−1+2θ − (Φn )−1+2θ ≥ C13 , for some constant C13 > 0, and ³ ´ kU n+1 − U n k ≤ C12 (Φn )θ − (Φn+1 )θ . Using these estimates, and considering three cases as in the proof of Theorem 3.4, we obtain the convergence rate (4.6). The proof is complete. ¤ Remark 4.5. The proof of Theorem 4.4 shows that we could replace assumption (4.2) by (4.1) together with the stability estimate (4.4). This could be easier to check in actual computations. Proof of Theorem 3.4 when ε = 0. Let µ > 0 be small enough so that 1 − cF ∆t/2 > µ. By Theorem 2.3, we have µ ¶ cF ∆t kU n+1 − U n k2 ∆t n+1 2 n+1 F (U )+ 1− −µ ≤ F (U n ) + kG k , (4.16) 2 ∆t 4µ for all n ≥ 0. We define, for all n ≥ 0, ∞ X Φn := F (U n ) − F (U ? ) + Cµ ∆t kGk+1 k2 , k=n

where Cµ > 1/(4µ). Let n ∈ N. By (4.16), we get ¶ µ ¶ µ kU n+1 − U n k2 1 cF ∆t n n+1 −µ + Cµ − ∆tkGn+1 k2 . Φ −Φ ≥ 1− 2 ∆t 4µ

20


Now, we replace Gn+1 by its expression given in (4.1) and we use the Cauchy-Schwarz inequality to obtain µ µ ¶ ¶ cF ∆t 1 kU n+1 − U n k2 n n+1 Φ −Φ ≥ 1− − µ + Cµ − (1 − β) 2 4µ ∆t µ ¶µ ¶ 1 1 1− ∆tk∇F (U n+1 )k2 , + Cµ − 4µ β for all β > 1. For β > 1 close enough to 1, we have µ n+1 ¶2 − U nk Φn − Φn+1 ? kU n+1 ≥β + k∇F (U )k , ∆t ∆t for some constant β ? > 0 small enough. For the rest of the proof, we argue exactly as in the proof of Theorem 4.4. ¤ Remark 4.6. For Theorem 3.4 in the case ε = 0, we have only used that F ∈ 1,1 C 1 (Rd , R) instead of F ∈ Cloc (Rd , R) (assumption (1)). This regularity assumption on F could be further relaxed by an appropriate definition of the L Ã ojasiewicz inequality (see [2] for the autonomous case). 5. Applications 5.1. A finite element discretization of the damped wave equation. We consider the asymptotically autonomous nonlinear heat or damped wave equation εutt + ut − ∆u + f 0 (u) = g(t) x ∈ Ω, t > 0,

(5.1)

where Ω is a bounded domain of RN with Lipschitz boundary, ε ≥ 0 and g ∈ L2 (R+ , L2 (Ω)) satisfies Z ∞ 1+δ sup t kg(s)k2L2 (Ω) ds < ∞, (5.2) t∈R+

t

for some constant δ > 0. Equation (5.1) is endowed with homogeneous Dirichlet, homogeneous Neumann, or periodic boundary conditions (in the latter case, Ω is a parallelepiped). Let V = H01 (Ω) in the first case, V = H 1 (Ω) in the second case, 1 (Ω) in the third case. We assume that the nonlinearity f : R → R is and V = Hper real analytic and satisfies |s|→∞

f (s) s2

λ1 , 2

(5.3)

f 00 (s) ≥ −cf 00 ,

(5.4)

lim inf

> −

where cf 00 ≥ 0 and λ1 :=

inf

(∇w, ∇w).

w∈V, kwkL2 (Ω) =1

(5.5)

1 (Ω), then Notice that if V = H01 (Ω), then λ1 > 0, and if V = H 1 (Ω) or V = Hper λ1 = 0. Let V h denote a finite dimensional subspace of V such that V h ⊂ V ∩ C 0 (Ω). The space V h is typically a space of conforming P k or Qk finite elements. The


21

(finite element) space discretization of (5.1) reads: let (uh,0 , v h,0 ) ∈ V h × V h and find uh : R+ → V h which solves ε(uhtt , ϕ) + (uht , ϕ) + (∇uh , ∇ϕ) + (f 0 (uh ), ϕ) = (g, ϕ), ∀ϕ ∈ V h , ∀t ≥ 0, (5.6) uh (0) = uh,0 ,

uht (0) = v h,0 .

(5.7)

Here, (·, ·) denotes the L2 (Ω)-scalar product. As we will see it below, the matrix version of (5.6) is a system of ordinary differential equations which has the form (1.1). The result of Chill and Jendoubi [10] shows that there exists uh,∞ ∈ V h such that uh (t) → uh,∞ as t → ∞ (in fact, their result also holds in infinite dimension for equation (5.1), with suitable growth assumptions on f ). The backward Euler scheme for (5.6) can be written: let uh,0 , uh,1 ∈ V h and for n ≥ 1 let uh,n+1 solve ε(

uh,n+1 − 2uh,n + uh,n−1 uh,n+1 − uh,n , ϕ) + ( , ϕ) δt2 δt Z tn+1 1 +(∇uh,n+1 , ∇ϕ) + (f 0 (uh,n+1 ), ϕ) = ( g(t)dt, ϕ), δt tn

(5.8)

for all ϕ ∈ V h , where δt > 0 is the time step and tj = jδt (j = 0, 1, . . . ). By an application of Theorem 3.4, we have: Theorem 5.1. Assume that the previous assumptions are satisfied and, in particular, ε ≥ 0, f is real analytic and satisfies (5.3)-(5.4) and g satisfies (5.2). If (uh,n )n∈N is a sequence in V h which complies with (5.8), and if δt > 0 is such that 1/δt > (cf 00 − λ1 )/2, then there exists uh,∞ ∈ V h such that limn→∞ uh,n = uh,∞ and (∇uh,∞ , ∇ϕ) + (f 0 (uh,∞ ), ϕ) = 0,

∀ϕ ∈ V h .

(5.9)

Proof. Let (ehi )1≤i≤N h denote an orthonormal basis of V h for the L2 (Ω) scalar prodh uct. Define the function F h : RN → R by h

N X F (W ) = (f ( wi ehi ), 1), h

h

∀W = (w1 , . . . , wN h )t ∈ RN ,

(5.10)

i=1

so that



 Nh X ∇F h (W ) = (f 0 ( wi ehi ), ehj ) i=1

h

∀W = (w1 , . . . , wN h )t ∈ RN .

1≤j≤N h

PN h

By seeking uh (t) = ³ ´ Ah = (∇ehi , ∇ehj )

h

∈ RN ,

h i=1 ui (t)ei

1≤i,j≤N

, h

and defining U = (u1 , . . . , uN h )t ,

³ ´ G(t) = (g(t), ehi )

1≤i≤N h

,

(5.11)

the matrix version of (5.6) reads εUtt (t) + Ut (t) + Ah U (t) + ∇F h (U (t)) = G(t),

t ≥ 0.

(5.12)

This is a second-order gradient-like system in the form (1.1) with 1 F (W ) := hAh W, W i + F h (W ). 2

(5.13)

22


Since f is real analytic and V h ⊂ C 0 (Ω) by assumption, the function F h defined by (5.10) is real analytic; F (W ), defined by (5.13), is also a real analytic function of the variables (w1 , . . . , wN h ), so that F satisfies the L Ã ojasiewicz inequality (3.1) near h ? N every point U ∈ R . Moreover, by Parseval’s inequality, we have kG(t)k2 ≤ kg(t)k2 ,

∀t ≥ 0,

so that by (5.2), we see that G satisfies (1.2). As a consequence, the sequence (Gn+1 )n , defined by (1.4), or equivalently, µ Z tn+1 ¶ 1 n+1 h G := ( g(t)dt, ei ) ∀n ≥ 0, δt tn 1≤i≤N h satisfies (2.6). h h ˜ 'w For all W ' wh ∈ RN , W ˜ h ∈ RN , we have ˜ ), W − W ˜ i = (∇(wh − w h∇F (W ) − ∇F (W ˜ h ), ∇(wh − w ˜ h )) +(f 0 (wh ) − f 0 (w ˜ h ), wh − w ˜h) ≥ (λ1 − cf 00 )kwh − w ˜ h k2L2 (Ω) , by (5.5) and (5.4). So F satisfies (2.3) with cF = max{cf 00 − λ1 , 0}. Moreover, by (5.3), we find κ1 f (s) ≥ − s2 − κ2 , ∀s ∈ R, 2 h for some constants κ1 < λ1 and κ2 ≥ 0, so that for all W ' wh ∈ RN , 1 F (W ) = (∇wh , ∇wh ) + (f (wh ), 1) 2 κ1 λ1 h kw kL2 (Ω) − kwh k2L(Ω) − κ2 |Ω| ≥ 2 2 (λ1 − κ1 ) ≥ kW k2 − κ2 |Ω|. (5.14) 2 In particular, F is coercive, i.e., limkW k→∞ F (W ) = ∞. Assumptions (1)-(5) of Theorem 3.4 are all satisfied (see also Remark 2.6), so there exists uh,∞ ∈ V h such that uh,n → uh,∞ . We obtain (5.9) by passing to the limit in (5.8). ¤ Remark 5.2. A typical choice for f is the double-well potential f (s) = (s2 − λ)2 /4 (with λ > 0). In this case, if Ω = B is a ball in R2 , and if we use Dirichlet boundary conditions, steady states for problem (5.1) satisfy ∆u = u3 − λu in B, u = 0 on ∂B.

(5.15) (5.16)

It is well known that if λ > λ1 , then problem (5.15)-(5.16) has a non trivial odd smooth solution [18]. By rotational invariance of the problem, from this solution we easily build a continuum of solutions. If we consider a discretization of this problem, such as (5.9), with Ω a polygonal approximation of B, we expect a large number of solutions, and convergence to equilibrium is not obvious. It would be reasonable to assume that the solutions to (5.9) are isolated, but such an assumption is impossible to check. Theorem 5.1 guarantees convergence to equilibrium without any knowledge on the structure of solutions for (5.9).


23

Remark 5.3. In Theorem 5.1, the restriction on δt depends only on λ1 ≥ 0 and cf 00 . In particular, it does not depend on the mesh step h (or V h ), and it does not depend on ε. 5.2. A discretization of the Swift-Hohenberg equation with inertial term. We consider the modified Swift-Hohenberg equation εutt + ut + (I + ∆)2 u + f 0 (u) = g(t),

x ∈ Ω, t ≥ 0,

(5.17)

where Ω is a bounded domain of RN with Lipschitz boundary, ε ≥ 0 and g satisfies (5.2). We assume homogeneous Neumann or periodic boundary conditions, and f : R → R is real analytic and satisfies (5.3)-(5.4) (with λ1 = 0). For equation (5.1) the theoretical picture in the continuous case is well known: results concerning the existence and uniqueness of solutions, existence of global attractors, convergence to single equilibria have been proved with various growth assumptions on the nonlinearity (see, e.g., [10, 35, 36] and references therein). On the contrary, the literature on equation (5.17) is scarce, but it seems that similar results can be derived, even more easily. For instance, in a standard Hilbert setting, we can handle nonlinearities without any growth condition in dimension d ≤ 3 since the natural phase space dictated by the basic energy identity is H 2 × L2 . However, let us mention that in the specific case ε = 0, a result related to the well-posedness and to the convergence to equilibrium has been proved in [20] for a quartic nonlinearity f and under suitable smallness assumption on g. In addition, in the case g ≡ 0, existence of global attractors have been analyzed in [30, 32], while existence of nontrivial stationary solutions has been treated in [9] (cf. also references therein). In the case ε = 0, equation (5.17) is known as the Swift-Hohenberg equation [34]. It is an L2 (Ω)-gradient flow of the functional ¸ Z · 1 2 E(u) := (∆u + u) + f (u) dx, Ω 2 and a typical choice for f is the double-well potential f (s) = (s2 −r)2 /4 (with r > 0). This functional has recently been used in Materials Science by Elder and Grant [12, 13] to modeling liquid-solid transition phenomena. Following this approach, Galenko et al. [15] added the inertial term εutt in equation (5.17), in order to extend the domain of applicability of the model. As previously, the set of steady states for (5.17) can contain a continuum, so that convergence to equilibrium is not obvious; for instance, if we assume periodic boundary conditions, it is easy to see that for r large enough, there exists a non constant steady state of (5.17). Thus, by translation invariance of the problem, there is a continuum of equilibria. A conforming finite element space discretization of (5.17) gives the following system (compare with (5.12)): εUtt (t) + Ut (t) + (I − Ah )2 U (t) + ∇F h (U (t)) = G(t), where

Fh

t ≥ 0,

(5.18)

is defined as previously by (5.10) and G by (5.11). The function 1 F (W ) = h(I − Ah )W, (I − Ah )W i + F h (W ) 2 satisfies (2.3) with cF = cf 00 . Moreover, F is coercive by (5.14). Thus, if the time step δt > 0 satisfies 1/δt > cf 00 /2, any solution (U n )n of the backward Euler scheme applied to (5.18) converges to an equilibrium as n tends to ∞.

24


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