Chevron pattern equations: exponential attractor and global stabilization
CCHEVRON PATTERN EQUATIONS: EXPONENTIALATTRACTOR AND GLOBAL STABILIZATION
H. KALANTAROVA ∗ , V. KALANTAROV † AND O. VANTZOS ‡ Abstract.
The initial boundary value problem for a nonlinear systemof equations modeling the chevron patterns is studied in one and twospatial dimensions. The existence of an exponential attractor and thestabilization of the zero steady state solution through application of afinite-dimensional feedback control is proved in two spatial dimensions.The stabilization of an arbitrary fixed solution is shown in one spatialdimension along with relevant numerical results. Introduction
We consider the following coupled system of equations introduced tomodel chevron patterns observed in the dielectric regime of electroconvec-tion in nematic liquid crystals(1.1) τ ∂ t A = A + ∆ A − φ A − | A | A − ic φ∂ y A + iβA∂ y φ, (1.2) ∂ t φ = D ∂ x φ + D ∂ y φ − hφ + φ | A | − c Im [ A ∗ ∂ y A ] , where τ > D > D > c ≥ c ≥ h ≥ β ∈ R are givenparameters, the complex valued function A ( A ∗ denotes its complex conju-gate), which describes the amplitude of the convection pattern and the realvalued function φ , which denotes the angle of the director from the x-axisare unknown functions. This system was proposed by Rossberg et al. [17],[18] to model the formation and evolution of patterns. In his work, Rossbergshows that chevron patterns are observed in simulations of (1.1), (1.2). Theapplication of liquid crystals in display devices, namely LCD screens andtheir use in electronics has made the study of the electro-optical effects inliquid crystals interesting both to researchers in academia and in industry.The rest of the paper is organized as follows. Section 2 is devoted topreliminary theorems and inequalities. In section 3, we prove the existenceof an exponential attractor in two space dimensions. This is an improve-ment of a result from [10], where the existence of a global attractor for thesystem (1.1)-(1.2) is shown. In section 4, we use the finite-dimensional feed-back control approach for stabilizing the zero solution again in two space Date : February 10, 2021.
Key words and phrases. chevron patterns, feedback stabilization, exponential attractor,Fourier modes, finite differences. a r X i v : . [ m a t h . A P ] F e b H. KALANTAROVA ∗ , V. KALANTAROV † AND O. VANTZOS ‡ dimensions. Finally, in section 5 we study chevron patterns in one spa-tial dimension and prove global exponential stabilization to any fixed, notnecessarily stationary, solution. Notation
Throughout this paper, R denotes the set of real numbers, Ω ⊂ R denotesa bounded domain with sufficiently smooth boundary denoted by ∂ Ω.( · , · ) and (cid:107)·(cid:107) denote the inner product and the norm induced by it in L (Ω),respectively. That is, for f, g ∈ L (Ω)( f, g ) := (cid:90) Ω f ( x, y ) g ∗ ( x, y ) dxdy, (cid:107) f (cid:107) := (cid:18)(cid:90) Ω | f ( x, y ) | dxdy (cid:19) / . We are also using the following notation V := L × L , V := H × H , V := H ∩ H × H ∩ H . Preliminaries
In order to keep this work self contained, this section provides a theoremand inequalities that we need to prove results in the future sections.
Theorem 2.1. ( [10] ) If c ∈ [0 , or c ≥ c , h > , the system of equa-tions (1.1) - (1.2) together with the following initial and boundary conditions (2.1) A (cid:12)(cid:12)(cid:12) t =0 = A , φ (cid:12)(cid:12)(cid:12) t =0 = φ , A (cid:12)(cid:12)(cid:12) ∂ Ω = 0 , φ (cid:12)(cid:12)(cid:12) ∂ Ω = 0 , where A , φ ∈ L (Ω) , has a unique weak solution (2.2) A, φ ∈ C ([0 , T ]; L (Ω)) ∩ L ([0 , T ]; H (Ω)) , ∀ T > , such that (2.3) (cid:107) φ ( t ) (cid:107) ≤ M , (cid:107) A ( t ) (cid:107) ≤ M , ∀ t > , and (2.4) (cid:90) T (cid:107)∇ φ ( t ) (cid:107) dt ≤ M T , (cid:90) T (cid:107)∇ A ( t ) (cid:107) dt ≤ M T , ∀ T > , where M denotes a generic constant depending only on (cid:107) A (cid:107) , (cid:107) φ (cid:107) and | Ω | ,and where M T is a generic constant, which depends also on T, besides (cid:107) A (cid:107) , (cid:107) φ (cid:107) and | Ω | . In other words this problem generates a compact semigroup S ( t ), t ≥ , in the phase space V . Moreover this semigroup has a globalattractor that belongs to V . • Young’s inequality
For each a, b > (cid:15) > ab ≤ (cid:15) a p p + 1 (cid:15) q/p b q q , where p, q > p + q = 1 . HEVRON PATTERN EQUATIONS 3 • For u ∈ H (Ω) ∩ H (Ω) the following estimates are valid(2.6) ν (cid:107) ∆ u (cid:107) ≤ (cid:107)L u (cid:107) ≤ ν (cid:107) ∆ u (cid:107) where L is a second order uniformly elliptic operator. • Poincar´e-Friedrichs (P-F) inequality (2.7) (cid:107) u (cid:107) ≤ λ − (cid:107)∇ u (cid:107) , ∀ u ∈ H (Ω)and the inequality(2.8) ∞ (cid:88) k = N +1 | ( u, w k ) | ≤ λ − N +1 (cid:107)∇ u (cid:107) , ∀ u ∈ H (Ω) , where 0 < λ ≤ λ ≤ . . . ≤ λ N +1 ≤ . . . are the eigenvalues of theLaplace operator, − ∆, under the homogeneous Dirichlet’s boundarycondition and w k , for k = 1 , , . . . are the corresponding eigenfunc-tions. • Ladyzhenskaya inequality (2.9) (cid:107) u (cid:107) L (Ω) ≤ / (cid:107) u (cid:107) / (cid:107)∇ u (cid:107) / which is valid for u ∈ H (Ω) with Ω ⊂ R . • The D Agmon inequality (2.10) max x ∈ [0 ,L ] | u ( x ) | ≤ C (cid:107) u (cid:107)(cid:107) ∂ x u (cid:107) , ∀ u ∈ H (0 , L ) . • The D Agmon inequality (2.11) max x ∈ Ω | u ( x ) | ≤ C (cid:107) u (cid:107)(cid:107) ∆ u (cid:107) , ∀ u ∈ H (Ω) ∩ H (Ω) , where Ω ⊂ R . • Gagliardo-Nirenberg inequality
Let Ω ⊂ R n and u ∈ W m,r (Ω) ∩ L q (Ω), 1 ≤ r, q ≤ ∞ . For any integer j , 0 ≤ j ≤ m and for any number ˆ a in the interval jm ≤ ˆ a ≤
1, set1 p − jn = ˆ a (cid:18) r − mn (cid:19) + (1 − ˆ a ) 1 q . If m − j − nr is a nonnegative integer, then(2.12) (cid:107) D j u (cid:107) L p (Ω) ≤ C (cid:107) u (cid:107) ˆ aW m,r (Ω) (cid:107) u (cid:107) − ˆ aL q (Ω) , where the constant C depends only on Ω, r , m , q , j and ˆ a .3. Exponential Attractor
In this section, we are going to show that the semigroup S ( t ) : V → V , t ≥
0, associated to the problem (1.1)-(1.2) and (2.1) has an exponentialattractor, i.e., there exists a set
M ⊂ V which satisfies the following con-ditions:(1) M ⊂ V is compact and has a finite fractal dimension,(2) it is positively invariant, S ( t ) M ⊂ M , ∀ t > H. KALANTAROVA ∗ , V. KALANTAROV † AND O. VANTZOS ‡ (3) M attracts each bounded set B ⊂ V with an exponential rate, i.e.,for each bounded set B ⊂ V dist ( S ( t ) B, M ) ≤ Q ( (cid:107) B (cid:107) V ) e − αt , t > , where the distance function is defined by dist ( S ( t ) B, M ) := sup a ∈ S ( t ) B inf b ∈M (cid:107) a − b (cid:107) V ,α > Q ( · ) are independent of B .For the literature about exponential attractors for various dissipative dy-namical systems we refer to [6], [7], [16] and references therein. To provethe existence of an exponential attractor of the semigroup generated by theproblem (1.1)-(2.1), we use the following theorem Theorem 3.1. ( [16] ) Let E and E be two Banach spaces such that E iscompactly embedded in E . Assume that S ( t ) : E → E is a semigroup, whichpossesses a compact absorbing set B in E : S ( t ∗ ) B ⊂ B for t ∗ > . Assume further that,a) There exists
K > such that (3.1) (cid:107) S ( t ∗ ) ξ − S ( t ∗ ) ξ (cid:107) E ≤ K (cid:107) ξ − ξ (cid:107) E , ∀ ξ , ξ ∈ B . b) The map ( t, ξ ) → S ( t ) ξ is H¨older continuous (or Lipschitz continuous)on [0 , t ∗ ] × B .Then the semigroup S ( t ) possesses an exponential attractor M in E ,which is a subset of B . We proceed by showing that the semigroup S ( t ) : V → V satisfies theconditions stated in Theorem 3.1. First, we show the existence of a compactabsorbing set in V for S ( t ). We recall the following dissipative estimates(3.2) ddt (cid:2) τ (cid:107) A ( t ) (cid:107) + δ (cid:107) φ ( t ) (cid:107) (cid:3) + k (cid:2) τ (cid:107) A ( t ) (cid:107) + δ (cid:107) φ ( t ) (cid:107) (cid:3) + δ (cid:2) (cid:107)∇ A ( t ) (cid:107) + D (cid:107)∇ φ ( t ) (cid:107) (cid:3) ≤ | Ω | and(3.3) ddt (cid:2) τ (cid:107)∇ A ( t ) (cid:107) + ( L φ ( t ) , φ ( t )) (cid:3) + ν (cid:107) ∆ φ (cid:107) + 2 − (cid:107) ∆ A (cid:107) ≤ C (cid:2) τ (cid:107)∇ A (cid:107) + ( L φ, φ ) + 1 (cid:3) (cid:2) τ (cid:107)∇ A (cid:107) + ( L φ, φ ) (cid:3) which are derived in [10, (10)] and [10, (38)], respectively. Here k := min { τ − , h } , δ := 2(1 − c ) / (2 + c ) where c < , (3.4) D := min { D , D } , −L φ := − D ∂ x φ − D ∂ y φ, (3.5)and C is a constant that depends on τ , ν , c , c , β , D , D and M , whichis defined in the statement of Theorem 2.1. It follows from the estimate HEVRON PATTERN EQUATIONS 5 (3.2) that, the set B := (cid:8) [ A, φ ] ⊂ V : τ (cid:107) A (cid:107) + δ (cid:107) φ (cid:107) ≤ k − | Ω | (cid:9) is a positively invariant absorbing set for the semigroup S ( t ), t ≥
0. Theestimate (3.2) also implies that(3.6) δ (cid:90) t +1 t (cid:2) τ (cid:107)∇ A ( s ) (cid:107) + ( L φ ( s ) , φ ( s )) (cid:3) ds ≤ | Ω | (1 + 2 k − ) , ∀ t > . Employing the uniform Gronwall lemma and the estimate (3.3), we deducefrom (3.6) that(3.7) τ (cid:107)∇ A ( t ) (cid:107) + ( L φ ( t ) , φ ( t )) ≤ R , ∀ t ≥ , where R = (1 + C ) | Ω | (1 + 2 k − ) δ e C | Ω | (1+2 k − ) /δ . Hence B ⊂ V , and since V is compactly embedded into V , B is compactin V .Next, we show that the condition a) is satisfied. Let [ A , φ ] and [ ˜ A , ˜ φ ]be arbitrary two elements of B . Then [ a, ψ ] =: [ ˜ A − A, ˜ φ − φ ] the differenceof [ A ( t ) , φ ( t )] = S ( t )[ A , φ ] and [ ˜ A ( t ) , ˜ φ ] = S ( t )[ ˜ A , ˜ φ ] is a solution of thesystem(3.8) τ ∂ t a = a + ∆ a − φ a − ( ˜ φ − φ ) ˜ A − | ˜ A | ˜ A + | A | A − ic [ ψ∂ y ˜ A + φ∂ y a ] + iβ [ a∂ y ˜ φ + A∂ y ψ ] , (3.9) ∂ t ψ = D ∂ x ψ + D ∂ y ψ − hψ + ψ | ˜ A | + φ ( | ˜ A | − | A | ) − c Im (cid:104) a ∗ ∂ y ˜ A + A ∗ ∂ y a (cid:105) under the homogeneous Dirichlet boundary conditions and the initial con-ditions a ( x,
0) = a ( x ) := ˜ A ( x ) − A ( x ) , ψ ( x ) = ˜ φ ( x ) − φ ( x ) , x ∈ Ω . Now, we multiply (3.8) by − ∆ a ∗ and (3.9) by −L ψ , add obtained relationsand use the Young inequality(3.10) 12 ddt [ τ (cid:107)∇ a (cid:107) + ( L ψ, ψ )] − (cid:107)∇ a (cid:107) + 18 (cid:107) ∆ a (cid:107) + 12 (cid:107)L ψ (cid:107) + h ( L ψ, ψ ) ≤ φ , | a | ) + 2( ψ | ˜ φ + φ | , | ˜ A | ) + 2( | ˜ A | , | a | )+ 2( | a | ( | ˜ A | + | A | ) , | A | ) + 8 c ( | ψ | , | ∂ y A | ) + 8 c ( | φ | , | ∂ y A | )+ 4 β ( | a | , | ∂ y ˜ φ | ) + 4 β ( | A | , | ∂ y ψ | ) + 2( ψ , | A | )+ 2( φ | a | , ( | ˜ A | + | A | ) ) + 2 c ( | a | , | ∂ y ˜ A | ) + 2 c ( | A | , | ∂ y a | ) . H. KALANTAROVA ∗ , V. KALANTAROV † AND O. VANTZOS ‡ Next, we estimate terms on the right hand side of (3.10) by employingYoung’s inequality, the Ladyzhenskaya and the Gagliardo-Nirenberg inequal-ities(3.11) 2( φ , | a | ) ≤ (cid:107) φ (cid:107) L (cid:107) a (cid:107) L ∞ ≤ C (cid:107) a (cid:107)(cid:107) ∆ a (cid:107)(cid:107)∇ φ (cid:107) ≤ ε (cid:107) ∆ a (cid:107) + C ( ε ) (cid:107) a (cid:107) (cid:107)∇ φ (cid:107) , (3.12) 2( ψ | ˜ φ + φ | , | ˜ A | ) ≤ (cid:107) φ (cid:107) L ∞ ( (cid:107) ˜ φ (cid:107) L + (cid:107) φ (cid:107) L ) (cid:107) ˜ A (cid:107) L ≤ C (cid:107) ψ (cid:107)(cid:107) ∆ ψ (cid:107) ( (cid:107)∇ ˜ φ (cid:107) + (cid:107)∇ φ (cid:107) ) (cid:107)∇ ˜ A (cid:107) ≤ ε (cid:107)L ψ (cid:107) + C ( ε )( (cid:107)∇ ˜ φ (cid:107) + (cid:107)∇ φ (cid:107) ) (cid:107)∇ ˜ A (cid:107) (cid:107) ψ (cid:107) , (3.13) 2( | ˜ A | , | a | ) ≤ ε (cid:107) ∆ a (cid:107) + C ( ε ) (cid:107) a (cid:107) (cid:107)∇ ˜ A (cid:107) , (3.14) 2( | a | ( | ˜ A | + | A | ) , | A | ) ≤ ε (cid:107) ∆ a (cid:107) + C ( ε )( (cid:107)∇ ˜ A (cid:107) + (cid:107)∇ A (cid:107) ) (cid:107)∇ A (cid:107) (cid:107) a (cid:107) , (3.15) 8 c ( | ψ | , | ∂ y A | ) ≤ C (cid:107) ψ (cid:107)(cid:107) ∆ ψ (cid:107)(cid:107)∇ A (cid:107) ≤ ε (cid:107)L ψ (cid:107) + C ( ε ) (cid:107) ψ (cid:107) (cid:107)∇ A (cid:107) . The rest of the terms on the right hand side of (3.10) can be estimatedsimilarly. Utilizing the estimates (3.11)-(3.15) in (3.10) and choosing 9 ε = , ε = we arrive at the inequality(3.16) ddt (cid:2) τ (cid:107)∇ a (cid:107) + ( L ψ, ψ ) (cid:3) + τ (cid:107)∇ a (cid:107) + h ( L ψ, ψ ) ≤ M (cid:2) (cid:107) a (cid:107) + (cid:107) ψ (cid:107) (cid:3) , where M depends only on (cid:107)∇ A (cid:107) , (cid:107)∇ φ (cid:107) , (cid:107)∇ ˜ A (cid:107) , (cid:107)∇ ˜ φ (cid:107) and t . The desiredsmoothing estimate (3.1) follows from the last inequality and the Lipschitzcontinuity of the semigroup with respect to [ A, φ ] in V established in [10,Theorem II.1].It remains to show that S ( t ) is H¨older continuous with respect to t , thatit satisfies the condition b). Multiplying (1.1) and (1.2) by ∂ t A ∗ and ∂ t φ respectively, and then using the estimates (3.2), (3.3) we deduce that (cid:90) T (cid:2) (cid:107) ∂ t A ( t ) (cid:107) + (cid:107) ∂ t φ ( t ) (cid:107) (cid:3) dt ≤ C T , ∀ T > , ∀ [ A, φ ] ∈ B , where C T depends only on T and B . Thus for each t , t ∈ [0 , T ] and[ A, φ ] ∈ B [ A ( t ) − A ( t ) , φ ( t ) − φ ( t )] = (cid:90) t t [ ∂ t A ( t ) , ∂ t φ ( t )] dt, HEVRON PATTERN EQUATIONS 7 and thanks to the Cauchy-Schwarz inequality we have (cid:107) [ A ( t ) − A ( t ) , φ ( t ) − φ ( t )] (cid:107) V = (cid:90) t t (cid:107) [ ∂ t A ( t ) , ∂ t φ ( t )] (cid:107) dt ≤ | t − t | (cid:90) T (cid:2) (cid:107) ∂ t A ( t ) (cid:107) + (cid:107) ∂ t φ ( t ) (cid:107) (cid:3) ≤ C T | t − t | . Hence all conditions of the Theorem 3.1 are satisfied. So we proved thefollowing theorem:
Theorem 3.2.
If the conditions of the Theorem 2.1 are satisfied, then thesemigroup S ( t ) : V → V , t ≥ possesses an exponential attractor. Feedback stabilization by finitely many Fourier modes
This section is devoted to the problem of global stabilization of the zerosteady state of the system (1.1)-(1.2) by finitely many Fourier modes. Thereare a number of papers on the problem of stabilization of solutions of non-linear PDE’s using finite-dimensional feedback controllers (see, e.g. [1], [3],[4], [5], [8], [9], [11], [15] and references therein). Our study of this questionis mainly inspired by the results obtained in [1], where the authors usedvarious types of finite-dimensional controllers to stabilize the zero steadystate solution to semilinear parabolic equations. Accordingly, we study thefollowing feedback control system τ ∂ t A = A + ∆ A − φ A − | A | A − ic φ∂ y A (4.1) + iβA∂ y φ − µ N (cid:88) k =1 ( A, w k ) w k ,∂ t φ = D ∂ x φ + D ∂ y φ − hφ + φ | A | − c Im [ A ∗ ∂ y A ] , (4.2)(4.3) A (cid:12)(cid:12) ∂ Ω = φ (cid:12)(cid:12) ∂ Ω = 0 , A (cid:12)(cid:12) t =0 = A , φ (cid:12)(cid:12) t =0 = φ , where A , φ ∈ L (Ω) are given functions and the conditions on µ > ddt (cid:2) τ (cid:107) A (cid:107) + δ (cid:107) φ (cid:107) (cid:3) − (cid:107) A (cid:107) + (cid:107) A (cid:107) L + δ D (cid:107)∇ φ (cid:107) + δ h (cid:107) φ (cid:107) + δ (cid:107)∇ A (cid:107) ≤ − µ N (cid:88) k =1 | ( A, w k ) | , where δ and D are as defined in (3.4)-(3.5), are essentially the same withthe proof of Theorem 2.1 in Section 2 and the estimate (2.9) in [10], respec-tively, which we will therefore omit. H. KALANTAROVA ∗ , V. KALANTAROV † AND O. VANTZOS ‡ Employing the inequality (2.8), we infer from (4.4) that(4.5) 12 ddt (cid:2) τ (cid:107) A (cid:107) + δ (cid:107) φ (cid:107) (cid:3) + ( µ − N (cid:88) k =1 | ( A, w k ) | + (cid:107) A (cid:107) L + δ D (cid:107)∇ φ (cid:107) + δ h (cid:107) φ (cid:107) + ( δ − λ − N +1 ) (cid:107)∇ A (cid:107) ≤ . Choosing µ ≥ N large enough such that λ − N +1 < δ and using thePoincar´e inequality, it follows from (4.5) that(4.6) 12 ddt (cid:2) τ (cid:107) A (cid:107) + δ (cid:107) φ (cid:107) (cid:3) + δ D (cid:107)∇ φ (cid:107) + δ h (cid:107) φ (cid:107) + δ (cid:107) A (cid:107) + (cid:107) A (cid:107) L ≤ , where δ := λ ( δ − λ − N +1 ). From this inequality we infer that12 ddt (cid:2) τ (cid:107) A (cid:107) + δ (cid:107) φ (cid:107) (cid:3) + m (cid:2) τ (cid:107) A (cid:107) + δ (cid:107) φ (cid:107) (cid:3) ≤ , where m = min { h + D λ , δ τ } . Thus we have(4.7) τ (cid:107) A ( t ) (cid:107) + δ (cid:107) φ ( t ) (cid:107) ≤ e − m t (cid:2) τ (cid:107) A (cid:107) + δ (cid:107) φ (cid:107) (cid:3) , which proves that all solutions of the controlled problem tend to the zerosteady state as t → ∞ with an exponential rate in L (Ω) whenever(4.8) µ ≥ λ − N +1 < δ . Finally, we would like to note that, the estimates obtained above allow usto claim the global unique solvability of the problem (4.1)-(4.3) in C (0 , T ; V ) ∩ L (0 , T ; V ) , ∀ T > . Moreover the smoothing estimate (3.16) is also valid for solutions of theproblem (4.1)-(4.3). Thus the following theorem holds true:
Theorem 4.1.
If the number N and µ satisfy the conditions (4.8) , then thesolution [ A ( t ) , φ ( t )] of the problem (4.1) - (4.3) tends to zero stationary statewith an exponential rate in V .
1D Chevron pattern equations
In this section, we study 1D version of the system (1.1)-(2.1), reduced toone spatial dimension by eliminating the dependence on y and taking x tolie in the interval Ω = [0 , L ].(5.1) τ ∂ t A − ∂ x A − A + | A | A + φ A = 0 ,∂ t φ − D ∂ x φ + hφ − | A | φ = 0 , x ∈ (0 , L ) , t > ,A (cid:12)(cid:12) x =0 = A (cid:12)(cid:12) x = L = φ (cid:12)(cid:12) x =0 = φ (cid:12)(cid:12) x = L = 0 ,A (cid:12)(cid:12) t =0 = A , φ (cid:12)(cid:12) t =0 = φ , HEVRON PATTERN EQUATIONS 9
We take the scalar product of the first equation in (5.1) with A ∗ , and thesecond equation with φ . Then after a series of integration by parts, we addthe resulting equations and obtain(5.2) 12 ddt (cid:2) τ (cid:107) A (cid:107) + (cid:107) φ (cid:107) (cid:3) + (cid:107) ∂ x A (cid:107) + D (cid:107) ∂ x φ (cid:107) −(cid:107) A (cid:107) + (cid:107) A (cid:107) L + h (cid:107) φ (cid:107) = 0 . Employing the Poincar´e inequality and the following inequality (cid:107) A (cid:107) ≤ L (cid:107) A (cid:107) L ((0 ,L )) , in (5.2), we get ddt (cid:2) τ (cid:107) A ( t ) (cid:107) + (cid:107) φ ( t ) (cid:107) (cid:3) + α (cid:2) τ (cid:107) A ( t ) (cid:107) + (cid:107) φ ( t ) (cid:107) (cid:3) + (cid:107) A (cid:107) L + 2 h (cid:107) φ (cid:107) ≤ L, where α = min { π τL + τ , π D L } . Then(5.3) τ (cid:107) A ( t ) (cid:107) + (cid:107) φ ( t ) (cid:107) ≤ e − αt (cid:2) τ (cid:107) A (cid:107) + (cid:107) φ (cid:107) (cid:3) + Lα , due to Gronwall lemma, and integrating the resulting inequality in t over(0 , T ), yields(5.4) (cid:90) T (cid:2) τ (cid:107) A ( t ) (cid:107) + (cid:107) φ ( t ) (cid:107) (cid:3) dt ≤ − e − αT α ( τ (cid:107) A (cid:107) + (cid:107) φ (cid:107) ) + LTα , ∀ T > Remark 5.1.
We can prove that the semigroup S ( t ) : V → V generatedby the problem (5.1) possesses an exponential attractor, using a similarargument to the one in section 3.It is easy to see that ddt Λ( t ) = − (cid:107) ∂ t A ( t ) (cid:107) − (cid:107) ∂ t φ ( t ) (cid:107) , whereΛ( t ) := (cid:107) ∂ x A ( t ) (cid:107) + D (cid:107) ∂ x φ ( t ) (cid:107) + 12 (cid:107) A ( t ) (cid:107) L +( φ ( t ) , | A ( t ) | ) −(cid:107) A ( t ) (cid:107) + h (cid:107) φ ( t ) (cid:107) . So Λ( t ) is a Lyapunov function for the system. Therefore the global attractorof the system consists of stationary states and trajectories joining them (if L (cid:29) W,
0] (when L (cid:29) W is astationary state of 1D Ginzburg-Landau equation: (cid:26) − W (cid:48)(cid:48) − W + | W | W = 0 , x ∈ (0 , L ) ,W (0) = W ( L ) = 0 , (see, e.g., [2]).Moreover the system possesses an inertial manifold (see, e.g., [12], [13]and references therein). ∗ , V. KALANTAROV † AND O. VANTZOS ‡ Feedback stabilization.
The zero solution of the system (5.1) is un-stable for L (cid:29)
1, which can easily be seen by setting φ ≡ A = O ( (cid:15) ),where (cid:15) (cid:28)
1. In this case, the solution of the linearization of the firstequation in (5.1) around zero τ ∂ t A − ∂ x A − A = 0 , has the form A = ∞ (cid:88) k =0 α k sin (cid:18) kπL x (cid:19) , where α k ( t ) = α k (0) e − (cid:16) k π − L τL (cid:17) t , which implies that for L (cid:29) t → ∞ , A does not go to zero. Motivatedby this observation, we study the feedback stabilization of the 1D systemby using the a priori estimates (5.3) and (5.4). Contrary to section 4 whereit is the zero steady state that is stabilized, in the 1D case we can stabilizeany time-dependent, not necessarily stationary, solution.We start with deriving the uniform estimates of solutions to (5.1). Multi-plying the first equation in (5.1) by − ∂ x A ∗ in L , and utilizing the inequality − Re ( | A | A, ∂ x A ) ≥ ( | A | , | ∂ x A | ) , gives us(5.5) τ ddt (cid:107) ∂ x A (cid:107) + (cid:107) ∂ x A (cid:107) − (cid:107) ∂ x A (cid:107) + ( | A | , | ∂ x A | ) + ( φ , | ∂ x A | ) ≤ | ( φ∂ x φ, A ∗ ∂ x A ) | ≤ ε ( | A | , | ∂ x A | ) + C ε ( | φ | , | ∂ x φ | ) . We estimate the last terms on the right hand side of (5.5) and the term (cid:107) ∂ x A (cid:107) , by using the Gagliardo-Nirenberg inequality (2.12) (cid:107) u (cid:107) L ≤ C (cid:107) u (cid:107) / (cid:107) ∂ x u (cid:107) / , (cid:107) ∂ x u (cid:107) L ≤ C (cid:107) u (cid:107) / (cid:107) ∂ x u (cid:107) / and Young’s inequality, as follows (cid:107) ∂ x A (cid:107) ≤ ε (cid:107) A (cid:107) + ε (cid:107) ∂ x A (cid:107) ,C ε ( | φ | , | ∂ x φ | ) ≤ C ε (cid:107) φ (cid:107) L (cid:107) ∂ x φ (cid:107) L ≤ C ε C (cid:107) φ (cid:107) / (cid:107) ∂ x φ (cid:107) / (5.6) ≤ ε (cid:107) ∂ x φ (cid:107) + C ε (cid:107) φ (cid:107) . So, we have(5.7) τ ddt (cid:107) ∂ x A (cid:107) + (1 − ε ) (cid:107) ∂ x A (cid:107) + (1 − ε )( | A | , | ∂ x A | ) ≤ ε (cid:107) ∂ x φ (cid:107) + C ε (cid:107) φ (cid:107) + 14 ε (cid:107) A (cid:107) . HEVRON PATTERN EQUATIONS 11
Next we multiply the second equation in (5.1) by − ∂ x φ in L :(5.8) 12 ddt (cid:107) ∂ x φ (cid:107) + D (cid:107) ∂ x φ (cid:107) + h (cid:107) ∂ x φ (cid:107) ≤ ( | A | , | ∂ x φ | )+ 2( | A || ∂ x A | , | φ || ∂ x φ | ) ≤ | A | , | ∂ x φ | ) + ( φ , | ∂ x A | )We estimate the first term on the right hand side of (5.8), again by usingGagliardo-Nirenberg inequality(5.9) 2( | A | , | ∂ x φ | ) ≤ (cid:107) A (cid:107) L (cid:107) ∂ x φ (cid:107) L ≤ C (cid:107) A (cid:107) (cid:107) ∂ x A (cid:107) (cid:107) φ (cid:107) (cid:107) ∂ x φ (cid:107) ≤ ε (cid:107) ∂ x A (cid:107) + ε (cid:107) ∂ x φ (cid:107) + C ( ε ) (cid:107) A (cid:107) + C ( ε ) (cid:107) φ (cid:107) . Combining (5.6) and (5.9) with (5.8), it follows that12 ddt (cid:107) ∂ x φ (cid:107) + ( D − ε ) (cid:107) ∂ x φ (cid:107) + h (cid:107) ∂ x φ (cid:107) ≤ ε (cid:107) ∂ x A (cid:107) + C ε (cid:107) φ (cid:107) + C ( ε ) (cid:107) A (cid:107) + C ( ε ) (cid:107) φ (cid:107) . Adding the above inequality to (5.7) we get12 ddt (cid:2) τ (cid:107) ∂ x A (cid:107) + (cid:107) ∂ x φ (cid:107) (cid:3) + (1 − ε ) (cid:107) ∂ x A (cid:107) + (1 − ε )( | A | , | ∂ x A | )+ ( D − ε ) (cid:107) ∂ x φ (cid:107) + h (cid:107) ∂ x φ (cid:107) ≤ C ε (cid:107) φ (cid:107) + 14 ε (cid:107) A (cid:107) + C ( ε ) (cid:107) A (cid:107) + C ( ε ) (cid:107) φ (cid:107) The resulting inequality after setting ε = , ε = D in the aboveinequality implies the dissipativity of the system in the phase space V .More precisely the following inequality holds true(5.10) ddt E ( t ) + d E ( t ) ≤ Q ( (cid:107) A (cid:107) ) + Q ( (cid:107) φ ) (cid:107) ) , where E ( t ) := τ (cid:107) ∂ x A (cid:107) + (cid:107) ∂ x φ (cid:107) , d = π L min { τ , D } and Q ( · ), Q ( · ) aremonotone functions.Assume that [ A, φ ] is an arbitrary given and possibly time-dependentsolution of the problem (5.1) and consider the following feedback controlsystem(5.11) τ ∂ t ˜ A − ∂ x ˜ A − ˜ A + | ˜ A | ˜ A + ˜ φ ˜ A = − µ N (cid:80) k =1 ( ˜ A, w k ) w k ( x ) ,∂ t ˜ φ − D ∂ x ˜ φ + h ˜ φ − | ˜ A | ˜ φ = 0 , x ∈ (0 , L ) , t > , ˜ A (cid:12)(cid:12) x =0 = ˜ A (cid:12)(cid:12) x = L = ˜ φ (cid:12)(cid:12) x =0 = ˜ φ (cid:12)(cid:12) x = L = 0 , ˜ A (cid:12)(cid:12) t =0 = ˜ A , ˜ φ (cid:12)(cid:12) t =0 = ˜ φ , ∗ , V. KALANTAROV † AND O. VANTZOS ‡ where w k ( x ) = sin( kπL x ) , k = 1 , , ... . Then the pair of functions [ a, ψ ] :=[ ˜ A − A, ˜ φ − φ ] is a solution of the problem(5.12) τ ∂ t a − ∂ x a − a + | ˜ A | ˜ A − | A | A + φ a + ( ˜ φ − φ ) ˜ A = − µ N (cid:80) k =1 ( a, w k ) w k ( x ) ,∂ t ψ − D ∂ x ψ + hψ − | A | ψ = ( | ˜ A | − | A | ) ˜ φ − µ N (cid:80) k =1 ( ψ, w k ) w k ( x ) , x ∈ (0 , L ) , t > ,a (cid:12)(cid:12) x =0 = a (cid:12)(cid:12) x = L = ψ (cid:12)(cid:12) x =0 = ψ (cid:12)(cid:12) x = L = 0 , ˜ a (cid:12)(cid:12) t =0 = a , ψ (cid:12)(cid:12) t =0 = ψ , where a = ˜ A − A , ψ = ˜ φ − φ . Taking the inner product of the firstequation in (5.12) with a ∗ and of the second equation with ψ , we get(5.13) τ ddt (cid:107) a (cid:107) + (cid:107) ∂ x a (cid:107) − (cid:107) a (cid:107) + ( φ , | a | ) + ( | ˜ A | , | a | )+ Re ( ˜ φ − φ , ˜ Aa ) ≤ ( | ˜ A || A | , | a | ) + ( | A | , | a | ) − µ N (cid:88) k =1 | ( a, w k ) | , (5.14) 12 ddt (cid:107) ψ (cid:107) + D (cid:107) ∂ x ψ (cid:107) + h (cid:107) ψ (cid:107) =( | A | , | ψ | ) + ( | A | − | ˜ A | , ˜ φψ ) − µ N (cid:88) k =1 ( ψ, w k ) . We use the Sobolev inequality (cid:107) u (cid:107) L ∞ ≤ (cid:107) ∂ x u (cid:107) , H¨older’s inequality andYoung’s inequality (2.5) to produce the following estimates(5.15) | ( φ − ˜ φ , ˜ Aa ) | ≤ ( | ψ || φ + ˜ φ | , | ˜ A || a | ) ≤
12 ( | a | , | A | ) + 12 ( | ψ | , | φ + ˜ φ | ) ≤ (cid:107) A (cid:107) L ∞ (cid:107) a (cid:107) + ( (cid:107) φ (cid:107) L ∞ + (cid:107) ˜ φ (cid:107) L ∞ ) (cid:107) ψ (cid:107) ≤ (cid:107) ∂ x A (cid:107) (cid:107) a (cid:107) + ( (cid:107) ∂ x φ (cid:107) + (cid:107) ∂ x ˜ φ (cid:107) ) (cid:107) ψ (cid:107) , ( | A | , | a | ) ≤ (cid:107) ∂ x A (cid:107) (cid:107) a (cid:107) , ( | ˜ A || A | , | a | ) ≤ ( (cid:107) ∂ x A (cid:107) + (cid:107) ∂ x ˜ A (cid:107) ) (cid:107) a (cid:107) , (5.16) ( | A | , ψ ) ≤ (cid:107) ∂ x A (cid:107) (cid:107) ψ (cid:107) , (5.17) | ( | A | − | ˜ A | , ˜ φψ ) | ≤ ( | a || A + ˜ A | , | ˜ φ || ψ | )(5.18) ≤
12 ( | A + ˜ A | , | a | ) + 12 ( | ˜ φ | , | ψ | ) ≤ ( (cid:107) ∂ x A (cid:107) + (cid:107) ∂ x ˜ A (cid:107) ) (cid:107) a (cid:107) + 12 (cid:107) ∂ x ˜ φ (cid:107) (cid:107) ψ (cid:107) . HEVRON PATTERN EQUATIONS 13
Adding (5.13) to (5.14) and using the estimates (5.16)-(5.18), we get(5.19) 12 ddt (cid:2) τ (cid:107) a (cid:107) + (cid:107) ψ (cid:107) (cid:3) + (cid:107) ∂ x a (cid:107) + D (cid:107) ∂ x ψ (cid:107) − (cid:107) a (cid:107) + h (cid:107) ψ (cid:107) ≤ (cid:18) (cid:107) ∂ x A (cid:107) + 2 (cid:107) ∂ x ˜ A (cid:107) (cid:19) (cid:107) a (cid:107) + (cid:18) (cid:107) ∂ x A (cid:107) + (cid:107) ∂ x φ (cid:107) + 32 (cid:107) ∂ x ˜ φ (cid:107) (cid:19) (cid:107) ψ (cid:107) − µ N (cid:88) k =1 | ( a, w k ) | − µ N (cid:88) k =1 | ( ψ, w k ) | . Since (cid:107) ∂ x ˜ A (cid:107) , (cid:107) ∂ x A (cid:107) , (cid:107) ∂ x φ (cid:107) , (cid:107) ∂ x ˜ ψ (cid:107) ≤ M , we have12 ddt (cid:2) τ (cid:107) a (cid:107) + (cid:107) ψ (cid:107) (cid:3) + (cid:107) ∂ x a (cid:107) + D (cid:107) ∂ x ψ (cid:107) − (cid:107) a (cid:107) + h (cid:107) ψ (cid:107) ≤ M ( (cid:107) a (cid:107) + (cid:107) ψ (cid:107) ) − µ N (cid:88) k =1 | ( a, w k ) | − µ N (cid:88) k =1 | ( ψ, w k ) | . We rewrite the above inequality in the following form(5.20) 12 ddt (cid:2) τ (cid:107) a (cid:107) + (cid:107) ψ (cid:107) (cid:3) + (cid:107) ∂ x a (cid:107) + D (cid:107) ∂ x ψ (cid:107) + ( µ − − M ) N (cid:88) k =1 | ( a, w k ) | + ( µ + h − M ) N (cid:88) k =1 | ( ψ, w k ) | ≤ (1 + 6 M ) ∞ (cid:88) k = N +1 | ( a, w k ) | + 6 M ∞ (cid:88) k = N +1 | ( ψ, w k ) | ≤ (1 + 6 M ) λ − N +1 (cid:107) ∂ x a (cid:107) + 6 M λ − N +1 (cid:107) ∂ x ψ (cid:107) . Assuming that(5.21) (1+6 M ) λ − N +1 ≤ , M λ − N +1 ≤ D , µ ≥ M , µ + h ≥ M and using the inequality (2.8) in (5.12), we derive the desired inequality ddt (cid:2) τ (cid:107) a (cid:107) + (cid:107) ψ (cid:107) (cid:3) + λ (cid:2) (cid:107) a (cid:107) + D (cid:107) ψ (cid:107) (cid:3) ≤ τ (cid:107) a ( t ) (cid:107) + (cid:107) ψ ( t ) (cid:107) ≤ (cid:2) τ (cid:107) a (cid:107) + (cid:107) ψ (cid:107) (cid:3) e − λ r t , where r := min { τ , D } .Hence we proved the following Theorem 5.2.
Let [ A ( t ) , φ ( t )] be a given solution of the problem (5.1) . If µ , µ , N and N are so large that (5.21) is satisfied then each solution ofthe controlled problem (5.11) tends to [ A ( t ) , φ ( t )] as t → ∞ in V with anexponential rate. ∗ , V. KALANTAROV † AND O. VANTZOS ‡ Remark 5.3.
Let us note that analog of the Theorem 5.2 is true also forthe 2D system of equations. It can be proved in a similar way thanks to theestimates (3.3), (3.7).5.2.
Numeric results.
We consider the following one-dimensional versionof the Chevron problem (4.1)–(4.2) over the interval 0 ≤ x ≤ L , where thedependence on the y -direction has been eliminated: τ ∂ t A − ∂ x A − A + | A | A + φ A = 0 , (5.22) ∂ t φ − D ∂ x φ + hφ − | A | φ = 0 , x ∈ (0 , L ) , t > , (5.23) A (cid:12)(cid:12) x =0 = A (cid:12)(cid:12) x = L = φ (cid:12)(cid:12) x =0 = φ (cid:12)(cid:12) x = L = 0 , (5.24) A (cid:12)(cid:12) t =0 = A , φ (cid:12)(cid:12) t =0 = φ , (5.25)In order to study the problem numerically, we introduce the following semi-implicit finite-differences scheme with uniform space and time discretizationsteps δx = L/N and δt > A ki ≈ A ( iδx, kδt ), for 0 ≤ i ≤ N : τ A k +1 i − A ki δt − A k +1 i − − A k +1 i + A k +1 i +1 δx + ( H k + ) i A k +1 i = ( H k − ) i A ki , (5.26) φ k +1 i − φ ki δt − D φ k +1 i − − φ k +1 i + φ k +1 i +1 δx + ( G k + ) i φ k +1 i = ( G k − ) i φ ki , (5.27) ( H k + ) i = | A ki | + | φ ki | , ( H k − ) i = 1 , (5.28) ( G k + ) i = h, ( G k − ) i = | A ki | , (5.29) A k +10 = A k +1 N = φ k +10 = φ k +1 N = 0 , (5.30) A i = A ( iδx ) , φ i = φ ( iδx ) , ≤ i ≤ N . (5.31)This leads to a linear system of the form: τδt ( A k +1 h − A kh ) + ∆ h A k +1 h + H k + A k +1 h = H k − A kh , (5.32) 1 δt ( φ k +1 h − φ kh ) + D ∆ h φ k +1 h + G k + φ k +1 h = G k − φ kh , (5.33)where the subscript h denotes a vector of the form f h = { f i } Ni =0 . Thematrix ∆ h is tridiagonal and positive definite, with diagonal entries (∆ h ) ii = δx and off-diagonal entries (∆ h ) i,i +1 = (∆ h ) i − ,i = − δx , reflecting theDirichlet boundary conditions at x = 0 and x = L . The matrices H k ± and G k ± are diagonal with the non-negative vectors H k ± and G k ± resp. inthe main diagonal, and therefore symmetric and positive semi-definite. Byconstruction, H k + − H k − = H kh = − | A kh | + | φ kh | and G k + − G k − = G kh = h − | φ kh | .Although a full analysis is beyond the scope of this paper, the behavior ofthe proposed scheme is captured by the following estimates for the growthof the L vector norm (cid:107) u h (cid:107) = (cid:80) Ni =0 | u i | and the discrete H semi-norm HEVRON PATTERN EQUATIONS 15 u † h ∆ h u h of the solution:(5.34) 12 (cid:18) − δtτ (cid:107) H k − (cid:107) ∞ (cid:19) (cid:107) A k +1 h (cid:107) + δtτ ( A k +1 h ) † ∆ h A k +1 h + δtτ ( A k +1 h ) † H k + A k +1 h ≤ (cid:18) δtτ (cid:107) H k − (cid:107) ∞ (cid:19) (cid:107) A kh (cid:107) , (5.35) 12 (cid:16) − δt (cid:107) G k − (cid:107) ∞ (cid:17) (cid:107) φ k +1 h (cid:107) + δtD ( φ k +1 h ) † ∆ h φ k +1 h + δt ( φ k +1 h ) T G k + φ k +1 h ≤ (cid:16) δt (cid:107) G k − (cid:107) ∞ (cid:17) (cid:107) φ kh (cid:107) . and the energy-type inequalities:(5.36) τ δt (cid:107) A k +1 h − A kh δt (cid:107) + 12 ( A k +1 h ) † ∆ h A k +1 h + 12 ( A k +1 h ) † H k +1 h A k +1 h ≤ δt (cid:107) A k +1 h (cid:107) (cid:107) H k +1 h − H kh δt (cid:107) ∞ + 12 ( A kh ) † ∆ h A kh + 12 ( A kh ) † H kh A kh (5.37) δt (cid:107) φ k +1 h − φ kh δt (cid:107) + D φ k +1 h ) T ∆ h φ k +1 h + 12 ( φ k +1 h ) T G k +1 h φ k +1 h ≤ δt (cid:107) φ k +1 h (cid:107) (cid:107) G k +1 h − G kh δt (cid:107) ∞ + D A kh ) T ∆ h φ kh + 12 ( φ kh ) T G kh φ kh The first two estimates follow from multiplying the equations (5.32)-(5.33)from the left with ( A k +1 h ) † and ( φ k +1 h ) T respectively. Likewise the other twoestimates follow from left multiplication with ( A k +1 h − A kh ) † and ( φ k +1 h − φ kh ) T .Note also the use of the infinity vector norm (cid:107) u h (cid:107) ∞ = max i | u i | .In matrix form, the system (5.32)-(5.33) can be written as M kA A k +1 h = L kA A kh , (5.38) M kφ φ k +1 h = L kφ φ kh , (5.39)with M kA = I + δtτ ∆ h + δtτ H k + , L kA = I + δtτ H k − (5.40) M kφ = I + δtD ∆ h + δt G k + , L kφ = I + δt G k − . (5.41)The matrices M kA and M kφ are symmetric, unconditionally positive-definiteand sparse, and hence the system (5.38)-(5.39) can be solved efficiently,either directly via a suitable factorization routine, or iteratively via for in-stance the Conjugate Gradient method. ∗ , V. KALANTAROV † AND O. VANTZOS ‡ For the stabilization with K discrete modes W jh ∈ R N , 1 ≤ j ≤ K stackedin an N × K matrix W ij = ( W jh ) i , we extend the scheme (5.32)-(5.33) to τδt ( A k +1 h − A kh ) + ∆ h A k +1 h + H k + A k +1 h = H k − A kh − µ WW T A k +1 h , (5.42) 1 δt ( φ k +1 h − φ kh ) + D ∆ h φ k +1 h + G k + φ k +1 h = G k − φ kh , (5.43)and correspondingly, in matrix form,( M kA + µδtτ WW T ) A k +1 h = L kA A kh , (5.44) M kφ φ k +1 h = L kφ φ kh . (5.45)The stabilized matrix M kA,W = M kA + µδtτ WW T is not sparse, but can stillbe inverted via an iterative method, such as CG, as a low-rank update ofa sparse matrix for the extra cost of K vector-vector products per matrix-vector product.Let 0 < λ < . . . < λ N be the eigenvalues of ∆ h and V jh correspondingeigenvectors, mutually orthogonal and normalized so that (cid:107) V jh (cid:107) = 1. Then(5.46) (cid:107) ( M kA ) − L kA V jh (cid:107) ≤ δtτ (cid:107) H k − (cid:107) ∞ δtτ λ j and likewise for φ . This implies that modes with λ j > (cid:107) H k − (cid:107) ∞ are damped(independent of the time step δt notably). For the stabilized system on theother hand,(5.47) (cid:107) ( M kA,W ) − L kA V jh (cid:107) ≤ δtτ (cid:107) H k − (cid:107) ∞ δtτ λ j + µδtτ (cid:107) W T V jh (cid:107) since the eigenvectors V jh constitute an orthonormal basis and threfore (cid:107) M kA,W V jh (cid:107) = N (cid:88) l =1 | ( V lh ) T M kA,W V jh | ≥ | ( V jh ) T M kA,W V jh | = (cid:16) δtτ λ j + µδtτ (cid:107) W T V jh (cid:107) (cid:17) , It follows that we have full damping if λ j + µ (cid:107) W T V jh (cid:107) > (cid:107) H k − (cid:107) ∞ for allmodes with λ j ≤ (cid:107) H k − (cid:107) ∞ .In the Chevron system (5.26)-(5.31), we have ( H k − ) i = 1, for all 0 ≤ i ≤ N ,and so (cid:107) H k − (cid:107) ∞ = 1 at all times t k . This implies that we can choose a fixed µ > W such that λ j + µ (cid:107) W T V jh (cid:107) > λ j < = 1, and then (cid:107) M kA,W V jh (cid:107) < V jh , and so (cid:107) A k +1 h (cid:107) = (cid:107) M kA,W A kh (cid:107) < (cid:107) A kh (cid:107) .On the other hand (cid:107) G k − (cid:107) ∞ = (cid:107)| A kh | (cid:107) ∞ ≤ (cid:107) A kh (cid:107) ≤ (cid:107) A h (cid:107) because of thestabilisation of A . So if in addition λ j + µ (cid:107) W T V jh (cid:107) > (cid:107) A h (cid:107) for any λ j ≤ (cid:107) A k (cid:107) , then (cid:107) φ k +1 h (cid:107) < (cid:107) φ kh (cid:107) . We conclude that: HEVRON PATTERN EQUATIONS 17 if µ, (cid:15) > and W ∈ R K × N are chosen such that λ j + µ (cid:107) W T V jh (cid:107) ≥ max(1 , (cid:107) A h (cid:107) )+ (cid:15) for all λ j ≤ max(1 , (cid:107) A h (cid:107) ) ,then (cid:107) A kh (cid:107) , (cid:107) φ kh (cid:107) → . It is worth noting that although the bounds above depend on the initialdata (cid:107) A h (cid:107) in order to secure the monotone decay of (cid:107) A kh (cid:107) , (cid:107) φ kh (cid:107) , thediscrete solution does eventually decay to 0 regardless of (cid:107) A h (cid:107) . Indeed, wehave already shown that A can be stabilised with a number of modes thatis independent of the initial data. If the bound above is not immediatelysatisfied, the destabilising term | A | φ in the second equation might be toostrong initially, but as | A | → φ also beginsto decay.Using discrete Fourier modes as W jh , i.e. W ji ∼ cos( jπL iδx ) and scaled sothat (cid:107) V jh (cid:107) = 1, which correspond to the eigenvalues λ j = ( jπL ) , we canmake the choice, albeit somewhat over-cautious, µ = max(1 , (cid:107) A h (cid:107) ) andkeep the first K = (cid:100)√ µ Lπ (cid:101) + 1 modes. In Fig. 1-2, we present the evolutionof a highly oscillatory initial condition, so that all modes are present withhigh probability, in the presence and absence resp. of stabilisation. Under nostabilisation, the solution in Fig. 1 tends towards the steady state | A | = √ h and φ = √ − h predicted by the dynamic analysis (modulo boundary layersat the two ends of the domain due to the Dirichlet boundary conditions).Applying full stabilisation with the number of modes prescribed above, weobserve in Fig. 2 monotone decay of the solution towards the zero state aspredicted. Remark 5.4.
Similar to how it is shown in the section 3 we can showthat the semigroup S ( t ) : V → V generated by the problem possesses anexponential attractor. ∗ , V. KALANTAROV † AND O. VANTZOS ‡ Re(A),Im(A),|A| at time 0 phi at time 0
Re(A),Im(A),|A| at time 10 phi at time 10
Re(A),Im(A),|A| at time 50 phi at time 50
Re(A),Im(A),|A| at time 90 phi at time 90 ||A||_2 vs time ||phi||_2 vs time
Figure 1.
Evolution of a highly-oscillatory initial conditiontowards a non-trivial steady state. The parameters are τ = D = 1 , L = 100 and h = 0 . HEVRON PATTERN EQUATIONS 19
Re(A),Im(A),|A| at time 0 phi at time 0
Re(A),Im(A),|A| at time 1 phi at time 1
Re(A),Im(A),|A| at time 10 phi at time 10
Re(A),Im(A),|A| at time 500 phi at time 500 ||A||_2 vs time ||phi||_2 vs time
Figure 2.
Full stabilisation of the solution of Fig. 1 withthe K = 147 Fourier modes prescribed by the stabilizationscheme in the text. The solution decays towards the zerostate. ∗ , V. KALANTAROV † AND O. VANTZOS ‡ References [1] A. Azouani and E. S. Titi,
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HEVRON PATTERN EQUATIONS 21 † Azerbaijan State Oil and Industry University, Baku, Azerbaijan ‡‡