Behaviour at infinite time of Nematic Phase Problem Mouhamadou - PowerPoint PPT Presentation

Behaviour at infinite time of Nematic Phase Problem Mouhamadou Samsidy Goudiaby + Supervisors Blanca Climent-Ezquerra ∗ , Francisco Guillén-Gonzales ∗ , + EDAN, Universidad de Sevilla, Spain. ∗ UFR SAT, LANI, Université Gaston Berger, Saint-Louis, Senegal.

What is a liquid crystal ? Nematic Model Behaviour at infinite time 1 What is a liquid crystal ? 2 Nematic Model 3 Behaviour at infinite time Some topics about liquid crystals ...

What is a liquid crystal ? Nematic Model Behaviour at infinite time 1 What is a liquid crystal ? 2 Nematic Model 3 Behaviour at infinite time Some topics about liquid crystals ...

What is a liquid crystal ? Nematic Model Behaviour at infinite time What is a liquid crystal ? Liquid crystals (LCs) are substances which exhibit an intermediate phase of matter that has properties between those of a conventional liquid and those of a solid crystal. Discovering of liquid crystals attributed to the Austrian botanist Friedrich Richard Reinitzer in 1888. Otto Lehmann, a German physicist solved the problem with description of a new state of matter midway between a liquid and a crystal. In 1991, Pierre-Gilles de Gennes, a French physicist received the Nobel Prize because of issues related to (LC). Some topics about liquid crystals ...

What is a liquid crystal ? Nematic Model Behaviour at infinite time What is a liquid crystal ? Some topics about liquid crystals ...

Some LC phases Nematic : no positional order but long-range orientational order. Smectic : well-defined layers and orientational order.

Nematic

Smectic A

Nematic-Smectic

What is a liquid crystal ? Nematic Model Behaviour at infinite time What is a liquid crystal ? LCs in nature Forming biological membrane, Protein solution generate silk of spider, DNA and polypeptides. Some technological applications Liquid crystal displays (LCD), Construction of optics blinds. Some topics about liquid crystals ...

What is a liquid crystal ? Nematic Model Behaviour at infinite time 1 What is a liquid crystal ? 2 Nematic Model 3 Behaviour at infinite time Some topics about liquid crystals ...

Nematic model Simplified model by [Lin’89], [Lin,Liu’95], [Lin,Liu’00] and [Coutand,Shkoller’01] Original equations by Ericksen and Leslie, during 1958-1968. Elements in LC model Studied d , an unit vectorial function modeling the orientation of the crystals molecules. Approximation by Ginzburg-Landau penalization, | d | = 1 is relaxed by | d | ≤ 1 using ǫ 2 ( | d | 2 − 1) d f ( d ) = 1 Oseen Frank energy (elastic energy) � 1 1 � 2 |∇ d | 2 + F ( d ) 4 ǫ 2 ( | n | 2 − 1) 2 is a potential � E e = , F ( n ) = Ω function of f ( n ) , i.e. f ( n ) = ∇ n F ( n ) Euler-Lagrange system w ≡ − ∆ d + f ( d ) = 0

Nematic model We assume the liquid crystal confined in an open bounded domain Ω ⊂ R N ( N = 2 or 3 ) with regular boundary ∂ Ω . u : Ω × [0 , + ∞ ) �→ R N is the flow velocity, p : Ω × [0 , + ∞ ) �→ R is the fluid pressure, d : Ω × [0 , + ∞ ) �→ R 3 is the orientation vector. We denote Q = (0 , + ∞ ) × Ω , � = (0 , + ∞ ) × ∂ Ω , the L p norm is denoted by | · | p , 1 ≤ p ≤ ∞ , and the H m norm by || · || m Conservation of angular momentum ∂ t d + ( u · ∇ ) d − ∆ d + f ( d ) = 0 , Conservation of linear momentum ∂ t u + ( u · ∇ ) u − ν ∆ u + ∇ p + ( ∇ d ) t ( − ∆ d + f ( d )) = 0 , ∇ · u = 0

What is a liquid crystal ? Nematic Model Behaviour at infinite time Nematic model Nematic Model  ∂ t u + ( u · ∇ ) u − ν ∆ u + ∇ p + ( ∇ d ) t ( − ∆ d + f ( d )) = 0 ,  ∇ · u = 0 (1) ∂ t d + ( u · ∇ ) d − ∆ d + f ( d ) = 0 ,  Initial conditions u ( x , 0) = u 0 ( x ) , d ( x , 0) = d 0 ( x ) in Ω . (2) Boundary conditions u ( x , t ) = 0 , d ( x , t ) = d 0 ( x ) on ∂ Ω × (0 , T ) (3) Some topics about liquid crystals ...

What is a liquid crystal ? Nematic Model Behaviour at infinite time Some results Time-independent boundary data for initial value problem F.H. Lin, C. Liu. Non-parabolic dissipative systems modeling the flow of liquid crystals, 1995. Time-dependent boundary data B. Climent-Ezquerra, F. Guillén-González, M. J. Moreno-Iraberte, Regularity and time-periodicity for a nematic liquid crystal model, 2009. F. Guillén-González, M. A. Rodríguez-Bellido, and M. A. Rojas-Medar, Sufficient conditions for regularity and uniqueness of a 3D nematic liquid crystal model, 2009. B. Climent-Ezquerra, F. Guillén-González, M. Rojas-Medar, Reproductivity for a nematic liquid crystal model, 2006. Some topics about liquid crystals ...

What is a liquid crystal ? Nematic Model Behaviour at infinite time Energy law The total energy E ( u ( t ) , d ( t )) = 1 2 + 1 � 2 | u ( t ) | 2 2 |∇ d ( t ) | 2 2 + F ( d ( t )) Ω E c ( u ( t )) = 1 2 | u ( t ) | 2 2 E e ( d ( t )) = 1 2 |∇ d ( t ) | 2 � 2 + Ω F ( d ( t )) Energy law d dt E ( u ( t ) , d ( t )) + ν |∇ u ( t ) | 2 2 + | − ∆ d ( t ) + f ( d ( t )) | 2 2 = 0 (4) Some topics about liquid crystals ...

What is a liquid crystal ? Nematic Model Behaviour at infinite time 1 What is a liquid crystal ? 2 Nematic Model 3 Behaviour at infinite time Some topics about liquid crystals ...

The initial-value problem at infinite time Definition (Weak solution). We say that ( u , d ) is a weak solution of (1)-(14) in (0 , + ∞ ) if ∇ · u = 0 , � � = 0 , � � = h , u � d � || ( u ( t ) , d ( t )) || 0 × 1 ≤ C 1 ∀ t ≥ 0 (5) � t e − γ t e γ s || ( u ( t ) , d ( t )) || 2 ∀ γ > 0 , 1 × 2 ds ≤ C 2 , ∀ t ≥ 0 , (6) 0 Definition (Strong solution). A weak solution ( u , d ) of (1)-(14) is a strong solution in (0 , + ∞ ) if || ( u ( t ) , d ( t )) || 1 × 2 ≤ C 3 ∀ t ≥ 0 (7) � t e − γ t e γ s || ( u ( t ) , d ( t )) || 2 ∀ γ > 0 , 2 × 3 ds ≤ C 4 , ∀ t ≥ 0 , (8) 0

What is a liquid crystal ? Nematic Model Behaviour at infinite time Main result (Asymptotic stability) Let Ω , d 0 be regular enough, with | d 0 | ≤ 1 in ¯ Ω , assume ( u 0 , d 0 ) ∈ H 1 × H 2 and || ( u 0 , d 0 ) || 1 × 2 ≤ C . If ( u ( t ) , d ( t )) is a strong solution of (1)-(14) in (0 , + ∞ ) , then the total enery E ( u ( t ) , d ( t )) ց E ∞ = E e (¯ d ) when t ↑ + ∞ , where ¯ d is a critical point of elastic energy, that is, a solution of the stationary problem � − ∆¯ d + f (¯ d ) = 0 in Ω (9) ¯ d | ∂ Ω = h . Moreover, ( u ( t ) , d ( t )) satisfies → 0 in L 2 and d ( t ) − → ¯ → 0 in H 1 d in H 2 . u ( t ) − 0 , ∆ d ( t ) − f ( d ( t )) − Some topics about liquid crystals ...

What is a liquid crystal ? Nematic Model Behaviour at infinite time First step (Theorem 1) Let Ω , d 0 be regular enough, with | d 0 | ≤ 1 in ¯ Ω , assume ( u 0 , d 0 ) ∈ H 1 × H 2 and || ( u 0 , d 0 ) || 1 × 2 ≤ M 0 . If ( u ( t ) , d ( t )) is a strong solution of (1)-(14) in (0 , + ∞ ) , then the total enery E ( u ( t ) , d ( t )) ց E ∞ ≥ 0 , as t ↑ + ∞ and → 0 in H 1 → 0 in L 2 (Ω) , as t ↑ + ∞ u ( t ) − 0 (Ω) , ∆ d ( t ) − f ( d ( t )) − Some topics about liquid crystals ...

What is a liquid crystal ? Nematic Model Behaviour at infinite time Sketch of the proof Weak estimation d dt E ( u ( t ) , d ( t )) + G ( u ( t ) , d ( t )) ≤ 0 Strong estimation d G ( u ( t ) , d ( t )) 3 + 1 � � dt G ( u ( t ) , d ( t )) ≤ C . where G ( u ( t ) , d ( t )) = ν |∇ u ( t ) | 2 2 + | − ∆ d ( t ) + f ( d ( t )) | 2 2 . Some topics about liquid crystals ...

Strong estimation General Framework ′ ( t ) + G ( t ) ≤ 0 E ( t ) , G ( t ) ≥ 0 , E a.e. t ∈ ( t 0 , + ∞ ) Then, E ∈ C b [ t 0 , + ∞ ) , is a decreasing function and there exists E ∞ ≥ 0 such that t → + ∞ E ( t ) = E ∞ . lim Let G ∈ L 1 ( t 0 , + ∞ ) be a function satisfying ′ ( t ) ≤ C ( G ( t ) 3 + 1) . Then, t → + ∞ G ( t ) = 0 . lim G B. Climent-Ezquerra, F. Guillén-González, M. J. Rodríguez-Bellido, Stability for nematic liquid crystals with stretching terms, 2009. General Framework + Weak estimation ⇒ E ( u ( t ) , d ( t )) ց E ∞ ≥ 0 , General Framework + Strong estimation � � ⇒ lim |∇ u ( t ) | 2 + | − ∆ d ( t ) + f ( d ( t )) | 2 = 0 , t → + ∞

What is a liquid crystal ? Nematic Model Behaviour at infinite time Second step (Theorem 2) E ∞ = E e (¯ d ) where ¯ d is a critical point of elastic energy, that is, a solution of the stationary problem � − ∆¯ d + f (¯ d ) = 0 in Ω (10) ¯ d | ∂ Ω = h . Some topics about liquid crystals ...

What is a liquid crystal ? Nematic Model Behaviour at infinite time ω -limit set Let S be the set � � S = (0 , d ) , / − ∆ d + f ( d ) = 0 in Ω d | ∂ Ω = h (11) The ω -limit set of ( u 0 , d 0 ) ∈ V × H 2 ⊂ L 2 × H 1 with respect to the strong solution in (0 , + ∞ ) of problem (1)-(14) is defined as follow u ( x ) , ¯ � ω (( u 0 , d 0 )) = (¯ d ( x )) , / there exist { t n } ր + ∞ such that d ( x )) in L 2 × H 1 � u ( x ) , ¯ ( u ( x , t n ) , d ( x , t n )) → (¯ Proposition ω (( u 0 , d 0 )) is a nonempty bounded subset in H 1 × H 2 and w (( u 0 , d 0 )) ⊂ S . Some topics about liquid crystals ...