Introduction A priori estimates Existence results Global attractor for δ > 0 Exponential attractor for δ > 0 On the Cahn-Hilliard equation with a chemical potential dependent mobility Riccarda Rossi (Universit` a di Brescia) joint work with Maurizio Grasselli (Politecnico di Milano) Alain Miranville (Universit´ e de Poitiers) Giulio Schimperna (Universit` a di Pavia) AIMS Eighth International Conference, Dresden, May 25–28, 2010 Riccarda Rossi On the Cahn-Hilliard equation with a chemical potential dependent mobility
Introduction A priori estimates Existence results Global attractor for δ > 0 Exponential attractor for δ > 0 The equation We consider the generalized (viscous) Cahn-Hilliard equation: χ t − ∆( α ( δ χ t − ∆ χ + φ ( χ ))) = 0 in Ω × (0 , T ) , Ω ⊂ R N , N = 1 , 2 , 3, a bdd smooth domain, (0 , T ) a time interval; ◮ ◮ α : D ( α ) ⊂ R → R is strictly increasing and differentiable; ◮ δ ≥ 0 a constant; ◮ Typically φ ( χ ) = χ 3 − χ (derivative of the double-well potential) Riccarda Rossi On the Cahn-Hilliard equation with a chemical potential dependent mobility
Introduction A priori estimates Existence results Global attractor for δ > 0 Exponential attractor for δ > 0 The equation We consider the generalized (viscous) Cahn-Hilliard equation: χ t − ∆( α ( δ χ t − ∆ χ + φ ( χ ))) = 0 in Ω × (0 , T ) , Ω ⊂ R N , N = 1 , 2 , 3, a bdd smooth domain, (0 , T ) a time interval; ◮ ◮ α : D ( α ) ⊂ R → R is strictly increasing and differentiable; ◮ δ ≥ 0 a constant; ◮ Typically φ ( χ ) = χ 3 − χ (derivative of the double-well potential) If α is linear , α ( r ) := κ r ∀ r ∈ R , ( κ > 0 � mobility ) Riccarda Rossi On the Cahn-Hilliard equation with a chemical potential dependent mobility
Introduction A priori estimates Existence results Global attractor for δ > 0 Exponential attractor for δ > 0 The equation We consider the generalized (viscous) Cahn-Hilliard equation: χ t − ∆( α ( δ χ t − ∆ χ + φ ( χ ))) = 0 in Ω × (0 , T ) , Ω ⊂ R N , N = 1 , 2 , 3, a bdd smooth domain, (0 , T ) a time interval; ◮ ◮ α : D ( α ) ⊂ R → R is strictly increasing and differentiable; ◮ δ ≥ 0 a constant; ◮ Typically φ ( χ ) = χ 3 − χ (derivative of the double-well potential) If α is linear , α ( r ) := κ r ∀ r ∈ R , ( κ > 0 � mobility ) for δ = 0 we have the Cahn-Hilliard equation χ t − κ ∆( − ∆ χ + φ ( χ )) = 0 in Ω × (0 , T ) , Riccarda Rossi On the Cahn-Hilliard equation with a chemical potential dependent mobility
Introduction A priori estimates Existence results Global attractor for δ > 0 Exponential attractor for δ > 0 The equation We consider the generalized (viscous) Cahn-Hilliard equation: χ t − ∆( α ( δ χ t − ∆ χ + φ ( χ ))) = 0 in Ω × (0 , T ) , Ω ⊂ R N , N = 1 , 2 , 3, a bdd smooth domain, (0 , T ) a time interval; ◮ ◮ α : D ( α ) ⊂ R → R is strictly increasing and differentiable; ◮ δ ≥ 0 a constant; ◮ Typically φ ( χ ) = χ 3 − χ (derivative of the double-well potential) If α is linear , α ( r ) := κ r ∀ r ∈ R , ( κ > 0 � mobility ) for δ > 0 we have the viscous Cahn-Hilliard equation χ t − κ ∆( δ χ t − ∆ χ + φ ( χ )) = 0 in Ω × (0 , T ) , proposed in [Novick-Cohen ’88] to account for viscosity effects in the phase separation in polymers . Riccarda Rossi On the Cahn-Hilliard equation with a chemical potential dependent mobility
Introduction A priori estimates Existence results Global attractor for δ > 0 Exponential attractor for δ > 0 The equation We consider the generalized (viscous) Cahn-Hilliard equation: χ t − ∆( α ( δ χ t − ∆ χ + φ ( χ ))) = 0 in Ω × (0 , T ) , Ω ⊂ R N , N = 1 , 2 , 3, a bdd smooth domain, (0 , T ) a time interval; ◮ ◮ α : D ( α ) ⊂ R → R is strictly increasing and differentiable; ◮ δ ≥ 0 a constant; ◮ Typically φ ( χ ) = χ 3 − χ (derivative of the double-well potential) If α is linear , α ( r ) := κ r ∀ r ∈ R , ( κ > 0 � mobility ) for δ > 0 we have the viscous Cahn-Hilliard equation χ t − κ ∆( δ χ t − ∆ χ + φ ( χ )) = 0 in Ω × (0 , T ) , proposed in [Novick-Cohen ’88] to account for viscosity effects in the phase separation in polymers . Both for δ = 0 and δ > 0: wide literature on well-posedness (for various variants of the model), long-time behaviour , dynamics of pattern formation.. Riccarda Rossi On the Cahn-Hilliard equation with a chemical potential dependent mobility
Introduction A priori estimates Existence results Global attractor for δ > 0 Exponential attractor for δ > 0 Gurtin’s generalized Cahn-Hilliard equation • M.E. Gurtin [Phys. D ’96] proposed a novel derivation of the Cahn-Hilliard equations, thus obtaining the generalized viscous Cahn-Hilliard equation ( χ t − div ( M ( Z ) ∇ w ) = 0 (GVCHE) w = δ ( Z ) χ t − ∆ χ + φ ( χ ) ◮ w chemical potential ◮ M mobility tensor (symmetric, positive definite) ◮ M = M ( Z ), δ = δ ( Z ), with constitutive variables: Z = ( χ, ∇ χ, χ t , w , ∇ w )! Riccarda Rossi On the Cahn-Hilliard equation with a chemical potential dependent mobility
Introduction A priori estimates Existence results Global attractor for δ > 0 Exponential attractor for δ > 0 Gurtin’s generalized Cahn-Hilliard equation • M.E. Gurtin [Phys. D ’96] proposed a novel derivation of the Cahn-Hilliard equations, thus obtaining the generalized viscous Cahn-Hilliard equation ( χ t − div ( M ( Z ) ∇ w ) = 0 (GVCHE) w = δ ( Z ) χ t − ∆ χ + φ ( χ ) ◮ w chemical potential ◮ M mobility tensor (symmetric, positive definite) ◮ M = M ( Z ), δ = δ ( Z ), with constitutive variables: Z = ( χ, ∇ χ, χ t , w , ∇ w )! • Several results [ Miranville & Bonfoh, Carrive, Cherfils, Grasselli, Pi´ etrus, Rakotoson, Rougirel, Schimperna, Zelik.. ]: well-posedness and long-time behaviour for variants of (GVCHE) (also in the anisotropic case) with periodic and Neumann b.c., and M ( Z ) = M , M ( χ ), constant δ . Riccarda Rossi On the Cahn-Hilliard equation with a chemical potential dependent mobility
Introduction A priori estimates Existence results Global attractor for δ > 0 Exponential attractor for δ > 0 Gurtin’s generalized Cahn-Hilliard equation • M.E. Gurtin [Phys. D ’96] proposed a novel derivation of the Cahn-Hilliard equations, thus obtaining the generalized viscous Cahn-Hilliard equation ( χ t − div ( M ( Z ) ∇ w ) = 0 (GVCHE) w = δ ( Z ) χ t − ∆ χ + φ ( χ ) ◮ w chemical potential ◮ M mobility tensor (symmetric, positive definite) ◮ M = M ( Z ), δ = δ ( Z ), with constitutive variables: Z = ( χ, ∇ χ, χ t , w , ∇ w )! • Several results [ Miranville & Bonfoh, Carrive, Cherfils, Grasselli, Pi´ etrus, Rakotoson, Rougirel, Schimperna, Zelik.. ]: well-posedness and long-time behaviour for variants of (GVCHE) (also in the anisotropic case) with periodic and Neumann b.c., and M ( Z ) = M , M ( χ ), constant δ . • Well-posedness and long-time behaviour for the standard Cahn-Hilliard eq. (viscous and non-viscous), with a concentration -dependent mobility tensor: [Barrett, Blowey, Bonetti, Colli, Dreyer, Gilardi, Elliott, Novick-Cohen, Garcke, Schimperna, Sprekels..] . Riccarda Rossi On the Cahn-Hilliard equation with a chemical potential dependent mobility
Introduction A priori estimates Existence results Global attractor for δ > 0 Exponential attractor for δ > 0 A chemical potential dependent mobility χ t − ∆( α ( δ χ t − ∆ χ + φ ( χ ))) = 0 in Ω × (0 , T ) , is a particular case of (GVCHE): ( χ t − div ( M ( Z ) ∇ w ) = 0 in Ω × (0 , T ), w = δ χ t − ∆ χ + φ ( χ ) with δ ( Z ) = δ , M ( Z ) = M ( w ) , (admissible, α ′ > 0!), i.e., a chemical potential -dependent mobility tensor!!!! • dettagli: conservazione della massa (natural boundary conditions) • • citare i lavori di Rossi su well-posedness results e long-time behaviour for two different I.B.V. corresponding to two choices of the mobility law α . Riccarda Rossi On the Cahn-Hilliard equation with a chemical potential dependent mobility
Introduction A priori estimates Existence results Global attractor for δ > 0 Exponential attractor for δ > 0 A chemical potential dependent mobility χ t − ∆( α ( δ χ t − ∆ χ + φ ( χ ))) = 0 in Ω × (0 , T ) , is a particular case of (GVCHE): χ t − div ( M ( Z ) ∇ ( δ χ t − ∆ χ + φχ )) = 0 in Ω × (0 , T ) M ( Z ) = M ( w ) := α ′ ( w ) I , with δ ( Z ) = δ , (admissible, α ′ > 0!), i.e., a chemical potential -dependent mobility tensor!! Riccarda Rossi On the Cahn-Hilliard equation with a chemical potential dependent mobility
Introduction A priori estimates Existence results Global attractor for δ > 0 Exponential attractor for δ > 0 A chemical potential dependent mobility χ t − ∆( α ( δ χ t − ∆ χ + φ ( χ ))) = 0 in Ω × (0 , T ) , is a particular case of (GVCHE): χ t − div ( M ( Z ) ∇ ( δ χ t − ∆ χ + φχ )) = 0 in Ω × (0 , T ) M ( Z ) = M ( w ) := α ′ ( w ) I , with δ ( Z ) = δ , (admissible, α ′ > 0!), i.e., a chemical potential -dependent mobility tensor!! • no-flux boundary conditions for χ and w : mass conservation for χ Riccarda Rossi On the Cahn-Hilliard equation with a chemical potential dependent mobility
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