Sharp interface limit of stochastic Cahn-Hilliard equation Rongchan - PowerPoint PPT Presentation

Sharp interface limit of stochastic Cahn-Hilliard equation Rongchan Zhu Beijing Institute of Technology/Bielefeld University Joint work with Lubomir Banas and Huanyu Yang [Banas/Yang/Z.: arXiv:1905.09182] [Yang/Z.:arXiv:1905.07216] Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 1 / 20

Table of contents Introduction 1 Singular noise for large σ 2 Weak approach and tightness for small σ 3 Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 2 / 20

Introduction Introduction The Cahn-Hilliard equation ∂ t u = ∆( − ∆ u + F ′ ( u )) , (1) on [0 , T ] × D with Neumann boundary condition. Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 3 / 20

Introduction Introduction The Cahn-Hilliard equation ∂ t u = ∆( − ∆ u + F ′ ( u )) , (1) on [0 , T ] × D with Neumann boundary condition. 4 ( u 2 − 1) 2 : the double-well potential F ( u ) = 1 Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 3 / 20

Introduction Introduction The Cahn-Hilliard equation ∂ t u = ∆( − ∆ u + F ′ ( u )) , (1) on [0 , T ] × D with Neumann boundary condition. 4 ( u 2 − 1) 2 : the double-well potential F ( u ) = 1 v = − ∆ u + F ′ ( u ) Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 3 / 20

Introduction Introduction The Cahn-Hilliard equation ∂ t u = ∆( − ∆ u + F ′ ( u )) , (1) on [0 , T ] × D with Neumann boundary condition. 4 ( u 2 − 1) 2 : the double-well potential F ( u ) = 1 v = − ∆ u + F ′ ( u ) well-known model to describe phase separation Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 3 / 20

Introduction Introduction The Cahn-Hilliard equation ∂ t u = ∆( − ∆ u + F ′ ( u )) , (1) on [0 , T ] × D with Neumann boundary condition. 4 ( u 2 − 1) 2 : the double-well potential F ( u ) = 1 v = − ∆ u + F ′ ( u ) well-known model to describe phase separation is also H − 1 -gradient flow of the energy functional E ( u ) := 1 � � |∇ u ( x ) | 2 dx + F ( u ( x )) dx , (2) 2 D D Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 3 / 20

Introduction Introduction The Cahn-Hilliard equation ∂ t u = ∆( − ∆ u + F ′ ( u )) , (1) on [0 , T ] × D with Neumann boundary condition. 4 ( u 2 − 1) 2 : the double-well potential F ( u ) = 1 v = − ∆ u + F ′ ( u ) well-known model to describe phase separation is also H − 1 -gradient flow of the energy functional E ( u ) := 1 � � |∇ u ( x ) | 2 dx + F ( u ( x )) dx , (2) 2 D D If u is a solution to equation (1), then d � |∇ v ( x ) | 2 dx ≤ 0 . dt E ( u ( t , · )) = − D Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 3 / 20

Introduction Introduction The Cahn-Hilliard equation ∂ t u = ∆( − ∆ u + F ′ ( u )) , (1) on [0 , T ] × D with Neumann boundary condition. 4 ( u 2 − 1) 2 : the double-well potential F ( u ) = 1 v = − ∆ u + F ′ ( u ) well-known model to describe phase separation is also H − 1 -gradient flow of the energy functional E ( u ) := 1 � � |∇ u ( x ) | 2 dx + F ( u ( x )) dx , (2) 2 D D If u is a solution to equation (1), then d � |∇ v ( x ) | 2 dx ≤ 0 . dt E ( u ( t , · )) = − D The minimizers of the energy (2) are the constant functions u ≡ 1 and u ≡ − 1, which represent the “pure phases” of the system. Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 3 / 20

Introduction Introduction In a large scale, let τ = ε 3 t , η = ε x , and u ε = u ( τ, η ), Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 4 / 20

Introduction Introduction In a large scale, let τ = ε 3 t , η = ε x , and u ε = u ( τ, η ), then u ε satisfies ∂ τ u ε = ∆ η ( − ε ∆ η u ε + 1 ε F ′ ( u ε )) , (3) Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 4 / 20

Introduction Introduction In a large scale, let τ = ε 3 t , η = ε x , and u ε = u ( τ, η ), then u ε satisfies ∂ τ u ε = ∆ η ( − ε ∆ η u ε + 1 ε F ′ ( u ε )) , (3) Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 4 / 20

Introduction Introduction In a large scale, let τ = ε 3 t , η = ε x , and u ε = u ( τ, η ), then u ε satisfies ∂ τ u ε = ∆ η ( − ε ∆ η u ε + 1 ε F ′ ( u ε )) , (3) v ε = − ε ∆ u ε + 1 ε F ′ ( u ε ): chemical potential Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 4 / 20

Introduction Introduction In a large scale, let τ = ε 3 t , η = ε x , and u ε = u ( τ, η ), then u ε satisfies ∂ τ u ε = ∆ η ( − ε ∆ η u ε + 1 ε F ′ ( u ε )) , (3) v ε = − ε ∆ u ε + 1 ε F ′ ( u ε ): chemical potential As ε → 0, (3) represent a long time behavior of (1) in a large scale. Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 4 / 20

Introduction Introduction In a large scale, let τ = ε 3 t , η = ε x , and u ε = u ( τ, η ), then u ε satisfies ∂ τ u ε = ∆ η ( − ε ∆ η u ε + 1 ε F ′ ( u ε )) , (3) v ε = − ε ∆ u ε + 1 ε F ′ ( u ε ): chemical potential As ε → 0, (3) represent a long time behavior of (1) in a large scale. The energy functional of (3) is E ε ( u ε ) := ε � |∇ u ε ( x ) | 2 dx + 1 � F ( u ε ( x )) dx , (4) 2 ε D D Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 4 / 20

Introduction Introduction In a large scale, let τ = ε 3 t , η = ε x , and u ε = u ( τ, η ), then u ε satisfies ∂ τ u ε = ∆ η ( − ε ∆ η u ε + 1 ε F ′ ( u ε )) , (3) v ε = − ε ∆ u ε + 1 ε F ′ ( u ε ): chemical potential As ε → 0, (3) represent a long time behavior of (1) in a large scale. The energy functional of (3) is E ε ( u ε ) := ε � |∇ u ε ( x ) | 2 dx + 1 � F ( u ε ( x )) dx , (4) 2 ε D D and satisfies d � |∇ v ε | 2 ≤ 0 . dt E ε ( u ε ) = − (5) D Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 4 / 20

Introduction Introduction In a large scale, let τ = ε 3 t , η = ε x , and u ε = u ( τ, η ), then u ε satisfies ∂ τ u ε = ∆ η ( − ε ∆ η u ε + 1 ε F ′ ( u ε )) , (3) v ε = − ε ∆ u ε + 1 ε F ′ ( u ε ): chemical potential As ε → 0, (3) represent a long time behavior of (1) in a large scale. The energy functional of (3) is E ε ( u ε ) := ε � |∇ u ε ( x ) | 2 dx + 1 � F ( u ε ( x )) dx , (4) 2 ε D D and satisfies d � |∇ v ε | 2 ≤ 0 . dt E ε ( u ε ) = − (5) D As ε → 0, F ( u ε ) → 0, Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 4 / 20

Introduction Introduction In a large scale, let τ = ε 3 t , η = ε x , and u ε = u ( τ, η ), then u ε satisfies ∂ τ u ε = ∆ η ( − ε ∆ η u ε + 1 ε F ′ ( u ε )) , (3) v ε = − ε ∆ u ε + 1 ε F ′ ( u ε ): chemical potential As ε → 0, (3) represent a long time behavior of (1) in a large scale. The energy functional of (3) is E ε ( u ε ) := ε � |∇ u ε ( x ) | 2 dx + 1 � F ( u ε ( x )) dx , (4) 2 ε D D and satisfies d � |∇ v ε | 2 ≤ 0 . dt E ε ( u ε ) = − (5) D As ε → 0, F ( u ε ) → 0, which implies u ε → − 1 + 2 1 E for some E ⊂ [0 , T ] × D . Γ t := ∂ E t is the interface. Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 4 / 20

Introduction Motion of Γ t u ε approximates to 1 on one region D + and to − 1 in D − , u ε → − 1 + 2 1 E Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 5 / 20

Introduction Motion of Γ t u ε approximates to 1 on one region D + and to − 1 in D − , u ε → − 1 + 2 1 E v ε → v , 2 ∂ t 1 E t = ∆ v . Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 5 / 20

Introduction Motion of Γ t u ε approximates to 1 on one region D + and to − 1 in D − , u ε → − 1 + 2 1 E v ε → v , 2 ∂ t 1 E t = ∆ v . ∆ v = 0 in D \ Γ t ; Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 5 / 20

Introduction Motion of Γ t u ε approximates to 1 on one region D + and to − 1 in D − , u ε → − 1 + 2 1 E v ε → v , 2 ∂ t 1 E t = ∆ v . ∆ v = 0 in D \ Γ t ; ∂ v ∂ n = 0 on ∂ D ; Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 5 / 20