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

1

Introduction

2

Singular noise for large σ

3

Weak approach and tightness for small σ

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 2 / 20

Introduction

Introduction

The Cahn-Hilliard equation ∂tu = ∆(−∆u + F ′(u)), (1)

• n [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 ∂tu = ∆(−∆u + F ′(u)), (1)

• n [0, T] × D with Neumann boundary condition.

F(u) = 1

4(u2 − 1)2: the double-well potential

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 3 / 20

Introduction

Introduction

The Cahn-Hilliard equation ∂tu = ∆(−∆u + F ′(u)), (1)

• n [0, T] × D with Neumann boundary condition.

F(u) = 1

4(u2 − 1)2: the double-well potential

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 ∂tu = ∆(−∆u + F ′(u)), (1)

• n [0, T] × D with Neumann boundary condition.

F(u) = 1

4(u2 − 1)2: the double-well potential

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 ∂tu = ∆(−∆u + F ′(u)), (1)

• n [0, T] × D with Neumann boundary condition.

F(u) = 1

4(u2 − 1)2: the double-well potential

v = −∆u + F ′(u) well-known model to describe phase separation is also H−1-gradient flow of the energy functional E(u) := 1 2

• D

|∇u(x)|2dx +

• D

F(u(x))dx, (2)

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 3 / 20

Introduction

Introduction

The Cahn-Hilliard equation ∂tu = ∆(−∆u + F ′(u)), (1)

• n [0, T] × D with Neumann boundary condition.

F(u) = 1

4(u2 − 1)2: the double-well potential

v = −∆u + F ′(u) well-known model to describe phase separation is also H−1-gradient flow of the energy functional E(u) := 1 2

• D

|∇u(x)|2dx +

• D

F(u(x))dx, (2) If u is a solution to equation (1), then d dt E(u(t, ·)) = −

• D

|∇v(x)|2dx ≤ 0.

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 3 / 20

Introduction

Introduction

The Cahn-Hilliard equation ∂tu = ∆(−∆u + F ′(u)), (1)

• n [0, T] × D with Neumann boundary condition.

F(u) = 1

4(u2 − 1)2: the double-well potential

v = −∆u + F ′(u) well-known model to describe phase separation is also H−1-gradient flow of the energy functional E(u) := 1 2

• D

|∇u(x)|2dx +

• D

F(u(x))dx, (2) If u is a solution to equation (1), then d dt E(u(t, ·)) = −

• D

|∇v(x)|2dx ≤ 0. 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 = ε3t, η = εx, and uε = 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 = ε3t, η = εx, and uε = u(τ, η), then uε satisfies ∂τuε = ∆η(−ε∆ηuε + 1 εF ′(uε)), (3)

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 4 / 20

Introduction

Introduction

In a large scale, let τ = ε3t, η = εx, and uε = u(τ, η), then uε satisfies ∂τuε = ∆η(−ε∆ηuε + 1 εF ′(uε)), (3)

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 4 / 20

Introduction

Introduction

In a large scale, let τ = ε3t, η = εx, and uε = u(τ, η), then uε satisfies ∂τuε = ∆η(−ε∆ηuε + 1 εF ′(uε)), (3) v ε = −ε∆uε + 1

εF ′(uε): 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 = ε3t, η = εx, and uε = u(τ, η), then uε satisfies ∂τuε = ∆η(−ε∆ηuε + 1 εF ′(uε)), (3) v ε = −ε∆uε + 1

εF ′(uε): 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 = ε3t, η = εx, and uε = u(τ, η), then uε satisfies ∂τuε = ∆η(−ε∆ηuε + 1 εF ′(uε)), (3) v ε = −ε∆uε + 1

εF ′(uε): chemical potential

As ε → 0, (3) represent a long time behavior of (1) in a large scale. The energy functional of (3) is Eε(uε) := ε 2

• D

|∇uε(x)|2dx + 1 ε

• D

F(uε(x))dx, (4)

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 4 / 20

Introduction

Introduction

In a large scale, let τ = ε3t, η = εx, and uε = u(τ, η), then uε satisfies ∂τuε = ∆η(−ε∆ηuε + 1 εF ′(uε)), (3) v ε = −ε∆uε + 1

εF ′(uε): chemical potential

As ε → 0, (3) represent a long time behavior of (1) in a large scale. The energy functional of (3) is Eε(uε) := ε 2

• D

|∇uε(x)|2dx + 1 ε

• D

F(uε(x))dx, (4) and satisfies d dt Eε(uε) = −

• D

|∇v ε|2 ≤ 0. (5)

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 4 / 20

Introduction

Introduction

In a large scale, let τ = ε3t, η = εx, and uε = u(τ, η), then uε satisfies ∂τuε = ∆η(−ε∆ηuε + 1 εF ′(uε)), (3) v ε = −ε∆uε + 1

εF ′(uε): chemical potential

As ε → 0, (3) represent a long time behavior of (1) in a large scale. The energy functional of (3) is Eε(uε) := ε 2

• D

|∇uε(x)|2dx + 1 ε

• D

F(uε(x))dx, (4) and satisfies d dt Eε(uε) = −

• D

|∇v ε|2 ≤ 0. (5) As ε → 0, F(uε) → 0,

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 4 / 20

Introduction

Introduction

In a large scale, let τ = ε3t, η = εx, and uε = u(τ, η), then uε satisfies ∂τuε = ∆η(−ε∆ηuε + 1 εF ′(uε)), (3) v ε = −ε∆uε + 1

εF ′(uε): chemical potential

As ε → 0, (3) represent a long time behavior of (1) in a large scale. The energy functional of (3) is Eε(uε) := ε 2

• D

|∇uε(x)|2dx + 1 ε

• D

F(uε(x))dx, (4) and satisfies d dt Eε(uε) = −

• D

|∇v ε|2 ≤ 0. (5) As ε → 0, F(uε) → 0, which implies uε → −1 + 21E for some E ⊂ [0, T] × D. Γt := ∂Et is the interface.

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 4 / 20

Introduction

Motion of Γt

uε approximates to 1 on one region D+ and to −1 in D−, uε → −1 + 21E

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 5 / 20

Introduction

Motion of Γt

uε approximates to 1 on one region D+ and to −1 in D−, uε → −1 + 21E v ε → v, 2∂t1Et = ∆v.

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 5 / 20

Introduction

Motion of Γt

uε approximates to 1 on one region D+ and to −1 in D−, uε → −1 + 21E v ε → v, 2∂t1Et = ∆v. ∆v = 0 in D \ Γt;

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 5 / 20

Introduction

Motion of Γt

uε approximates to 1 on one region D+ and to −1 in D−, uε → −1 + 21E v ε → v, 2∂t1Et = ∆v. ∆v = 0 in D \ Γt;

∂v ∂n = 0 on ∂D;

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 5 / 20

Introduction

Motion of Γt

uε approximates to 1 on one region D+ and to −1 in D−, uε → −1 + 21E v ε → v, 2∂t1Et = ∆v. ∆v = 0 in D \ Γt;

∂v ∂n = 0 on ∂D;

Formally, t

• D

∂t1Etψ = −1 2 t

• D+ ∇v∇ψ − 1

2 t

• D− ∇v∇ψ

= 1 2 t

• D+ div(∇vψ) + 1

2 t

• D− div(∇vψ)

= 1 2 t

• Γt

(∂nv + − ∂nv −)ψ.

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 5 / 20

Introduction

Hele-Shaw model

Formally derived by [Pego: 1989] and rigorous proved by [Alikakos, Bates,Chen: 1994]: v ε → v, (v, Γ) solves the following free boundary problem:                    ∆v = 0 in D \ Γt, t > 0, ∂v ∂n = 0 on ∂D, v= 2 3H on Γt, V = 1 2(∂nv + − ∂nv −) on Γt, (6) H: mean curvature of Γt; V: normal velocity of Γ

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 6 / 20

Introduction

Hele-Shaw model

Formally derived by [Pego: 1989] and rigorous proved by [Alikakos, Bates,Chen: 1994]: v ε → v, (v, Γ) solves the following free boundary problem:                    ∆v = 0 in D \ Γt, t > 0, ∂v ∂n = 0 on ∂D, v= 2 3H on Γt, V = 1 2(∂nv + − ∂nv −) on Γt, (6) H: mean curvature of Γt; V: normal velocity of Γ

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 6 / 20

Introduction

Approximate solution to the deterministic Hele-Shaw

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 7 / 20

Introduction

Approximate solution to the deterministic Hele-Shaw

Let (v, Γ) be a smooth solution to deterministic Hele-Shaw model (6).

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 7 / 20

Introduction

Approximate solution to the deterministic Hele-Shaw

Let (v, Γ) be a smooth solution to deterministic Hele-Shaw model (6). [Alikakos, Bates, Chen 1994]: construct a pair (uε

A, v ε A) such that

   ∂tuε

A = ∆v ε A in DT,

v ε

A = 1

εF ′(uε

A) − ε∆uε A + r ε A in DT,

(7)

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 7 / 20

Introduction

Approximate solution to the deterministic Hele-Shaw

Let (v, Γ) be a smooth solution to deterministic Hele-Shaw model (6). [Alikakos, Bates, Chen 1994]: construct a pair (uε

A, v ε A) such that

   ∂tuε

A = ∆v ε A in DT,

v ε

A = 1

εF ′(uε

A) − ε∆uε A + r ε A in DT,

(7) Γt is the zero level set of uε

A(t)

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 7 / 20

Introduction

Approximate solution to the deterministic Hele-Shaw

Let (v, Γ) be a smooth solution to deterministic Hele-Shaw model (6). [Alikakos, Bates, Chen 1994]: construct a pair (uε

A, v ε A) such that

   ∂tuε

A = ∆v ε A in DT,

v ε

A = 1

εF ′(uε

A) − ε∆uε A + r ε A in DT,

(7) Γt is the zero level set of uε

A(t)

r ε

AC(DT ) εK−2.

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 7 / 20

Introduction

Approximate solution to the deterministic Hele-Shaw

Let (v, Γ) be a smooth solution to deterministic Hele-Shaw model (6). [Alikakos, Bates, Chen 1994]: construct a pair (uε

A, v ε A) such that

   ∂tuε

A = ∆v ε A in DT,

v ε

A = 1

εF ′(uε

A) − ε∆uε A + r ε A in DT,

(7) Γt is the zero level set of uε

A(t)

r ε

AC(DT ) εK−2.

v ε

A − v εC(DT ) ε.

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 7 / 20

Introduction

Approximate solution to the deterministic Hele-Shaw

Let (v, Γ) be a smooth solution to deterministic Hele-Shaw model (6). [Alikakos, Bates, Chen 1994]: construct a pair (uε

A, v ε A) such that

   ∂tuε

A = ∆v ε A in DT,

v ε

A = 1

εF ′(uε

A) − ε∆uε A + r ε A in DT,

(7) Γt is the zero level set of uε

A(t)

r ε

AC(DT ) εK−2.

v ε

A − v εC(DT ) ε.

For x with d(x, Γt) > Cε, |uε

A(t, x) − 1| ε

• r

|uε

A(t, x) + 1| ε.

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 7 / 20

Introduction

Approximate solution to the deterministic Hele-Shaw

Let (v, Γ) be a smooth solution to deterministic Hele-Shaw model (6). [Alikakos, Bates, Chen 1994]: construct a pair (uε

A, v ε A) such that

   ∂tuε

A = ∆v ε A in DT,

v ε

A = 1

εF ′(uε

A) − ε∆uε A + r ε A in DT,

(7) Γt is the zero level set of uε

A(t)

r ε

AC(DT ) εK−2.

v ε

A − v εC(DT ) ε.

For x with d(x, Γt) > Cε, |uε

A(t, x) − 1| ε

• r

|uε

A(t, x) + 1| ε.

uε − uε

AC(DT ) → 0.

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 7 / 20

Singular noise for large σ

Stochastic Cahn-Hilliard equation

           duε = ∆v εdt + εσdW in DT, v ε = 1 εF ′(uε) − ε∆uε in DT, ∂uε ∂n = ∂v ε ∂n = 0, (t, x) ∈ [0, T] × ∂D

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 8 / 20

Singular noise for large σ

Stochastic Cahn-Hilliard equation

           duε = ∆v εdt + εσdW in DT, v ε = 1 εF ′(uε) − ε∆uε in DT, ∂uε ∂n = ∂v ε ∂n = 0, (t, x) ∈ [0, T] × ∂D [Antonopoulou, Bl¨

• mker, Karali 2018]: W is a trace-class Wiener process and

σ > 23

3 , the limit is the same as the deterministic case.

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 8 / 20

Singular noise for large σ

Stochastic Cahn-Hilliard equation

           duε = ∆v εdt + εσdW in DT, v ε = 1 εF ′(uε) − ε∆uε in DT, ∂uε ∂n = ∂v ε ∂n = 0, (t, x) ∈ [0, T] × ∂D [Antonopoulou, Bl¨

• mker, Karali 2018]: W is a trace-class Wiener process and

σ > 23

3 , the limit is the same as the deterministic case.

Idea of proof: Compare uε and uε

A + Itˆ

• ’s formula.

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 8 / 20

Singular noise for large σ

Stochastic Cahn-Hilliard equation

           duε = ∆v εdt + εσdW in DT, v ε = 1 εF ′(uε) − ε∆uε in DT, ∂uε ∂n = ∂v ε ∂n = 0, (t, x) ∈ [0, T] × ∂D [Antonopoulou, Bl¨

• mker, Karali 2018]: W is a trace-class Wiener process and

σ > 23

3 , the limit is the same as the deterministic case.

Idea of proof: Compare uε and uε

A + Itˆ

• ’s formula.

Problems: Singular noise or Small σ ⇒? Stochastic Hele shaw

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 8 / 20

Singular noise for large σ

Singular noise: space-time white noise

In our case, W is an L2-cylindrical Wiener process.

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 9 / 20

Singular noise for large σ

Singular noise: space-time white noise

In our case, W is an L2-cylindrical Wiener process. Let Rε = uε − uε

A,

dRε = −ε∆2Rεdt + 1 ε∆

• F ′(uA

ε + Rε) − F ′(uA ε )

• dt + ∆r A

ε dt + εσdWt;

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 9 / 20

Singular noise for large σ

Singular noise: space-time white noise

In our case, W is an L2-cylindrical Wiener process. Let Rε = uε − uε

A,

dRε = −ε∆2Rεdt + 1 ε∆

• F ′(uA

ε + Rε) − F ′(uA ε )

• dt + ∆r A

ε dt + εσdWt;

Da prato-Debussche’s trick: Z ε

t := εσ t 0 e−(t−s)ε∆2dWs, Y ε := Rε − Z ε

satisfies: ∂tY ε = −ε∆2Y ε + 1 ε∆

• F ′(uA

ε + Y ε + Z ε) − F ′(uA ε )

• + ∆r A

ε ;

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 9 / 20

Singular noise for large σ

Singular noise: space-time white noise

In our case, W is an L2-cylindrical Wiener process. Let Rε = uε − uε

A,

dRε = −ε∆2Rεdt + 1 ε∆

• F ′(uA

ε + Rε) − F ′(uA ε )

• dt + ∆r A

ε dt + εσdWt;

Da prato-Debussche’s trick: Z ε

t := εσ t 0 e−(t−s)ε∆2dWs, Y ε := Rε − Z ε

satisfies: ∂tY ε = −ε∆2Y ε + 1 ε∆

• F ′(uA

ε + Y ε + Z ε) − F ′(uA ε )

• + ∆r A

ε ;

E(Z εC(DT )) ε(σ− 1

4 )−; Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 9 / 20

Singular noise for large σ

Singular noise: space-time white noise

In our case, W is an L2-cylindrical Wiener process. Let Rε = uε − uε

A,

dRε = −ε∆2Rεdt + 1 ε∆

• F ′(uA

ε + Rε) − F ′(uA ε )

• dt + ∆r A

ε dt + εσdWt;

Da prato-Debussche’s trick: Z ε

t := εσ t 0 e−(t−s)ε∆2dWs, Y ε := Rε − Z ε

satisfies: ∂tY ε = −ε∆2Y ε + 1 ε∆

• F ′(uA

ε + Y ε + Z ε) − F ′(uA ε )

• + ∆r A

ε ;

E(Z εC(DT )) ε(σ− 1

4 )−;

Stopping time argument yields that Theorem 1 [Banas, Yang, Z. 19] For σ > 107

12 , RεL3

t L3 x converges to 0 in probability. Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 9 / 20

Singular noise for large σ

Singular noise: space-time white noise

In our case, W is an L2-cylindrical Wiener process. Let Rε = uε − uε

A,

dRε = −ε∆2Rεdt + 1 ε∆

• F ′(uA

ε + Rε) − F ′(uA ε )

• dt + ∆r A

ε dt + εσdWt;

Da prato-Debussche’s trick: Z ε

t := εσ t 0 e−(t−s)ε∆2dWs, Y ε := Rε − Z ε

satisfies: ∂tY ε = −ε∆2Y ε + 1 ε∆

• F ′(uA

ε + Y ε + Z ε) − F ′(uA ε )

• + ∆r A

ε ;

E(Z εC(DT )) ε(σ− 1

4 )−;

Stopping time argument yields that Theorem 1 [Banas, Yang, Z. 19] For σ > 107

12 , RεL3

t L3 x converges to 0 in probability.

⇒ the limit is the same as the deterministic case

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 9 / 20

Singular noise for large σ

Singular noise: Conservative noise

This method can be applied to duε,h = ∆

• −ε∆uε,h + 1

ε

• F ′(uε,h) − 3cε

h,tuε,h

dt + εσ∇ · dW h

t ,

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 10 / 20

Singular noise for large σ

Singular noise: Conservative noise

This method can be applied to duε,h = ∆

• −ε∆uε,h + 1

ε

• F ′(uε,h) − 3cε

h,tuε,h

dt + εσ∇ · dW h

t ,

W h = W ∗ ρh

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 10 / 20

Singular noise for large σ

Singular noise: Conservative noise

This method can be applied to duε,h = ∆

• −ε∆uε,h + 1

ε

• F ′(uε,h) − 3cε

h,tuε,h

dt + εσ∇ · dW h

t ,

W h = W ∗ ρh cε

h,t is renormalization constant

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 10 / 20

Singular noise for large σ

Singular noise: Conservative noise

This method can be applied to duε,h = ∆

• −ε∆uε,h + 1

ε

• F ′(uε,h) − 3cε

h,tuε,h

dt + εσ∇ · dW h

t ,

W h = W ∗ ρh cε

h,t is renormalization constant

|cε

h,t| ε2σ−1| log h|

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 10 / 20

Singular noise for large σ

Singular noise: Conservative noise

This method can be applied to duε,h = ∆

• −ε∆uε,h + 1

ε

• F ′(uε,h) − 3cε

h,tuε,h

dt + εσ∇ · dW h

t ,

W h = W ∗ ρh cε

h,t is renormalization constant

|cε

h,t| ε2σ−1| log h|

Global well-posedness has been obtained in [R¨

• ckner, Yang, Z. 18].

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 10 / 20

Singular noise for large σ

Singular noise: Conservative noise

This method can be applied to duε,h = ∆

• −ε∆uε,h + 1

ε

• F ′(uε,h) − 3cε

h,tuε,h

dt + εσ∇ · dW h

t ,

W h = W ∗ ρh cε

h,t is renormalization constant

|cε

h,t| ε2σ−1| log h|

Global well-posedness has been obtained in [R¨

• ckner, Yang, Z. 18].

Let Rε,h = uε,h − uε

A,

Theorem 2 [Banas, Yang, Z. 19] Assume εθ h2. Then for σ > 26

3 + θ, Rε,hL3

t L3 x converges

to 0 in probability.

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 10 / 20

Singular noise for large σ

Singular noise: Conservative noise

This method can be applied to duε,h = ∆

• −ε∆uε,h + 1

ε

• F ′(uε,h) − 3cε

h,tuε,h

dt + εσ∇ · dW h

t ,

W h = W ∗ ρh cε

h,t is renormalization constant

|cε

h,t| ε2σ−1| log h|

Global well-posedness has been obtained in [R¨

• ckner, Yang, Z. 18].

Let Rε,h = uε,h − uε

A,

Theorem 2 [Banas, Yang, Z. 19] Assume εθ h2. Then for σ > 26

3 + θ, Rε,hL3

t L3 x converges

to 0 in probability. ⇒ the limit is the same as the deterministic case

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 10 / 20

Singular noise for large σ

Small σ?

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 11 / 20

Singular noise for large σ

Small σ?

[Antonopoulou, Bl¨

• mker, Karali 2018]: Conjecture: σ = 1 formally derive

stochastic Hele Shaw model

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 11 / 20

Singular noise for large σ

Small σ?

[Antonopoulou, Bl¨

• mker, Karali 2018]: Conjecture: σ = 1 formally derive

stochastic Hele Shaw model Difficult to prove convergence for small σ by the approximate solution uε

A;

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 11 / 20

Singular noise for large σ

Small σ?

[Antonopoulou, Bl¨

• mker, Karali 2018]: Conjecture: σ = 1 formally derive

stochastic Hele Shaw model Difficult to prove convergence for small σ by the approximate solution uε

A;

Only possible to prove the convergence to deterministic Hele-Shaw model;

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 11 / 20

Singular noise for large σ

Small σ?

[Antonopoulou, Bl¨

• mker, Karali 2018]: Conjecture: σ = 1 formally derive

stochastic Hele Shaw model Difficult to prove convergence for small σ by the approximate solution uε

A;

Only possible to prove the convergence to deterministic Hele-Shaw model; Difficulty: Noise is not regular enough w.r.t. time t.

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 11 / 20

Weak approach and tightness for small σ

Main results for σ ≥ 1/2

We consider the stochastic Cahn-Hilliard equation on a bounded smooth open domain D ⊂ Rd (d = 2, 3):                duε = ∆v εdt + εσdWt, (t, x) ∈ [0, T] × D, v ε = −ε∆uε(t) + 1 εF ′(uε(t)), (t, x) ∈ [0, T] × D, ∂uε ∂n = ∂v ε ∂n = 0, (t, x) ∈ [0, T] × ∂D, uε(0, x) = uε

0(x),

x ∈ D. (8)

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 12 / 20

Weak approach and tightness for small σ

Main results for σ ≥ 1/2

We consider the stochastic Cahn-Hilliard equation on a bounded smooth open domain D ⊂ Rd (d = 2, 3):                duε = ∆v εdt + εσdWt, (t, x) ∈ [0, T] × D, v ε = −ε∆uε(t) + 1 εF ′(uε(t)), (t, x) ∈ [0, T] × D, ∂uε ∂n = ∂v ε ∂n = 0, (t, x) ∈ [0, T] × ∂D, uε(0, x) = uε

0(x),

x ∈ D. (8) Main results [Yang Z. 19] For σ ≥ 1/2, in radial symmetric case the limit is the deterministic Hele Shaw model

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 12 / 20

Weak approach and tightness for small σ

Main results for σ ≥ 1/2

We consider the stochastic Cahn-Hilliard equation on a bounded smooth open domain D ⊂ Rd (d = 2, 3):                duε = ∆v εdt + εσdWt, (t, x) ∈ [0, T] × D, v ε = −ε∆uε(t) + 1 εF ′(uε(t)), (t, x) ∈ [0, T] × D, ∂uε ∂n = ∂v ε ∂n = 0, (t, x) ∈ [0, T] × ∂D, uε(0, x) = uε

0(x),

x ∈ D. (8) Main results [Yang Z. 19] For σ ≥ 1/2, in radial symmetric case the limit is the deterministic Hele Shaw model ⇒ Conjecture in [Antonopoulou, Bl¨

• mker, Karali 2018] may not ture.

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 12 / 20

Weak approach and tightness for small σ

Main results for σ ≥ 1/2

We consider the stochastic Cahn-Hilliard equation on a bounded smooth open domain D ⊂ Rd (d = 2, 3):                duε = ∆v εdt + εσdWt, (t, x) ∈ [0, T] × D, v ε = −ε∆uε(t) + 1 εF ′(uε(t)), (t, x) ∈ [0, T] × D, ∂uε ∂n = ∂v ε ∂n = 0, (t, x) ∈ [0, T] × ∂D, uε(0, x) = uε

0(x),

x ∈ D. (8) Main results [Yang Z. 19] For σ ≥ 1/2, in radial symmetric case the limit is the deterministic Hele Shaw model ⇒ Conjecture in [Antonopoulou, Bl¨

• mker, Karali 2018] may not ture.

We conjecture in general case for σ ≥ 1/2, the limit is the deterministic Hele Shaw model.

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 12 / 20

Weak approach and tightness for small σ

Idea of Proof: Lyapunov property

Recall Eε(uε) := ε 2

• D

|∇uε(x)|2dx + 1 ε

• D

F(uε(x))dx.

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 13 / 20

Weak approach and tightness for small σ

Idea of Proof: Lyapunov property

Recall Eε(uε) := ε 2

• D

|∇uε(x)|2dx + 1 ε

• D

F(uε(x))dx. By Itˆ

• ’s formula

dEε(uε) = DEε(uε), duε + ε2σ 2 Tr(QD2Eε(uε))dt = − ∇v ε, ∇v εdt + ε2σ+1 2 Tr(−∆Q)dt + ε2σ−1 2 Tr(F ′′(uε)Q)dt + εσv ε, dWt.

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 13 / 20

Weak approach and tightness for small σ

Idea of Proof: Lyapunov property

Recall Eε(uε) := ε 2

• D

|∇uε(x)|2dx + 1 ε

• D

F(uε(x))dx. By Itˆ

• ’s formula

dEε(uε) = DEε(uε), duε + ε2σ 2 Tr(QD2Eε(uε))dt = − ∇v ε, ∇v εdt + ε2σ+1 2 Tr(−∆Q)dt + ε2σ−1 2 Tr(F ′′(uε)Q)dt + εσv ε, dWt. Lemma 3 (Lyapunov property) Assume Tr(−∆Q) < ∞ and sup0<ε<1 Eε(uε

0) < E0 then

there exists ε0 ∈ (0, 1) such that for any ε ∈ (0, ε0] and any p ≥ 1, E sup

t∈[0,T]

Eε(t)p (ε2σ−1 + E0)p, E T ∇v ε2

L2dt

p (ε2σ−1 + E0)p.

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 13 / 20

Weak approach and tightness for small σ

Idea of Proof: Tightness for σ ≥ 1

2

Lemma 4 Assume σ ≥ 1

• 2. For any β ∈ (0, 1

12)

E

• uεC β([0,T];L2)
• 1

For any δ > 0, there exists a constant C ≡ C(δ, T) > 0, such that P T v ε(t)2

H1dt ≤ C

• ≥ 1 − δ.

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 14 / 20

Weak approach and tightness for small σ

Idea of Proof: Tightness for σ ≥ 1

2

Lemma 4 Assume σ ≥ 1

• 2. For any β ∈ (0, 1

12)

E

• uεC β([0,T];L2)
• 1

For any δ > 0, there exists a constant C ≡ C(δ, T) > 0, such that P T v ε(t)2

H1dt ≤ C

• ≥ 1 − δ.

Tightness + Skorohod theorem ⇒ Theorem 5 [Y, Zhu: 19] For any σ ≥ 1

2, P − a.s. ω,

uε → −1 + 21E in C([0, T], L2

w),

v ε → v in L2

w(0, T; H1)

1E ∈ L∞(0, T; BV )

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 14 / 20

Weak approach and tightness for small σ

The Limit equation for σ ≥ 1

2

Since duε = ∆v εdt + εσdWt,

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 15 / 20

Weak approach and tightness for small σ

The Limit equation for σ ≥ 1

2

Since duε = ∆v εdt + εσdWt, ε → 0, in a weak sense 2∂t1E = ∆v.

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 15 / 20

Weak approach and tightness for small σ

The Limit equation for σ ≥ 1

2

Since duε = ∆v εdt + εσdWt, ε → 0, in a weak sense 2∂t1E = ∆v. ⇒ weak solutions to the deterministic model

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 15 / 20

Weak approach and tightness for small σ

The Limit equation for σ ≥ 1

2

Since duε = ∆v εdt + εσdWt, ε → 0, in a weak sense 2∂t1E = ∆v. ⇒ weak solutions to the deterministic model ⇒ rigorously proved in radial symmetric case and conjectured in general case to the deterministic Hele Shaw model:                    ∆v = 0 in D \ Γt, t > 0, ∂v ∂n = 0 on ∂D, v= 2 3H on Γt, V = 1 2(∂nv + − ∂nv −) on Γt. (9)

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 15 / 20

Weak approach and tightness for small σ

The Limit equation for σ ≥ 1

2

Since duε = ∆v εdt + εσdWt, ε → 0, in a weak sense 2∂t1E = ∆v. ⇒ weak solutions to the deterministic model ⇒ rigorously proved in radial symmetric case and conjectured in general case to the deterministic Hele Shaw model:                    ∆v = 0 in D \ Γt, t > 0, ∂v ∂n = 0 on ∂D, v= 2 3H on Γt, V = 1 2(∂nv + − ∂nv −) on Γt. (9) ⇒ The only possibility to converge to ”stochastic Hele-Shaw model” is σ = 0.

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 15 / 20

Weak approach and tightness for small σ

Equation driven by “smeared” noise

We consider the following random PDE:                    ∂uε ∂t = ∆v ε + εσξε

t ,

(t, x) ∈ [0, T] × D, v ε = −ε∆uε(t) + 1 εF ′(uε(t)), (t, x) ∈ [0, T] × D, ∂uε ∂n = ∂v ε ∂n = 0, (t, x) ∈ [0, T] × ∂D, uε(0, x) = uε

0(x),

x ∈ D, (10) where ξε → dW

dt .

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 16 / 20

Weak approach and tightness for small σ

Equation driven by “smeared” noise

We consider the following random PDE:                    ∂uε ∂t = ∆v ε + εσξε

t ,

(t, x) ∈ [0, T] × D, v ε = −ε∆uε(t) + 1 εF ′(uε(t)), (t, x) ∈ [0, T] × D, ∂uε ∂n = ∂v ε ∂n = 0, (t, x) ∈ [0, T] × ∂D, uε(0, x) = uε

0(x),

x ∈ D, (10) where ξε → dW

dt .

Theorem 6 [Y, Zhu 19]: For any σ ≥ 0, P − a.s. ω, uε → −1 + 21E in C([0, T], L2

w),

v ε → v in L2

w(0, T; H1),

1E ∈ L∞(0, T; BV ).

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 16 / 20

Weak approach and tightness for small σ

Equation driven by “smeared” noise

We consider the following random PDE:                    ∂uε ∂t = ∆v ε + εσξε

t ,

(t, x) ∈ [0, T] × D, v ε = −ε∆uε(t) + 1 εF ′(uε(t)), (t, x) ∈ [0, T] × D, ∂uε ∂n = ∂v ε ∂n = 0, (t, x) ∈ [0, T] × ∂D, uε(0, x) = uε

0(x),

x ∈ D, (10) where ξε → dW

dt .

Theorem 6 [Y, Zhu 19]: For any σ ≥ 0, P − a.s. ω, uε → −1 + 21E in C([0, T], L2

w),

v ε → v in L2

w(0, T; H1),

1E ∈ L∞(0, T; BV ). σ > 0 ⇒ deterministic model (9).

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 16 / 20

Weak approach and tightness for small σ

Stochastic Hele-Shaw model

For σ = 0, ∂tuε = ∆v ε + ξε.

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 17 / 20

Weak approach and tightness for small σ

Stochastic Hele-Shaw model

For σ = 0, ∂tuε = ∆v ε + ξε. Main results for σ = 0 [Y, Zhu: 19] the limit equation (weak formula for stochastic Hele Shaw model): 2d1Et = ∆vdt + dWt.

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 17 / 20

Weak approach and tightness for small σ

Stochastic Hele-Shaw model

For σ = 0, ∂tuε = ∆v ε + ξε. Main results for σ = 0 [Y, Zhu: 19] the limit equation (weak formula for stochastic Hele Shaw model): 2d1Et = ∆vdt + dWt. In radial symmetric case, v = 2 3H on Γt.

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 17 / 20

Weak approach and tightness for small σ

Stochastic Hele-Shaw model

For σ = 0, ∂tuε = ∆v ε + ξε. Main results for σ = 0 [Y, Zhu: 19] the limit equation (weak formula for stochastic Hele Shaw model): 2d1Et = ∆vdt + dWt. In radial symmetric case, v = 2 3H on Γt. Formally

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 17 / 20

Weak approach and tightness for small σ

Stochastic Hele-Shaw model

For σ = 0, ∂tuε = ∆v ε + ξε. Main results for σ = 0 [Y, Zhu: 19] the limit equation (weak formula for stochastic Hele Shaw model): 2d1Et = ∆vdt + dWt. In radial symmetric case, v = 2 3H on Γt. Formally −∆v = ξ := dWt

dt

in D \ Γt;

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 17 / 20

Weak approach and tightness for small σ

Stochastic Hele-Shaw model

For σ = 0, ∂tuε = ∆v ε + ξε. Main results for σ = 0 [Y, Zhu: 19] the limit equation (weak formula for stochastic Hele Shaw model): 2d1Et = ∆vdt + dWt. In radial symmetric case, v = 2 3H on Γt. Formally −∆v = ξ := dWt

dt

in D \ Γt; Let ˆ v = v + ∆−1ξ t

• D

∂t1Etψ = −1 2 t

• D+ ∇ˆ

v∇ψ − 1 2 t

• D− ∇ˆ

v∇ψ = 1 2 t

• Γt

(∂nˆ v + − ∂nˆ v −)ψ.

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 17 / 20

Weak approach and tightness for small σ

Stochastic Hele-Shaw model

For σ = 0, we proved in radial symmetric case, that the sharp interface limit of (10) is the weak formula of the following ”stochastic Hele-Shaw” model:                      ∆vdt = −dWt in D \ Γt, t > 0, ∂v ∂n = 0 on ∂D, v = 2 3H on Γt, Vdt = 1 2 ∂ ∂n

• Γt

(vdt + ∆−1dWt), (11) where H: mean curvature ∂ ∂n

• Γt

f = ∂f + ∂n − ∂f − ∂n .

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 18 / 20

Weak approach and tightness for small σ

Further plan

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 19 / 20

Weak approach and tightness for small σ

Further plan

Rigorous meaning of stochastic Hele-Shaw model (11);

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 19 / 20

Weak approach and tightness for small σ

Further plan

Rigorous meaning of stochastic Hele-Shaw model (11); Approximate solution to (11);

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 19 / 20

Weak approach and tightness for small σ

Further plan

Rigorous meaning of stochastic Hele-Shaw model (11); Approximate solution to (11); Rigorous proof of convergence to (11) in general case;

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 19 / 20

Weak approach and tightness for small σ

Further plan

Rigorous meaning of stochastic Hele-Shaw model (11); Approximate solution to (11); Rigorous proof of convergence to (11) in general case; Singular noise for small σ.

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 19 / 20

Weak approach and tightness for small σ

N D Alikakos, P W Bates, and Xinfu Chen. Convergence of the Cahn-Hilliard Equation to the Hele-Shaw Model. Archive for Rational Mechanics and Analysis, 128(2):165–205, 1994. D C Antonopoulou, D Bl¨

• mker, and G D Karali.

The sharp interface limit for the stochastic Cahn-Hilliard equation. Annales de l’Institut Henri Poincar´ e Probabilit´ es et Statistiques, 54(1):280–298, 2018. Lubomir Banas, Huanyu Yang, and Rongchan Zhu. Sharp interface limit of stochastic Cahn-Hilliard equation with singular noise. arXiv.org, May 2019. Xinfu Chen. Global asymptotic limit of solutions of the Cahn-Hilliard equation. Journal of Differential Geometry, 44(2):262–311, 1996. Giuseppe Da Prato and Arnaud Debussche. Stochastic Cahn-Hilliard equation. Nonlinear Analysis: Theory, Methods & Applications, 26(2):241–263, 1996. J E Hutchinson and Y Tonegawa. Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory. Calculus of Variations and Partial Differential Equations, 10(1):49–84, January 2000.

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 19 / 20

Weak approach and tightness for small σ

R L Pego. Front Migration in the Nonlinear Cahn-Hilliard Equation. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 422(1863):261–278, April 1989. Michael R¨

• ckner, Huanyu Yang, and Rongchan Zhu.

Conservative stochastic 2-dimensional Cahn-Hilliard equation. February 2018. Matthias R¨

• ger and Reiner Schaetzle.

On a modified conjecture of De Giorgi. Mathematische Zeitschrift, 254(4):675–714, December 2006. Matthias R¨

• ger and Yoshihiro Tonegawa.

Convergence of phase-field approximations to the Gibbs-Thomson law. Calculus of Variations and Partial Differential Equations, 32(1):111–136, 2008.

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 20 / 20

Weak approach and tightness for small σ

Thank you!

Rongchan Zhu (Beijing Institute of Technology) stochastic Cahn-Hilliard equation July 29, 2019 20 / 20