Yang, Huanyu: Stochastic Cahn-Hilliard Equations and Their Sharp Interface Limits. 2019
Inhalt
- Preface
- Introduction and Main Results
- The deterministic case
- The stochastic case
- Well-posedness for stochastic Cahn-Hilliard equation
- Sharp interface limit for big >0
- Sharp interface limit for small 0
- Structure of the thesis
- Preliminary
- Conservative stochastic 2-dimensional Cahn-Hilliard equation
- Notations and preliminaries
- The Linear Equation and Wick Powers
- The Solution to the Shifted Equation
- Relation to the solution given by Dirichlet forms
- Solution given by Dirichlet forms
- Relation between the two solutions
- Markov uniqueness in the restricted sense
- Stationary solution
- Ergodicity
- Sharp interface limit of stochastic Cahn-Hilliard equation with singular noise
- Notations and preliminaries
- The sharp interface limit for space-time white noise
- The proof of the Main Theorem
- The decomposition of the equation for the error
- Estimate for Z
- Local-in-time estimate for Y up to T on the set
- Final step: Globalization TT
- Sharp interface limit for conservative noise
- Weak solutions to the sharp interface limit of stochastic Cahn-Hilliard equations
- Preliminary
- Basic notations and assumptions
- Definition of a weak solution to the limit of equation (5.1)
- Main results for Q-Wiener noise
- Remarks on the definition of weak solutions
- Convergence
- Lyapunov functional E and basic estimates
- Estimates for {u}
- Estimates for {v}
- Tightness
- Proof of Theorem 5.3
- The case that =12
- Case of radial symmetry for 12
- Proof of Theorem 5.4
- The case for ``smeared" noise
- Bibliography