Absolute continuity and numerical approximation of stochastic Cahn–Hilliard equation with unbounded noise diffusion
In this article, we consider the absolute continuity and numerical approximation of the solution of the stochastic Cahn–Hilliard equation with unbounded noise diffusion. We first obtain the Hölder continuity and Malliavin differentiability of the solution of the stochastic Cahn–Hilliard equation by...
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Published in | Journal of Differential Equations Vol. 269; no. 11; pp. 10143 - 10180 |
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Abstract | In this article, we consider the absolute continuity and numerical approximation of the solution of the stochastic Cahn–Hilliard equation with unbounded noise diffusion. We first obtain the Hölder continuity and Malliavin differentiability of the solution of the stochastic Cahn–Hilliard equation by using the strong convergence of the spectral Gakerkin approximation. Then we prove the existence and strict positivity of the density function of the law of the exact solution for the stochastic Cahn–Hilliard equation with sublinear growth diffusion coefficient, which fills a gap for the existed result when the diffusion coefficient satisfies a growth condition of order 1/3<α<1. To approximate the density function of the exact solution, we propose a full discretization based on the spatial spectral Galerkin approximation and the temporal drift implicit Euler scheme. Furthermore, a general framework for deriving the strong convergence rate of the full discretization is developed based on the variation approach and the factorization method. Consequently, we obtain the sharp mean square convergence rates in both time and space via Sobolev interpolation inequalities and semigroup theories. To the best of our knowledge, this is the first result on the convergence rate of full discretizations for the considered equation. |
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AbstractList | In this article, we consider the absolute continuity and numerical approximation of the solution of the stochastic Cahn–Hilliard equation with unbounded noise diffusion. We first obtain the Hölder continuity and Malliavin differentiability of the solution of the stochastic Cahn–Hilliard equation by using the strong convergence of the spectral Gakerkin approximation. Then we prove the existence and strict positivity of the density function of the law of the exact solution for the stochastic Cahn–Hilliard equation with sublinear growth diffusion coefficient, which fills a gap for the existed result when the diffusion coefficient satisfies a growth condition of order 1/3<α<1. To approximate the density function of the exact solution, we propose a full discretization based on the spatial spectral Galerkin approximation and the temporal drift implicit Euler scheme. Furthermore, a general framework for deriving the strong convergence rate of the full discretization is developed based on the variation approach and the factorization method. Consequently, we obtain the sharp mean square convergence rates in both time and space via Sobolev interpolation inequalities and semigroup theories. To the best of our knowledge, this is the first result on the convergence rate of full discretizations for the considered equation. |
Author | Cui, Jianbo Hong, Jialin |
Author_xml | – sequence: 1 givenname: Jianbo surname: Cui fullname: Cui, Jianbo email: jcui82@gatech.edu organization: School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA – sequence: 2 givenname: Jialin surname: Hong fullname: Hong, Jialin email: hjl@lsec.cc.ac.cn organization: LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China |
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Cites_doi | 10.1137/17M1154904 10.1137/110828150 10.1023/A:1008686922032 10.1080/10451120290019195 10.1093/imanum/dry052 10.1063/1.1744102 10.1016/0362-546X(94)00277-O 10.1137/17M1121627 10.1137/18M1215554 10.1214/19-AOP1345 10.1016/j.jde.2017.05.002 10.2307/3318542 10.1016/j.jde.2018.10.034 10.1016/j.jde.2018.05.004 10.1016/0167-2789(84)90180-5 10.1016/j.jde.2015.10.004 10.1016/j.apnum.2017.09.010 10.1016/j.spa.2018.02.008 |
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Keywords | Stochastic Cahn–Hilliard equation Numerical approximation 35R60 Strong convergence rate 60H07 Unbounded noise diffusion Malliavin calculus 60H15 60H35 |
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References | Cahn, Hilliard (br0060) 1958; 2 Bréhier, Cui, Hong (br0050) 2019; 39 Becker, Jentzen (br0040) 2019; 129 Chai, Cao, Zou, Zhao (br0090) 2018; 124 Nualart (br0210) 2006 Cui, Hong (br0100) Qi, Wang (br0220) Cardon-Weber (br0070) 2001; 7 Cui, Hong, Sun (br0150) Kovács, Larsson, Mesforush (br0190) 2011; 49 Novick-Cohen, Segel (br0200) 1984; 10 Bally, Pardoux (br0030) 1998; 9 Da Prato, Debussche (br0160) 1996; 26 Furihata, Kovács, Larsson, Lindgren (br0170) 2018; 56 Cui, Hong (br0110) 2018; 56 Cui, Hong, Liu, Zhou (br0140) 2019; 266 Cardon-Weber (br0080) 2002; 72 Cui, Hong, Liu (br0130) 2017; 263 Antonopoulou, Farazakis, Karali (br0010) 2018; 265 Cui, Hong (br0120) 2019; 57 Hutzenthaler, Jentzen (br0180) 2020; 48 Antonopoulou, Karali, Millet (br0020) 2016; 260 Cardon-Weber (10.1016/j.jde.2020.07.007_br0080) 2002; 72 Cui (10.1016/j.jde.2020.07.007_br0110) 2018; 56 Cui (10.1016/j.jde.2020.07.007_br0140) 2019; 266 Cahn (10.1016/j.jde.2020.07.007_br0060) 1958; 2 Cui (10.1016/j.jde.2020.07.007_br0120) 2019; 57 Bréhier (10.1016/j.jde.2020.07.007_br0050) 2019; 39 Antonopoulou (10.1016/j.jde.2020.07.007_br0020) 2016; 260 Antonopoulou (10.1016/j.jde.2020.07.007_br0010) 2018; 265 Furihata (10.1016/j.jde.2020.07.007_br0170) 2018; 56 Becker (10.1016/j.jde.2020.07.007_br0040) 2019; 129 Qi (10.1016/j.jde.2020.07.007_br0220) Hutzenthaler (10.1016/j.jde.2020.07.007_br0180) 2020; 48 Kovács (10.1016/j.jde.2020.07.007_br0190) 2011; 49 Chai (10.1016/j.jde.2020.07.007_br0090) 2018; 124 Cui (10.1016/j.jde.2020.07.007_br0150) Novick-Cohen (10.1016/j.jde.2020.07.007_br0200) 1984; 10 Cui (10.1016/j.jde.2020.07.007_br0100) Cardon-Weber (10.1016/j.jde.2020.07.007_br0070) 2001; 7 Da Prato (10.1016/j.jde.2020.07.007_br0160) 1996; 26 Bally (10.1016/j.jde.2020.07.007_br0030) 1998; 9 Cui (10.1016/j.jde.2020.07.007_br0130) 2017; 263 Nualart (10.1016/j.jde.2020.07.007_br0210) 2006 |
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Title | Absolute continuity and numerical approximation of stochastic Cahn–Hilliard equation with unbounded noise diffusion |
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