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Weak Solutions to the Cahn-Hilliard Equation with Degenerate Diffusion Mobility in ℝN

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Abstract

This paper is concerned with a popular form of Cahn-Hilliard equation which plays an important role in understanding the evolution of phase separation. We get the existence and regularity of a weak solution to nonlinear parabolic, fourth order Cahn-Hilliard equation with degenerate mobility M(u) = um(1 − u)m which is allowed to vanish at 0 and 1. The existence and regularity of weak solutions to the degenerate Cahn-Hilliard equation are obtained by getting the limits of Cahn-Hilliard equation with non-degenerate mobility. We explore the initial value problem with compact support and obtain the local non-negative result. Further, the above derivation process is also suitable for the viscous Cahn-Hilliard equation with degenerate mobility.

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Acknowledgements

The authors are grateful to reviewers and for their time and comments.

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Authors and Affiliations

  1. College of Mathematics, Physics and Statistics, Shanghai University of Engineering Science, Shanghai, 201620, P. R. China

    Ji Hui Wu

  2. College of Sciences, Shanghai University, Shanghai, 200444, P. R. China

    Ji Hui Wu

  3. College of Mechanical Engineering, Beijing University of Technology, Beijing, 100124, P. R. China

    Lei Lu

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  1. Ji Hui Wu
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  2. Lei Lu
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Correspondence to Ji Hui Wu or Lei Lu.

Additional information

Supported by NSF of Shanghai University of Engineering Science (Grant No. 0244-E3-0507-19-05155)

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Wu, J.H., Lu, L. Weak Solutions to the Cahn-Hilliard Equation with Degenerate Diffusion Mobility in ℝN. Acta. Math. Sin.-English Ser. 35, 1629–1654 (2019). https://doi.org/10.1007/s10114-019-8318-4

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  • DOI: https://doi.org/10.1007/s10114-019-8318-4

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