Global-in-time Gevrey regularity solutions for the functionalized Cahn-Hilliard equation

Kelong Cheng; Cheng Wang; Steven M. Wise; Zixia Yuan

doi:10.3934/dcdss.2020186

Article Contents

2020, Volume 13, Issue 8: 2211-2229. Doi: 10.3934/dcdss.2020186

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Global-in-time Gevrey regularity solutions for the functionalized Cahn-Hilliard equation

1.
School of Science, Southwest University of Science and Technology, Mianyang, Sichuan 621010, China
2.
Mathematics Department, The University of Massachusetts, North Dartmouth, MA 02747, USA
3.
Mathematics Department, The University of Tennessee, Knoxville, TN 37996, USA
4.
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China

^* Corresponding author: swise1@utk.edu
^* Corresponding author: swise1@utk.edu

Received: March 31, 2018

Early access: November 2019

Published: May 2020

C. Wang was supported by NSF grant DMS-1418689. S.M. Wise was supported by NSF grants DMS-1418692 and DMS-1719854

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Abstract

The existence and uniqueness of Gevrey regularity solutions for the functionalized Cahn-Hilliard (FCH) and Cahn-Hilliard-Willmore (CHW) equations are established. The energy dissipation law yields a uniform-in-time $ H^2 $ bound of the solution, and the polynomial patterns of the nonlinear terms enable one to derive a local-in-time solution with Gevrey regularity. A careful calculation reveals that the existence time interval length depends on the $ H^3 $ norm of the initial data. A further detailed estimate for the original PDE system indicates a uniform-in-time $ H^3 $ bound. Consequently, a global-in-time solution becomes available with Gevrey regularity.

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Mathematics Subject Classification: Primary: 35K35, 35K55.

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