Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

Diffeomorphic shape evolution coupled with a reaction-diffusion PDE on a growth potential


Authors: Dai-Ni Hsieh, Sylvain Arguillère, Nicolas Charon and Laurent Younes
Journal: Quart. Appl. Math.
MSC (2020): Primary 35R37, 35K57, 35Q92
DOI: https://doi.org/10.1090/qam/1600
Published electronically: August 24, 2021
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Abstract: This paper studies a longitudinal shape transformation model in which shapes are deformed in response to an internal growth potential that evolves according to an advection reaction diffusion process. This model extends prior works that considered a static growth potential, i.e., the initial growth potential is only advected by diffeomorphisms. We focus on the mathematical study of the corresponding system of coupled PDEs describing the joint dynamics of the diffeomorphic transformation together with the growth potential on the moving domain. Specifically, we prove the uniqueness and long time existence of solutions to this system with reasonable initial and boundary conditions as well as regularization on deformation fields. In addition, we provide a few simple simulations of this model in the case of isotropic elastic materials in 2D.


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Additional Information

Dai-Ni Hsieh
Affiliation: Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, MD 21218
MR Author ID: 1036115
ORCID: 0000-0002-1365-4877
Email: dnhsieh@jhu.edu

Sylvain Arguillère
Affiliation: Laboratoire Paul Painlevé, University of Lille, France
ORCID: 0000-0002-8916-0291
Email: sylvain.arguillere@univ-lille.fr

Nicolas Charon
Affiliation: Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, MD 21218
MR Author ID: 1045405
ORCID: 0000-0002-6032-247X
Email: charon@cis.jhu.edu

Laurent Younes
Affiliation: Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, MD 21218
MR Author ID: 238511
Email: laurent.younes@jhu.edu

Received by editor(s): April 7, 2021
Received by editor(s) in revised form: July 12, 2021
Published electronically: August 24, 2021
Additional Notes: The third author acknowledges the support of the NSF through the grant DMS-1945224.
Article copyright: © Copyright 2021 Brown University