Nonlocal-to-Local Convergence of Cahn–Hilliard Equations : Neumann Boundary Conditions and Viscosity Terms

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2021
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Davoli, Elisa
Scarpa, Luca
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Archive for Rational Mechanics and Analysis. Springer. 2021, 239(1), pp. 117-149. ISSN 0003-9527. eISSN 1432-0673. Available under: doi: 10.1007/s00205-020-01573-9
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We consider a class of nonlocal viscous Cahn-Hilliard equations with Neumann boundary conditions for the chemical potential. The double-well potential is allowed to be singular (e.g. of logarithmic type), while the singularity of the convolution kernel does not fall in any available existence theory under Neumann boundary conditions. We prove well-posedness for the nonlocal equation in a suitable variational sense. Secondly, we show that the solutions to the nonlocal equation converge to the corresponding solutions to the local equation, as the convolution kernels approximate a Dirac delta. The asymptotic behaviour is analyzed by means of monotone analysis and Gamma convergence results, both when the limiting local Cahn-Hilliard equation is of viscous type and of pure type.

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ISO 690DAVOLI, Elisa, Luca SCARPA, Lara TRUSSARDI, 2021. Nonlocal-to-Local Convergence of Cahn–Hilliard Equations : Neumann Boundary Conditions and Viscosity Terms. In: Archive for Rational Mechanics and Analysis. Springer. 2021, 239(1), pp. 117-149. ISSN 0003-9527. eISSN 1432-0673. Available under: doi: 10.1007/s00205-020-01573-9
BibTex
@article{Davoli2021Nonlo-55533,
  year={2021},
  doi={10.1007/s00205-020-01573-9},
  title={Nonlocal-to-Local Convergence of Cahn–Hilliard Equations : Neumann Boundary Conditions and Viscosity Terms},
  number={1},
  volume={239},
  issn={0003-9527},
  journal={Archive for Rational Mechanics and Analysis},
  pages={117--149},
  author={Davoli, Elisa and Scarpa, Luca and Trussardi, Lara}
}
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