The separation property for 2D Cahn-Hilliard equations: Local, nonlocal and fractional energy cases
نویسندگان
چکیده
We study the separation property for Cahn-Hilliard type equations with constant mobility and (physically relevant) singular potentials in two dimensions. That is, any solution initial finite energy stays uniformly away from pure phases $ \pm 1 a certain time on. Beyond its physical interest, this plays crucial role to achieve high order Sobolev analytic regularity of solutions analyze their longtime behavior. In local case, we streamline known arguments by exploiting inequality obtain direct entropy estimates. nonlocal provide new proof based on De Giorgi estimates rather than Alikakos-Moser argument. Finally, spectral-fractional prove nonlinear fractional index s\in (0,1) filling gap between first-order (local) zero-order (nonlocal) cases. all aforementioned cases, our proofs neither make use Trudinger-Moser nor assumptions involving third derivative entropy, as previous contributions. particular, they apply more general class Flory-Huggins (Boltzmann-Gibbs) logarithmic density. Besides, methods present series technical advantages, which can be useful analysis important systems that couple other (e.g., reaction-diffusion and/or Navier-Stokes systems) well stochastic counterparts.
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Article history: Received 10 May 2013 Received in revised form 18 July 2014 Accepted 2 August 2014 Available online 8 August 2014
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ژورنال
عنوان ژورنال: Discrete and Continuous Dynamical Systems
سال: 2023
ISSN: ['1553-5231', '1078-0947']
DOI: https://doi.org/10.3934/dcds.2023010