Davoli Elisa, Scarpa Luca, Trussardi Lara
Institute of Analysis and Scientific Computing, TU Wien, Wiedner Hauptstrasse 8-10, 1040 Vienna, Austria.
Institut für Mathematik, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.
Arch Ration Mech Anal. 2021;239(1):117-149. doi: 10.1007/s00205-020-01573-9. Epub 2020 Oct 4.
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.
我们考虑一类具有化学势的诺伊曼边界条件的非局部粘性Cahn-Hilliard方程。双阱势可以是奇异的（例如对数型），而卷积核的奇异性在诺伊曼边界条件下不属于任何现有的存在性理论范畴。我们在合适的变分意义下证明了非局部方程的适定性。其次，我们表明，当卷积核逼近狄拉克δ函数时，非局部方程的解收敛到局部方程的相应解。通过单调分析和伽马收敛结果分析了渐近行为，包括极限局部Cahn-Hilliard方程为粘性型和纯型的情况。
