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Degenerate nonlocal Cahn-Hilliard equations: well-posedness, regularity and local asymptotics. (English) Zbl 1467.45017

The authors develop a well-posedness theory for nonlocal Cahn-Hilliard equations with a convection term and a singular kernel of the form \[ K_{\epsilon}(x,y)=\displaystyle\frac{\rho_{\varepsilon}(|x-y|)}{|x-y|^2}, \] for \(\epsilon>0\). Here \((\rho_{\epsilon})_{\epsilon}\) is a suitable sequence of mollifiers. Their results are based on a suitable approximation of the nonlinearity, a fixed point argument for the existence analysis for the approximating equations, some uniform estimates, and nontrivial compactness and monotonicity arguments. The authors prove the convergence of solutions for the nonlocal convective Cahn-Hilliard equation with singular kernel to solutions of the associated local one. The regularity analysis for the solutions of an evolution problem which provides a nonlocal variant of the Cahn-Hilliard PDE is finally presented.

MSC:

45K05 Integro-partial differential equations
35K25 Higher-order parabolic equations
35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
76R05 Forced convection
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References:

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