On the separation property and the global attractor for the nonlocal Cahn-Hilliard equation in three dimensions
Andrea Giorgini
Abstract
Abstract We consider the nonlocal Cahn-Hilliard equation with constant mobility and singular potential in three dimensional bounded and smooth domains. This model describes phase separation in binary fluid mixtures. Given any global solution (whose existence and uniqueness are already known), we prove the so-called instantaneous and uniform separation property: any global solution with initial finite energy is globally confined (in the $$L^\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> </mml:math> metric) in the interval $$[-1+\delta ,1-\delta ]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>[</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>δ</mml:mi> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>δ</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> on the time interval $$[\tau ,\infty )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> for any $$\tau >0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>τ</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , where $$\delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> only depends on the norms of the initial datum, $$\tau $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> </mml:math> and the parameters of the system. We then exploit such result to improve the regularity of the global attractor for the dynamical system associated to the problem.