Regularity and Long-Time Behavior of Global Weak Solutions to a Coupled Cahn–Hilliard System: The Off-Critical Case

We consider a diffuse interface model that describes the macro- and micro-phase separation processes of a polymer mixture. The resulting system consists of a Cahn–Hilliard equation and a Cahn–Hilliard–Oono type equation endowed with the singular Flory–Huggins potential. For the initial boundary value problem in a bounded smooth domain of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>d</mml:mi> <mml:mo>∈</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:math> ) with homogeneous Neumann boundary conditions for the phase functions as well as chemical potentials, we study the regularity and long-time behavior of global weak solutions in the off-critical case, that is, the mass is not conserved during the micro-phase separation of diblock copolymers. By investigating an auxiliary system with viscous regularizations, we show that every global weak solution regularizes instantaneously for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>t</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:math> . In two dimensions, we obtain the instantaneous strict separation property under a mild growth condition on the first derivative of potential functions near pure phases <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mo>±</mml:mo> <mml:mn>1</mml:mn> </mml:math> , while in three dimensions, we establish the eventual strict separation property for sufficiently large time. Finally, we prove that every global weak solution converges to a single equilibrium as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>t</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mo>+</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:math> .