On well-posedness and decay of strong solutions for a coupled Cahn–Hilliard system

In this paper, we consider the Cauchy problem for a coupled Cahn–Hilliard system in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> </m:math> R³ . This system can be seen as the stationary system of a novel thermodynamically consistent three-phase model. The small initial data global existence of strong solutions are proved. Moreover, the optimal decay rates of the higher-order spatial derivatives of the strong solution are established, the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mover accent="true"> <m:mrow> <m:mi>H</m:mi> </m:mrow> <m:mo>̇</m:mo> </m:mover> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mi>s</m:mi> </m:mrow> </m:msup> </m:math> Ḣ⁻ˢ <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mrow> <m:mfenced close=")" open="("> <m:mrow> <m:mn>0</m:mn> <m:mo>≤</m:mo> <m:mi>s</m:mi> <m:mo><</m:mo> <m:mfrac> <m:mrow> <m:mn>3</m:mn> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:mfrac> </m:mrow> </m:mfenced> </m:mrow> </m:math> (0≤ s< 3/2) negative Sobolev norms is shown to be preserved along time evolution and enhance the decay rates.