What does this research mean for the field?

The coupled Cahn–Hilliard system exhibits global existence of strong solutions for small initial data and demonstrates optimal decay rates of higher-order spatial derivatives of these solutions. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

This research aims to analyze the Cauchy problem for a coupled Cahn–Hilliard system and prove the existence and decay properties of its solutions.

March 4, 2026Advances in Nonlinear Analysis0 citationsOpen Access

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

NDNing DuanNortheastern University YWYinghao WangNortheastern University XZXiaopeng ZhaoNortheastern University

Key Points

This research aims to analyze the Cauchy problem for a coupled Cahn–Hilliard system and prove the existence and decay properties of its solutions.
Investigation of a coupled Cahn–Hilliard system in three-dimensional space
Proof of existence of strong solutions for small initial data
Establishment of optimal decay rates for higher-order spatial derivatives
Analysis of the preservation of negative Sobolev norms over time
Global existence of strong solutions is demonstrated for small initial data
Optimal decay rates of higher-order spatial derivatives are established
Negative Sobolev norms are shown to be preserved as time evolves

Abstract

Abstract In this paper, we consider the Cauchy problem for a coupled Cahn–Hilliard system in R 3 R^3. 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 H ̇ − s {H}^-s 0 ≤ s 3 2 (0 s< 32) negative Sobolev norms is shown to be preserved along time evolution and enhance the decay rates.

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