Global existence of solutions to a non-isentropic compressible diffuse interface model

Chunxiang Kong; Xiaofang Feng; Chunxiang Kong; Xiaofang Feng

doi:10.3934/math.2026700

AIMS Mathematics

2026, Volume 11, Issue 6: 17093-17114. doi: 10.3934/math.2026700

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Research article

Global existence of solutions to a non-isentropic compressible diffuse interface model

Chunxiang Kong ^{1
,
,},
Xiaofang Feng ²

1.
College of Mathematics and Computer Science, Yan'an University, Yan'an, Shanxi 716000, China
2.
College of Mathematics and Computer Science, Yan'an University, Yan'an, Shanxi 716000, China; Email: 3431647524@qq.com

Received: 17 January 2026 Revised: 15 May 2026 Accepted: 19 May 2026 Published: 12 June 2026
MSC : 35Q30, 36C20, 76T30

In this paper, we investigate the existence of solutions to a class of compressible Navier–Stokes/Allen–Cahn systems describing the motion of a mixture of two viscous compressible fluids. Using the energy method and refined interpolation inequalities, we establish the global existence of solutions in the Sobolev spaces $ H^{i} $ for $ i = 2, 4 $. The system features a temperature-dependent heat conductivity $ k(\theta) = \theta^\beta $ with $ \beta > 0 $, which characterizes the flow of a two-phase immiscible, thermally viscous, compressible fluid mixture. The main technical difficulty of this work lies in the complicated estimates induced by high-order partial derivatives in the proof of solution regularity.
- Navier–Stokes/Allen–Cahn system,
- global existence,
- uniqueness,
- non-isentropic,
- compressible diffuse interface
Citation: Chunxiang Kong, Xiaofang Feng. Global existence of solutions to a non-isentropic compressible diffuse interface model[J]. AIMS Mathematics, 2026, 11(6): 17093-17114. doi: 10.3934/math.2026700

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Abstract

In this paper, we investigate the existence of solutions to a class of compressible Navier–Stokes/Allen–Cahn systems describing the motion of a mixture of two viscous compressible fluids. Using the energy method and refined interpolation inequalities, we establish the global existence of solutions in the Sobolev spaces $ H^{i} $ for $ i = 2, 4 $. The system features a temperature-dependent heat conductivity $ k(\theta) = \theta^\beta $ with $ \beta > 0 $, which characterizes the flow of a two-phase immiscible, thermally viscous, compressible fluid mixture. The main technical difficulty of this work lies in the complicated estimates induced by high-order partial derivatives in the proof of solution regularity.

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