GLOBAL SOLUTIONS TO 1D COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND FREE-BOUNDARY*

doi:10.1007/s10473-024-0111-5

Abstract

Abstract: This paper is concerned with the Navier-Stokes/Allen-Cahn system, which is used to model the dynamics of immiscible two-phase flows. We consider a 1D free boundary problem and assume that the viscosity coefficient depends on the density in the form of $\eta(\rho)=\rho^\alpha$. The existence of unique global $H^{2m}$-solutions $(m\in\mathbb N)$ to the free boundary problem is proven for when $0<\alpha<\frac14$. Furthermore, we obtain the global $C^\infty$-solutions if the initial data is smooth.

Key words: Navier-Stokes/Allen-Cahn system, density-dependent viscosity, free boundary, global solutions

CLC Number:

35A02

Shijin Ding, Yinghua Li, Yu Wang. GLOBAL SOLUTIONS TO 1D COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND FREE-BOUNDARY^*[J].Acta mathematica scientia,Series B, 2024, 44(1): 195-214.

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GLOBAL SOLUTIONS TO 1D COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND FREE-BOUNDARY^*

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