(Non-)convergence of solutions of the convective Allen–Cahn equation

Helmut Abels ()
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Helmut Abels: University of Regensburg

Partial Differential Equations and Applications, 2022, vol. 3, issue 1, 1-11

Abstract: Abstract We consider the sharp interface limit of a convective Allen–Cahn equation, which can be part of a Navier–Stokes/Allen–Cahn system, for different scalings of the mobility $$m_\varepsilon =m_0\varepsilon ^\theta $$ m ε = m 0 ε θ as $$\varepsilon \rightarrow 0$$ ε → 0 . In the case $$\theta >2$$ θ > 2 we show a (non-)convergence result in the sense that the concentrations converge to the solution of a transport equation, but they do not behave like a rescaled optimal profile in normal direction to the interface as in the case $$\theta =0$$ θ = 0 . Moreover, we show that an associated mean curvature functional does not converge to the corresponding functional for the sharp interface. Finally, we discuss the convergence in the case $$\theta =0,1$$ θ = 0 , 1 by the method of formally matched asymptotics.

Keywords: Two-phase flow; Diffuse interface model; Allen–Cahn equation; Sharp interface limit; 76T99; 35Q30; 35Q35; 35R35; 76D05; 76D45 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s42985-021-00140-5

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