| Abstract: |
| In this talk, I will report a recent work on the sharp interface limit of a coupled Navier--Stokes/Allen--Cahn system
in a three dimensional, bounded and smooth domain, when a parameter $\varepsilon > 0$ that is proportional to
the thickness of the diffuse interface tends to zero, rigorously. The argument is based on the method of rigorous matched asymptotic expansions. In particular, we obtain optimal estimates for the linearized error system in $L^2$-Sobolev type spaces, which are second order in space of $\mathbf{w}$ and third order in space for $u$, with suitable weights. This provides better discription of the system close to the interface, which leads us to extend previous results from two dimensions to three dimensional case. |
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