Abstract:
The existence of classical solutions was established in [(∗)] Tani A. and Tani H., “Two-phase radial viscous fingering problem in a Hele-Shaw cell with surface tension, I: Classical solvavility,’ Mat. Zametki SVFU, 31, No. 4, 82–105 (2024), for the two-phase radial viscous fingering problem in a Hele-Shaw cell under the surface tension (the original two-phase problem) by means of parabolic regularization with a small parameter ε (> 0) in the time-derivative terms and the non-homogeneous terms (the parabolic regularized two-phase problem), vanishing along some subsequence {εn}n∈N of {ε > 0}. In this paper we prove the uniqueness of classical solutions to the original two-phase problem. This gives the improvement to the convergence result in [(∗)]: the convergence of the full sequence {ε > 0}, not the subsequence {εn}n∈N, of classical solutions of the parabolic regularized two-phase problem to those of the original two- phase problem. Similar results for some one-phase problem have been already studied in Tani H., “Classical solvability of the radial viscous fingering problem in a Hele-Shaw cell with surface tension,” Sib. J. Pure Appl. Math., 16, 79–92 (2016) (the existence) and in Tani A. and Tani H., “On the uniqueness of the classical solution of the radial viscous fingering problem in a Hele-Shaw cell with surface tension,” J. Appl. Mech. Tech. Phys., 65, No. 5 (2024) (the uniqueness).
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