On the uniqueness of the classical solutions of the radial viscous fingering problems in a Hele-Shaw cell
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 14 (2021) no. 4, pp. 475-482.

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In [9, 10] we established the existence of classical solutions to two-phase and one-phase radial viscous fingering problems, respectively, in a Hele-Shaw cell by the parabolic regularization and by vanishing the coefficient of the derivative with respect to time in a parabolic equation. In this paper we show the uniqueness of such solutions to the respective problems.
Keywords: classical solution, radial viscous fingering.
Mots-clés : unique existence
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Atusi Tani; Hisasi Tani. On the uniqueness of the classical solutions of the radial viscous fingering problems in a Hele-Shaw cell. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 14 (2021) no. 4, pp. 475-482. http://geodesic.mathdoc.fr/item/JSFU_2021_14_4_a8/

[1] B.V. Bazaliĭ, “On estimates for the solution of a model conjugation problem in the theory of problems with a free boundary”, Differ. Uravn., 33 (1997), 1374–1381 (in Russian) | MR | Zbl

[2] B.V. Bazaliĭ, “On a proof of the classical solvability of the Hele-Shaw problem with a free surface”, Ukr. Mat. Zh., 50 (1998), 1452–1462 (in Russian) | DOI | MR | Zbl

[3] H.S. Hele-Shaw, “The flow of water”, Nature, 58 (1898), 33–36

[4] D.A. Kessler, J. Koplik, H. Levine, “Pattern selection in fingering growth phenomena”, Adv. Phys., 37 (1988), 255–339 | DOI

[5] P.G. Saffman, G.I. Taylor, “The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid”, Proc. Royal Soc. London. Ser. A, 245 (1958), 312–329 | MR | Zbl

[6] V.A. Solonnikov, “General boundary value problems for systems elliptic in the sense of A. Douglis and L. Nirenberg. I”, Izv. Akad. Nauk SSSR Ser. Mat., 28 (1964), 665–706 ; “General boundary value problems for systems elliptic in the sense of A. Douglis and L. Nirenberg. II”, Trudy Mat. Inst. Steklov, 92, 1966, 233–297 (in Russian) | MR | MR | Zbl

[7] V.A. Solonnikov, “On boundary value problems for linear parabolic systems of differential equations of general form”, Trudy Mat. Inst. Steklov, 83, 1965, 3–163 (in Russian) | MR

[8] A. Tani, “Two-phase free boundary problem for compressible viscous fluid motion”, J. Math. Kyoto Univ., 24 (1984), 243–267 | MR | Zbl

[9] A. Tani, H. Tani, “Classical solvability of the two-phase radial viscous fingering problem in a Hele-Shaw cell”, Mathematical Fluid Dynamics, Present and Future, Springer Proc. Math. Stat., 183, eds. Y. Shibata, Y. Suzuki, 2016, 317–348 | MR | Zbl

[10] A. Tani, H. Tani, “Classical solvability of the radial viscous fingering problem in a Hele-Shaw cell”, Math. Notes NEFU, 25 (2018), 79–92 | DOI | MR