On~the transient motion of an isolated volume of viscous incompressible fluid

V. A. Solonnikov

Izvestiya. Mathematics , Tome 31 (1988) no. 2, pp. 381-405.

Résumé

Solvability locally in time is proved for a free-boundary problem describing the evolution of a finite mass of viscous incompressible fluid in $\mathbf R^n$, $n=2,3$, without taking surface tension into account. Sufficient conditions for the extension of a solution to the infinite time interval $t>0$ are given. A solution is obtained in the space $W_p^{2,1}$ with $p>n$. Bibliography: 19 titles.

Bibliographie

[1] Solonnikov V. A., “Razreshimost zadachi o dvizhenii vyazkoi neszhimaemoi zhidkosti, ogranichennoi svobodnoi poverkhnostyu”, Izv. AN SSSR. Ser. matem., 41:6 (1977), 1388–1424 | MR | Zbl

[2] Beale J. T., “The initial value problem for the Navier–Stokes equations with a free boundary”, Comm. Pure Appl. Math., 31:3 (1981), 359–392 | DOI | MR

[3] Beale J. T., “Large-time regularity of viscous surface waves”, Arch. Rat. Mech. Anal., 84:4 (1984), 307–352 | DOI | MR | Zbl

[4] Allain G., “Small-time existence for the Navier–Stokes equations with a free surface”, Ecole Polytechnique. Rapport interne, no. 135, 1985, 1–24

[5] Solonnikov V. A., “Razreshimost zadachi ob evolyutsii izolirovannogo ob'ema vyazkoi neszhimaemoi kapillyarnoi zhidkosti”, Zap. nauchn. semin. LOMI AN SSSR, 140, 1984, 179–186 | MR | Zbl

[6] Bytev V. O., “Neustanovivsheesya dvizhenie koltsa vyazkoi neszhimaemoi zhidkosti so svobodnymi granitsami”, PMTF, 1970, no. 3, 82–88 | MR

[7] Lavrenteva O. M., Dvizhenie vraschayuschegosya koltsa vyazkoi kapillyarnoi zhidkosti, Dep. v VINITI 27.11.84 No 7562-84, M., 1984, 2–50

[8] Korn A., “Über einige Ungleichungen, welche in der Theorie der elastischen und elektrischen Schwingungen eine Rolle spielen”, Bull. Intern. Cracovie Akad. Umiejet. (Cl. Sci. Math. Nat.), 1909, 705–724 | Zbl

[9] Friedrichs K. O., “On the boundary value problems of the theory of elasticity and Korn's inequality”, Ann. of Math., 48:2 (1947), 441–471 | DOI | MR | Zbl

[10] Fichera G., “Existence theorems in elasticity”, Handbuch der Physik, S. Flügge and C. Truesdell editors, 6a/2 (1972), 347–424

[11] Mosolov P. P., Myasnikov V. P., “Dokazatelstvo neravenstva Korna”, Dokl. AN SSSR, 201:1 (1971), 36–39 | MR | Zbl

[12] Horgan C. O., Payne L. E., “On inequalities of Korn, Friedrichs and Babuška-Aziz”, Arch. Rat. Mech. Anal., 82:2 (1983), 165–179 | DOI | MR | Zbl

[13] Ladyzhenskaya O. A., Solonnikov V. A., “O nekotorykh zadachakh vektornogo analiza i obobschennykh postanovkakh kraevykh zadach dlya uravneniya Nave–Stoksa”, Zap. nauchn. semin. LOMI AN SSSR, 59, 1976, 81–116 | Zbl

[14] Solonnikov V. A., “O razreshimosti vtoroi nachalno-kraevoi zadachi dlya lineinoi nestatsionarnoi sistemy uravnenii Nave–Stoksa”, Zap. nauchn. semin. LOMI AN SSSR, 69, 1977, 200–218 | MR | Zbl

[15] Solonnikov V. A., “Otsenki resheniya odnoi nachalno-kraevoi zadachi dlya lineinoi nestatsionarnoi sistemy uravnenii Nave–Stoksa”, Zap. nauchn. semin. LOMI AN SSSR, 59, 1976, 178–254 | MR | Zbl

[16] Solonnikov V. A., “Otsenki reshenii nestatsionarnoi linearizovannoi sistemy uravnenii Nave–Stoksa”, Tr. Matem. in-ta im. V. A. Steklova AN SSSR, 70, 1964, 213–317 | MR | Zbl

[17] Mogilevskii I. Sh., “Otsenki reshenii obschei nachalno-kraevoi zadachi dlya lineinoi nestatsionarnoi sistemy uravnenii Nave-Stoksa v ogranichennoi oblasti”, Tr. Matem. in-ta im. V. A. Steklova AN SSSR, 159, 1983, 61–94 | MR

[18] Ilin V. P., “O skhodimosti posledovatelnostei funktsii v nekotorykh funktsionalnykh prostranstvakh”, Uspekhi matem. nauk, 12:1 (1957), 192–195 | MR

[19] Sakhaev Sh., Solonnikov V. A., “Otsenki reshenii kraevykh zadach magnitnoi gidrodinamiki”, Tr. Matem. in-ta im. V. A. Steklova AN SSSR, 127, 1975, 76–92 | Zbl