On a steady three-dimensional noncompact free boundary value problem for the Navier–Stokes equations

K. I. Pileckas   ; L. Zaleskis

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 34, Tome 306 (2003), pp. 134-164

Résumé

A steady three-dimensional flow of a viscous incompressible fluid with a noncompact free boundary above a fixed unbounded bottom is studied. It is assumed that the motion of the fluid is generated by sources and sinks situated in a bounded part of the bottom and having zero total flux. The existence for small data of the unique solution to this problem is proved and the asymptotics of the solution is constructed.

Bibliographie

[1] D. K. Faddeev, B. Z. Vulich, V. A. Solonnikov, N. N. Uralceva, “Selected topics in analysis and higher algebra”, Sobolev spaces, Ch. 3, Leningrad, 1981, 130–190

[2] L. V. Kapitanskii, K. Pileckas, “Certain problems of vector analysis”, Zap. Nauchn. Semin. LOMI, 138, 1984, 65–85 | MR

[3] O. A. Ladyzhenskaya, V. G. Osmolovskii, “On the free surface of a fluid layer over a solid sphere”, Vestnik Leningrad. Univ. Math., 13 (1976), 25–30 | Zbl

[4] V. G. Osmolovskii, “The free surface of a drop in a symmetric force field”, Zap. Nauchn. Semin. LOMI, 52, 1975, 160–174 | MR | Zbl

[5] S. A. Nazarov, K. Pileckas, “On noncompact free boundary problems for the plane stationary Navier–Stokes equations”, J. für Reine und Angewandte Math., 438 (1993), 103–141 | MR | Zbl

[6] S. A. Nazarov, K. Pileckas, “On the solvability of the Stokes and Navier–Stokes problem in the domains that are layer-like at infinity”, J. Math. Fluid Mech., 1:1 (1999), 78–116 | DOI | MR | Zbl

[7] S. A. Nazarov, K. Pileckas, “The asymptotic properties of the solutions to the Stokes problem in domains that are layer-like at infinity”, J. Math. Fluid Mech., 1:2 (1999), 131–167 | DOI | MR | Zbl

[8] S. A. Nazarov, K. Pileckas, “On the Fredholm property of the Stokes operator in a layer-like domain”, Z. Anal. Anwendungen, 20:1 (2001), 155–182 | MR | Zbl

[9] K. Pileckas, “Existence of solutions for the Navier–Stokes equations having an infinite dissipation of energy in a class of domains with noncompact boundaries”, Zap. Nauchn. Semin. LOMI, 110, 1981, 180–202 | MR

[10] K. Pileckas, “Solvability of a problem of plane motion of a viscous incompressible fluid with noncompact free boundary”, Diff. Eqs. Appl., 30 (1981), 57–96 | Zbl

[11] K. Pileckas, “On the problem of motion of heavy viscous incompressible fluid with noncompact free boundary”, Litovskii Mat. Sb., 28:2 (1988), 315–333 | MR

[12] K. Pileckas, “On a plane motion of a viscous incompressible capillary liquid with a noncompact free boundary”, Arch. Mech., 41:2–3 (1989), 329–342 | MR | Zbl

[13] K. Pileckas, “The example of nonuniqeness of the solution to a noncompact free boundary problem for the stationary Navier–Stokes system”, Diff. Eqs. Appl., 42 (1988), 59–64 | MR | Zbl

[14] K. Pileckas, “Asymptotics of solutions of the stationary Navier–Stokes system of equations in a domain of layer type”, Mat. Sb., 193:12 (2002), 69–104 | MR

[15] V. V. Puchnachov, “The plane stationary problem with a free boundary for the Navier–Stokes equations”, Z. Prikl. Meh. i Tehn. Fiz., 3 (1972), 91–102 | MR

[16] D. Sattinger, “On the free surface of a viscous fluid”, Proc. R. Soc. London A, 349 (1976), 183–204 | DOI | MR | Zbl

[17] J. Socolowsky, “Solvability of a two-dimensional problem for the motion of two viscous incompressible liquids with non-compact free boundaries”, Zap. Nauchn. Semin. LOMI, 163, 1987, 146–153 | MR | Zbl

[18] V. A. Solonnikov, V. E. Shchadilov, “On a boundary value problem for a stationary system of Navier–Stokes equations”, Trudy Steklov Mat. Inst., 125, 1973, 196–210 | MR | Zbl

[19] V. A. Solonnikov, “Solvability of the problem of plane motion of heavy viscous incompressible capillary liquid partially filling a container”, Izv. Akad. Nauk. SSSR, Ser. Mat., 43 (1979), 203–236 | MR | Zbl

[20] V. A. Solonnikov, “On a free boundary problem for the system of Navier–Stokes equations”, Trudy S. L. Sobolev Sem., 2, 1978, 127–140 | MR | Zbl

[21] V. A. Solonnikov, “Solvability of a three-dimensional boundary value problem with a free surface for the stationary Navier–Stokes system”, Zap. Nauchn. Semin. LOMI, 84, 1979, 252–285 | MR | Zbl

[22] V. A. Solonnikov, “On the Stokes equations in domains with nonsmooth boundaries and on a viscous incompressible flow with a free surface”, Collège de France Seminar, V. III (Paris, 1980/1981), Res. Notes in Math., 70, Pitman, Boston, Mass., 1982, 340–423 | MR

[23] V. A. Solonnikov, “Solvability of the problem on the effluence of a viscous incompressible fluid into an open basin”, Trudy Steklov Mat. Inst., 179, 1988, 174–202 | MR

[24] V. A. Solonnikov, “On the solvability of some two-dimensional quasistationary free boundary problems for the Navier–Stokes equations with moving contact point”, Zap. Nauchn. Semin. POMI, 206, 1993, 119–126 | MR

[25] V. A. Solonnikov, “On the problems with free boundaries and moving contact points for two-dimensional stationary Navier–Stokes equations”, Zap. Nauchn. Semin. POMI, 213, 1994, 179–205 | MR | Zbl

[26] V. A. Solonnikov, “Some free boundary problems for the Navier–Stokes equations with moving contact points and lines”, Partial differential equations: models in physics and biology, 82, eds. G. Lumer, S. Nicaise, B.-W. Schulze, 1994, 329–350 | MR | Zbl

[27] V. A. Solonnikov, “On some free boundary problems for the Navier–Stokes equations with moving contact points and lines”, Math. Annal., 302 (1995), 743–772 | DOI | MR | Zbl

[28] V. A. Solonnikov, “Free boundary problems for the Navier–Stokes equations with moving contact points”, Free boundary problems: theory and applications, Pitman Research Notes in Math. Series, 323, 1995, 213–214 | MR

[29] V. A. Solonnikov, “Solvability of the free boundary problem governing a viscous incompressible flow through an aperture”, Algebra Analiz, 9:2 (1997), 169–191 | MR | Zbl

[30] V. A. Solonnikov, “Solvability of the two-dimensional free boundary problem for the Navier–Stokes equations for the limiting values of the contact angle”, Quaderni di matematica, 2 (1998), 163–210 | MR | Zbl

[31] V. A. Solonnikov, “Solvability of two stationary free boundary problems for the Navier–Stokes equations”, Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8), 1 (1998), 283–342 | MR

[32] V. A. Solonnikov, “Stationary free boundary problems for the Navier–Stokes equations”, Advanced topics in theoretical fluid mechanics, Pitman Research Notes Math. Series, 392, 1998, 147–212 | MR | Zbl

[33] V. A. Solonnikov, “On the problem of a steady fall of a drop in a liquid medium”, J. Math. Fluid Mech., 1 (1999), 326–355 | DOI | MR | Zbl