Free boundary problems for nonstationary Navier-Stokes equations

Dissertationes Mathematicae

Volumn , Issue 424, 2004, Pages 1-135

  Zadrzyńska, Ewa ^a  

^a WARSAW UNIVERSITY OF TECHNOLOGY   (Poland)

Author keywords

Compressible viscous fluid; Free boundary; Global existence; Incompressible viscous fluid; Local existence; Stability

Indexed keywords

EID: 84893508500     PISSN: 00123862     EISSN: 17306310     Source Type: Journal    
DOI: 10.4064/dm424-0-1     Document Type: Article

References (94)

1. [Al1] G. Allain, Un probléme de Navier-Stokes avec surface libre et tension superćielle, Ann. Fac. Sci. Toulouse 7 (1985), 29-56.
2. [Al2] G. Allain, Small-time existence for the Navier-Stokes equations with a free surface, Appl. Math. Optim. 16 (1987), 37-50.
3. [B1] J. T. Beale, The initial value problem for the Navier-Stokes equations with a free boundary, Comm. Pure Appl. Math. 31 (1980), 359-392.
4. [B2] J. T. Beale, Large time regularity of viscous surface waves, Arch. Rat. Mech. Anal. 84 (1984), 307-352.
5. [BNis] J. T. Beale and Y. Nishida, Large-time behavior of viscous surface waves, in: Lecture Notes in Numer. Appl. Anal. 8, Springer, 1985, 1-14.
6. [BesIlN] O. V. Besov, V. P. Il'in and S. M. Nikol'skií, Integral Representations of Functions and Imbedding Theorems, Nauka, Moscow, 1975 (in Russian).
7. [BurZaj] M. Burnat and W. M. Zaja{ogonek}czkowski, On local motion of a compressible barotropic viscous uid with the boundary slip condition, Topol. Methods Nonlinear. Anal. 10 (1997), 195-223.
8. [D1] B. Ducomet, Hydrodynamical models of gaseous stars, Rev. Math. Phys. 8 (1996), 957-1000.
9. [D2] B. Ducomet, On the stability of a stellar structure in one dimension, Math. Models Methods Appl. Sci. 6 (1996), 365-384.
10. [D3] B. Ducomet, On the stability of a stellar structure in one dimension II: The reactive case, Math. Model. Numer. Anal. 31 (1997), 381-407.
11. [D4] B. Ducomet, On the stability of a stellar structure in one dimension II: On the stability of a stellar structure in one dimension II: Some stability results for reactive Navier-Stokes-Poisson systems, in: Evolution Equations: Existence, regularity and singularities, Banach Center Publ. 52, Inst. Math., Polish Acad. Sci., Warszawa, 2000, 83-118.
12. [FBen] H. Fujita-Yashima and R. Benabidallah, Equation á symétrie sphérique d'un gaz visqueux et calorifére avec la surface libre, Ann. Mat. Pura Appl. 168 (1995), 75-117.
13. [FPadNov] H. Fujita-Yashima, M. Padula and A. Novotny, Equation monodimensionnelle d'un gaz visqueux et calorifére avec des conditions initiales moins restrictives, Ricerche Mat. 42 (1993), 199-248.
14. [Gol] K. K. Golovkin, On equivalent norms for fractional spaces, Trudy Mat. Inst. Steklov. 66 (1962), 364-383 (in Russian).
15. [J] S. Jiang, On initial boundary value problems for a viscous heat-conducting onedimensional real gas, J. Diff. Equations 110 (1994), 157-181.
16. [JinPad] B. J. Jin and M. Padula, On existence of compressible viscous ow in a horizontal layer with free upper surfaces, Universitá degli Studi di Ferrara, Dipartimento di Matematica, preprint n. 317, 2002.
17. [Kaw] B. Kawohl, Global existence of large solutions to initial boundary value problems for a viscous heat-conducting one-dimensional real gas, J. Diff. Equations 58 (1985), 76-103.
18. [Kaz1] A. V. Kazhikov, The global correctness of boundary-value problems for model equations of viscous gas, Dinamika Sploshnoí Sredy 21 (1975), 18-47 (in Russian).
19. [Kaz2] A. V. Kazhikov, On the global solvability of one-dimensional boundary value problems for the equations of viscous heat-conductive gas, Dinamika Sploshnoí Sredy 21 (1975), 24 (1976), 45-61 (in Russian).
20. [LadSol] O. A. Ladyzhenskaya and V. A. Solonnikov, On some problems of vector analysis and generalized formulations of boundary problems for Navier-Stokes equations, Zap. Nauchn. Sem. LOMI 59 (1976), 81-116 (in Russian).
21. [LadSolUr] O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, Nauka, Moscow, 1967 (in Russian).
22. [LagSol] M. V. Lagunova and V. A. Solonnikov, Nonstationary problem of thermocapillary convection, Stability Appl. Anal. Contin. Media 1 (1991), 47-72.
23. [LanLif] L. Landau and E. Lifschitz, Hydrodynamics, Nauka, Moscow, 1986 (in Russian); English transl.: Fluid Mechanics, Pergamon Press, Oxford, 1987.
24. [LedWal] P. Ledoux and Th. Walraven, Variable stars, in: Handbuch der Physik LI, Springer, Berlin, 1985, 353-604.
25. [LMag] J. L. Lions and E. Magenes, Nonhomogeneous Boundary Value Problems and Applications, Springer, Berlin, 1972.
26. [Mar] J. Marcinkiewicz, Sur les multiplicateurs des séries de Fourier, Studia Math. 8 (1939), 78-91.
27. [Mik] S. G. Mikhlin, Multidimensional Singular Integrals and Integral Equations, Pergamon Press, 1965.
28. 2l;l for solutions of a boundary value problem for the nonstationary Stokes system with surface tension in boundary condition, in: Proc. Conf. on Navier-Stokes Equations (Oberwolfach, 1991).
29. [MogSol] I. Sh. Mogilevskií and V. A. Solonnikov, On the solvability of a free boundary problem for the Navier-Stokes equations in the Hölder spaces of functions, in: Nonlinear Analysis. A Tribute in Honour of Giovanni Prodi, Quaderni, Pisa 1991, 257-272.
30. [M] P. B. Mucha, Compressible Navier-Stokes system in 1-D, Math. Methods Appl. Sci. 24 (2001), 607-622.
31. [MZaj1] P. B. Mucha and W. M. Zaja{ogonek}czkowski, On local existence of solutions of the free boundary problem for an incompressible viscous self-gravitating uid motion, Appl. Math. (Warsaw) 27 (2000), 319-333.
32. p-framework, Studia Math. 143 (2000), 75-101.
33. [Nag] T. Nagasawa, On the outer pressure problem of the one-dimensional polytropic ideal gas, Japan J. Appl. Math. 5 (1988), 53-85.
34. [Nis] T. Nishida, Equations of uid dynamics{pipe}free surface problems, Comm. Pure Appl. Math. 39 (1986), 221-238.
35. [NisTerW] T. Nishida, Y. Teramoto and H. A. Win, Navier-Stokes ow down an inclined plane: downward periodic motion, J. Math. Kyoto Univ. 33 (1993), 787-801.
36. [Ok] M. Okada, Free boundary value problems for the equation of one-dimensional motion of viscous gas, Japan J. Appl. Math. 6 (1989), 161-177.
37. [OkMak] M. Okada and T. Makino, Free boundary problem for the equation of sherically symmetric motion of viscous gas, Japan J. Industr. Appl. Math. 10 (1993), 219-235.
38. [Scw] B. Schweizer, Free boundary uid systems in a semigroup approach and oscillatory behaviour, SIAM J. Math. Anal. 28 (1997), 1135-1157.
39. [S1] P. Secchi, On the uniqueness of motion of viscous gaseous stars, Math. Methods Appl. Sci. 13 (1990), 391-404.
40. [S2] P. Secchi, On the motion of gaseous stars in the presence of radiation, Comm. Partial Diff. Equations 15 (1990), 185-204.
41. [S3] P. Secchi, On the evolution equations of viscous gaseous stars, Ann. Scuola Norm. Sup. Pisa 18 (1991), 295-318.
42. [SVal] P. Secchi and A. Valli, A free boundary problem for compressible viscous uid, J. Reine Angew. Math. 341 (1983), 1-31.
43. [Ser] J. Serrin, Mathematical principles of classical uid mechanics, in: Handbuch der Physik, VIII/1, Springer, Berlin, 1959, 125-263.
44. [Sol1] V. A. Solonnikov, On boundary value problems for linear parabolic systems of differential equations of general type, Trudy Mat. Inst. Steklov. 83 (1965) (in Russian); English transl.: Proc. Steklov Inst. Math. 83 (1967).
45. [Sol2] V. A. Solonnikov, Estimates of a solution of an initial-boundary value problem for the linear nonstationary Navier-Stokes system, Zap. Nauchn. Sem. LOMI 59 (1976), 178-254 (in Russian).
46. [Sol3] V. A. Solonnikov, On the solvability of the second initial-boundary value problem for the linear nonstationary Navier-Stokes system, Zap. Nauchn. Sem. LOMI 59 (1976), 69 (1977), 200-218 (in Russian); English transl.: J. Soviet Math. 10 (1978), 141-193.
47. [Sol4] V. A. Solonnikov, Solvability of a problem on the motion of a viscous incompressible uid bounded by a free surface, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), 1388-1424 (in Russian); English transl.: Math. USSR-Izv. 11 (1977), 1323-1358.
48. [Sol5] V. A. Solonnikov, Solvability of the evolution problem for an isolated mass of a viscous incompressible capillary liquid, Zap. Nauchn. Sem. LOMI 140 (1984), 179-186 (in Russian); English transl.: J. Soviet Math. 32 (1986), 223-238.
49. [Sol6] V. A. Solonnikov, On an unsteady ow of a finite mass of a liquid bounded by a free surface, Zap. Nauchn. Sem. LOMI 152 (1986), 137-157 (in Russian); English transl.: J. Soviet Math. 40 (1988), 672-686.
50. [Sol7] V. A. Solonnikov, Free boundary problems and problems in non compact domains for the Navier-Stokes equations, in: Proc. Inter. Congress of Math. (Berkeley, 1986), 1113-1122.
51. [Sol8] V. A. Solonnikov, On an unsteady motion of an isolated volume of a viscous incompressible uid, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), 1065-1087 (in Russian); English transl.: Math. USSR-Izv. 31 (1988), 381-405.
52. [Sol9] V. A. Solonnikov, On an evolution of an isolated volume of a viscous incompressible capillary uid for large time values, Vestnik Leningrad. Univ. 1987, no. 3, 49-55 (in Russian).
53. [Sol10] V. A. Solonnikov, On nonstationary motion of a finite isolated mass of self-gravitating uid, Algebra i Analiz 1 (1989), 207-249 (in Russian); English transl.: Leningrad Math. J. 1 (1990), 227-276.
54. [Sol11] V. A. Solonnikov, On an initial-boundary value problem for the Stokes system arising in the study of a problem with a free boundary, Trudy Mat. Inst. Steklov. 188 (1990), 150-188 (in Russian); English transl.: Proc. Steklov Inst. Math. 188 (1991), 191-239.
55. [Sol12] V. A. Solonnikov, On an evolution problem of thermocapillary convection, Le Matematiche 66 (1991), 449-460.
56. [Sol13] V. A. Solonnikov, Solvability of the problem of evolution of a viscous incompressible uid bounded by a free surface on a finite time interval, Algebra i Analiz 3 (1991), 222-257; English transl.: St. Petersburg Math. J. 3 (1992), 189-220.
57. [Sol14] V. A. Solonnikov, On an evolution problem of thermocapillary convection, in: Internat. Ser. Numer. Math. 106, Birkhäuser, 1992, 301-317.
58. [SolT1] V. A. Solonnikov and A. Tani, A problem with free boundary for Navier-Stokes equations for a compressible uid in the presence of surface tension, Zap. Nauchn. Sem. LOMI 182 (1990), 142-148 (in Russian).
59. [SolT2] V. A. Solonnikov and A. Tani, Free boundary problem for a viscous compressible ow with the surface tension, in: Constantine Carathéodory: An International Tribute, T. M. Rassias (ed.), vol. 2, 1991, 1270-1303.
60. [SolT3] V. A. Solonnikov and A. Tani, Evolution free boundary problem for equations of motion of viscous compressible barotropic liquid, in: Lecture Notes in Math. 1530, Springer, 1992, 30-55.
61. [SolT4] V. A. Solonnikov and A. Tani, Equilibrium figures of slowly rotating viscous compressible barotropic capillary liquid, Adv. Math. Sci. Appl. 2 (1993), 139-145.
62. [St] G. Ströhmer, About the decay of surface waves on viscous uids without surface tension, Banach Center Publ. 60, Inst. Math., Polish Acad. Sci., 2003, 55-72.
63. [StZaj] G. Ströhmer and W. M. Zaja{ogonek}czkowski, Local existence of solutions of the free boundary problem for the equations of compressible barotropic viscous self-gravitating uids, Appl. Math. (Warsaw) 26 (1999), 1-31.
64. [Syl] D. Sylvester, Large time existence of small viscous surface waves without surface tension, Comm. Partial Diff. Equations 15 (1990), 823-903.
65. [TanT] N. Tanaka and A. Tani, Large-time existence of surface waves of compressible viscous uid, Proc. Japan Acad. Ser. A 69 (1993), 230-233.
66. [T1] A. Tani, On the free boundary value problem for compressible viscous uid motion, J. Math. Kyoto Univ. 21 (1981), 839-859.
67. [T2] A. Tani, Small-time existence for the three-dimensional incompressible Navier-Stokes equations with a free surface, Arch. Rat. Mech. Anal. 133 (1996), 299-331.
68. [TItTan] A. Tani, S. Itoh and N. Tanaka, The initial value problem for the Navier-Stokes equations with general slip boundary condition, Adv. Math. Sci. Appl. 4 (1994), 51-69.
69. [TTan] A. Tani and N. Tanaka, Large-time existence of surface waves in incompressible viscous uids with or without surface tension, Arch. Rat. Mech. Anal. 130 (1995), 303-314.
70. [Ter1] Y. Teramoto, The initial value problem for a viscous incompressible ow down an inclined plane, Hiroshima Math. J. 15 (1985), 619-643.
71. [Ter2] Y. Teramoto, On the Navier-Stokes ow down an inclined plane, J. Math. Kyoto Univ. 32 (1992), 593-619.
72. [Wag] A. Wagner, Nonstationary Marangoni convection, Appl. Math. (Warsaw) 26 (1999), 195-220.
73. [Z1] E. Zadrzýnska, On nonstationary motion of a fixed mass of a general viscous compressible heat conducting capillary uid bounded by a free boundary, Appl. Math. (Warsaw) 26 (1999), 25 (1999), 489-511.
74. [Z2] E. Zadrzýnska, Free boundary problem for a viscous heat-conducting ow with surface tension, Topol. Methods Nonlin. Anal. 19 (2002), 313-338.
75. [ZZaj1] E. Zadrzýnska and W. M. Zaja{ogonek}czkowski, On local motion of a general compressible viscous heat conducting uid bounded by a free surface, Ann. Polon. Math. 59 (1994), 133-170.
76. [ZZaj2] E. Zadrzýnska and W. M. Zaja{ogonek}czkowski, Conservation laws in free boundary problems for viscous compressible heatconducting uids, Bull. Polish Acad. Sci. Tech. Sci. 42 (1994), 197-207.
77. [ZZaj3] E. Zadrzýnska and W. M. Zaja{ogonek}czkowski, On global motion of a compressible heat conducting uid bounded by a free surface, Acta Appl. Math. 37 (1994), 221-231.
78. [ZZaj4] E. Zadrzýnska and W. M. Zaja{ogonek}czkowski, On a differential inequality for equations of a viscous compressible heat conducting uid bounded by a free surface, Ann. Polon. Math. 61 (1995), 141-188.
79. [ZZaj5] E. Zadrzýnska and W. M. Zaja{ogonek}czkowski, Conservation laws in free boundary problems for viscous compressible heat conducting capillary uids, Bull. Polish Acad. Sci. Tech. Sci. 43 (1995), 423-444.
80. [ZZaj6] E. Zadrzýnska and W. M. Zaja{ogonek}czkowski, On the global existence theorem for a free boundary problem for equations of a viscous compressible heat conducting uid, Ann. Polon. Math. 63 (1996), 199-221.
81. [ZZaj7] E. Zadrzýnska and W. M. Zaja{ogonek}czkowski, On a differential inequality for a viscous compressible heat conducting capillary uid bounded by a free surface, Ann. Polon. Math. 63 (1996), 65 (1996), 23-53.
82. [ZZaj8] E. Zadrzýnska and W. M. Zaja{ogonek}czkowski, On the global existence theorem for a free boundary problem for equations of a viscous compressible heat conducting capillary uid, J. Appl. Anal. 2 (1996), 125-169.
83. [ZZaj9] E. Zadrzýnska and W. M. Zaja{ogonek}czkowski, Local existence of solutions of a free boundary problem for equations of compressible viscous heat-conducting uids, Appl. Math. (Warsaw) 25 (1998), 179-220.
84. [ZZaj10] E. Zadrzýnska and W. M. Zaja{ogonek}czkowski, On nonstationary motion of a fixed mass of a viscous compressible barotropic uid bounded by a free boundary, Colloq. Math. 79 (1999), 283-310.
85. [ZZaj11] E. Zadrzýnska and W. M. Zaja{ogonek}czkowski, Local existence of solutions of a free boundary problem for equations of compressible viscous heat-conducting capillary uids, J. Appl. Anal. 6 (2000), 227-250.
86. [ZZaj12] E. Zadrzýnska and W. M. Zaja{ogonek}czkowski, On some inequalities for solutions of equations describing the motion of a viscous compressible heat-conducting capillary uid bounded by a free surface, Appl. Math. (Warsaw) 28 (2001), 31-53.
87. [ZZaj13] E. Zadrzýnska and W. M. Zaja{ogonek}czkowski, On an inequality for a free boundary problem for equations of viscous compressible heat-conducting capillary uid, Appl. Math. (Warsaw) 28 (2001), 29 (2002), 399-438.
88. [ZZaj14] E. Zadrzýnska and W. M. Zaja{ogonek}czkowski, Differential inequalities for general uid motions bounded by a free surface, J. Appl. Anal., to appear.
89. [ZZaj15] E. Zadrzýnska and W. M. Zaja{ogonek}czkowski, On nonstationary motion of a fixed mass of a general uid bounded by a free surface, Banach Center Publ. 60, Inst. Math., Polish Acad. Sci., 2003, 253-266.
90. [Zaj1] W. M. Zaja{ogonek}czkowski, On an initial-boundary value problem for the parabolic system which appears in free boundary problems for compressible Navier-Stokes equations, Dissertationes Math. 304 (1990).
91. [Zaj2] W. M. Zaja{ogonek}czkowski, On local motion of a compressible barotropic viscous uid bounded by a free surface, in: Partial Differential Equations, Banach Center Publ. 27, Inst. Math., Polish Acad. Sci., 1992, 511-553.
92. [Zaj3] W. M. Zaja{ogonek}czkowski, On nonstationary motion of a compressible viscous uid bounded by a free surface, Dissertationes Math. 324 (1993).
93. [Zaj4] W. M. Zaja{ogonek}czkowski, On nonstationary motion of a compressible barotropic viscous capillary uid bounded by a free surface, SIAM J. Math. Anal. 25 (1994), 1-84.
94. [Zaj5] W. M. Zaja{ogonek}czkowski, Existence of local solutions for free boundary problems for viscous compressible barotropic uids, Ann. Polon. Math. 60 (1995), 255-287.