On estimates of solutions of the non-stationary Stokes problem in anisotropic Sobolev spaces and on estimates for the resolvent of the Stokes operator

Russian Mathematical Surveys

Volumn 58, Issue 2, 2003, Pages 331-365

  Solonnikov, V A ^a  

^a STEKLOV MATHEMATICAL INSTITUTE   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0042655204     PISSN: 00360279     EISSN: 14684829     Source Type: Journal    
DOI: 10.1070/RM2003v058n02ABEH000613     Document Type: Article

References (39)

1. K. K. Golovkin and O. A. Ladyzhenskaya, "Solutions of non-stationary boundary value problems for the Navier-Stokes equations", Trudy Mat. Inst. Steklov. 59 (1960), 100-114. (Russian)
2. K. K. Golovkin and V. A. Solonnikov, "The first boundary value problem for the nonstationary Navier-Stokes equations", Dokl. Akad. Nauk SSSR 140 (1961), 287-290; . English transl., Soviet. Math. Dokl. 2 (1961), 1188-1193.
3. K. K. Golovkin and V. A. Solonnikov, "The first boundary value problem for the nonstationary Navier-Stokes equations", Dokl. Akad. Nauk SSSR 140 (1961), 287-290; . English transl., Soviet. Math. Dokl. 2 (1961), 1188-1193.
4. V. A. Solonnikov, "Estimates of the solutions of a non-stationary linearized system of Navier-Stokes equations", Trudy Mat. Inst. Steklov. 70 (1964), 213-317; English transl., Amer. Math. Soc. Transl. (2) 75 (1968), 1-116.
5. V. A. Solonnikov, "Estimates of the solutions of a non-stationary linearized system of Navier-Stokes equations", Trudy Mat. Inst. Steklov. 70 (1964), 213-317; English transl., Amer. Math. Soc. Transl. (2) 75 (1968), 1-116.
6. V. A. Solonnikov, "Estimates of the solutions of the nonstationary Navier-Stokes system", Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 38 (1973), 153-231; English transl., J. Soviet. Math. 8 (1977), 467-523.
7. V. A. Solonnikov, "Estimates of the solutions of the nonstationary Navier-Stokes system", Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 38 (1973), 153-231; English transl., J. Soviet. Math. 8 (1977), 467-523.
8. r spaces", Math. Z. 178 (1981), 297-329.
9. Y. Giga and H. Sohr, "On the Stokes operator in exterior domains", J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 (1989), 103-130.
10. p-estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains", J. Funct. Anal. 102 (1991), 72-94.
11. p-spaces", Math. Z. 106 (1987), 415-425.
12. W. Borchers and W. Varnhorn, "On the boundedness of the Stokes semigroup in two-dimensional exterior domains", Math. Z. 213 (1993), 275-299.
13. P. Maremonti and V. A. Solonnikov, "On estimates for the solutions of the nonstationary Stokes problem in anisotropic Sobolev spaces with a mixed norm", Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 222 (1995), 124-150; English transl., J. Math. Sci. (New York) 87 (1997), 3859-3877.
14. P. Maremonti and V. A. Solonnikov, "On estimates for the solutions of the nonstationary Stokes problem in anisotropic Sobolev spaces with a mixed norm", Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 222 (1995), 124-150; English transl., J. Math. Sci. (New York) 87 (1997), 3859-3877.
15. P. Maremonti and V. A. Solonnikov, "On nonstationary Stokes problem in exterior domains", Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 24 (1997), 395-449.
16. p-estimates for solutions to the initial-boundary value problem for the generalized Stokes equations in a half-space", Probl. Mat. Anal. 20 (2000), 243-270; English transl., J. Math. Sci. (New York) 102 (2000), 4523-4543.
17. p-estimates for solutions to the initial-boundary value problem for the generalized Stokes equations in a half-space", Probl. Mat. Anal. 20 (2000), 243-270; English transl., J. Math. Sci. (New York) 102 (2000), 4523-4543.
18. p-estimates for solutions to the initial-boundary value problem for the generalized Stokes system of equations in a bounded domain", Probl. Mat. Anal. 21 (2000), 211-254; English transl., J. Math. Sci. (New York) 105 (2001), 2448-2484.
19. p-estimates for solutions to the initial-boundary value problem for the generalized Stokes system of equations in a bounded domain", Probl. Mat. Anal. 21 (2000), 211-254; English transl., J. Math. Sci. (New York) 105 (2001), 2448-2484.
20. G. A. Seregin, "Some estimates near the boundary for solutions to the nonstationary linearized Navier-Stokes equations", Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 271 (2000), 204-223; English transl., J. Math. Sci. (New York) 115 (2003), 2820-2831.
21. G. A. Seregin, "Some estimates near the boundary for solutions to the nonstationary linearized Navier-Stokes equations", Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 271 (2000), 204-223; English transl., J. Math. Sci. (New York) 115 (2003), 2820-2831.
22. V. A. Solonnikov, "Estimates of solutions of the Stokes equations in Sobolev spaces with a mixed norm", Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 288 (2002), 204-231; English transl. in J. Math. Sci. (New York) (to appear).
23. V. A. Solonnikov, "Estimates of solutions of the Stokes equations in Sobolev spaces with a mixed norm", Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 288 (2002), 204-231; English transl. in J. Math. Sci. (New York) (to appear).
24. p-estimates for solutions to the initial-boundary value problem for the generalized Stokes equations in a half-space", Probl. Mat. Anal. 20 (2000), 243-270; English transl., J. Math. Sci. (New York) 102 (2000), 4523-4543.
25. Ya. S. Bugrov, "Functional spaces with mixed norm", Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 1137-1158; English transl., Math. USSR-Izv. 5 (1972), 1145-1167.
26. H. Sohr, The Navier-Stokes equations. An elementary functional analytic approach, Birkhäuser, Basel 2001.
27. R. Farwig and H. Sohr, "The stationary and nonstationary Stokes system in exterior domains with nonzero divergence and nonzero boundary values", Math. Methods Appl. Sci. 17 (1994), 269-291.
28. p", SIAM J. Math. Anal. 12 (1981), 201-228.
29. O. V. Besov, V. P. Il'in, and S. M. Nikol'skii, Integral representations of functions and imbedding theorems, Nauka, Moscow 1975; English transl., vols. I, II, Wiley, New York 1978, 1979.
30. A. P. Calderón and A. Zygmund, "On singular integrals", Amer. J. Math. 78 (1956), 289-309.
31. V. A. Solonnikov, "Estimates of solutions of an initial-boundary value problem for the linear nonstationary Navier-Stokes system", Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 59 (1976), 178-254; English transl., J. Soviet Math. 10 (1978), 336-393.
32. V. A. Solonnikov, "Estimates of solutions of an initial-boundary value problem for the linear nonstationary Navier-Stokes system", Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 59 (1976), 178-254; English transl., J. Soviet Math. 10 (1978), 336-393.
33. V. A. Solonnikov, "The solvability of the second initial-boundary value problem for the linear, time-dependent system of Navier-Stokes equations", Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 69 (1977), 200-218; English transl., J. Soviet Math. 10 (1978), 141-155.
34. V. A. Solonnikov, "The solvability of the second initial-boundary value problem for the linear, time-dependent system of Navier-Stokes equations", Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 69 (1977), 200-218; English transl., J. Soviet Math. 10 (1978), 141-155.
35. V. A. Solonnikov, "The solvability of the second initial-boundary value problem for the linear, time-dependent system of Navier-Stokes equations", Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 69 (1977), 200-218; English transl., J. Soviet Math. 10 (1978), 141-155.
36. V. A. Solonnikov, "The solvability of the second initial-boundary value problem for the linear, time-dependent system of Navier-Stokes equations", Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 69 (1977), 200-218; English transl., J. Soviet Math. 10 (1978), 141-155.
37. P. Maremonti and V. A. Solonnikov, "An estimate for the solutions of the Stokes system in exterior domains", Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 180 (1990), 105-120; English transl., J. Math. Sci. (New York) 68 (1994), 229-239.
38. P. Maremonti and V. A. Solonnikov, "An estimate for the solutions of the Stokes system in exterior domains", Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 180 (1990), 105-120; English transl., J. Math. Sci. (New York) 68 (1994), 229-239.
39. M. E. Bogovskii, "The solution of some problems of vector analysis related to the operators div and grad", Trudy Sem. Sobolev. 1 (1980), 5-40. (Russian)