L3,∞-solutions of the Navier-Stokes equations and backward uniqueness

Russian Mathematical Surveys

Volumn 58, Issue 2, 2003, Pages 211-250

  Escauriaza, L ^a   Seregin, G ^a   Sverak V ^a  

^a NONE

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0042154193     PISSN: 00360279     EISSN: 14684829     Source Type: Journal    
DOI: 10.1070/RM2003v058n02ABEH000609     Document Type: Article

References (53)

1. H. Beirao da Veiga, "On the smoothness of a class of weak solutions to the Navier-Stokes equations", J. Math. Fluid Mech. 2 (2000), 315-323.
2. L. Caffarelli, R.-V. Kohn, and L. Nirenberg, "Partial regularity of suitable weak solutions of the Navier-Stokes equations", Comm. Pure Appl. Math. 35 (1982), 771-831.
3. L. Escauriaza, "Carleman inequalities and the heat operator", Duke Math. J. 104 (2000), 113-127.
4. L. Escauriaza and P. J. Fernández, "Unique continuation for parabolic operators", Ark. Mat. (to appear).
5. L. Escauriaza, G. Seregin, and V. Šverák, "On backward uniqueness for parabolic equations", Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 288 (2002), 100-103; English transl. in J. Math. Sci. (New York) (to appear).
6. L. Escauriaza, G. Seregin, and V. Šverák, "On backward uniqueness for parabolic equations", Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 288 (2002), 100-103; English transl. in J. Math. Sci. (New York) (to appear).
7. L. Escauriaza, G. Seregin, and V. Šverák, "On backward uniqueness for parabolic equations", Arch. Ration. Mech. Anal. (to appear).
8. L. Escauriaza, G. Seregin, and V. Šverák, "Backward uniqueness for the heat operator in a half-space", Algebra i Analiz 15:1 (2003) (to appear); English transl. to appear in St. Petersburg Math. J.
9. L. Escauriaza, G. Seregin, and V. Šverák, "Backward uniqueness for the heat operator in a half-space", Algebra i Analiz 15:1 (2003) (to appear); English transl. to appear in St. Petersburg Math. J.
10. L. Escauriaza and L. Vega, "Carleman inequalities and the heat operator. II", Indiana Univ. Math. J. 50 (2001), 1149-1169.
11. G. P. Galdi, "An introduction to the Navier-Stokes initial-boundary value problem", Fundamental Directions in Mathematical Fluid Mechanics (G. P. Galdi et al., eds.), Birkhäuser, Basel 2000, pp. 1-70.
12. p and regularity of weak solutions of the Navier-Stokes system", J. Differential Equations 62 (1986), 186-212.
13. p-estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains", J. Funct. Anal. 102 (1991), 72-94.
14. E. Hopf, "Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen", Math. Nachr. 4 (1951), 213-231.
15. L. Hörmander, Linear partial differential operators, Springer-Verlag, Berlin-Göttingen-Heidelberg 1963; Russian transl., Mir, Moscow 1965.
16. O. A. Ladyzhenskaya, Mathematical problems of the dynamics of viscous incompressible fluids, Gos. Izdat. Fiz.-Mat. Lit., Moscow 1961; 2nd rev. aug. ed. of English transl., The Mathematical theory of viscous incompressible flow, Gordon and Breach, New York-London 1969.
17. O. A. Ladyzhenskaya, Mathematical problems of the dynamics of viscous incompressible fluids, Gos. Izdat. Fiz.-Mat. Lit., Moscow 1961; 2nd rev. aug. ed. of English transl., The Mathematical theory of viscous incompressible flow, Gordon and Breach, New York-London 1969.
18. O. A. Ladyzhenskaya, "On uniqueness and smoothness of generalized solutions to the Navier-Stokes equations", Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 5 (1967), 169-185; English transl., Sem. Math. V. A. Steklov Math. Inst. Leningrad 5 (1969), 60-66.
19. O. A. Ladyzhenskaya, "On uniqueness and smoothness of generalized solutions to the Navier-Stokes equations", Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 5 (1967), 169-185; English transl., Sem. Math. V. A. Steklov Math. Inst. Leningrad 5 (1969), 60-66.
20. O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, 2nd rev. aug. ed., Nauka, Moscow 1970. (Russian)
21. O. A. Ladyzhenskaya and G. A. Seregin, "On partial regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations", J. Math. Fluid Mech. 1 (1999), 356-387.
22. O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural'tseva, Linear and quasi-linear equations of parabolic type, Nauka, Moscow 1967; English transl., Amer. Math. Soc., Providence, RI 1968.
23. J. Leray, "Sur le mouvement d'un liquide visqueux emplissant l'espace", Acta Math. 63 (1934), 193-248.
24. F.-H. Lin, "A new proof of the Caffarelli-Kohn-Nirenberg theorem", Comm. Pure Appl. Math. 51 (1998), 241-257.
25. m with applications to weak solutions", Math. Z. 187 (1984), 471-480.
26. H. Kozono, "Uniqueness and regularity of weak solutions to the Navier-Stokes equations", Lecture Notes Numer. Appl. Anal. 16 (1998), 161-208.
27. H. Kozono and H. Sohr, "Remark on uniqueness of weak solutions to the Navier-Stokes equations", Analysis 16 (1996), 255-271.
28. 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.
29. 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.
30. K. Masuda, "On the analyticity and the unique continuation theorem for solutions of the Navier-Stokes equations", Proc. Japan Acad. 43 (1967), 827-832.
31. K. Masuda, "Weak solutions of the Navier-Stokes equations", Tôhoku Math. J. (2) 36 (1984), 623-646.
32. S. Micu and E. Zuazua, "On the lack of null-controllability of the heat equation on the half space", Portugal. Math. 58 (2001), 1-24.
33. J. Nečas, M. Růzička, and V. Šverák, "On Leray's self-similar solutions of the Navier-Stokes equations", Acta Math. 176 (1996), 283-294.
34. 3)", J. Math. Fluid Mech. 1 (1999), 309-325.
35. G. Prodi, "Un teorema di unicità per le equazioni di Navier-Stokes", Ann. Mat. Pura Appl. (4) 48 (1959), 173-182.
36. V. Scheffer, "Partial regularity of solutions to the Navier-Stokes equations", Pacific J. Math. 66 (1976), 535-552.
37. V. Scheffer, "Hausdorff measure and the Navier-Stokes equations", Comm. Math. Phys. 55 (1977), 97-112.
38. V. Scheffer, "The Navier-Stokes equations in a bounded domain", Comm. Math. Phys. 73 (1980), 1-42.
39. V. Scheffer, "Boundary regularity for the Navier-Stokes equations in a half-space", Comm. Math. Phys. 85 (1982), 275-299.
40. G. A. Seregin, "Interior regularity for solutions to the modified Navier-Stokes equations", J. Math. Fluid Mech. 1 (1999), 235-281.
41. G. A. Seregin, "On the number of singular points of weak solutions to the Navier-Stokes equations", Comm. Pure Appl. Math. 54 (2001), 1019-1028.
42. G. A. Seregin, "Local regularity of suitable weak solutions to the Navier-Stokes equations near the boundary", J. Math. Fluid Mech. 4 (2002), 1-29.
43. G. A. Seregin, "Differentiability properties of weak solutions to the Navier-Stokes equations", Algebra i Analiz 14:1 (2002), 194-237; English transl., St. Petersburg Math. J. (to appear).
44. G. A. Seregin, "Differentiability properties of weak solutions to the Navier-Stokes equations", Algebra i Analiz 14:1 (2002), 194-237; English transl., St. Petersburg Math. J. (to appear).
45. G. A. Seregin and V. Šverák, "Navier-Stokes equations with lower bounds on pressure", Arch. Ration. Mech. Anal. 183 (2002), 65-86.
46. G. Seregin and V. Šverák, "The Navier-Stokes equations and backward uniqueness", Nonlinear Problems in Mathematical Physics and Related Topics, II. In Honor of Professor O. A. Ladyzhenskaya (M. Sh. Birman et al., eds.), Kluwer/Plenum, New York 2002, pp. 359-370.
47. J. Serrin, "On the interior regularity of weak solutions of the Navier-Stokes equations", Arch. Ration. Mech. Anal. 9 (1962), 187-195.
48. J. Serrin, "The initial value problem for the Navier-Stokes equations", Nonlinear Problems (R. Langer, ed.), Univ. of Wisconsin Press, Madison 1963, pp. 69-98.
49. H. Sohr and W. von Wahl, "On the singular set and the uniqueness of weak solutions of the Navier-Stokes equations", Manuscripta Math. 49 (1984), 27-59.
50. V. A. Solonnikov, "Estimates of the solutions of a nonstationary 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.
51. V. A. Solonnikov, "Estimates of the solutions of a nonstationary 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.
52. M. Struwe, "On partial regularity results for the Navier-Stokes equations", Comm. Pure Appl. Math. 41 (1988), 437-458.
53. D. Tataru, Carleman estimates, unique continuation, and applications, http://math.berkeley.edu/̃tataru/ucp.html.