JOURNAL BROWSE
Search
Advanced SearchSearch Tips
LOCAL REGULARITY CRITERIA OF THE NAVIER-STOKES EQUATIONS WITH SLIP BOUNDARY CONDITIONS
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
 Title & Authors
LOCAL REGULARITY CRITERIA OF THE NAVIER-STOKES EQUATIONS WITH SLIP BOUNDARY CONDITIONS
Bae, Hyeong-Ohk; Kang, Kyungkeun; Kim, Myeonghyeon;
  PDF(new window)
 Abstract
We present regularity conditions for suitable weak solutions of the Navier-Stokes equations with slip boundary data near the curved boundary. To be more precise, we prove that suitable weak solutions become regular in a neighborhood boundary points, provided the scaled mixed norm with 3/p + 2/q = 2, < is sufficiently small in the neighborhood.
 Keywords
Navier-Stokes equations;slip boundary data;suitable weak solution;
 Language
English
 Cited by
 References
1.
S. N. Antontsev and H. B. de Oliveira, Navier-Stokes equations with absorption under slip boundary conditions: existence, uniqueness and extinction in time, Kyoto Conference on the Navier-Stokes Equations and their Applications, 21-41, RIMS Kkyroku Bessatsu, B1, Res. Inst. Math. Sci. (RIMS), Kyoto, 2007.

2.
H. Bae, H. Choe, and B. Jin, Pressure representation and boundary regularity of the Navier-Stokes equations with slip boundary condition, J. Differential Equations 244 (2008), no. 11, 2741-2763. crossref(new window)

3.
H. Beirao da Veiga, On the regularity of flows with Ladyzhenskaya shear-dependent viscosity and slip or nonslip boundary conditions, Comm. Pure Appl. Math. 58 (2005), no. 4, 771-831.

4.
L. Caffarelli, R. Kohn, and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math. 35 (1982), no. 6, 771-831. crossref(new window)

5.
H. Choe and J. L. Lewis, On the singular set in the Navier-Stokes equations, J. Funct. Anal. 175 (2000), no. 2, 348-369. crossref(new window)

6.
L. Escauriaza, G. Seregin, and V. Sverak, $L^{3,{\infty}}$-solutions of Navier-Stokes equations and backward uniqueness, Russian Math. Surveys 58 (2003), no. 2, 211-250. crossref(new window)

7.
E. Fabes, B. Jones, and N. Riviere, The initial value problem for the Navier-Stokes equations with data in $L^p$, Arch. Ration. Mech. Anal. 45 (1972), 222-248. crossref(new window)

8.
Y. Giga, Solutions for semilinear parabolic equations in $L^p$ and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations 62 (1986), 186-212. crossref(new window)

9.
S. Gustafson, K. Kang, and T.-P. Tsai, Regularity criteria for suitable weak solutions of the Navier-Stokes equations near the boundary, J. Differential Equations 226 (2006), no. 2, 594-618. crossref(new window)

10.
S. Gustafson, K. Kang, and T.-P. Tsai, Interior regularity criteria for suitable weak solutions of the Navier-Stokes equations, Comm. Math. Phys. 273 (2007), no. 1, 161-176. crossref(new window)

11.
E. Hopf, Uber die Anfangswertaufgabe fur die hydrodynamischen Grundgleichungen, Math. Nachr. 4 (1950), 213-231. crossref(new window)

12.
S. Itoh and A. Tani, The initial value problem for the non-homogeneous Navier-Stokes equations with general slip boundary condition, Proc. Roy. Soc. Edinburgh Sect. A 130 (2000), no. 4, 827-835. crossref(new window)

13.
K. Kang, On boundary regularity of the Navier-Stokes equations, Comm. Partial Differential equations 29 (2004), no. 7-8, 955-987. crossref(new window)

14.
J. Kim and M. Kim, Local regularity of the Navier-Stokes equations near the curved boundary, J. Math. Anal. Appl. 363 (2010), no. 1, 161-173. crossref(new window)

15.
O. A. Ladyzenskaja, On the uniqueness and smoothness of generalized solutions of the Navier-Stokes equations, Zapiski Scient. Sem. LOMI 5 (1967), 169-185.

16.
O. A. Ladyzenskaja and G. A. Seregin, On partial regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations, J. Math. Fluid Mech. 1 (1999), no. 4, 356-387. crossref(new window)

17.
O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, Amer. Math. Soc., 1968.

18.
J. Leray, Essai sur le mouvement d'un fluide visqueux emplissant l'espace, Acta Math. 63 (1934), 193-248. crossref(new window)

19.
F. Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem, Comm. Pure Appl. Math. 51 (1998), no. 3, 241-257. crossref(new window)

20.
P. Maremonti, Some theorems of existence for solutions of the Navier-Stokes equations with slip boundary conditions in half-space, Ric. Mat. 40 (1991), no. 1, 81-135.

21.
T. Ohyama, Interior regularity of weak solutions of the time-dependent Navier-Stokes equation, Proc. Japan Acad. 36 (1960), 273-277. crossref(new window)

22.
G. Prodi, Un teorama di unicita per le equazioni di Navier-Stokes, Ann. Mat. Pura Appl. 48 (1959), 173-182. crossref(new window)

23.
V. Scheffer, Partial regularity of solutions to the Navier-Stokes equations, Pacific J. Math. 66 (1976), no. 2, 535-552. crossref(new window)

24.
V. Scheffer, Hausdorff measure and the Navier-Stokes equations, Comm. Math. Phys. 55 (1977), no. 2, 97-112. crossref(new window)

25.
V. Scheffer, The Navier-Stokes equations on a bounded domain, Comm. Math. Phys. 73 (1980), no. 1, 1-42. crossref(new window)

26.
V. Scheffer, Boundary regularity for the Navier-Stokes equations in a half-space, Comm. Math. Phys. 85 (1982), no. 2, 275-299. crossref(new window)

27.
G. A. Seregin, Local regularity of suitable weak solutions to the Navier-Stokes equations near the boundary, J. Math. Fluid Mech. 4 (2002), no. 1, 1-29. crossref(new window)

28.
G. A. Seregin, Some estimates near the boundary for solutions to the non-stationary linearized Navier-Stokes equations, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 271 (2000), Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 31, 204-223, 317; translation in J. Math. Sci. (N. Y.) 115 (2003) no. 6, 2820-2831.

29.
G. A. Seregin, T. N. Shilkin, and V. A. Solonnikov, Boundary partial regularity for the Navier-Stokes equations, J. Math. Sci. 132 (2006), no. 3, 339-358. crossref(new window)

30.
J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal. 9 (1962), 187-195.

31.
R. Shimada, On the $L^p$-$L^q$ maximal regularity for Stokes equations with Robin boundary condition in a bounded domain, Math. Methods Appl. Sci. 30 (2007), no. 3, 257-289. crossref(new window)

32.
H. Sohr, Zur Regularitatstheorie der instationaren Gleichungen von Navier-Stokes, Math. Z. 184 (1983), no. 3, 359-375. crossref(new window)

33.
V. A. Solonnikov, Estimates of solutions of the Stokes equations in S. L. Sobolev spaces with mixed norm, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 288 (2002), Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 32, 204-231, 273-274; translation in J. Math. Sci. (N. Y.) 123 (2004), no. 6, 4637-4653.

34.
V. A. Solonnikov and V. E. Scadilov, A certain boundary value problems for the stationary system of Navier-Stokes equations, Tr. Mat. Inst. Steklova 125 (1973), 196-210; translation in On a boundary value problems for the stationary system of Navier-Stokes equations, Proc. Steklov Inst. Math. 125 (1973), 186-199.

35.
M. Struwe, On partial regularity results for the Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), no. 4, 437-458. crossref(new window)

36.
S. Takahashi, On interior regularity criteria for weak solutions of the Navier-Stokes equations, Manuscripta Math. 69 (1990), no. 3, 237-254. crossref(new window)