JOURNAL BROWSE
Search
Advanced SearchSearch Tips
REGULARITY OF SOLUTIONS OF 3D NAVIER-STOKES EQUATIONS IN A LIPSCHITZ DOMAIN FOR SMALL DATA
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
 Title & Authors
REGULARITY OF SOLUTIONS OF 3D NAVIER-STOKES EQUATIONS IN A LIPSCHITZ DOMAIN FOR SMALL DATA
Jeong, Hyo Suk; Kim, Namkwon; Kwak, Minkyu;
  PDF(new window)
 Abstract
We consider the global existence of strong solutions of the 3D incompressible Navier-Stokes equations in a bounded Lipschitz do-main under Dirichlet boundary condition. We present by a very simple argument that a strong solution exists globally when the product of norms of the initial velocity and the gradient of the initial velocity and , norm of the forcing function are small enough. Our condition is scale invariant and implies many typical known global existence results for small initial data including the sharp dependence of the bound on the volumn of the domain and viscosity. We also present a similar result in the whole domain with slightly stronger condition for the forcing.
 Keywords
Navier-Stokes equations;global existence;strong solution;
 Language
English
 Cited by
 References
1.
R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.

2.
J. D. Avrin, Large-eigenvalue global existence and regularity results for the Navier- Stokes equations, J. Differential Equations 127 (1996), no.2, 365-390. crossref(new window)

3.
R. Brown and Z. Shen, Estimates for the Stokes operator in Lipschitz domains, Indiana Univ. Math. J. 44 (1995), no. 4, 1183-1206.

4.
H. J. Choe and K. Hideo, The Stokes problem for Lipschitz domains, Indiana Univ. Math. J. 51 (2002), no. 5, 1235-1259. crossref(new window)

5.
J. Y. Chemin, I. Gallagher, and M. Paicu, Global regularity for some classes of large solutions to the Navier-Stokes equations, Ann. of Math. (2) 173 (2011), no. 2, 983-1012. crossref(new window)

6.
I. Chueshov, G. Raugel, and A. M. Rekalo, Interface boundary value problem for the Navier-Stokes equations in thin two-layer domains, J. Differential Equations 208 (2005), no. 2, 449-493. crossref(new window)

7.
P. Constantin and C. Foias, Navier-Stokes Equations, University of Chicago Press, Chicago, 1988.

8.
P. Deuring and W. von Wahl, Strong solutions of the Navier-Stokes system in Lipschitz bounded domains, Math. Nachr. 171 (1995), 111-148. crossref(new window)

9.
H. Fujita and T. Kato, On the Navier-Stokes initial value problem, Arch. Rational Mech. Anal. 16 (1964), 269-315. crossref(new window)

10.
L. T. Hoang, Incompressible fluids in thin domains with Navier friction boundary conditions (I), J. Math. Fluid Mech. 12 (2010), no. 3, 435-472. crossref(new window)

11.
L. T. Hoang and G. R. Sell, Navier-Stokes equations with Navier boundary conditions for an oceanic model, J. Dynam. Differential Equations 22 (2010), no. 3, 563-616. crossref(new window)

12.
E. Hopf, Uber die Anfangswertaufgabe fur die hydrodynamischen Grudgleichungen, Math. Nachr. 4 (1951), 213-231.

13.
D. Iftimie, The 3D Navier-Stokes equations seen as a perturbation of the 2D Navier- Stokes equations, Bull. Soc. Math. France 127 (1999), no. 4, 473-517. crossref(new window)

14.
D. Iftimie and G. Raugel, Some results on the Navier-Stokes equations in thin 3D domains, J. Differential Equations 169 (2001), no. 2, 281-331. crossref(new window)

15.
D. Iftimie, G. Raugel, and G. R. Sell, Navier-Stokes equations in thin 3D domains with Navier boundary conditions, Indiana Univ. Math. J. 56 (2007), no. 3, 1083-1156. crossref(new window)

16.
M. Kwak and N. Kim, Global existence for 3D Navier-Stokes equations in a thin periodic domain, J. Korean Soc. Ind. Appl. Math. 15 (2011), no. 2, 143-150.

17.
I. Kukavica and M. Ziane, Regularity of the Navier-Stokes equation in a thin periodic domain with large data, Discrete Contin. Dyn. Syst. 16 (2006), no. 1, 67-86. crossref(new window)

18.
J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math. 63 (1934), no. 1, 193-248. crossref(new window)

19.
M. Mitrea and S. Monniaux, The nonlinear Hodge-Navier-Stokes equations in Lipschitz domains, Differential Integral Equations 22 (2009), no. 3-4, 339-356.

20.
S. Monniaux, On uniqueness for the Navier-Stokes system in 3D-bounded Lipschitz domains, J. Funct. Anal. 195 (2002), no. 1, 1-11. crossref(new window)

21.
S. Montgomery-Smith, Global regularity of the Navier-Stokes equations on thin three dimensional domains with periodic boundary conditions, Electron. J. Differential Equations 1999 (1999), no. 19, 1-19.

22.
M. Paicu and Z. Zhang, Global regularity for the Navier-Stokes equations with some classes of large initial data, Anal. PDE 4 (2011), no. 1, 95-113. crossref(new window)

23.
G. Raugel and G. R. Sell, Navier-Stokes equations on thin 3D domains. I. Global attractors and global regularity of solutions, J. Amer. Math. Soc. 6 (1993), no. 3, 503-568.

24.
G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Math. Sciences 143, Springer, Berlin, 2002.

25.
R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS Regional Conference Series, No. 66, SIAM, Philadelphia, 1995.

26.
R. Temam and M. Ziane, Navier-Stokes equations in three-dimensional thin domains with various boundary conditions, Adv. Differential Equations 1 (1996), no. 4, 499-546.

27.
K. Wang, On global regularity of incompressible Navier-Stokes equations in ${\mathbb{R}}^3$, Commun. Pure Appl. Anal. 8 (2009), no. 3, 1067-1072. crossref(new window)