Regularity of solutions of 3D Navier-Stokes equations in a Lipschitz domain for small data
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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 L2 norms of the initial velocity and the gradient of the initial velocity and Lp,2, p ≥ 4 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.
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 L2 norms of the initial velocity and the gradient of the initial velocity and Lp,2, p ≥ 4 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.
참고문헌 (27)
- R. A. Adams, "Sobolev Spaces" (1975), Academic Press
- J. D. Avrin, "Large-eigenvalue global existence and regularity results for the Navier- Stokes equations", J. Differential Equations, 127(2), pp.365~390 (1996)
- R. Brown, "Estimates for the Stokes operator in Lipschitz domains", Indiana Univ. Math. J, 44(4), pp.1183~1206 (1995)
- H. J. Choe, "The Stokes problem for Lipschitz domains", Indiana Univ. Math. J, 51(5), pp.1235~1259 (2002)
- J. Y. Chemin, "Global regularity for some classes of large solutions to the Navier-Stokes equations", Ann. of Math, 173(2), pp.983~1012 (2011)
- I. Chueshov, "Interface boundary value problem for the Navier-Stokes equations in thin two-layer domains", J. Differential Equations, 208(2), pp.449~493 (2005)
- P. Constantin, "Navier-Stokes Equations" (1988), University of Chicago Press
- P. Deuring, "Strong solutions of the Navier-Stokes system in Lipschitz bounded domains", Math. Nachr, 171, pp.111~148 (1995)
- H. Fujita, "On the Navier-Stokes initial value problem", Arch. Rational Mech. Anal, 16, pp.269~315 (1964)
- L. T. Hoang, "Incompressible fluids in thin domains with Navier friction boundary conditions (I)", J. Math. Fluid Mech, 12(3), pp.435~472 (2010)
- L. T. Hoang, "Navier-Stokes equations with Navier boundary conditions for an oceanic model", J. Dynam. Differential Equations, 22(3), pp.563~616 (2010)
- E. Hopf, "Uber die Anfangswertaufgabe fur die hydrodynamischen Grudgleichungen", Math. Nachr, 4, pp.213~231 (1951)
- D. Iftimie, "The 3D Navier-Stokes equations seen as a perturbation of the 2D Navier- Stokes equations", Bull. Soc. Math. France, 127(4), pp.473~517 (1999)
- D. Iftimie, "Some results on the Navier-Stokes equations in thin 3D domains", J. Differential Equations, 169(2), pp.281~331 (2001)
- D. Iftimie, "Navier-Stokes equations in thin 3D domains with Navier boundary conditions", Indiana Univ. Math. J, 56(3), pp.1083~1156 (2007)
- 곽민규, "GLOBAL EXISTENCE FOR 3D NAVIER-STOKES EQUATIONS IN A THIN PERIODIC DOMAIN", Journal of the Korean Society for Industrial and Applied Mathematics, 15(2), pp.143~150 (2011), 한국산업응용수학회
- I. Kukavica, "Regularity of the Navier-Stokes equation in a thin periodic domain with large data", Discrete Contin. Dyn. Syst, 16(1), pp.67~86 (2006)
- J. Leray, "Sur le mouvement d’un liquide visqueux emplissant l’espace", Acta Math, 63(1), pp.193~248 (1934)
- M. Mitrea, "The nonlinear Hodge-Navier-Stokes equations in Lipschitz domains", Differential Integral Equations, 22(3), pp.339~356 (2009)
- S. Monniaux, "On uniqueness for the Navier-Stokes system in 3D-bounded Lipschitz domains", J. Funct. Anal, 195(1), pp.1~11 (2002)
- S. Montgomery-Smith, "Global regularity of the Navier-Stokes equations on thin three dimensional domains with periodic boundary conditions", Electron. J. Differential Equations, 1999(19), pp.1~19 (1999)
- M. Paicu, "Global regularity for the Navier-Stokes equations with some classes of large initial data", Anal. PDE, 4(1), pp.95~113 (2011)
- G. Raugel, "Navier-Stokes equations on thin 3D domains. I. Global attractors and global regularity of solutions", J. Amer. Math. Soc, 6(3), pp.503~568 (1993)
- G. R. Sell, "Dynamics of Evolutionary Equations, In Applied Math. Sciences 143" (2002), Springer
- R. Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis", CBMS Regional Conference Series (1995)
- R. Temam, "Navier-Stokes equations in three-dimensional thin domains with various boundary conditions", Adv. Differential Equations, 1(4), pp.499~546 (1996)
- K. Wang, "On global regularity of incompressible Navier-Stokes equations in R3", Commun. Pure Appl. Anal, 8(3), pp.1067~1072 (2009)