The Uniqueness of Nondecaying Solutions for the Navier-Stokes Equations
Tóm tắt
The uniqueness of solutions of the Navier-Stokes equations in the whole space is established when the velocity field is bounded and the pressure field is a BMO-valued locally integrable-in-time function for bounded initial data. Here the velocity field may not decay at space infinity. Although there are a few results concerning uniqueness without the decay assumption, our result is new and applicable for solutions constructed by solving the integral equations.
Tài liệu tham khảo
Cannon, J.R., Knightly, G.H.: A note on Cauchy problem for the Navier-Stokes equations. SIAM J. Appl. Math. 18, 614–644 (1970)
Cannone, M.: Ondelettes, Paraproduits et Navier-Stokes. Diclerot Editeur, Arts et Sciences, Paris-New York-Amsterdam, 1995
Carpio, A.: Large-time behavior in incompressible Navier-Stokes equations. SIAM J. Math. Anal. 27, 449–475 (1996)
Cazenave, T., Haraux, A.: An Introduction to Semilinear Evolution Equations. Oxford Lecture Series in Mathematics and Its Applications 13, Clarendon Press, Oxford University Press, 1998
Galdi, G.P., Maremonti, P.: A Uniqueness theorem for viscous fluid motions in exterior domains. Arch. Rational. Mech. Anal. 91, 375–384 (1986)
Giga, Y.: Solutions for semilinear parabolic equations in L p and regularity of weak solutions of the Navier-Stokes systems. J. Differential Equations 62, 186–212 (1986)
Giga, Y., Inui, K., Kato, J., Matsui, S.: Remarks on the uniqueness of bounded solutions of the Navier-Stokes equations. Nonlinear Anal. 47, 4151–4156 (2001)
Giga, Y., Inui, K., Matsui, S.: On the Cauchy problem for the Navier-Stokes equations with nondecaying initial data. Quaderni di Matematica 4, 27–67 (1999)
Giga, Y., Matsui, S., Sawada, O.: Global existence of two-dimensional Navier-Stokes flow with nondecaying initial velocity. J. Math. Fluid Mech. 3, 302–315 (2001)
Giga, Y., Matsui, S., Shimizu, Y.: On estimates in Hardy spaces for the Stokes flow in a half space. Math. Z. 231, 383–396 (1999)
Giga, Y., Miyakawa, T.: Solutions in L r to the Navier-Stokes initial value problem. Arch. Rational. Mech. Anal. 89, 267–281 (1985)
Kato, T.: Strong L p-solutions of the Navier-Stokes equations in R n with applications to weak solutions. Math. Z. 187, 471–480 (1984)
Knightly, G.H.: A Cauchy problem for the Navier-Stokes equations. SIAM J. Math. Anal. 3, 506–511 (1972)
Kim, N., Chae, D.: On the uniqueness of the unbounded classical solutions of the Navier-Stokes and associated equations. J. Math. Anal. Appl. 186, 91–96 (1994)
Okamoto, H.: A uniqueness theorem for the unbounded classical solutions of nonstationary Navier-Stokes equations in R 3. J. Math. Anal. Appl. 181, 473–482 (1994)
Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory integrals. Princeton University Press, 1993
Uchiyama, A.: A constructive proof of the Fefferman-Stein decomposition of BMO R n). Acta Math. 148, 215–241 (1982)