Long time uniform stability for solutions of $n$-dimensional Navier-Stokes equations
Author:
Linghai Zhang
Journal:
Quart. Appl. Math. 57 (1999), 283-315
MSC:
Primary 35Q30; Secondary 35B40, 76D05, 76E99
DOI:
https://doi.org/10.1090/qam/1686191
MathSciNet review:
MR1686191
Full-text PDF Free Access
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- Ling Hai Zhang, Decay estimates for the solutions of some nonlinear evolution equations, J. Differential Equations 116 (1995), no. 1, 31β58. MR 1317702, DOI https://doi.org/10.1006/jdeq.1995.1028
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P. Constantin and C. Fefferman, Direction of vorticity and the problem of global regularity for the Navier-Stokes equations, Indiana University Mathematics Journal 42, 775β789 (1993)
B. Guo and L. Zhang, Decay of solutions to magnetohydrodynamics equations in two space dimensions, Proceedings of the Royal Society of London Ser. A 449, 79β91 (1995)
J. G. Heywood, The Navier-Stokes equations, on the existence, regularity and decay of solutions, Indiana University Mathematics Journal 29, 639β681 (1980)
T. Kato, Strong $L^{p}$-solutions of the Navier-Stokes equations in $R^{m}$, with applications to weak solutions, Math. Z. 187, 471β480 (1984)
R. Kajikiya and T. Miyakawa, On $L^{2}$-decay of weak solutions of the Navier-Stokes equations in $R^{n}$, Math. Z. 192, 135β148 (1986)
R. Pego, Stability in systems of conservation laws with dissipation, Lectures in Applied Math. 23, 345β357 (1986)
R. Pego, P. Smereka, and M. I. Weinstein, Oscillatory instability of traveling waves for a KdV-Burgers equation, Physica D 67, 45β65 (1993)
F. Weissler, The Navier-Stokes initial value problems in $L^{p}$, Arch. Rational Mech. Analysis 74, 219β230 (1980)
M. Wiegner, Decay results for weak solutions of the Navier-Stokes equations on $R^{n}$, J. London Math. Soc. 35, 303β313 (1987)
M. Wiegner, Decay and stability in $L^{p}$ for strong solutions of the Cauchy problem for the Navier-Stokes equations, in βThe Navier-Stokes Equations, Theory and Numerical Methods", edited by J. G. Heywood et al, Springer-Verlag, New York, Lecture Notes in Math. 1431, 95β99 (1990)
M. Wiegner, Decay of the ${L_\infty }$-norm of solutions of the Navier-Stokes equations in unbounded domains, Acta Appl. Math. 37, 215β219 (1994)
M. Schonbek, Decay of solutions of parabolic conservation laws, Comm. Partial Differential Equations 7, 449β473 (1980)
M. Schonbek, $L^{2}$ decay for weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Analysis 88, 209β222 (1985)
M. Schonbek, Large time behavior of solutions to the Navier-Stokes equations, Comm. Partial Differential Equations 11, 733β763 (1986)
P. Secchi, $L^{2}$ stability for weak solutions of the Navier-Stokes equations in $R^{3}$, Indiana University Mathematics Journal 36, 685β691 (1987)
J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Analysis 9, 187β195 (1962)
J. Serrin, The initial value problems for the Navier-Stokes equations, in βNonlinear Problemsβ (R. E. Langer, ed.), University of Wisconsin Press, Madison, 1963, pp. 69β98
E. M. Stein, Singular integrals and differentiability properties of functions, Princeton University Press, 1970
R. Temam, Navier-Stokes equations, theory and numerical analysis, North-Holland, Amsterdam and New York, 1979
H. Beirao da Veiga, Existence and asymptotic behavior for strong solutions of the Navier-Stokes equations in the whole space, Indiana University Mathematics Journal 36, 149β166 (1987)
H. Beirao da Veiga and P. Secchi, $L^{p}$-stability for the strong solutions of the Navier-Stokes equations in the whole space, Arch. Rational Mech. Anal. 98, 65β69 (1987)
L. Zhang, Initial value problem for a nonlinear parabolic equation with singular integral-differential term, Acta Math. Appl. Sinica 8, 367β376 (1992)
L. Zhang, Decay estimates for the solutions of some nonlinear evolution equations, J. Differential Equations 116, 31β58 (1995)
L. Zhang, Sharp rate of decay of solutions to 2-dimensional Navier-Stokes equations, Comm. Partial Differential Equations 20, 119β127 (1995)
L. Zhang, Decay estimates of solutions to the initial value problems for a generalized nonlinear Korteweg-de Vries equation, Chinese Ann. Math. 16A, 22β32 (1995) [in Chinese]
Y. Zhou and B. Guo, The periodic boundary value problem and the initial value problem for the generalized Korteweg-de Vries systems of higher order, Acta Math. Sinica 27, 154β176 (1984)
Y. Zhou and B. Guo, Initial value problems for a nonlinear singular integral-differential equation of deep water, in βPartial Differential Equations", edited by S. S. Chern, Springer-Verlag, New York, Lecture Notes in Math. 1306, 278β290 (1988)
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