zbMATH — the first resource for mathematics

Lower bounds of rates of decay for solutions to the Navier-Stokes equations. (English) Zbl 0739.35070
The author gives explicit lower bounds for the \(L^ 2\)-decay of solutions to the Navier-Stokes equations in two and three space dimensions \[ u_ t-\Delta u+(u\cdot\nabla)u+\nabla p=0,\quad \hbox{div} u=0\quad \hbox{ in }\mathbb{R}^ n\times(0,\infty),\quad u(0)=u_ 0\hbox{ in }\mathbb{R}^ n. \] She especially concentrates on the case that the Fourier transform of the initial data vanishes at the origin. Under some additional conditions on \(u_ 0\) she is able to show that \[ | u(\centerdot,t) |^ 2 _{L^ 2(\mathbb{R}^ n)} \geq C(1+t)^{- \alpha(n)} \] where \(\alpha(n)=n/2+1\). Roughly speaking, the result is proved as follows: let \(v\) denote the solution of the heat equation with initial data \(u(\centerdot,t_ 0)\), where the time \(t_ 0\) is chosen in such a way as to ensure that \(v\) decays at a slow rate. Then it is shown that \(u\) cannot decay faster than \(v\).

35Q30 Navier-Stokes equations
35K55 Nonlinear parabolic equations
Full Text: DOI
[1] 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. · Zbl 0509.35067 · doi:10.1002/cpa.3160350604 · doi.org
[2] Ryuji Kajikiya and Tetsuro Miyakawa, On \?² decay of weak solutions of the Navier-Stokes equations in \?\(^{n}\), Math. Z. 192 (1986), no. 1, 135 – 148. · Zbl 0607.35072 · doi:10.1007/BF01162027 · doi.org
[3] O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Second English edition, revised and enlarged. Translated from the Russian by Richard A. Silverman and John Chu. Mathematics and its Applications, Vol. 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969. · Zbl 0184.52603
[4] Maria Elena Schonbek, \?² decay for weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal. 88 (1985), no. 3, 209 – 222. · Zbl 0602.76031 · doi:10.1007/BF00752111 · doi.org
[5] -, Large time behavior of solutions to the Navier-Stokes equations, Comm. Partial Differential Equations 11 (1986), 753-763.
[6] Michael Wiegner, Decay results for weak solutions of the Navier-Stokes equations on \?\(^{n}\), J. London Math. Soc. (2) 35 (1987), no. 2, 303 – 313. · Zbl 0652.35095 · doi:10.1112/jlms/s2-35.2.303 · doi.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.