Nonlinear Analysis: Theory, Methods & Applications
Asymptotic behavior for the Navier–Stokes equations with nonzero external forces☆
Introduction
We study the asymptotic behavior in the weighted of solutions for the Navier–Stokes equations with external forces in the whole space : Here, is given initial data. The velocity and the pressure are unknown.
The decay problem for weak solutions of the Navier–Stokes equations was first proposed by Leray [15] for the Cauchy problem in . Kato [14] obtained temporal decay rates for strong solutions, for the first time. Schonbek [19], [20], [21], [22] worked on the temporal decay problem in . She obtained the lower and the upper bounds. In [20], she showed that if , , and the average of the initial data is nonzero, then In [21], [22], it was shown that, if the average is zero, , and under some restrictions on , then for . See also Miyakawa and Schonbek [17] for the lower bound.
Borchers and Miyakawa [7] studied the decay problem in half spaces . They obtained that if , then provided . For example, if then the decay rate is . For , they obtained that . We [3] showed that the decay rate of -norm of the solutions for the Navier–Stokes equations in the half space is if and .
For the spatial decay, Farwig and Sohr [9], [10] showed the spatial decays for the exterior problems. He and Xin [12] showed that if and , there exists a class of weak solutions satisfying and also that if , then there is a class of weak solutions satisfying for all , . If for some , then the right-hand side of the above inequality can be replaced by a constant independent of . Schonbek and Schonbek [23] studied the decay properties of for , when is smooth. We [4] showed the following. Let . Assume that , , and . Then there is a weak solution of (1.1) satisfying the following inequality for all ; Interpolating with the temporal decays, we may obtain the temporal-spatial decay rates. Miyakawa [16] obtained pointwise upper bounds of the Navier–Stokes flows in . In [5], we obtained the lower bounds as in [21], [22], [17]. However, we include weights for the temporal-spatial decays for . The upper bound parts are estimated in several papers, for example [4], [11], [12]. For exterior domains, refer to [4], [1]. Modifying methods in [12], we improved the rates for exterior domains in [4].
All of the above are decay estimates without external forces. With external forces, Wiegner [24] and Ogawa [18] estimated temporal decays. In this paper, we estimate the decay rates of Stokes solutions and of the Navier–Stokes solutions with external forces.
In Section 2, we provide an example of external forces which indicates slow decays. Then, we obtain the temporal-spatial decays for the Stokes flow, and in Section 3 we obtain the decays with weight for the weak solutions of the Navier–Stokes equations.
Section snippets
Decay rate of solutions for the Stokes equations
In this section we obtain the decay rate for the Stokes equations in the whole space . If the initial data and the external force are divergence free in , then solution for the Stokes equations is reduced to that of the heat equations with initial data and inhomogeneous term . The main focus in this section is the estimation on since the term concerning initial data is estimated in [5].
If the external force is not divergence free, it can be decomposed in the form ,
Decay rates with weight for the Navier–Stokes equations
In this section we consider the decay rates for weak solutions with weight for the Navier–Stokes equations. For , since the strong solution is still unknown, we should consider the approximate solutions , , of (1.1) with initial data , of the following equations: where is a retarded mollification of .
We recall that the retarded mollification of is defined by
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2009, Annali dell'Universita di Ferrara
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This work was supported by Korea–France STAR program (2007).