On Two-Dimensional Navier-Stokes Flows with Rotational Symmetries

Abstract

Abstract. Navier-Stokes flows are found on R 2 that decay in time more rapidly than observed in general. The decay rate is determined in accordance with the order of symmetry with respect to the action of dihedral groups of orthogonal transformations. Contrary to the previous work [13], the basic existence result is proved with no restriction on the size of initial data. Our result extends that of [3] under di¤erent assumptions on the initial data. Unlike [3], the proofs are all carried out without using estimates for the associated vorticity transport equations. Key Words and Phrases. Navier-Stokes system, Cauchy problem, Group symmetry, Asymptotic behavior, Weighted estimate.2000 Mathematics Subject Classification Numbers. 35Q30, 76D05. Statement of the resultsThis paper studies the Cauchy problem for the Navier-Stokes system in R 2 :Here, x ¼ ðx 1 ; x 2 Þ A R 2 , t b 0, q t ¼ q=qt; u ¼ uðx; tÞ ¼ ðu k ðx; tÞÞ 2 k¼1 and p ¼ pðx; tÞ denote unknown velocity and pressure, respectively; and a ¼ aðxÞ is a given initial velocity. The kinematic viscosity is normalized. The standard notation is used for di¤erential operators and function spaces and the summation convention is employed for repeated indices.We are interested in finding a (unique) solution u to (1.1) that satisfies kuðtÞk r a cð1 þ tÞ Àðmþ1Þ=2Àð1À1=rÞ ð1 a r a yÞ ð1:2Þ for some m A N U f0g. (Here and in what follows k Á k r denotes the L r -norm.) In view of the structure of solutions to parabolic equations, it is reasonable to guess that such solutions would decay su‰ciently rapidly as t þ jxj ! y; and our purpose is to specify a class of initial data that provide us with solutions satisfying (1.2).

On weighted-norm estimates for nonstationary incompressible Navier–Stokes flows in a 3D exterior domain

“…See also [19] for solutions with some symmetries. The balance relation (1.8) agrees with that for solutions of the linear heat equation on R n .…”

Decay properties of solutions to the incompressible magnetohydrodynamics equations in a half space

“…This result remains valid in the case of the half space R n C , by applying for the L p L q estimates of Stokes semigroup established in [3]. Here, small condition is unnecessary if n D 2, see [25,39] for examples. So from here and on, we always assume that problem (1.1) possesses a unique strong solution.…”

Inhomogeneous Incompressible Viscous Flows With Slowly Varying Initial Data

“…This section is devoted to the 2D case and can be of independent interest. We generalize the decay in time estimates obtained in the case of homogeneous Navier-Stokes equation by Wiegner in [33] (see also the works [6,19,30,31] and see [13] for the application of this method to a singular perturbed 2D Navier-Stokes system). We remark that to obtain this optimal time decay estimate for v h , we need to use a completely new formulation (see (3.22) below) of the inhomogeneous Navier-Stokes system.…”