On \(L^ 2\) decay of weak solutions of the Navier-Stokes equations in \(\mathbb R^ n\)
From the authors' Introduction: ''We consider the initial value problem for the Navier-Stokes equations in \({\mathbb{R}}^ n\), \[ \partial u/\partial t+u\cdot \nabla u-\Delta u+\nabla p=0\quad in\quad {\mathbb{R}}^ n\times (0,\infty), \] \[ \nabla \cdot u=0\quad in\quad {\mathbb{R}}^ n\times (0,\infty),\quad u(x,0)=a(x)\quad in\quad {\mathbb{R}}^ n, \] where \(n\geq 2\), \(u=(u^ i)^ n_{i=1}\) and p denote, respectively, unknown velocity and pressure, while \(a=(a^ i)^ n_{i=1}\) is a given initial velocity....Our purpose in this paper is to improve and generalize the result of \textit{M. E. Schonbek} [Arch. Ration. Mech. Anal. 88, 209-222 (1985; Zbl 0602.76039)] by the systematic use of a modified version of her method. In fact, for all \(n\geq 2\) and \(a\in L^ 2\) with \(\nabla \cdot a=0\), we shall show that the weak solution... tends to zero in \(L^ 2\) as \(t\to \infty\). Moreover, we show that if a is in \(L^ 2\cap L^ r\) for some \(1\leq r<2\), then the corresponding solution decays in \(L^ 2\) like \(t^{-(n/r-n/2)/2}\)''.
scientific article; zbMATH DE number 3981695