\(L^ 2\) decay for the Navier-Stokes flow in halfspaces
The paper deals with the problem of existence of a weak solution of the Navier-Stokes equations in halfspaces which converges in the space \(L^ 2\) to 0 as time goes to infinity. It is shown that for each initial data in \(L^ 2\) there exists a weak solution with above property. Moreover, the authors prove that if the initial velocity belongs to \(L^ 2\cap L^ r\) for some \(1\leq r<2\), then the corresponding weak solution tends to 0 like \(t^{-n(1/r-1/2)/2}\), which is the decay rate in \(L^ 2\) of the solution of the Stokes equations with the same initial function. In proving the above results, the authors employ the argument developed by \textit{M. E. Schonbek} [Arch. Ration. Mech. Anal. 88, 209-222 (1985; Zbl 0602.76031)]. The key estimate is the following: \[ (1)\quad \| E(\lambda)P(w\cdot \nabla)u\| _{L^ 2}\leq C\lambda ^{(n+2)/4}\| w\| _{L^ 2}\| u\| _{L^ 2}\quad for\quad \lambda >0, \] where P is the orthogonal projection on \(L^ 2\) into its solenoidal subspace; \(n\geq 2\) is the space dimension; and E(\(\lambda)\) is the spectral measure associated with the Stokes operator. The authors prove (1) by a systematic use of the inequalities of Poincaré-Sobolev type for the fractional powers of the Stokes operator in halfspaces, which are the consequences from the boundedness and analyticity of the semigroup generated by the operator in \(L^ p\), and the invariance of the halfspace under the transformations: \(x\to \lambda x\), \(\lambda >0\).
scientific article; zbMATH DE number 4019622
scientific article