$$L^1-$$ decay of higher-order norms of solutions to the Navier–Stokes equations in the upper-half space

Abstract

The aim of this article devotes to establishing the \(L^1\)-decay of cubic order spatial derivatives of solutions to the Navier–Stokes equations, which is a long-time challenging problem. To solve this problem, new tools have to be found to overcome these main difficulties: \(L^1-L^1\) estimate fails for the Stokes flow; the projection operator \(P:\,L^1(\mathbb {R}^n_+)\rightarrow L^1_\sigma (\mathbb {R}^n_+)\) becomes unbounded; the steady Stokes’s estimates does not work any more in \(L^1(\mathbb {R}^n_+)\). We first give the asymptotic behavior with weights of negative exponent for the Stokes flow and Navier–Stokes equations in \(L^1(\mathbb {R}^n_+)\), and these are also independent of interest by themselves. Secondly, we decompose the convection term into two parts, and translate the unboundedness of projection operator into studying an \(L^1\)-estimate for an elliptic problem with homogeneous Neumann boundary conditions, which is established by using the weighted estimates of the Gaussian kernel’s convolution. Finally, a crucial new formula is given for the fundamental solution of the Laplace operator, which is employed for overcoming the strong singularity in studying the cubic order spatial derivatives in \(L^1(\mathbb {R}^n_+)\).