Spatial Decay/Asymptotics in the Navier–Stokes Equation

We discuss the occurrence of spatial asymptotic expansions of solutions to the Navier–Stokes equation on $$ {\mathbb{R}} ^d$$ . In particular, we prove that the Navier–Stokes equation is locally well-posed in a class of weighted Sobolev and asymptotic spaces. The solutions depend analytically on the initial data and time and (generically) develop nontrivial asymptotic terms as $$|x|\to\infty$$ . In addition, the solutions have a spatial smoothing property that depends on the order of the asymptotic expansion. DOI 10.1134/S1061920824601812