Forward self-similar solutions of the Navier-Stokes equations in R^n

Abstract

We investigate forward self-similar solutions to the incompressible Navier--Stokes equations with homogeneous initial data $u_0(x)=\sigma(x/|x|)/|x|$, $x\in\mathbb R^n$, focusing on the critical four-dimensional case. For the existence theory in dimensions $3\leq n\leq5$, we develop a Galerkin scheme for the Leray profile problem, featuring a decomposition of the self-similar data and nonlocal heat kernel effects across spatial scales. Exploiting the structural properties of the data and the coercivity of the Leray operator, we derive uniform \textit{a priori} estimates, pass to the limit, and obtain weak forward self-similar solutions. In the 4D case, we prove new Stokes-type estimates for the linear Leray system. The inverse Leray operator admits an integral representation as a singular integral operator; its $L^p$-boundedness follows from Calder\'on--Zygmund theory. Combined with a 4D compactness estimate for the convective term, this closes the regularity iteration. For spatial decay, we reformulate the problem in weighted spaces. The non-integrability of the low-frequency kernel causes a logarithmic loss under $\sigma\in W^{1,\infty}(\mathbb S^3)$. With $\sigma\in C^{1,\gamma}(\mathbb S^3)$, $0<\gamma<1$, frequency-localized estimates remove this loss and yield sharp decay.