On the local Smoothness of Solutions of the Navier-Stokes Equations

TL;DR

The paper analyzes the Cauchy problem for incompressible Navier–Stokes in $\mathbb{R}^d$ with initial data $a\in L^d(\mathbb{R}^d)$, focusing on the local smoothing effects of the flow and how regularity propagates in time. It proves that for any finite $T>0$ and any nonnegative integers $m,n$, the weighted derivatives satisfy $t^{m+n/2} D_t^m \nabla_x^n u \in L^{d+2}(\mathbb{R}^d\times(0,T))$, provided the solution stays in $L^{d+2}_{x,t}$; this yields local smoothing and, for small data, global decay. The authors develop a mild-solution framework using a heat-kernel decomposition $u = U - B(u,u)$ (or equivalently $u = U + B(v,v))$ and construct a contraction in a weighted Banach space $X$, establishing existence and the stated estimates via a fixed-point argument. Higher regularity in time and space is obtained by induction on $m$, Calderón–Zygmund estimates for the Stokes kernel, and an iterative bootstrap (including Grönwall-type arguments) to extend the smoothing from a short interval to all $t>0$ for small initial data, resulting in decay and full smoothness of the mild solution.

Abstract

We consider the Cauchy problem for incompressible Navier-Stokes equations $u_t+u\nabla_xu-\Delta u+\nabla p=0, div u=0 in R^d \times R^+$ with initial data $a\in L^d(R^d)$, and study in some detail the smoothing effect of the equation. We prove that for $T<\infty$ and for any positive integers $n$ and $m$ we have $t^{m+n/2}D^m_tD^{n}_x u\in L^{d+2}(R^d\times (0,T))$, as long as the $\|u\|_{L^{d+2}_{x,t}(R^d\times (0,T))}$ stays finite.

Theorems & Definitions (20)

• Theorem 2.1
• Remark 2.2
• Remark 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 3.1
• proof
• Remark 3.2
• Remark 4.1
• Lemma 4.2