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Parabolic-scalings on large-time behavior of the incompressible Navier--Stokes flow

Masakazu Yamamoto

TL;DR

The paper develops a parabolic-scaling framework to characterize the large-time asymptotics of incompressible Navier–Stokes flow in $\mathbb{R}^n$ up to the $2n$-th order, revealing both power-law decay terms and logarithmic evolutions. By combining Taylor renormalization with vorticity-based analysis and renormalization techniques, it constructs unique profile functions $U_m$ and $K_m$ that capture the drift and higher-order nonlinear effects. The expansion leverages vorticity moments and the Biot–Savart law, and identifies when logarithmic terms arise from higher-order nonlinear contributions $\mathcal{I}_p$ with $n+3\le p\le 2n+2$. The results extend prior Escobedo–Zuazua-type expansions, provide sharp decay behavior for the remainder, and establish the uniqueness of the parabolically scaled expansion, highlighting the role of drift in generating logarithmic evolutions.

Abstract

Through asymptotic expansion, the large-time behavior of incompressible Navier--Stokes flow in $n$-dimensional whole space is depicted. Especially, from their parabolic scalings, large-time behaviors of any terms on the expansion are clarified. The parabolic scalings also guarantee the uniqueness of the expansion. In the preceding work, the expansion with the $n$th order has already been derived. They also predicted that the flow has some logarithmic evolutions in higher-order decay. In this paper, an asymptotic expansion with $2n$th order is presented. Furthermore, logarithmic evolutions are discovered.

Parabolic-scalings on large-time behavior of the incompressible Navier--Stokes flow

TL;DR

The paper develops a parabolic-scaling framework to characterize the large-time asymptotics of incompressible Navier–Stokes flow in up to the -th order, revealing both power-law decay terms and logarithmic evolutions. By combining Taylor renormalization with vorticity-based analysis and renormalization techniques, it constructs unique profile functions and that capture the drift and higher-order nonlinear effects. The expansion leverages vorticity moments and the Biot–Savart law, and identifies when logarithmic terms arise from higher-order nonlinear contributions with . The results extend prior Escobedo–Zuazua-type expansions, provide sharp decay behavior for the remainder, and establish the uniqueness of the parabolically scaled expansion, highlighting the role of drift in generating logarithmic evolutions.

Abstract

Through asymptotic expansion, the large-time behavior of incompressible Navier--Stokes flow in -dimensional whole space is depicted. Especially, from their parabolic scalings, large-time behaviors of any terms on the expansion are clarified. The parabolic scalings also guarantee the uniqueness of the expansion. In the preceding work, the expansion with the th order has already been derived. They also predicted that the flow has some logarithmic evolutions in higher-order decay. In this paper, an asymptotic expansion with th order is presented. Furthermore, logarithmic evolutions are discovered.

Paper Structure

This paper contains 9 sections, 3 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Notations.
  3. Preliminaries
  4. Proof of main results
  5. Derivation of the expansion
  6. Uniqueness of the expansion
  7. Approximation with low order is already well known
  8. Biot--Savart laws for the approximations
  9. Conservation-laws of the approximations

Key Result

Theorem 1.1

Let $\omega_0 \in L^1 (\mathbb{R}^n) \cap L^\infty (\mathbb{R}^n),~ |x|^{2n+1} \omega_0 \in L^1 (\mathbb{R}^n)$ and $\int_{\mathbb{R}^n} x^\alpha \omega_0$$dx = 0$ for $|\alpha| \le 1$. Assume that the solutions $u$ of NS for $a^j = - \nabla (-\Delta)^{-1} \cdot \omega_0^{\star j}$ and $\omega$ of v for $\lambda > 0$, and as $t \to +\infty$ for $1 \le q \le \infty$. In addition, if $|x|^{2n+2} \o

Theorems & Definitions (11)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Claim 3.1
  • proof
  • Claim 3.2
  • proof
  • Claim 3.3
  • ...and 1 more