Non-Algebraic Decay for Solutions to the Navier-Stokes Equations

TL;DR

This work extends Wiegner's classical algebraic-decay results for the Navier–Stokes equations to non-algebraic decay profiles in two dimensions by comparing the nonlinear Leray solution $u$ with the corresponding heat-flow $v$, under forcing and initial data in natural $L^p$ spaces. Using a Duhamel-based decomposition, energy estimates, and vorticity arguments, the authors prove $\|u(t)\|=O(\Phi(t))$ and $\|u(t)-v(t)\|=o(\Phi(t))$ under general decay controls $\Phi$ with specific monotonicity and integrability properties, and sharpen these to $\|u(t)-v(t)\|=O(\Phi(t)^2)$ when $\Phi$ satisfies a stronger $L^{2}$-type condition (Ass-Phi2), or to $\|u(t)-v(t)\|=O(t^{-1})$ if $\Phi\in L^{2}(\mathbb{R}^{+})$. In 2D this closes a gap in Wie–Gern's theorem and aligns with recent CPZ-type results showing algebraic-decay Leray solutions are meager, providing robust quantitative decay profiles even for slowly decaying data and forcing.

Abstract

Around forty years ago, Michael Wiegner provided, in a seminal paper, sharp algebraic decay rates for solutions of the Navier--Stokes equations, showing that these solutions behave asymptotically like the solutions of the heat equation with the same data as $t\to+\infty$, in the $L^2$-norm, up to some critical decay rate. In the present paper, we close a gap that appears in the conclusion of Wiegner's theorem in the 2D case, for solutions with non-algebraic decay rate.

Theorems & Definitions (16)

• Theorem 1.1: L. Brandolese, C. Perusato, and P. Zingano, 2024 CPZ
• Theorem 1.2: M. Wiegner, 1987 Wie87
• Theorem 1.3
• Remark 1.4
• Remark 1.5
• Remark 1.6: The higher dimensional case
• Lemma 2.1
• proof
• Remark 2.2
• proof : Proof of Assertion i) of Theorem \ref{['th:intro']}