Stability of time-periodic solutions to the Navier-Stokes-Fourier system

TL;DR

The paper addresses the existence and stability of time-periodic solutions for the 3D Navier-Stokes-Fourier system in $\mathbb{R}^3$ under a small time-periodic forcing, and provides decay rates for perturbations in $L^p$. It introduces a slow-decay Besov framework, notably the homogeneous Besov space $\dot{B}^{1/2}_{2,\infty}$ intersected with $\dot{H}^k$, together with a modified energy functional that couples density, momentum, and temperature terms. The main results prove the existence and uniqueness of a time-periodic solution for small forcing and global stability, along with quantitative time-decay estimates for perturbations around the time-periodic state. The approach blends a priori Besov-space estimates, local-in-time construction, a semigroup analysis of the linearized problem, and a Duhamel-type control of nonlinear terms.

Abstract

We prove the existence and stability of a time-periodic solution to the Navier-Stokes-Fourier system in the three-dimensional whole space when a time-periodic external force is sufficiently small. The time decay estimate of the perturbation around the time-periodic solution is derived under the assumption that an initial perturbation is small and belongs to $L^p$ for some $1\leq p\leq 2$.

Theorems & Definitions (23)

• Theorem 1.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Proposition 3.1
• Proposition 3.2
• Lemma 3.3