Optimal Decay Estimates for the Radially Symmetric Compressible Navier-Stokes Equations

TL;DR

The paper addresses the large-time behavior of radially symmetric solutions to the barotropic compressible Navier–Stokes equations in $\mathbb{R}^3$ and proves an optimal decay rate for the nonlinear part by decomposing the solution into linear and nonlinear components within a Besov-space framework. It shows that the nonlinear portion decays faster under radial symmetry, allowing the full solution to inherit the same decay rate as the low-frequency part of the curl-free linear problem, with a complementary lower bound establishing sharpness. The main results give precise $L^p$ decay for the product $a(t)u(t)$ and corresponding Besov-space bounds, along with a weighted scalar-system decay, all under small initial-data assumptions in critical Besov norms. This work extends linear-decay sharpness to the nonlinear regime and provides a rigorous mechanism for understanding the long-time dynamics of radially symmetric compressible flows, with potential implications for stability and numerical approximations in fluid dynamics.

Abstract

We examine the large-time behaviour of solutions to the compressible Navier-Stokes equations under the assumption of radial symmetry. In particular, we calculate a fast time-decay estimate of the norm of the nonlinear part of the solution. This allows us to obtain a bound from below for the time-decay of the solution in $L^\infty$, proving that our decay estimate in that space is sharp. The decay rate is the same as that of the linear problem for curl-free flow. We also obtain an estimate for a scalar system related to curl-free solutions to the compressible Navier-Stokes equations in a weighted Lebesgue space.

Theorems & Definitions (28)

• Proposition 1.1
• Definition 1
• Proposition 1.2
• Proposition 1.3
• Remark 1.4
• Proposition 1.5
• Theorem 1.6
• Remark 1.7
• Theorem 1.8
• Definition 2