Compressible Navier--Stokes equations with a potential force: global well-posedness and optimal time-decay rates for arbitrarily large L² initial data

2026/08/01 by Jinkai Ni, Luqi Wang, Zhipeng Zhang
Mathematics · #math.AP #msc:35Q30 #msc:35B40 #msc:76N15

paper · pdf

35 pages. All comments are welcome

arxiv created 2026/08/01 · arxiv updated 2026/08/04

Abstract

We study the Cauchy problem for the three-dimensional barotropic compressible Navier--Stokes equations with a time-independent potential force near a spatially nonconstant stationary state. The potential is controlled in unweighted homogeneous Besov spaces; in particular, no polynomial spatial-weight condition involving (1+|x|)^j∇^jϕ is imposed. For initial data relative to the stationary state that are sufficiently small in H^\frac12-δ∩ H³, we establish the existence and uniqueness of a global strong solution in H³, while allowing the initial L² norm to be arbitrarily large. If the initial data are bounded in B^s_2,∞ for s∈[-\frac32,-1), then the solution and its first spatial derivative decay at the optimal rates (1+t)^-(k-s)/(2) with k=0 and 1, respectively. The analysis relies on refined homogeneous energy estimates and a frequency-localized description for the dissipative and asymptotic structures of the system.

Citations

• Optimal Decay Estimates for the Radially Symmetric Compressible Navier-Stokes Equations
• Optimal decay of compressible Navier-Stokes equations with or without potential force
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• Unique solvability of the initial boundary value problems for compressible viscous fluids