Compressible Navier--Stokes equations with a potential force: global well-posedness and optimal time-decay rates for arbitrarily large $L^2$ initial data

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 $(1+|x|{)}^j{\nabla }^j\varphi$ is imposed. For initial data relative to the stationary state that are sufficiently small in ${\dot{H}}^{\frac{1}{2}-\delta }\cap {\dot{H}}^3$, we establish the existence and uniqueness of a global strong solution in $H^3$, while allowing the initial $L^2$ norm to be arbitrarily large. If the initial data are bounded in ${\dot{B}}_{2,\infty }^s$ for $s\in [-\frac{3}{2},-1)$, then the solution and its first spatial derivative decay at the optimal rates $(1+t{)}^{-\frac{k-s}{2}}$ 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.

36 pages. All comments are welcome