On Liouiville Type Theorem for the 3D Isentropic Navier-Stokes System without D-condition

In this paper, we establish Liouville-type theorems for the steady compressible Navier-Stokes system. Assuming a smooth solution $u \in L^p(\mathbb{R}^3)$, $3 \le p \le \frac{9}{2}$, with bounded density, one obtains $u \equiv0$. This generalizes the result of Li-Yu \cite{Li-Yu} by removing the Dirichlet condition $\int_{\mathbb{R}^3} |\nabla u|^2 \, dx < \infty$. If $\frac{9}{2} < p < 6$, Liouville-type theorem holds under the additional oscillation condition for momentum $\rho u \in \dot{B}^{\frac{3}{p} - \frac{3}{2}}_{\infty,\infty}(\mathbb{R}^3)$. For the marginal case $u \in L^6(\mathbb{R}^3)$, the oscillation condition can be replaced by $\rho u \in BMO^{-1}(\mathbb{R}^3)$. We also present results in Morrey-type spaces: $u \in \dot{M}^{s,6}(\mathbb{R}^3)$ and $\rho u \in \dot{M}_w^{q,3}(\mathbb{R}^3)$ for $2 \le s \le 6$ and $\frac{3}{2} < q \le 3$.