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 $u\equiv 0$. This generalizes the result of Li-Yu \cite{Li-Yu} by removing the Dirichlet condition ${\int }_{{\mathbb{R}}^3}\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{2}dx<\mathrm{\infty }$. If $\frac{9}{2}<p<6$, Liouville-type theorem holds under the additional oscillation condition for momentum $\rho u\in {\dot{B}}_{\mathrm{\infty },\mathrm{\infty }}^{\frac{3}{p}-\frac{3}{2}}({\mathbb{R}}^3)$. For the marginal case $u\in L^6({\mathbb{R}}^3)$, the oscillation condition can be replaced by $\rho u\in BM{O}^{-1}({\mathbb{R}}^3)$. 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 $\frac{3}{2}<q\le 3$.