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

Abstract: 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 (ρu \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 (ρu \in BMO^{-1}(\mathbb{R}^3)). We also present results in Morrey-type spaces: (u \in \dot{M}^{s,6}(\mathbb{R}^3)) and (ρu \in \dot{M}_w^{q,3}(\mathbb{R}^3)) for (2 \le s \le 6) and (\frac{3}{2} < q \le 3).