2025/10/12 by Bang, Jeaheang, Gui, Changfeng, Liu, Hao +2
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2510.10488
We prove that the steady incompressible Navier-Stokes equations with any given (-3)-homogeneous, locally Lipschitz external force on ℝn∖\0\, 4≤ n≤ 16, have at least one (-1)-homogeneous solution which is scale-invariant and regular away from the origin. The global uniqueness of the self-similar solution is obtained as long as the external force is small. The key observation is to exploit a nice relation between the radial component of the velocity and the total head pressure under the self-similarity assumption. It plays an essential role in establishing the energy estimates. If the external force has only the nonnegative radial component, we can prove the existence of (-1)-homogeneous solutions for all n≥ 4. The regularity of the solution follows from integral estimates of the positive part of the total head pressure, which is due to the maximum principle and a ``dimension-reduction" effect arising from the self-similarity.