2024/12/10 by Bang, Jeaheang, Gui, Changfeng, Liu, Hao +2
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2412.07283
This paper is concerned with self-similar solutions of the steady Navier-Stokes system in a two-dimensional sector with the no-slip boundary condition. We give necessary and sufficient conditions in terms of the angle of the sector and the flux to guarantee the existence of self-similar solutions of a given type. We also investigate the uniqueness and non-uniqueness of flows with a given type, which not only give rigorous justifications for some statements in \citeRosenhead40 but also show that some numerical computations in \citeRosenhead40 may not be precise. The non-uniqueness result is a new phenomenon for these flows. As a consequence of the classification of self-similar solutions in the half-space, we characterize the leading order term of the steady Navier-Stokes system in an aperture domain when the flux is small. The main approach is to study the ODE system governing self-similar solutions, where the detailed properties of both complete and incomplete elliptic functions have been investigated.