Stability and bifurcation of Navier-Stokes equations in an annular domain with mixed boundary conditions
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912532324679680 |
|---|---|
| author | Hou, Zhibo Li, Liang Wang, Quan |
| author_facet | Hou, Zhibo Li, Liang Wang, Quan |
| contents | We study the existence and stability of non-trivial steady-state solutions to the two-dimensional incompressible Navier-Stokes equations in an annular domain $Ω= B(0,b) \setminus \overline{B(0,a)}$ with radii $b>a>0$.The outer boundary $\partial B(0, b)$ is subject to the free condition, while the inner boundary $\partial B(0, a)$ obeys a Navier-slip condition with effective slip length $α> 0$. Our main results are fourfold. First, we establish global-in-time strong solutions and derive a sharp energy estimate that underpins the subsequent nonlinear instability analysis. Second, for $α> 0$, we compute an explicit critical viscosity $μ_c:=μ_c(α, a,b,μ)$ that separates qualitatively different dynamical behaviours. Third, we precisely characterize the stability properties of the trivial solution in three distinct regimes. The zero solution is globally asymptotically stable in $H^2$ if $μ> μ_c$. If $μ= μ_c$, we prove an alternative theorem that completely describes the local dynamics near the trivial state. If $μ< μ_c$, the trivial solution is nonlinearly unstable in every $L^p (p \geq 1)$.Finally, we demonstrate that for $μ< μ_c$, the system undergoes a pitchfork bifurcation that generates an infinite family of non-trivial steady states. For generic choices of $(α,b)$, this bifurcation is supercritical; a measurable subset of parameter space yields subcritical transitions. Notably, all bifurcating solutions share the same topological pattern-a single row of counter-rotating vortices-despite their mathematical non-uniqueness. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_07694 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stability and bifurcation of Navier-Stokes equations in an annular domain with mixed boundary conditions Hou, Zhibo Li, Liang Wang, Quan Analysis of PDEs We study the existence and stability of non-trivial steady-state solutions to the two-dimensional incompressible Navier-Stokes equations in an annular domain $Ω= B(0,b) \setminus \overline{B(0,a)}$ with radii $b>a>0$.The outer boundary $\partial B(0, b)$ is subject to the free condition, while the inner boundary $\partial B(0, a)$ obeys a Navier-slip condition with effective slip length $α> 0$. Our main results are fourfold. First, we establish global-in-time strong solutions and derive a sharp energy estimate that underpins the subsequent nonlinear instability analysis. Second, for $α> 0$, we compute an explicit critical viscosity $μ_c:=μ_c(α, a,b,μ)$ that separates qualitatively different dynamical behaviours. Third, we precisely characterize the stability properties of the trivial solution in three distinct regimes. The zero solution is globally asymptotically stable in $H^2$ if $μ> μ_c$. If $μ= μ_c$, we prove an alternative theorem that completely describes the local dynamics near the trivial state. If $μ< μ_c$, the trivial solution is nonlinearly unstable in every $L^p (p \geq 1)$.Finally, we demonstrate that for $μ< μ_c$, the system undergoes a pitchfork bifurcation that generates an infinite family of non-trivial steady states. For generic choices of $(α,b)$, this bifurcation is supercritical; a measurable subset of parameter space yields subcritical transitions. Notably, all bifurcating solutions share the same topological pattern-a single row of counter-rotating vortices-despite their mathematical non-uniqueness. |
| title | Stability and bifurcation of Navier-Stokes equations in an annular domain with mixed boundary conditions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2508.07694 |