Steady-state bifurcation and local stability of 2D Burgers’ equations with external resistance terms

We investigate the steady-state bifurcation and local stability of the two-dimensional Burgers’ equation with external resistance terms λu under Dirichlet and Neumann boundary conditions, respectively. First, we prove that the system undergoes bifurcation at the first simple eigenvalue under Dirichlet boundary condition and at the first double eigenvalue under Neumann boundary condition. Furthermore, we establish specific criteria for the occurrence of subcritical and supercritical bifurcations in the system under different homogeneous boundary conditions, analyze the regularity and local stability of the bifurcation solutions under Neumann boundary condition. To validate the theoretical results, we apply numerical analysis to a problem analogous to Rayleigh-Bénard convection between two parallel plates. On the one hand, the solution structures from \({\bar{u}}_{1}\) to \({\bar{u}}_{4}\) indicate flow patterns corresponding to the regular convection rolls; in contrast, \({\bar{u}}_{5}\) to \({\bar{u}}_{8}\) represent complex flow patterns coupled in the x - and y -directions corresponding to the cross rolls or spiral structures in the convection problem. On the other hand, for \(0<\gamma <\frac{5c\pi }{2\sqrt{21-3\pi \sqrt{2}}}\) , the system exhibits a subcritical bifurcation, implying that complex flow patterns emerge before the parameter λ reaches the critical threshold \(\lambda _{0}=\nu \) ; for \(\gamma >\frac{5c\pi }{2\sqrt{21-3\pi \sqrt{2}}}\) , a supercritical bifurcation occurs, and stable flow patterns emerge only when λ attains \(\lambda _{0}=\nu \) .