Steady-state bifurcation and regularity of nonlinear Burgers equation with mean value constraint

In this paper, we focus on the steady-state bifurcation problem of the nonlinear Burgers equation within a bounded domain, considering both homogeneous Dirichlet boundary conditions and homogeneous Neumann boundary conditions with a mean value constraint. Unlike previous studies, we develop an enhanced turbulence model by incorporating nonlinear higher-order terms (such as $u^2$ and $u^3$) and linear source terms $\lambda u$ into the one-dimensional Burgers equation. Our steady-state bifurcation analysis establishes for the first time how the coupled forward energy cascade and inverse energy transfer mechanisms collectively govern the dynamics of initial flow instability. By combining the spectral theorem for a linear compact operator with the normalized Lyapunov–Schmidt reduction method and the implicit function theorem, we derive the complete criterion for the critical bifurcation condition, the explicit form of the bifurcation solution, and its regularity.