Critical mean field equations for equilibrium turbulence with sign-changing prescribed functions
Linlin Sun, Xiaobao Zhu
TL;DR
The paper analyzes the critical mean field equation for equilibrium turbulence on a compact surface with unit area, allowing sign-changing prescribed functions $h_1,h_2$ and critical parameters $ρ_1=8π$, $ρ_2∈(0,8π]$ or $ρ_1=ρ_2=8π$. Using a refined Brezis–Merle-type analysis and a perturbation-variational strategy, it establishes sufficient Ding–Jost–Li–Wang-type conditions for existence in the critical regime, extending Zhou’s results to the sign-changing setting. A detailed blow-up analysis, neck estimates, and sharp lower bounds for the energy are combined with carefully constructed test functions to rule out blow-up under the stated conditions. Consequently, minimizers exist for the critical functionals and solve the Euler–Lagrange equations, shedding light on equilibrium turbulence models with vortices of mixed sign and enabling extensions to non-constant prescribed data.
Abstract
Let $(M,g)$ be a compact Riemann surface with unit area. We investigate the mean field equation for equilibrium turbulence: \begin{align} \begin{cases} -Δu = ρ_1\left(\frac{h_1e^{u}}{\int_Mh_1e^udv_g}-1\right) - ρ_2\left(\frac{h_2e^{-u}}{\int_Mh_2e^{-u}dv_g}-1\right), \\ \int_Mudv_g=0, \end{cases} \end{align} where $ρ_1=8π$ and $ρ_2\in(0,8π]$ are parameters, and $h_1, h_2$ are smooth functions on $M$ that are positive somewhere. By employing a refined Brezis-Merle type analysis, we establish sufficient conditions of Ding-Jost-Li-Wang type for the existence of solutions to this equation in critical cases, particularly when $h_1$ and $h_2$ may change signs. Our results extend Zhou's existence theorems (Nonlinear Anal. 69 (2008), no.~8, 2541--2552) for the case $h_1=h_2\equiv 1$.