Ohtsuka, Hiroshi; Suzuki, Takashi Mean field equation for the equilibrium turbulence and a related functional inequality. (English) Zbl 1109.26014 Adv. Differ. Equ. 11, No. 3, 281-304 (2006). The paper deals with the following mean field equation for equilibrium turbulence on a two-dimensional compact orientable Riemannian manifold \((M,g)\) without boundary: \[ -\Delta_g v=\lambda_1 \Big(\frac{e^v}{\int_M e^v\;dv_g}-\frac{1}{| M| }\Big)- \lambda_2 \Big(\frac{e^v}{\int_M e^v\;dv_g}-\frac{1}{| M| }\Big),\quad \int_M v\;dv_g=0, \] where \(\Delta_g, dv_g\) and \(| M| \) are the Laplace-Beltrami operator, the volume form, and the volume of \(M\), respectively, and \(\lambda_1, \lambda_2\) are nonnegative constants. The authors develop blow-up analysis and establish a functional inequality of Trudinger-Moser type. Reviewer: Gian Luigi Forti (Milano) Cited in 2 ReviewsCited in 26 Documents MSC: 26D10 Inequalities involving derivatives and differential and integral operators 35B40 Asymptotic behavior of solutions to PDEs 35J20 Variational methods for second-order elliptic equations 35J60 Nonlinear elliptic equations 35Q35 PDEs in connection with fluid mechanics 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 49K20 Optimality conditions for problems involving partial differential equations 76F99 Turbulence Keywords:Trudinger-Moser inequality PDFBibTeX XMLCite \textit{H. Ohtsuka} and \textit{T. Suzuki}, Adv. Differ. Equ. 11, No. 3, 281--304 (2006; Zbl 1109.26014)