Let (M,g) be a compact Riemann surface with unit area. We investigate the mean field equation for equilibrium turbulence <Equation ID="Equ1"> <EquationNumber>(0.1)</EquationNumber> <EquationSource Format="TEX">$\left\{\begin{aligned}&-\Delta u=\rho_1\left(\frac{h_1 \rm{e}^u}{\int_M h_1 \rm{e}^u \, dv_g}-1\right)-\rho_2\left(\frac{h_2 \rm{e}^{-u}}{\int_M h_2 \rm{e}^{-u} \, dv_g}-1\right),\\&\int_M u \, dv_g = 0.\end{aligned}\right.$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mo>{</mo> <mtable> <mtr> <mtd> <mspace width="thinmathspace" /> </mtd> <mtd> <mi /> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>=</mo> <msub> <mi>ρ</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>h</mi> <mn>1</mn> </msub> <msup> <mrow> <mi mathvariant="normal">e</mi> </mrow> <mi>u</mi> </msup> </mrow> <mrow> <msub> <mo>∫</mo> <mi>M</mi> </msub> <msub> <mi>h</mi> <mn>1</mn> </msub> <msup> <mrow> <mi mathvariant="normal">e</mi> </mrow> <mi>u</mi> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msub> <mi>v</mi> <mi>g</mi> </msub> </mrow> </mfrac> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>−</mo> <msub> <mi>ρ</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>h</mi> <mn>2</mn> </msub> <msup> <mrow> <mi mathvariant="normal">e</mi> </mrow> <mrow> <mo>−</mo> <mi>u</mi> </mrow> </msup> </mrow> <mrow> <msub> <mo>∫</mo> <mi>M</mi> </msub> <msub> <mi>h</mi> <mn>2</mn> </msub> <msup> <mrow> <mi mathvariant="normal">e</mi> </mrow> <mrow> <mo>−</mo> <mi>u</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msub> <mi>v</mi> <mi>g</mi> </msub> </mrow> </mfrac> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mspace width="thinmathspace" /> </mtd> <mtd> <mi /> <msub> <mo>∫</mo> <mi>M</mi> </msub> <mi>u</mi> <mspace width="thinmathspace" /> <mi>d</mi> <msub> <mi>v</mi> <mi>g</mi> </msub> <mo>=</mo> <mn>0.</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" /> </mrow> </math></EquationSource> </Equation> where ρ1 = 8π and ρ2 ∈ (0, 8π] are parameters, and h1 and h2 are smooth functions on M that are positive somewhere. By employing refined Brezis-Merle type analysis, we establish sufficient conditions of Ding-Jost-Li-Wang type for the existence of solutions to (<InternalRef RefID="Equ1">0.1</InternalRef>) in critical cases, particularly when h1 and h2 may change signs. Our results extend Zhou’s existence theorems (Zhou (2008)) for the case h1 = h2 ≡ 1.

Critical mean field equations for equilibrium turbulence with sign-changing prescribed functions

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发表日期：

2026年5月26日

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Linlin Sun,
Xiaobao Zhu

摘要

Let (M,g) be a compact Riemann surface with unit area. We investigate the mean field equation for equilibrium turbulence (0.1) $\left\{\begin{aligned}&-\Delta u=\rho_1\left(\frac{h_1 \rm{e}^u}{\int_M h_1 \rm{e}^u \, dv_g}-1\right)-\rho_2\left(\frac{h_2 \rm{e}^{-u}}{\int_M h_2 \rm{e}^{-u} \, dv_g}-1\right),\\&\int_M u \, dv_g = 0.\end{aligned}\right.$ ${\begin{matrix} - Δ u = ρ_{1} (\frac{h_{1} e^{u}}{\int_{M} h_{1} e^{u} d v_{g}} - 1) - ρ_{2} (\frac{h_{2} e^{- u}}{\int_{M} h_{2} e^{- u} d v_{g}} - 1), \\ \int_{M} u d v_{g} = 0. \end{matrix}$ where ρ₁ = 8π and ρ₂ ∈ (0, 8π] are parameters, and h₁ and h₂ are smooth functions on M that are positive somewhere. By employing refined Brezis-Merle type analysis, we establish sufficient conditions of Ding-Jost-Li-Wang type for the existence of solutions to (0.1) in critical cases, particularly when h₁ and h₂ may change signs. Our results extend Zhou’s existence theorems (Zhou (2008)) for the case h₁ = h₂ ≡ 1.