Critical mean field equations for equilibrium turbulence with sign-changing prescribed functions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910961837801472 |
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| author | Sun, Linlin Zhu, Xiaobao |
| author_facet | Sun, Linlin Zhu, Xiaobao |
| contents | 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 $h_1=h_2\equiv 1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_16414 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Critical mean field equations for equilibrium turbulence with sign-changing prescribed functions Sun, Linlin Zhu, Xiaobao Analysis of PDEs 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 $h_1=h_2\equiv 1$. |
| title | Critical mean field equations for equilibrium turbulence with sign-changing prescribed functions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2505.16414 |