Homogeneous steady states for the generalized surface quasi-geostrophic equations
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908733447077888 |
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| author | Abe, Ken Gómez-Serrano, Javier Jeong, In-Jee |
| author_facet | Abe, Ken Gómez-Serrano, Javier Jeong, In-Jee |
| contents | We consider homogeneous (stationary self-similar) solutions to the generalized surface quasi-geostrophic (gSQG) equations parametrized by the constant $0<s<1$, representing the 2D Euler equations ($s=1$), the SQG equations $(s=1/2)$, and stationary equations ($s=0$); namely, solutions whose stream function $ψ$ and advected scalar $ω$ are of the form \begin{align*} ψ=\frac{w(θ)}{r^β},\quad ω=\frac{g(θ)}{r^{β+2s}}, \end{align*} in polar coordinates $(r,θ)$ with parameter $β\in \mathbb{R}$. We classify homogeneous steady states across the full parameter space, and we identify the limiting singular regimes assuming an odd symmetric profile $(w,g)$ with Fourier modes larger than $m_0\geq 1$. Specifically, we show existence of such solutions for $-m_0-2s<β<-2s$ and $0<β<m_0+2$ ($1/2-s<β< m_0+2$ for $0<s<1/2$) and non-existence of such solutions for $-2s\leq β\leq 0$. The main result provides examples of self-similar solutions which belong to critical and supercritical regimes for the local well-posedness of the gSQG equations for $0<s<1$ and the first examples of self-similar solutions for the SQG equations and the more singular equations $0<s\leq 1/2$ in the stationary setting. We also complement our findings with a numerical illustration of the solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_03009 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Homogeneous steady states for the generalized surface quasi-geostrophic equations Abe, Ken Gómez-Serrano, Javier Jeong, In-Jee Analysis of PDEs We consider homogeneous (stationary self-similar) solutions to the generalized surface quasi-geostrophic (gSQG) equations parametrized by the constant $0<s<1$, representing the 2D Euler equations ($s=1$), the SQG equations $(s=1/2)$, and stationary equations ($s=0$); namely, solutions whose stream function $ψ$ and advected scalar $ω$ are of the form \begin{align*} ψ=\frac{w(θ)}{r^β},\quad ω=\frac{g(θ)}{r^{β+2s}}, \end{align*} in polar coordinates $(r,θ)$ with parameter $β\in \mathbb{R}$. We classify homogeneous steady states across the full parameter space, and we identify the limiting singular regimes assuming an odd symmetric profile $(w,g)$ with Fourier modes larger than $m_0\geq 1$. Specifically, we show existence of such solutions for $-m_0-2s<β<-2s$ and $0<β<m_0+2$ ($1/2-s<β< m_0+2$ for $0<s<1/2$) and non-existence of such solutions for $-2s\leq β\leq 0$. The main result provides examples of self-similar solutions which belong to critical and supercritical regimes for the local well-posedness of the gSQG equations for $0<s<1$ and the first examples of self-similar solutions for the SQG equations and the more singular equations $0<s\leq 1/2$ in the stationary setting. We also complement our findings with a numerical illustration of the solutions. |
| title | Homogeneous steady states for the generalized surface quasi-geostrophic equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2510.03009 |