Weiwei Ao, J. Dávila, Manuel del Pino, M. Musso, Juncheng Wei
{"title":"广义无粘面拟地转方程的移动解和旋转解","authors":"Weiwei Ao, J. Dávila, Manuel del Pino, M. Musso, Juncheng Wei","doi":"10.1090/TRAN/8406","DOIUrl":null,"url":null,"abstract":"For the generalized surface quasi-geostrophic equation $$\\left\\{ \\begin{aligned} & \\partial_t \\theta+u\\cdot \\nabla \\theta=0, \\quad \\text{in } \\mathbb{R}^2 \\times (0,T), \\\\ & u=\\nabla^\\perp \\psi, \\quad \\psi = (-\\Delta)^{-s}\\theta \\quad \\text{in } \\mathbb{R}^2 \\times (0,T) , \\end{aligned} \\right. $$ $0<s<1$, we consider for $k\\ge1$ the problem of finding a family of $k$-vortex solutions $\\theta_\\varepsilon(x,t)$ such that as $\\varepsilon\\to 0$ $$ \\theta_\\varepsilon(x,t) \\rightharpoonup \\sum_{j=1}^k m_j\\delta(x-\\xi_j(t)) $$ for suitable trajectories for the vortices $x=\\xi_j(t)$. We find such solutions in the special cases of vortices travelling with constant speed along one axis or rotating with same speed around the origin. In those cases the problem is reduced to a fractional elliptic equation which is treated with singular perturbation methods. A key element in our construction is a proof of the non-degeneracy of the radial ground state for the so-called fractional plasma problem $$(-\\Delta)^sW = (W-1)^\\gamma_+, \\quad \\text{in } \\mathbb{R}^2, \\quad 1<\\gamma < \\frac{1+s}{1-s}$$ whose existence and uniqueness have recently been proven in \\cite{chan_uniqueness_2020}.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2020-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"26","resultStr":"{\"title\":\"Travelling and rotating solutions to the generalized inviscid surface quasi-geostrophic equation\",\"authors\":\"Weiwei Ao, J. Dávila, Manuel del Pino, M. Musso, Juncheng Wei\",\"doi\":\"10.1090/TRAN/8406\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For the generalized surface quasi-geostrophic equation $$\\\\left\\\\{ \\\\begin{aligned} & \\\\partial_t \\\\theta+u\\\\cdot \\\\nabla \\\\theta=0, \\\\quad \\\\text{in } \\\\mathbb{R}^2 \\\\times (0,T), \\\\\\\\ & u=\\\\nabla^\\\\perp \\\\psi, \\\\quad \\\\psi = (-\\\\Delta)^{-s}\\\\theta \\\\quad \\\\text{in } \\\\mathbb{R}^2 \\\\times (0,T) , \\\\end{aligned} \\\\right. $$ $0<s<1$, we consider for $k\\\\ge1$ the problem of finding a family of $k$-vortex solutions $\\\\theta_\\\\varepsilon(x,t)$ such that as $\\\\varepsilon\\\\to 0$ $$ \\\\theta_\\\\varepsilon(x,t) \\\\rightharpoonup \\\\sum_{j=1}^k m_j\\\\delta(x-\\\\xi_j(t)) $$ for suitable trajectories for the vortices $x=\\\\xi_j(t)$. We find such solutions in the special cases of vortices travelling with constant speed along one axis or rotating with same speed around the origin. In those cases the problem is reduced to a fractional elliptic equation which is treated with singular perturbation methods. A key element in our construction is a proof of the non-degeneracy of the radial ground state for the so-called fractional plasma problem $$(-\\\\Delta)^sW = (W-1)^\\\\gamma_+, \\\\quad \\\\text{in } \\\\mathbb{R}^2, \\\\quad 1<\\\\gamma < \\\\frac{1+s}{1-s}$$ whose existence and uniqueness have recently been proven in \\\\cite{chan_uniqueness_2020}.\",\"PeriodicalId\":8445,\"journal\":{\"name\":\"arXiv: Analysis of PDEs\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2020-08-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"26\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv: Analysis of PDEs\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1090/TRAN/8406\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv: Analysis of PDEs","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/TRAN/8406","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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