TY - JOUR

T1 - Travelling and rotating solutions to the generalized inviscid surface quasi-geostrophic equation

AU - Ao, Weiwei

AU - Davila Bonczos, Juan

AU - Del Pino, Manuel

AU - Musso, Monica

AU - Wei, Juncheng

PY - 2021/6/9

Y1 - 2021/6/9

N2 - For the generalized surface quasi-geostrophic equation$$\left\{\begin{aligned}& \partial_t \theta+u\cdot \nabla \theta=0,\quad \text{in } \R^2 \times (0,T),\\& u=\nabla^\perp \psi,\quad\psi = (-\Delta)^{-s}\theta\quad \text{in } \R^2 \times (0,T) ,\end{aligned}\right.$$$0<s<1$, we consider for $k\ge1$ the problem of finding a family of$k$-vortex solutions $\theta_\ve(x,t)$ such that as $\ve\to 0$$$\theta_\ve(x,t) \rightharpoonup \sum_{j=1}^k m_j\delta(x-\xi_j(t))$$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$$ (-\Delta)^sW = (W-1)^\gamma_+ , \quad \text{in } \R^2, \quad 1<\gamma < \frac{1+s}{1-s}$$whose existence and uniqueness have recently been proven in \cite{chan_uniqueness_2020}.

AB - For the generalized surface quasi-geostrophic equation$$\left\{\begin{aligned}& \partial_t \theta+u\cdot \nabla \theta=0,\quad \text{in } \R^2 \times (0,T),\\& u=\nabla^\perp \psi,\quad\psi = (-\Delta)^{-s}\theta\quad \text{in } \R^2 \times (0,T) ,\end{aligned}\right.$$$0<s<1$, we consider for $k\ge1$ the problem of finding a family of$k$-vortex solutions $\theta_\ve(x,t)$ such that as $\ve\to 0$$$\theta_\ve(x,t) \rightharpoonup \sum_{j=1}^k m_j\delta(x-\xi_j(t))$$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$$ (-\Delta)^sW = (W-1)^\gamma_+ , \quad \text{in } \R^2, \quad 1<\gamma < \frac{1+s}{1-s}$$whose existence and uniqueness have recently been proven in \cite{chan_uniqueness_2020}.

U2 - 10.1090/tran/8406

DO - 10.1090/tran/8406

M3 - Article

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

ER -