论文标题
向广义的无粘性表面准整形方程式旅行和旋转解决方案
Travelling and rotating solutions to the generalized inviscid surface quasi-geostrophic equation
论文作者
论文摘要
对于广义的表面准晶格方程$$ \ weft \ {\ oken {Aligned}＆\partial_tθ+u \ cdot \ cdot \nablaθ= 0，\ quad \ quad \ text {in} （-Δ）^{ - s}θ\ Quad \ text {in} \ Mathbb {r}^2 \ times（0，t），\ end {aligned} \ right。 $$ $ 0 <s <1 $，我们考虑$ k \ ge1 $找到$ k $ - 沃尔特克斯解决方案的家庭的问题$θ_\ varepsilon（x，t）$，以至于$ \ varepsilon \ to 0 $ \ varepsilon \ to 0 $ $ $ $ $ $ $ thepsilon（x，x，x，x，x，x，x，x，x，x，t） M_Jδ（X-ξ_j（t））$$用于涡流的合适轨迹$ x =ξ_j（t）$。在特殊情况下，我们发现了这种解决方案，沿着一个轴以恒定速度行驶或以相同速度旋转的旋转速度行驶。在这种情况下，问题将减少为分数椭圆方程，该方程用奇异的扰动方法处理。我们构造中的一个关键要素是证明径向基态的非变分状况，即所谓的分数等离子体问题$$（ - δ）^sw =（w-1）^γ_+，\ quad \ quad \ text {in} \ mathbb {in} \ mathbb {r}已在\ cite {chan_iniqueness_2020}中得到证明。
For the generalized surface quasi-geostrophic equation $$\left\{ \begin{aligned} & \partial_t θ+u\cdot \nabla θ=0, \quad \text{in } \mathbb{R}^2 \times (0,T), \\ & u=\nabla^\perp ψ, \quad ψ= (-Δ)^{-s}θ\quad \text{in } \mathbb{R}^2 \times (0,T) , \end{aligned} \right. $$ $0<s<1$, we consider for $k\ge1$ the problem of finding a family of $k$-vortex solutions $θ_\varepsilon(x,t)$ such that as $\varepsilon\to 0$ $$ θ_\varepsilon(x,t) \rightharpoonup \sum_{j=1}^k m_jδ(x-ξ_j(t)) $$ for suitable trajectories for the vortices $x=ξ_j(t)$. 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 $$(-Δ)^sW = (W-1)^γ_+, \quad \text{in } \mathbb{R}^2, \quad 1<γ< \frac{1+s}{1-s}$$ whose existence and uniqueness have recently been proven in \cite{chan_uniqueness_2020}.
