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

Abstract

For the generalized surface quasi-geostrophic equation { a m p ; ∂ t θ + u ⋅ ∇ θ = 0 , in R 2 × ( 0 , T ) , a m p ; u = ∇ ⊥ ψ , ψ = ( − Δ ) − s θ in R 2 × ( 0 , T ) , \begin{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 . \end{equation*} 0 > s > 1 0>s>1 , we consider for k ≥ 1 k\ge 1 the problem of finding a family of k k -vortex solutions θ ε ( x , t ) \theta _\varepsilon (x,t) such that as ε → 0 \varepsilon \to 0 θ ε ( x , t ) ⇀ ∑ j = 1 k m j δ ( x − ξ j ( t ) ) \begin{equation*} \theta _\varepsilon (x,t) \rightharpoonup \sum _{j=1}^k m_j\delta (x-\xi _j(t)) \end{equation*} for suitable trajectories for the vortices x = ξ j ( t ) 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 ( − Δ ) s W = ( W − 1 ) + γ , in R 2 , 1 > γ > 1 + s 1 − s \begin{equation*} (-\Delta )^sW = (W-1)^\gamma _+ , \quad \text {in } \mathbb {R}^2, \quad 1>\gamma > \frac {1+s}{1-s} \end{equation*} whose existence and uniqueness have recently been proven in Chan, del Mar González, Huang, Mainini, and Volzone [Calc. Var. Partial Differential Equations 59 (2020), p. 42].

Vortex collapses for the Euler and Quasi-Geostrophic models

“…Corotating patch solutions with two patches [17] and N patches forming an N -fold symmetrical pattern [11] were recently exhibited with bifurcation argument. The C 1 analogous of these solutions has also been investigated independently recently in [1]. Another recent independent result [14] also build corotating solutions with N patches using variational argument.…”

“…The main interest of this new system is that Theorem 2.3 can be reformulated using only the differences y ij . 1) . Assume that for all i ∈ {1 .…”

KAM theory for active scalar equations

“…However, certain blow-up scenarios have been ruled out in [30,31] and solutions with exponential growth (not excluding blow up in finite time) are shown to exist in [57]. On the other hand, examples of a non-trivial global in time smooth solutions have been recently constructed in [2,25,50] with different approaches related to bifurcation theory/assisted computer proof and variational principle. In the setting of weak solutions, it is known that for Euler equations Yudovich solutions exist globally in time and they are unique, see [85].…”

On the existence of vortex-wave systems to inviscid gSQG equation

“…The second equation expresses the fact that each point vortex x i (t) moves by the velocity 'generated' by the background vorticity (the term ∇ ⊥ G s * θ(x i , t)) and the other l − 1 vortices (the term j =i κ j ∇ ⊥ G s (x i , x j ) ). If κ i = 0, i = 1, • • • , l, then the system reduces to the vorticity form of the modified or generalized surface quasi-geostrophic equation, which has been extensively studied; see [1,12] for example. If the background vorticity vanishes, then the system becomes the Kirchhoff-Routh equation, which is a model describing the motion of l concentrated vortices: see [15,17,23] for example.…”

“…Using the modified finite-dimensional reduction method, we are able to construct solutions concentrating at a given non-degenerate critical point of the Kirchhoff-Routh function, even if this is a saddle point. Moreover, one can intuitively see the concrete form of the solution, which is helpful for analyzing the nature of the solution; see [1] for example. 1.2.…”