Travelling and rotating solutions to the generalized inviscid surface quasi-geostrophic equation
Authors: Ao Weiwei , Dávila Juan , del Pino Manuel , Musso Monica , Wei Juncheng 
Published: March 2021

For the generalized surface quasi-geostrophic equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout Enlarged left-brace 1st Row 1st Column Blank 2nd Column a m p semicolon partial-differential Subscript t Baseline theta plus u dot nabla theta equals 0 comma in double-struck upper R squared times left-parenthesis 0 comma upper T right-parenthesis comma 2nd Row 1st Column Blank 2nd Column a m p semicolon u equals nabla Superscript up-tack Baseline psi comma psi equals left-parenthesis negative normal upper Delta right-parenthesis Superscript negative s Baseline theta in double-struck upper R squared times left-parenthesis 0 comma upper T right-parenthesis comma EndLayout">

{

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

<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="theta Subscript epsilon Baseline left-parenthesis x comma t right-parenthesis right harpoon with barb up sigma-summation Underscript j equals 1 Overscript k Endscripts m Subscript j Baseline delta left-parenthesis x minus xi Subscript j Baseline left-parenthesis t right-parenthesis right-parenthesis">

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis negative normal upper Delta right-parenthesis Superscript s Baseline upper W equals left-parenthesis upper W minus 1 right-parenthesis Subscript plus Superscript gamma Baseline comma in double-struck upper R squared comma 1 greater-than gamma greater-than StartFraction 1 plus s Over 1 minus s EndFraction">

(
−
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].