This article deals with the vortex patch problem of the two-dimensional Euler–Boussinesq system. This system couples the incompressible Euler equation for the velocity and a transport diffusion for the temperature with rough initial data of the Yudovich type.

1.
H.
Abidi
and
T.
Hmidi
, “
On the global well-posedness for Boussinesq system
,”
J. Differ. Equations
233
(
1
),
199
220
(
2007
).
2.
H.
Bahouri
,
J.-Y.
Chemin
, and
R.
Danchin
,
Fourier Analysis and Nonlinear Partial Differential Equations
(
Springer-Verlage Berlin Heidelberg
,
2011
).
3.
J. T.
Beale
,
T.
Kato
, and
A.
Majda
, “
Remarks on the breakdom of smooth solutions for the 3-D Euler equations
,”
Commun. Math. Phys.
94
,
61
66
(
1984
).
4.
J. M.
Bony
, “
Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires
,”
Ann. Sci. Ec. Norm. Super.
14
,
209
246
(
1981
).
5.
D.
Chae
, “
Global regularity for the 2D Boussinesq equations with partial viscous terms
,”
Adv. Math.
203
,
497
513
(
2006
).
6.
J.-Y.
Chemin
,
Perfect Incompressible Fluids
(
Oxford University Press
,
1998
).
7.
D.
Danchin
and
M.
Paicu
, “
Le théoreme de Leray et le théoreme de Fujita-Kato pour le system de Boussinesq partiellement visqueux
,”
Bull. Soc. Math. France
136
,
261
309
(
2008
).
8.
D.
Danchin
and
M.
Paicu
, “
Global well-posedness issues for the inviscid Boussinesq system with Yudovich’s type data
,”
Commun. Math. Phys.
290
(
1
),
1
14
(
2009
).
9.
Z.
Hassainia
and
T.
Hmidi
, “
On the inviscid Boussinesq system with rough initial data
,”
J. Math. Anal. Appl.
430
,
777
809
(
2015
).
10.
X.
Liu
,
M.
Wang
, and
Z.
Zhang
, “
Local well-posedness and blow-up criterion of the Boussinesq equations in critical Besov spaces
,”
J. Math. Fluid Mech.
12
,
280
3292
(
2010
).
11.
C.
Miao
and
Z.
Zhang
, “
On the global well-posedness for the Boussinesq systemwith horizontal dissipation
,”
Commun. Math. Phys.
321
(
1
),
33
67
(
2013
).
12.
H.
Meddour
and
M.
Zerguine
, “
Optimal rate of convergence in stratified Boussinesq system
,”
J. Dynamics Partial Diff. Eq.
15
(
2017
).
13.
T.
Hmidi
and
M.
Zerguine
, “
Vortex patch problem for stratified Euler equations
,”
Commun. Math. Sci.
12
(
8
),
1541
1563
(
2014
).
14.
T.
Kato
and
G.
Ponce
, “
Well posedness of the Euler and Navier-Stokes equations in the Lebesque spaces Lsp(R2).
,”
Rev. Mat. Iberoam.
2
(
1-2
),
73
88
(
1986
).
15.
H. C.
Pak
and
Y. J.
Park
, “
Existence of solution for the Euler equations in a critical Besov space B,11(Rn).
,”
Commun. Partial Differ. Equations
29
(
7-8
),
1149
1166
(
2004
).
16.
H.
Von Helmholtz
, “
Uber integral der hydrodynamischen gleichungen, welche der wirbelbewegung entsprechen
,”
J. Reine Angew. Math.
55
,
25
55
(
1858
).
17.
M.
Vishik
, “
Hydrodynamics in Besov spaces
,”
Arch. Ration. Mech. Anal.
145
,
197
214
(
1998
).
18.
W.
Wolibner
, “
Un théoreme sur l’existence du mouvement plan d’un fluid parfait, homogègne, incompressible, pendant un temps infiniment long
,”
Math. Z.
37
(
1
),
698
726
(
1933
).
19.
M.
Zerguine
, “
The regular vortex patch for stratified Euler equations with critical fractional dissipation
,”
J. Evol. Equations
15
,
667
698
(
2015
).
20.
Y.
Zhou
, “
Local well-posedness for the incompressible Euler equations in the critical Besov spaces
,”
Ann. Inst. Fourier
54
(
3
),
773
786
(
2004
).
You do not currently have access to this content.