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.
REFERENCES
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.
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 .
,” 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 .
,” 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
).© 2019 Author(s).
2019
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