TY - JOUR
T1 - The low Mach number limit for the isentropic Euler system with axisymmetric initial data
AU - Hmidi, Taoufik
PY - 2013/4
Y1 - 2013/4
N2 - This paper is devoted to the study of the low Mach number limit for the isentropic Euler system with axisymmetric initial data without swirl. In the first part of the paper we analyze the problem corresponding to the subcritical regularities, that is Hs with s> 5/2. Taking advantage of the Strichartz estimates and using the special structure of the vorticity we show that the lifespan Tε of the solutions is bounded below by 1/ε, where ε denotes the Mach number. Moreover, we prove that the incompressible parts converge to the solution of the incompressible Euler system when the parameter ε goes to zero. In the second part of the paper we address the same problem but for the Besov critical regularity B 5/22,1. This case turns out to be more subtle because of at least two features. The first one is related to the Beale-Kato-Majda criterion which is not known to be valid for rough regularities. The second one concerns the critical aspect of the Strichartz estimate L1 TL∞ for the acoustic parts (Δ-1div v ε,cε): it scales in the space variables like the space of the initial data.
AB - This paper is devoted to the study of the low Mach number limit for the isentropic Euler system with axisymmetric initial data without swirl. In the first part of the paper we analyze the problem corresponding to the subcritical regularities, that is Hs with s> 5/2. Taking advantage of the Strichartz estimates and using the special structure of the vorticity we show that the lifespan Tε of the solutions is bounded below by 1/ε, where ε denotes the Mach number. Moreover, we prove that the incompressible parts converge to the solution of the incompressible Euler system when the parameter ε goes to zero. In the second part of the paper we address the same problem but for the Besov critical regularity B 5/22,1. This case turns out to be more subtle because of at least two features. The first one is related to the Beale-Kato-Majda criterion which is not known to be valid for rough regularities. The second one concerns the critical aspect of the Strichartz estimate L1 TL∞ for the acoustic parts (Δ-1div v ε,cε): it scales in the space variables like the space of the initial data.
KW - axisymmetrix flows
KW - critical Besov spaces
KW - incompressible limit
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U2 - 10.1017/S1474748012000746
DO - 10.1017/S1474748012000746
M3 - Article
AN - SCOPUS:84874383486
SN - 1474-7480
VL - 12
SP - 335
EP - 389
JO - Journal of the Institute of Mathematics of Jussieu
JF - Journal of the Institute of Mathematics of Jussieu
IS - 2
ER -