Incompressible limit for the 2D isentropic Euler system with critical initial data

2012/03/17 by Taoufik Hmidi, Hmidi, Taoufik, Samira Sulaiman +1
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.1203.3871

22 pages

arxiv created 2012/03/17 · arxiv updated 2012/03/20

Abstract

We study in this paper the low Mach number limit for the 2d isentropic Euler system with ill-prepared initial data belonging to the critical Besov space B_2,1². By combining Strichartz estimates with the special structure of the vorticity we prove that the lifespan goes to infinity as the Mach number goes to zero. We also prove the strong convergence in the space of the initial data B_2,1² of the incompressible parts to the solution of the incompressible Euler system. There are at least two main difficulties: the first one concerns the Beale-Kato-Majda criterion which is not known to work for rough regularities. However, the second one is related to the critical aspect of the quantity \Vert(\textnormaldiv v_ε,∇ c_ε)\Vert_L_T¹L^∞ which has the scale of B_2,1² in the space variable.

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