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The incompressible limit in $$L^p$$ type critical spaces. (English) Zbl 1354.35096
Summary: This paper aims at justifying the low Mach number convergence to the incompressible Navier-Stokes equations for viscous compressible flows in the ill-prepared data case. The fluid domain is either the whole space, or the torus. A number of works have been dedicated to this classical issue, all of them being, to our knowledge, related to $$L^2$$ spaces and to energy type arguments. In the present paper, we investigate the low Mach number convergence in the $$L^p$$ type critical regularity framework. More precisely, in the barotropic case, the divergence-free part of the initial velocity field just has to be bounded in the critical Besov space $$\dot{B}^{d/p-1}_{p,r}\cap\dot{B}^{-1}_{\infty,1}$$ for some suitable $$(p,r)\in [2,4]\times [1,+\infty].$$ We still require $$L^2$$ type bounds on the low frequencies of the potential part of the velocity and on the density, though, an assumption which seems to be unavoidable in the ill-prepared data framework, because of acoustic waves. In the last part of the paper, our results are extended to the full Navier-Stokes system for heat conducting fluids.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 76D05 Navier-Stokes equations for incompressible viscous fluids 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 76Q05 Hydro- and aero-acoustics 42B25 Maximal functions, Littlewood-Paley theory 35B40 Asymptotic behavior of solutions to PDEs 35B45 A priori estimates in context of PDEs
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