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A minicourse on the low Mach number limit. (English) Zbl 1160.35060

This work consists actually of lecture notes of a short course dealing with low Mach number limits of heat-conducting non-isentropic compressible viscous flows in an infinite domain. The continuity equation, Navier-Stokes equations and heat conduction equations are expressed in non-dimensional forms by introducing Mach number, Reynolds number and Peclet number that measure, respectively, effects of compressibility, viscosity and heat conduction. The main interest is to investigate convergence of classical solutions for these equations as a singular limit to solutions for an incompressible flow as the Mach number tends to zero.
The existence and uniqueness of solutions under given initial conditions are studied and various singular limits are obtained by employing the approach of singular perturbations and a myriad of norm relations derived by exploring diverse properties of Sobolev spaces of square integrable functions with integer and non-integer orders. Special emphasis is put on perfect gases in which the internal energy is proportional to the absolute temperature and general gases in which the internal energy is a given function of the absolute temperature. In both cases it is assumed that gases obey Mariotte’s law relating the pressure, the density and the temperature of the gas. It is observed that the order of the entropy plays an important part in the details of the analysis.

MSC:

35Q30 Navier-Stokes equations
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35B25 Singular perturbations in context of PDEs
35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
76N15 Gas dynamics (general theory)
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