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We consider the free boundary problem for the Euler--Fourier system thatdescribes the motion of compressible, inviscid and heat-conducting fluids. Theeffect of surface tension is neglected and there is no heat flux across thefree boundary. We prove the local well-posedness of the problem in Lagrangiancoordinates under the Taylor sign condition. The solution is produced as thelimit of solutions to a sequence of tangentially-smoothed approximate problems,where the so-called corrector is crucially introduced beforehand in thetemperature equation so that the approximate initial data satisfying thecorresponding compatibility conditions can be constructed. To overcome thestrong coupling effect between the Euler part and the Fourier part in solvingthe linearized approximate problem, the temperature equation is furtherregularized by a pseudo-parabolic equation. Moreover, we prove the uniformestimates with respect to the Mach number of the solutions to the free-boundaryEuler--Fourier system with large temperature variations, which allow us tojustify the convergence towards the free-boundary inviscid low Mach numberlimit system by the strong compactness argument.

Well-posedness and Low Mach Number Limit of the Free Boundary Problem for the Euler--Fourier System