On the singular incompressible limit of inviscid compressible fluids
The author studies the convergence properties of solutions to the compressible inviscid Euler equations when the Mach number \(\lambda^{-1}\) (which measures the compressibility) goes to zero. It is assumed that the initial velocity is bounded in \(H^3\) and the initial pressure decays like \(\lambda^{-1}\) in \(H^3\). The result is then that the solution exists in \(H^3\) for a time interval \(T\) independent of \(\lambda\), and that one has weak compactness in the limit to the incompressible Euler equations (or strong convergence if one assumes strong convergence of the initial data). The main tool employed is the energy method for symmetric hyperbolic systems. The time regularity deteriorates as \(\lambda\to \infty\) (every time derivative loses a \(\lambda\)) but this turns out not to affect the uniformity of the energy estimates in \(\lambda\).
scientific article; zbMATH DE number 1484688