A compactness framework is formulated for the incompressible limit of approximate solutions with weak uniform bounds with respect to the adiabatic exponent for the steady Euler equations for compressible fluids in any dimension. One of our main observations is that the compactness can be achieved by using only natural weak estimates for the mass conservation and the vorticity. Another observation is that the incompressibility of the limit for the homentropic Euler flow is directly from the continuity equation, while the incompressibility of the limit for the full Euler flow is from a combination of all the Euler equations. As direct applications of the compactness framework, we establish two incompressible limit theorems for multidimensional steady Euler flows through infinitely long nozzles, which lead to two new existence theorems for the corresponding problems for multidimensional steady incompressible Euler equations.

In this paper, for compressible Euler equations in multiple space dimensions, we prove the breakdown of classical solutions with a large class of initial data by tracking the propagation of radially symmetric expanding wave including compression. The singularity formation considered in this paper is corresponding to the finite time shock formation. We also provide some new global sup-norm estimates on velocity and density functions for classical solutions and construct the corresponding classical solutions. All results in this paper have no restriction on the size of solutions and hence are large data results.

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