Both turbulence and shock formation in inviscid flows are prone to high wave number mode generations. This continuous generation of high wavemodes results in an energy cascade to ever smaller scales in turbulence and/or creation of shocks in compressible flows. This high wavenumber problem is often remedied by the addition of a viscous term in both compressible and incompressible flows. A regularization technique for the Burgers equation (Norgard and Mohseni J. Phys. A, 2008, and SIAM J. Multiscale Modeling and Simulation, 2009) was recently reported. The proposed inviscid regularization was extended to one-dimensional compressible Euler equations in 2009 (Norgard and Mohseni, submitted to SIAM J. Multiscale Modeling and Simulation, 2009). This investigation presents a formal derivation of these equations from basic principles. Our previous results are extended to multidimensional compressible and incompressible Euler equations. We define an observable divergence based on fluxes calculated from observable quantities at a desired scale. An observable divergence theorem is then proved and applied in the derivation of the regularized equations. It is shown that the derived equations reduce to inviscid Leray flow model in the limit of incompressibility. It is expected that this technique simultaneously regularize shocks and turbulence for compressible and incompressible flows.