Radon measures as solutions of the Cauchy problem for evolution equations

Abstract

We consider a standard Navier–Stokes system on the n-D torus $${\mathbb {T}}^n={\mathbb {R}}^n/{\mathbb {Z}}^n$$ , valid when the flow is not very compressible and the temperature does not vary too much. We construct a sequence of approximate solutions that tend to satisfy the equations in a weak sense for arbitrary physical initial conditions. By weak compactness, we obtain Radon measure solutions in density and momentum when the velocity of the flow is finite, as numerically observed in all tests, and the absence of void regions in the flow. We notice that the method also applies without viscosity and complements results on other systems of evolution equations such as the systems of isothermal and isentropic gas flows considered in Colombeau (Z Angew Math Phys 66(5):2575–2599, 2015).