Remarks on weak solutions to the Navier-Stokes equations for 2-D compressible isothermal fluids with spherically symmetric initial data

Abstract

We prove the existence of global weak solutions to the Cauchy problem for the Navier-Stokes equations of 2-dimensional compressible isothermal fluids when ρ 0 and m 0 are spherically symmetric, ρ 0 ∈ L 1 n L M , and m 0 /ρ 0 ∈ L 2 , where ρ 0 and m 0 are the initial density and momentum respectively, L M is the Orlicz space over R 2 with M = M(s) = (1 + s )ln(1 + s)-s. The proof is based on a uniform L 2 ([0, T], L∞ loc (R 2 )) estimate of the velocity in the spherically symmetrical approximate solutions and a compactness lemma which gives a compactness result concerning H n/2 and L M in R n .