Global existence of the radially symmetric solutions of the Navier–Stokes equations for the isentropic compressible fluids

Abstract

We study the isentropic compressible Navier–Stokes equations with radially symmetric data in an annular domain. We first prove the global existence and regularity results on the radially symmetric weak solutions with non‐negative bounded densities. Then we prove the global existence of radially symmetric strong solutions when the initial data ρ0, u0 satisfy the compatibility condition for some radially symmetric g ∈ L2. The initial density ρ0 needs not be positive. We also prove some uniqueness results on the strong solutions. Copyright © 2004 John Wiley & Sons, Ltd.

On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities

“…He showed that there is no global classical solution to the Cauchy problem for the isentropic compressible Navier-Stokes equations with compactly supported initial density and velocity. On the other hand, Choe and the second author [5] obtained a global existence result on radially symmetric strong solutions of the isentropic compressible Navier-Stokes equations in bounded and unbounded annular domains. Hence it is very likely that the methods in this paper and [5] can be combined to prove the global existence of radially symmetric classical solutions with nonnegative densities.…”

“…On the other hand, Choe and the second author [5] obtained a global existence result on radially symmetric strong solutions of the isentropic compressible Navier-Stokes equations in bounded and unbounded annular domains. Hence it is very likely that the methods in this paper and [5] can be combined to prove the global existence of radially symmetric classical solutions with nonnegative densities. This issue will be studied in a separated paper.…”

Global classical large solutions to 1D compressible Navier–Stokes equations with density-dependent viscosity and vacuum

“…On the existence and regularity of weak solutions with density connecting to vacuum continuously in 1D, please refer to [20]. During the pass two decades, Salvi, Choe, Kim and Jiang et al made great progress towards the local or global existence of strong (classical) solutions with vacuum, see [27,2,4,5,7,3]. Particularly, Choe and Kim in [5] showed that the radially symmetric strong solutions exist globally in time for γ 2 and N 2.…”

Existence and uniqueness of solutions for a class of non-Newtonian fluids with singularity and vacuum

“…They showed in [12] that if Ω is a bounded domain in R 3 , π = π(·) ∈ C 2 [0, ∞), ρ 0 ∈ H 2 (Ω), u 0 ∈ H 1 0 (Ω) ∩ H 2 (Ω) and the compatibility condition −μ u 0 + ∇π(ρ 0 ) = ρ 1/2 0 g, for some g ∈ L 2 (Ω), (1.3) is satisfied, then there exists a unique local strong solution (ρ, u) to the initial boundary value problem. Independently of their work, H. Choe and H. Kim [3,13], Y. Cho, H. Choe and H. Kim [4] proved a similar existence result when Ω is either a bounded domain or the whole space and the compatibility condition (1.3) is satisfied. This is somewhat surprising because in the context of the compressible Navier-Stokes or Euler equations, there have been many works concerning the existence of solution with compactly supported initial data.…”