Global Weak Solutions to the Compressible Navier-Stokes Equations in the Exterior Domain with Spherically Symmetric Data
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Global Weak Solutions to the Compressible Navier-Stokes Equations in the Exterior Domain with Spherically Symmetric Data Lingyu Jiang · Chao Wang
Received: 11 December 2011 / Accepted: 22 February 2012 / Published online: 8 March 2012 © Springer Science+Business Media B.V. 2012
Abstract In this paper, we prove the global existence of weak solutions to the full compressible Navier-Stokes equations in the domain exterior to a ball in Rn (n = 2, 3) and with spherically symmetric data. Keywords Compressible Navier-Stokes equations · Weak solutions · Spherically symmetric
1 Introduction We consider the global existence of weak solutions to the full compressible Naiver-Stokes equations with spherically symmetric data and in the domain exterior to a ball in two and three space dimensions. In this case, the equations which describe the motion of the compressible gases can be written as follows ⎧ ρv = 0, (t, r) ∈ [0, ∞) × [a, +∞), ∂t ρ + ∂r (ρv) + n−1 ⎪ r ⎪ ⎪ ⎪ 2 ⎪ 2 rv ⎨ ∂t (ρv) + ∂r (ρv ) + ∂r (ρθ ) + (n−1)ρv = 2μ(∂r2 v + (n−1)∂ − (n−1)v ), r r r2 (1.1) 3 (n−1)ρv c (n−1)ρθv 1 1 ⎪ ⎪ ∂t (cv ρθ + 2 ρv 2 ) + ∂r (cv ρθ v + 2 ρv 3 + ρθ v) + 2r + v r ⎪ ⎪ ⎪ ⎩ rθ rv = 2μ (n−1)v∂ + kθ ∂r2 θ + 2μ(vvr )r − ρθv(n−1) + kθ (n−1)∂ , r r r together with the initial data (ρ, v, θ )|t=0 = (ρ0 , v0 , θ0 ), L. Jiang Department of Mathematics, Central University of Finances and Economics, Beijing 100081, China e-mail: [email protected] C. Wang () Academy of Mathematics & Systems Science, The Chinese Academy of Sciences, Beijing 100190, China e-mail: [email protected]
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L. Jiang, C. Wang
and boundary conditions v(a, t) = 0,
θr (a, t) = 0,
(1.2)
where a is a given positive constant, ρ, v, θ denote the density, velocity and temperature of the compressible fluids respectively. Here we define the pressure function P = ρθ , which describes the motion of ideal gases, the heat conductive coefficient kθ is a positive constant, and the bulk viscosity coefficient λ = 0 (The case for λ = 0 can be followed along the same line as the case for λ = 0). One may check [2, 11] and the references therein for the derivation of the equations and the related mathematical results. In what follows, we denote ∞
f (r) p r n−1 dr < ∞ , Lp [a, ∞) = f | a
with the norm · Lp
∞ = ( a | · |p r n−1 dr)1/p . And we assume that the initial data satisfy
0 < c1 ≤ ρ0 ≤ c2 < ∞, ρ0 − 1 ∈ L1 [a, ∞) , v0 ∈ L2 [a, ∞) , θ0 − 1 ∈ L2 [a, ∞) , θ0 c > 0.
(1.3)
The Navier-Stokes equations for compressible fluids have been studied by many authors. The question concerning the global existence and the time-asymptotic behavior of solutions for large initial data has been largely solved in one dimension (see e.g. [1, 9, 10, 14]). The mathematical theory, however, is far from being complete in more than one dimension. In the case of sufficiently small initial data, there is an extensive literature on the global existence and the asymptotic behavior of solutions which is originated by the papers of Matsumura and Nishida [12, 13] (also see [6])
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