Global weak solution to 3D compressible flows with density-dependent viscosity and free boundary

In this paper, we obtain the global weak solution to the 3D spherically symmetric compressible isentropic Navier-Stokes equations with arbitrarily large, vacuum data and free boundary when the shear viscosity $\mu$ is a positive constant and the bulk viscosity $\lambda(\rho)=\rho^\beta$ with $\beta>0$. The analysis of the upper and lower bound of the density is based on some well-chosen functionals. In addition, the free boundary can be shown to expand outward at an algebraic rate in time.


1.
Introduction. The compressible isentropic Navier-Stokes equation in R 3 is described by ρ t + div(ρU) = 0, (ρU) t + div(ρU ⊗ U) + ∇P = µ∆U + ∇((µ + λ(ρ))divU, where ρ, U and P (ρ) are the density, the velocity and the pressure respectively. The pressure is given by P (ρ) = ρ γ , (γ > 1) Here, it is assumed that There are a large number of literatures on the well-posedness theories of the compressible Navier-Stokes equation (1) when both viscosities are positive constants. It is well-known that the global well-posedness theory for the one-dimension case is rather satisfactory, see [12,29,25,26] and the references therein. On multidimensional case, the local well-posedness theory of classical solutions was established in the absence of vacuum (see [33,17,38,31]) and the global well-posedness theory of classical solutions was obtained for initial data close to a non-vacuum key uniform estimates away from the symmetry center are established in section 4, In this section, these estimates do not depend on . Based on these, in section 5, we take the limits to obtain the global existence of weak solutions of the original system. Notation.
Let Ω be a domain in R 3 and m be an integer 1 ≤ p ≤ ∞. By W m,p (Ω) (W m,p 0 (Ω)) we denote the usual Sobolev space defined over Ω. For simplicity we also use the following abbreviations: The same letter C (sometimes used as C(X) to emphasize the dependence of C on X) will denote various positive constants.
Definition 2.1 (Weak solution). (ρ(t, r), u(t, r), a(t)) with ρ ≥ 0 a.e. is said to be a weak solution to the free surface problem (3)- (6) on Ω T × [0, T ], provided that it holds that 1) 2) For any The free boundary condition (5) is satisfied in the sense of trace.
(d) The density ρ ∈ C([0, ∞); W 1,∞ [0, a(t)) * ). Also, ρ(·, t) ≡ 0 inF C , and if ρu is taken to be zero inF c , then the weak form of the mass equation holds for test function φ ∈ C 1 (Ω t × [0, t]): The main ideas of the proof of Theorem 2.2 are motivated by the paper of Hoff-Jenssen [10] where they studied the spherically and cylindrically symmetric nonbarotropic flows with large data and forces, and established the global existence of weak solutions to the compressible nonbarotropic Navier-Stokes equations in the "fluid region".
3. Global existence of approximate FBVP. Consider a modified FBVP problem for Eq. (3) with the following initial data and boundary conditions for any fixed small enough > 0: where For simplicity of notation, we denote (ρ , u ) as (ρ, u) in the following if without confusions.
In order to estimate ρ, we first establish the following technical result. and Then there exists a constant C( , T ), such that Proof. We observe that (ρu) t − ϕ(r, t) r = 0, Thus, we have complete the proof of Lemma 3.4.
Proof. Let F (ψ, ρ) be a function of ψ, ρ to be determined. We can compute that Choose F satisfying the following equation Now we solve the above first-order hyperbolic partial differential equation. Set G(ρ, ψ) = ρF (ρ, ψ). Then G satisfies the equation By the characteristic method, we can solve that with f (·) being any smooth function. Here, we choose f (z) = e 1 2µ z , and thus Then we can get from (29) that Integrating the above ordinary differential equation over [0, t] with respect to τ along the particle path X(τ ; t, x) defined by it holds that Thus, one has ρ(r, t) ≤ ρ 0 (X(0; r, t))e 1 2µ ψ(X(0;r,t),0)+ For this, let F 1 (ψ, ρ) be a function of ψ, ρ to be determined later. We can compute that Choose F 1 (ψ, ρ) satisfying the following equation then we can solve that where g(·) is any smooth function. At the same time, by (32), it holds that Therefore, one has Integrating the above ordinary differential equation over [0, t] with respect to τ along the particle path X(τ ; t, x), it holds that Thus, there exist positive constant c(ε, T ) such that Therefore, we completed the proof of Lemma 3.5.
It is convenient to deal with the FBVP (3) in the Lagrangian coordinates. For simplicity we assume that a0 ρ 0 r 2 dr = 1, which implies a(t) The free boundary value problem (3) and (15)- (16) are changed to , with the initial data and boundary conditions given by where r = r(x, τ ) is defined by and the fixed boundary x = 1 corresponds to the free boundary a(τ ) = r(1, τ ) in Eulerian form determined by Energy estimation of Lemma 3.1 has follow form in Lagrangian coordinates.
Lemma 3.7 (Entropy estimation). Under the same assumptions as Lemma 3.1, then Proof. From (33), one has Multiplying above equation by u + r 2 2µ ln ρ + ρ β β x , integrating the result with respect to x over [0, 1], we get d dt In the sequel, we derive bounds for each term on the right-hand side of (40) as follows: By the upper bound of the density, one has and then and Substituting (41) and (42) into (40), we get Then, from Gronwall's inequality one has Using Lemma3.2 and Lemma3.5, we get Thus one has  Proof. Multiplying (33) 2 by u τ and integrating, we have
Taking the inner product of (53) with u τ over [0, 1] where we have used the following estimate Using Gronwall's inequality to (53), we obtain Consequently by (33) and (37), we have T 0 (a (τ )) 2 dτ ≤ C(T, ), The proof of proposition 1. First, we can change Proposition 1 in the Lagrangian coordinates. Next, with the help of Lemmas 3.1-3.9 for (ρ, u, a) and continuity argument, the global strong solution to the FBVP (3) and (15)-(16) under the assumptions of Proposition 1 can be shown by standard argument as in [34]. 4. Uniform estimates away from symmetry center. In order to obtain the global weak solutions containing the symmetric center r = 0, we need further the uniform-in-estimates to pass the limit → 0+. For this, the idea is originated by Hoff [13] and Hoff-Jessen [10]. Roughly speaking, for any given h > 0, define the particle path r h (t) by Thus r ε h (t) is the position at time t of a fixed fluid particle. So, given h > 0 there is a positive constant C independent of ε and T , such that . Then we consider the problem on the region r ∈ [r h (t), a(t)). Due to the fact that r h (t) ≥ c(h) with the positive constant c(h) independent of , we can get some uniformly interior estimates in for the solution on the region r ∈ [r h (t), a(t)). Therefore, we can pass the limit ε → 0 for any fixed h > 0 and then take h → 0 to get a weak solution. However, due to the bulk viscosity here depends on the density, the analysis in [10] can not be applied directly since we do not have by the elementary energy estimates. Thus we can not get the uniform estimates for the cut-off function along the particle path as in [10]. In order to overcome this difficulty, we first use the Lagrangian transformation to get the uniform interior estimates for the solution. Then we can change back to the Eulerian coordinates to pass to the limit → 0 and then h → 0 to get the global weak solution.
Thus the Lagrangian coordinates transformation translates the domain In this section, we derive some desired uniform estimates for (ρ , u , a ) to the modified FBVP (3) and (15)- (16). To simplify the presentation, we drop the superscipt .
Proof. Thanks to the fact Ch We shall make repeated use of a cut-off function which is convected with the flow and which vanishes near the origin. So we construct a cut-off function φ(x) satisfies Now we introduce the higher-order functional for a given solution: let σ(t) = min{1, t} and define For which corresponding to the following form in Euler coordinates: where Proof. Multiplying (33) 2 by σφ 2 u τ , one has In the sequel, we derive bounds for K j (1 ≤ j ≤ 3) on the right-hand side of (59). (60) (62) (63) Substituting (60) − (64) into (59), we obtain Taking J = (r 2 u) x , we have Thus, σφ(r 2 u) xx L 2 dt + 1 ≤ 1 2 σφ 2 |(r 2 u) xx | 2 dxdt + C(h, T ) σφ 2 u 2 t dxdt + 1 , Thus, we have complete the proof of Lemma 4.3.

5.
Proof of Theorem 2.2. By virtue of the a priori estimates derived in Sections 4, we are now able to prove our main theorem by taking appropriate limits in a manner analogous to that in [10].