Global weak solutions to the two-dimensional Navier-Stokes equations of compressible heat-conducting flows with symmetric data and forces

We prove the global existence of weak solutions to the Navier-Stokes equations of compressible heat-conducting fluids in two spatial dimensions with initial data and external forces which are large and spherically symmetric. The solutions will be obtained as the limit of the approximate solutions in an annular domain. We first derive a number of regularity results on the approximate physical quantities in the"fluid region", as well as the new uniform integrability of the velocity and temperature in the entire space-time domain by exploiting the theory of the Orlicz spaces. By virtue of these a priori estimates we then argue in a manner similar to that in [Arch. Rational Mech. Anal. 173 (2004), 297-343] to pass to the limit and show that the limiting functions are indeed a weak solution which satisfies the mass and momentum equations in the entire space-time domain in the sense of distributions, and the energy equation in any compact subset of the"fluid region".


Introduction
The two-dimensional Navier-Stokes equations of compressible heat-conducting fluids express the conservation of mass, and the balance of momentum and energy, which can be written as follows in Eulerian coordinates.
In the spherically symmetric case, namely, where r = r(x) := |x|, the system (1.1)-(1.3) takes the form: (̺u) t + (̺u 2 ) r + ̺u 2 r + P r (̺, θ) − ν u r + u r r = ̺f, (1.8) (̺θ) t + (̺uθ) r + ̺uθ r − κ θ rr + θ r r + P (̺, θ) u r + u r = Q, (1.9) where ν := λ + 2µ, Q = ν u r + u r 2 − 2µ r ∂ r u 2 ≥ 0. (1.10) The boundary and initial conditions become u = θ r = 0 at r = R, (1.11) (̺, u, θ)| t=0 = (̺, u 0 , θ 0 ), 0 < r < R. (1.12) The main purpose of this paper is to prove the global existence of weak solutions to the problem (1.1)-(1.5) when the initial data and external forces are large and spherically symmetric. Our work is motivated by the paper of Hoff and Jenssen [9] 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". In the entire space-time domain, however, the momentum equation in [9, Thorem 1.1] only holds weakly with a nonstandard interpretation of the viscosity terms as distributions. A natural question is to ask whether the momentum equation holds in the standard sense of distributions. A positive answer was given recently by Zhang, Jiang and Xie [17] for a screw pinch model arisen from plasma physics when the heat conductivity κ satisfies certain growth conditions. In the present paper, based on some new uniform global estimates of u and θ (see Lemma 2.4) which are established by applying the theory of the Orlicz spaces, we can give a positive result for the two-dimensional Navier-Stokes equations of compressible heat-conducting fluids (1.1)-(1.3), improving therefore the result of [9].
We now give the precise statement of our assumptions and results. The external force f is assumed to satisfy for each T > 0. The initial data (̺ 0 , u 0 , θ 0 ) are assumed to satisfy in Ω, (1.14) where S is the entropy density in the form of with Ψ(s) = s − logs − 1. We point out here that there are no smallness or regularity conditions imposed on f and (̺ 0 , u 0 , θ 0 ). Under the conditions (1.13)-(1.16), we shall prove the following existence theorem on spherically symmetric solutions to the problem (1.1)-(1.5).
Combining the global a priori estimates derived in Subsection 2.3, we shall prove Theorem 1.1 in Section 4 by the convergence argument similar to that in [9]. For this purpose, we consider the approximate solutions (̺ j , u j , θ j ) of the problem (1.1)-(1.5) in the annular regions Ω j := {x | ε j < r(x) < R}, where ε j is a sequence of positive inner radii tending to 0. Since the 1/r singularity in the equations (1.7)-(1.9) plays no role at the stage when ε j is fixed and positive, the global existence of approximate solutions (̺ j , u j , θ j ) for (1.1)-(1.5) can thus be shown in a manner similar to that in [9,17]. However, to pass to the limit as j → ∞ and to show the global existence of weak solutions to the original problem (1.1)-(1.5), we need some ε j -independent a priori estimates. This will be done in Sections 2 and 3. We first prove the global estimates in Section 2, where we derive the standard energy-entropy estimates in Subsection 2.1, and apply these estimates to establish a new uniform integrability of the approximate solutions in the entire spacetime domain by exploiting the theory of Orlicz spaces in Subsections 2.2 and 2.3, which is crucial in the proof of (d) of Theorem 1.1. Then, in Section 3, we list the well-known the pointwise bounds for ̺ j and θ j as consequences of the energy and entropy estimates. These pointwise bounds are independent of ε j , but only away from the origin of Lagrangian space. More precisely, as in [9], for any given h > 0 we define the particle position r j h (t) by from which and the standard energy-entropy estimates it follows that there exists a positive constant C(h), depending only on h > 0, such that r j h (t) ≥ C(h) > 0. With this observation, we can obtain that for any fixed h > 0 and T > 0, there is a positive constant C(T, h), depending only on h, T and the initial data, such that Applying these pointwise bounds, we can get a number of higher-order energy estimates for the approximate solutions in Subsection 3.2, which are also independent of ε j and only away from the origin of Lagrangian space. These ε j -independent bounds enable us to define the "fluid region" F (see (a) of Theorem 1.1) and to obtain the uniform Hölder continuity of the quantities on the compact subsets of F ∩ {t > 0} (see (b) of Theorem 1.1). Finally, all the assertions of (a)-(f) indicated in Theorem 1.1 will be proved in Section 4 by the convergence arguments adapted from Hoff and Jenssen's paper [9]. We note that the final step of this argument provides a sort of a posteriori validation that the equations (1.7)-(1.9) are indeed the correct forms of the general system (1.1)-(1.3) in the symmetric case considered here. As pointed out in [9], we still do not have sufficient information to infer that r(t) ≡ 0, nor do we know whether solutions exist for which r = 0. The analysis simply shows that r(t) may be positive, and that, if it is, a vacuum state of radius r(t) centered at the origin. In any case, the total mass is conserved in the spherical case, as is clear from (c) of Theorem 1.1, and the total momentum is zero because of symmetry.
We show in (e) only that the energy equation holds on the support of ̺, rather than in the entire space-time domain (0, ∞) × Ω. This is partly due to that we cannot obtain higher global regularity of θ and u. We may regard the restriction in (5) that the test function be supported in F as reasonable, since there is no fluid outside F , and the model is not really valid there. Additionally, the failure of the analysis to detect whether or not energy is lost ((f) of Theorem 1.1) calls into question the adequacy of the mass, energy, and entropy bounds in Lemma 2.1, which are the only known (global) a priori bounds in the multidimensional case now.
We end this section by mentioning some related existence results for large data in the multidimensional case. The global existence of weak solutions was first shown by Lions [16] for isentropic flows under the assumption that the specific heat ratio γ > 3n/(n + 2) where n = 2, 3 denotes the spatial dimension. Then, by using the curl-div lemma to delicately derive certain compactness, and applying Lions' idea and a technique from [12], Feireisl, Novotný and Petzeltová [4,6] extended Lions' existence result to the case γ > n/2. For any 1 ≤ γ ≤ n/2, a global weak solution still exists when the initial data have certain symmetry (e.g., spherical, or axisymmetric symmetry), see [8], [11]- [12]. For non-isentropic flows, the global existence for general data is still not available. Recently, under certain growth conditions upon the pressure, viscosity and heat-conductivity (i.e., radiative gases), Feireisl, et al. obtained the global existence of the so-called "variational solutions" in the sense that the energy equation is replaced by an energy inequality, see [3] for example. However, this result excludes the case of ideal gases unfortunately. The global existence of a solution for large data in the non-isentropic case needs further study.

Global Estimates
In this section we derive a priori global estimates for any smooth (approximate) solution (̺ ε , u ε , θ ε ) of (1.7)-(1.9) together with additional boundary conditions: We assume that the initial data and force are smooth and satisfy the bounds (1.13)-(1.15) with constants independent of ε and We refer to Section 4 for a brief discussion on the existence of such approximate solutions. As discussed in Section 1, we shall eventually take a sequence of inner radii ε j → 0 to prove Theorem 1.1. Since ε > 0 is fixed for the time being, we suppress the dependence on j.

Energy and Entropy Estimates
We start with the following lemma which states the standard energy and entropy estimates for these approximate solutions. (1.16) and Q is given in (1.10).
Proof. The bounds (2.2)-(2.4) are the standard energy estimates which follow directly from the equations (1.7)-(1.9), the boundary conditions and the assumption (1.13) on the external force.

Excursion to Theory of the Orlicz Spaces
Before deriving the global estimates on the temperature and velocity, we recall some wellknown results concerning the Orlicz spaces (see, for example, [1,15] for details), which are often used to investigate the 2D compressible Navier-Stokes equations (see [13,10,2] for example).
where the real-valued function φ defined on [0, ∞) has the following properties φ is right continuous and nodecreasing on [0, ∞).
We define Then Ψ is a Young's function as well. We call Ψ the complementary Young's function to Φ. If Φ is complimentary to Ψ, then Ψ is complimentary to Φ.

Definition 2.2 (Orlicz spaces).
Let Ω be a domain in R n and let Φ be a Young function. The Orlicz class K Φ (Ω) is the set of all (equivalent classes modulo equality a.e. in Ω of ) measure functions u defined on Ω that satisfy Ω Φ(|u(x)|)dx < ∞. The Orlicz space L Φ (Ω) is the linear hull of the Orlicz class K Φ (Ω), that is, the smallest vector space that contains K Φ (Ω). The functional . It is called the Luxembourg norm. Thus, L Φ (Ω) is a Banach space with respect to the Luxembourg norm. Definition 2.3 (Cone condition). Let y be a nonzero vector in R n . Let ∠(x, y) be the angle between the position vector x and y. For given such y, h > 0, and k satisfying 0 < k ≤ π, the set is called a finite cone of height h, axis direction y and aperture angle k with vertex at the origin. Ω ⊂ R n satisfies the cone condition if there exists a finite cone Λ, such that each x ∈ Ω is the vertex of a finite cone Λ x contained in Ω and congruent to Λ. (see [13,3. Appendix]).

Global Estimates on the Temperature and Velocity
Now, we are in a position derive the global estimates on temperature and velocity. First, we define̺ and make use of (2.2)-(2.4) and the definition of Luxemburg norm · L M (Ω) to deduce that there exists a constant C 1 (T ), such that
2) The other one is the revised version Sobolev embedding in two dimensions which will be used to derive bounds of the temperature. This is an idea due to Lions who ever used similar embedding to derive the global integrability of the temperature in the proof of the existence of weak solutions to the stationary problems for the full compressible Navier-Stokes equations in a bounded domain Ω ⊂ R 2 [16, (6.204) in Section 6.11]. These two auxiliary results are formulated in the following two lemmas.

Lemma 2.2 (Generalized Korn-Poincaré inequality).
Let Ω ⊂ R 2 be a bounded domain satisfying the cone condition. Assume that v ∈ W 1,2 (Ω), and ̺ ≥ 0 satisfies (2.12) Then there is a constant C 3 depending solely on C 1 and C 2 , such that Proof. We prove the lemma by contradiction. Suppose that the conclusion of Lemma 2.2 be false, then there would be a sequence {̺ n } ∞ n=1 of non-negative functions satisfying (2.12) and a sequence {v n } ∞ n=1 ⊂ W 1,2 (Ω), such that v n L 2 (Ω)) ≥ C n ∇v n L 2 (Ω) + Ω ̺ n |v n |dx , C n → +∞. (2.13) Setting w n = v n v n −1 L 2 (Ω) , making use of (2.6) and (2.13), we find that (2.14) In view of the hypothesis (2.12), we see that Then, for any q ≥ 1, there is a constant C 2 depending solely on q, C 1 and Ω, such that Proof. We use (2.19) and (2.6) to infer that Easily, it suffices to consider the case Λ = 0. By the definition of the Luxemburg norm · L N (Ω) , we obtain This completes the proof of Lemma 2.3.

Local Estimates
In order to taking to the limit as ε → 0, we will need further uniform bounds of higher order derivatives. Such bounds will be obtained away from the origin of Lagrangian space in the following sense. Define a curve r ε h (t) for h ≥ 0 by is the position at time t of a fixed fluid particle. Furthermore, an easy estimate, based on Jensen's inequality and boundedness of R ε ̺ ε Ψ(̺ −1 ε )rdr in (2.4) (see (1.16)), shows that h → 0 at a uniform rate as r ε h (t) → 0. That is, given h > 0, there is a positive constant C = C(h) independently of ε and T , such that Using (3.2), we can derive pointwise bounds for the approximate density and temperature, which are valid away from the origin h = 0 of Lagrangian space, but independent of ε. The idea of deriving the pointwise boundedness was used first by Kazhikov and Shelukhin [14], and later adapted by Frid and Shelukhin [7], and by Hoff and Jenssen [9] in a nontrivial way to show a pointwise boundedness similar to that given in Lemma 3.1 below. Lemma 3.1 (Pointwise bounds). Given h > 0 and T > 0, there is a constant C = (T, h), independent of ε, such that, if r ε h (t) is given by (3.1), then Proof. Taking n = 2 and m = 1 in the proof of [9, Lemma 2], we immediately obtain Lemma 3.1.
Next, we shall make use of a cut-off function which is convected with the flow and vanishes near the origin. The cut-off function is constructed as follows: For given ε and h, we can fix a smooth, increasing function φ 0 (r) with φ 0 (r) ≡ 0 on [0, r h (0)], 0 < φ 0 (r) ≤ 1 on (r h (0), 2r h (0)) and φ 0 (r) ≡ 1 on [2r h (0), R], and then define φ(t, r) to be the solution of the problem We choose φ 0 so that Thus, we can easily show that this boundedness persists for all time, i.e., We shall take p so large that the exponent on the right-hand side of (3.4) is close to one. Notice that here we have suppressed the dependence of φ on ε and h. As in [9], we now introduce three functionals of higher-order derivatives for (u ε r , θ ε r ): φ(t, r) u ε r + u ε r 2 (t, r)rdr where σ(t) = min{1, t}, "dot" denotes the convective derivative ∂ t + u∂ r , and we have again suppressed the dependence on ε and h for simplicity. Thus, we have the following estimates. Proof. This lemma can be shown following the same procedure as in the proof of [9, Lemmas 4 and 6] with taking m = 1, v = 0 and w = 0. We should point out here that in the proof one should make use of Lemma 2.1, Lemma 3.1, (3.3) and (3.4).
As the end of this section, we give some uniform integrability estimates. To describe these, we define the strictly increasing, convex function G by It is easy to see that for each fixed c the function r → ω(r, c) is continuous and increasing on (0, ∞), and that lim r→0 ω(r, c) = 0.
(a) Given b > 0 and T > 0, there is a constant C = C(T, b), such that   By virtue of the a priori estimates derived in Sections 2 and 3, we are now able to prove our main theorem by taking appropriate limits in a manner analogous to that in [9].

Convergence of the Approximate Solutions
By the a priori estimates established in Lemmas 3.1-3.2, 2.1 and 2.4, we have the following three propositions, which imply that there is a subsequence (ε j , δ j ) → (0, 0), such that the approximate solutions and their associated particle paths are convergent.

Weak Forms of the Navier-Stokes Equations
We now turn to the proof that the limiting functions are indeed a weak solution of the Navier-Stokes equations in [0, ∞) × Ω in the sense of Theorem 1.1.
First, the limiting functions ̺, u and θ have been defined in the fluid region F but not elsewhere. We therefore define ̺, ̺u and ̺θ to be identically zero in the vacuum region F c . As in Section 1 we let r(x) = |x| and define the velocity vector u : [0, ∞) ×Ω by u(t, x) = u(t, r) x r . (4.5) Abusing notation slightly, we also write ̺(t, x) and θ(t, x) in place of ̺(t, r(x)) and θ(t, r(x)). Similar notation applies to the approximate solutions, for which we now write u j in place of u ε j ,δ j , etc.