Global strong solution to the two dimensional nonhomogeneous incompressible heat conducting Navier-Stokes flows with vacuum

In this paper, we prove the unique global strong solution for the two dimensional nonhomogeneous incompressible heat conducting Navier-Stokes flows when the initial density can contain vacuum states, as long as the initial data satisfies some compatibility condition. Furthermore, our main result improves all the previous results where the initial density is strictly positive. The main ingredient of the proof is to use some critical Sobolev inequality of logarithmic type, which were originally due to Brezis-Gallouet in [ 3 ] and Brezis-Wainger in [ 4 ], some regularity properties of Stokes system and some delicate energy estimates for nonhomogeneous incompressible heat conducting flows.

If there is no temperature field, i.e., θ ≡ 0, (1) reduces to nonhomogeneous Navier-Stokes equations. There is huge literature about the well-posedness of solution to nonhomogeneous Navier-Stokes equations in multi-dimensional when initial density ρ 0 is bounded away from 0. We will recall some results for multi-dimensional nonhomogeneous Navier-Stokes equations. Indeed, Antontsev and Kazhikov [1,20] established the global existence of weak solutions. Later, Antontsev, Kazhikov and Monakhov [2] obtained the local existence and uniqueness of strong solutions. Furthermore, they proved that the local strong solution is global in two dimensions. When the initial density allows vacuum, the global existence of weak solution to nonhomogeneous Navier-Stokes equations was established by Simon [29], see also [24]. Choe and Kim [6] constructed a local strong solution as long as the initial data satisfied some compatibility conditions. Huang-Wang [17,15] and Zhang [34] established the global existence of strong solutions on bounded domain of R 3 , when the initial data satisfied some suitable small conditions. Recently, He, Li and Lü [11] obtain the global existence of strong solutions in R 3 to nonhomogeneous Navier-Stokes equations with density-dependent viscosity and vacuum, provided that the initial velocity is suitably small. However, the global existence of strong or smooth solutions is still an open problem in three dimensions space. Therefore, it is important and reasonable to study the mechanism of blowup and structure of possible singularities of strong solutions to the nonhomogeneous Navier-Stokes equations. If T * < ∞ is the maximal time of existence of a strong solution, Kim [21] proved the Serrin type criterion, namely, where r, s satisfy 2 s + n r = 1, n < r ≤ ∞, n is the dimension of the domain and L r w is weak L r space. We go back to the nonhomogeneous heat conduction system (1). The local existence of strong solutions was proved by Choe and Kim [5]. However, the global existence of strong solutions for heat conducting viscous incompressible fluids (1)- (3) is still open to two dimensions case. Therefore, one question came out naturally, wether the local strong solutions blows up in finite time. Similar arguments as [21], one can get the same criterion (4) for two dimensional nonhomogeneous incompressible heat conducting Navier-Stokes flows. In particular, for two dimensions case, it says that u L 2 (0,T ;L ∞ ) is uniformly bounded, then the local solution is global. Indeed, this is the main aim of this paper.
Throughout this paper, for 1 ≤ r ≤ ∞ and integer k ≥ 0, we denote the standard Lebesgue and Sobolev spaces as follows, Set the material derivative of f as follows.
Then, the strong solutions to the initial boundary value problem (1)-(3), are defined as follows.
Our main result of this paper can be stated as follows.
Theorem 1.2. For constant r > 2, assume that the initial data (ρ 0 , u 0 , θ 0 ) satisfy and the compatibility conditions and for some P 0 ∈ D 1,2 and g 1 , g 2 ∈ L 2 . Then there exists a global strong solution (ρ, u, θ) of the system (1)-(3). Remark 1. Along the same arguments as [5], we can obtain the local existence of strong solutions with vacuum to the system (1)-(3) in a two dimensional bounded domain. Therefore, the maximal time T * is well-defined. We will verify Theorem 1.2 by the contradiction arguments, and obtain the local strong solution does not blow up in finite time. Throughout this paper, we will concentrate on establishing global estimate for the density, velocity and temperature field. Furthermore, the approach can also be adapted to deal with the periodic case. In particular, it would be interesting to study the Cauchy problem in R 2 .
Remark 2. Indeed, for the two dimensions case, the key ingredient of the analysis here is a critical Sobolev inequality of logarithmic type, which was stated in Lemma 4 in the next section. The general version in multi-dimension n ≥ 2 is as follows, ln(e + f 2 L 2 (0,T ;W 1,q ) ) , for q > n. The energy inequality tells us u L 2 (0,T ;H 1 0 ) is uniformly bounded. This together with logarithmic Sobolev inequality gives the uniform estimates of u L 2 (0,T ;L ∞ ) . However, in higher-dimensional case n ≥ 3, it seems to be difficult to obtain the estimates u L 2 (0,T ;W 1,n ) , which is the main obstacle to extend the results in multi-dimension n ≥ 3.
The rest of this paper is organized as follows. In section 2, we will recall some known facts and elementary inequalities that will be used later. We will prove the Theorem 1.2 in Section 3.

2.
Preliminaries. In this section, we will recall some elementary lemmas and inequalities that will be used later. We start with the local existence of strong solutions, which can be established in the same manner as [5]. Proposition 1. (Local existence of strong solutions) Assume that the initial data (ρ 0 , u 0 , θ 0 ) satisfy (6)- (8). Then there exist a positive constant T 0 and unique strong solution (ρ, u, θ) to the initial boundary value problem (1)-(3) on Ω × (0, T 0 ).
Next, we will introduce the so-called Gagliardo-Nirenberg inequality, its proof can be founded in [26,27].
provided that and m − j − 2 r is not a nonnegative integer. If m − j − 2 r is a nonnegative integer (9) holds with ϑ = j m . Remark 4. Indeed, the inequality (9) was originated from the reference [26] by L. Nirenberg in 1959. In particular, the following inequality will be used in the next section.
Some regularity results for the Stokes equations were sated as follows. For its proof, refer to [10].
Let Ω be a bounded domain of R 2 of class C 1,1 . For any r ∈ (1, ∞), if F ∈ L r , there exists some positive constant C depending only on r such that the unique weak solution (u, P ) ∈ H 1 0 × H 1 to the following Stokes system in Ω, in Ω, satisfy Finally, in order to obtain the bound of u L 2 (0,T ;L ∞ ) , we introduce a critical Sobolev inequality of logarithmic type, which were originally due to Brezis-Gallouet in [3] and Brezis-Wainger in [4], and extended by Ozawa in [28]. For its proof, refer to [16,22,30].
Then there exists a constant C depending only on p such that 3. Proof of the main results. In view of Proposition 1, there exists T * > 0 such that (ρ, u, θ) be the strong solution to (1)- (3) on Ω × (0, T * ) with initial data (ρ 0 , u 0 , θ 0 ) satisfying (6)-(8). We will establish some necessary a priori bounds for strong solutions (ρ, u, θ) and extend the local strong solutions beyond T * . First, due to divu = 0, we establish the following upper uniform estimates of the density.
Lemma 3.1. There exists a positive constant C such that and sup Here and after, C 0 and C denote generic positive constants depending only on µ, c υ , T * and the initial data.
Proof. Indeed, since divu = 0, we obtain ρ L 1 ∩L ∞ = ρ 0 L 1 ∩L ∞ . The particle path can be defined as follows: It follows from the continuity equation that the density can be expressed by the formula ρ(x, t) = ρ 0 (y (0; x, t)) , which together with ρ 0 ≥ 0 implies (13). Hence, we complete the proof of Lemma 3.1.
Next, we will give the standard energy estimates as follows.
Proof. Using the standard maximum principle in [8] to (1) 3 together with θ 0 ≥ 0, we obtain inf Integrating (1) 3 with the spatial variable x and integrating by parts gives Multiplying (1) 2 by u and integrating the resulting equation over Ω yield to Thus, adding (18) multiplied by 2 to (17), we obtain this together with Gronwall's inequality gives (16). Thus, we complete the proof of Lemma 3.2.
The key estimates on ∇u will be given in the following lemma. To prove that, we will make use of critical Sobolev inequality of logarithmic type, which was stated in Lemma 2.3.
Proof. Multiplying the momentum equation (1) 2 by u t , integrating the resulting equation over Ω, integrating by parts and using Young's inequality yield to µ 2 Let Then, it follows from (20) and Gronwall's inequality that, for every 0 ≤ s ≤ T < T * , one has Now, we will estimate the term u L 2 (s,T ;L ∞ ) .
Using Lemma 2.3, we obtain that where we have used Poincaré inequality and the following facts due to (9) and (11). This together with Poincaré inequality and Hölder inequality gives In view of the energy estimate (16), we can choose s close enough to T * , such that then for s < T < T * yield to This finishes the proof of Lemma 3.2.
Remark 5. Unfortunately, the critical Sobolev inequality of logarithmic type (12) can not be adapted to deal with the 2D Cauchy problem, and it seems difficult to estimate u L 2 (R 2 ) .
We improve the regularity of the temperature θ as follows.
The boundedness of √ ρu t L 2 and ∇θ L 2 will be proved as follows. The proofs are the similar as Appendix A in [13] to the compressible case. To convenience to the readers, we sketch the detail proofs as follows.
Finally, we will give the second spatial derivatives of u and the L q -estimate (q > 2) of the first spatial derivative of ρ.
Lemma 3.7. Under the assumptions in Theorem 1.2, and let q > 2 be defined in Theorem 1.2, we have that for any 0 ≤ T < T * .
Remark 6. In particular, for the convenience of the readers, we give a detailed proof to the regularity of terms ρu and ρθ from the estimates on √ ρu t and ρθ t as follows.