MASS CONCENTRATION PHENOMENON TO THE 2D CAUCHY PROBLEM OF THE COMPRESSIBLE NAVIER-STOKES EQUATIONS

. In this paper, we consider the global strong solutions to the Cauchy problem of the compressible Navier-Stokes equations in two spatial dimensions with vacuum as far ﬁeld density. It is proved that the strong solutions exist globally if the density is bounded above. Furthermore, we show that if the solutions of the two-dimensional (2D) viscous compressible ﬂows blow up, then the mass of the compressible ﬂuid will concentrate on some points in ﬁnite time.


1.
Introduction. This paper concerns the Cauchy problem of the 2D compressible Navier-Stokes equations: where x = (x 1 , x 2 ) ∈ R 2 is the spatial coordinate and t ≥ 0 is the time. The unknown functions ρ, u = (u 1 , u 2 ) and P = P (ρ) denote the density, velocity field and pressure, respectively. The aim of the present paper is to show the existence of the global strong solutions and the formation of singularities in finite time. The equation of state is given by with A being a positive constant and γ > 1. The real parameters µ and λ are the shear viscosity and the bulk viscosity coefficients, respectively, which satisfy the following physical restrictions: System (1) will be investigated with initial conditions and the far-field conditions (ρ, u)(x, t) → (0, 0) as |x| → ∞.
There are huge literatures on the global regularity criterion for the multi-dimensional compressible Navier-Stokes equations. In the absence of vacuum for the initial density, the local well-posedness theory of classical solutions has been well developed, see [11,23,24] and the references therein. When vacuum is allowed for the initial density, it can be proved that the strong solutions exist locally in time if one adds suitable compatibility conditions [3,4,5]. For large initial data which may contain vacuum, the major breakthrough is due to Lions [22] (see also Feireisl [10] and Jiang-Zhang [17,18]), which proved that weak solutions exist globally in time when the exponent γ is suitably large. However, the uniqueness of such weak solutions remains completely open even in the 2D case. To our knowledge, Vaigant-Kazhikhov [29] proved the first result of the global well-posedness to the periodic boundary problem with large initial data, under the assumption that the initial density is uniformly away from vacuum and viscous coefficients µ is constant, λ = ρ β with β > 3. More recently, Huang-Li [13] and Jiu-Wang-Xin [19] have extended the global well-posedness of strong solutions for periodic case to the Cauchy problem of 2D Navier-Stokes equations with vacuum when the viscosity coefficients depend on the density ρ and β is suitably large. But, if both the shear and bulk viscosities are positive constants, the global well-posedness of classical solutions to the Cauchy problem of 2D baratropic compressible Navier-Stokes system with vacuum and large initial data remains open.
On the other hand, due to the results of Xin [32], who showed that there is no global smooth solutions to the Cauchy problem (1) at least in 1D, as long as the initial density has compact support. That is to say, the Navier-Stokes equations may produce singularities in finite time. Therefore, it is important to study the mechanism of blowup and structure of possible singularities to the compressible Navier-Stokes equation. Starting with the pioneering works by Beal-Kato-Madja [1] and Serrin [26], many articles have dedicated to provide sufficient conditions for the global regularity of incompressible and compressible viscous fluids (see [6,7,8,9,12,14,15,16,27,28,30,31] and the references therein). There are a series of blowup criteria for the 2D compressible Navier-Stokes equations, especially, Sun-Zhang [28] gave a blowup criterion for strong solutions in a bounded domain of R 2 . More precisely, they proved that if T * is the life span of the strong solution to system (1), then lim T →T * ρ L ∞ (0,T ;L ∞ ) = ∞. The goal of this paper is to extend the blowup criterion for the 2D compressible Navier-Stokes equations in bounded domain to unbounded domain (Cauchy problem). The main difficulty lies in the fact that the Brezis-Waigner's inequality [2] fails for the 2D Cauchy problem, and it seems difficult to estimate u L q (R 2 ) for any q > 1. One way to overcome this difficulty is to estimate the momentum ρu instead of the velocity u, since ρ decays for large x, the momentum ρu decays faster than u itself. Furthermore, we use the variant of Gagliardo-Nirenberg inequality and a finer estimate for the convective term ρu · ∇u. Moreover, the high order estimates on ρ and u will not be improved as that in the bounded domain case. Hence, we introduce the Hardy-type inequality to control the L q -norm of ρu. Furthermore, the initial density vacuum is allowed in this paper.
Before stating the main result, we first introduce the following simplified notations Without loss of generality, we assume that the initial density ρ 0 satisfies The local strong solutions for the 2D Navier-Stokes system with vacuum was obtained in [21]. Here we write down one of those results in R 2 .
The main result of this paper can be stated as follows.
Remark 1. Indeed, in view of continuity equation which implies that as long as the velocity field is regular, we can define the characteristic line as dy ds (s; x, t) = u(y(s; x, t), s), y(t; x, t) = x, and the density can be denoted as This implies that if the singularity of the solution to the compressible Navier-Stokes equations (1) formulates in finite time T * , there may hold for the density: (1) The density may concentrate, namely, (2) Vacuum states may appear in the non-vacuum region: there exist some x 1 ∈ R 2 and x 1 (t) satisfying (3) Vacuum states may vanish: there exist some x 0 ∈ R 2 and x 0 (t) satisfying Then one may ask: which one or some of (12)- (14) will happen when the singularity formulates? In view of the mass conservation equation (1) 1 and (6), Theorem 1.1 gives an answer to this question and shows that the mass of the fluid will concentrate on some points before other cases (2) and (3) happen.

Remark 2.
The approach can also be adapted to deal with the bounded domain in R 2 . In particular, it would be interesting to study whether the upper bound of density is a necessary condition.
The remain of this paper is organized as follows. Some important inequalities and auxiliary lemmas will be given in Section 2. We prove the main result Theorem 1.1 in Section 3.

2.
Preliminaries. In this section, some elementary lemmas will be used later. One of which is the variant of Gagliardo-Nirenberg inequality, the proof is referred to [25].
and constant C depends on q, m and r.
Next, the material derivativeḟ , the effective viscous flux G, and the vorticity w are defined as follows.
Therefore, there are two key elliptic system of G and w.
where we have used (1) 2 and (16). From the standard L p -estimate of elliptic system (17), we have the following estimate.
Lemma 2.2. Let (G, w) be a strong solutions of (17). Then there exists a generic positive constant C depending only on µ, λ and p, such that where 1 < p < ∞.
Remark 4. In particular, taking p 0 ∈ (1, 2), using Hölder's inequality, we have In addition, in order to estimate ∇u L p , we introduce the following inequality, which is crucial to the estimate in 2D Cauchy problem (see [19] for the detailed proof).
There exists a constant C depending only on p, such that The following Hardy-type inequality plays a crucial role in the estimate, the proof of which can be found in [21]. Lemma 2.4. Letx and η 0 be as in (7) and holds for positive constants M 1 and M 2 . Then there exists a positive constant C depending on M 1 , M 2 , γ, N 1 and η 0 such that for any v ∈D 1,2 {v ∈ H 1 loc (R 2 ) | ∇v ∈ L 2 }. Furthermore, for ε > 0 and η > 0, there exists a positive constant C depending on ε, η, M 1 , M 2 , γ, N 1 and η 0 such that every function v ∈D 1,2 satisfies Finally, in order to estimate the term ∇u L ∞ , we introduce the following Beal-Kato-Majda-type inequality, which was first proved in [1] when divu = 0. The proof for a general situation can be found in [20].
3. Proof of the main results. Let (ρ, u) be a strong solution of (1)-(3) on R 2 × [0, T * ). We will prove our main result by contradiction arguments. Otherwise, for some sufficiently large 1 < s 0 < ∞, one has In addition, in view of (6), the mass conservation equation (1) First, from the standard energy estimate for (ρ, u), it is easy to obtain the boundedness of ∇u L 2 (0,T ;L 2 ) .
Proof. First, the proof of (26) is standard. Multiplying the momentum equation by u and using the first equation in (1), we can obtain the estimate (26).
which together with (25) imply, for N 1 suitably large. The proof of Lemma 3.1 is completed.
Next, there is an important estimate about ρu, the similar arguments of the following estimates come from [28]. , such that for any T ∈ [0, T * ).
With the help of Lemma 2.2 and Lemma 2.3, we can prove the following key estimate on ∇u.
Hence, choosing ε sufficiently small, combining (41) with (42) yields d dt and noting that Thus, by using Gronwall's inequality, (43) yields which together with (21) and (26) gives Next, we can improve the regularity estimates on ρ and u.
Let p → ∞, the proof of Lemma 3.5 has finished.

Remark 6.
For the convenience of the readers, we state a detailed proof to the regularity of terms √ ρu t and ∇u t from the estimates onu as follows.
Furthermore, multiplying (69) by u t and integrating the resulting equation over = − ρu · ∇u · utdx − ρu · ∇ut ·udx − ρut · ∇u · utdx + Ptdivutdx Proof of Theorem 1.1. With the estimates in Lemma 3.1-Lemma 3.7 and local existence in Proposition 1, we can extend the local strong solutions of (ρ, u) beyond T * in the same way as that in [15], which contradicts the maximality of T * , we thus complete the proof.