Existence and a blow-up criterion of solution to the 3D compressible Navier-Stokes-Poisson equations with finite energy

We study the low-energy solutions to the 3D compressible Navier-Stokes-Poisson equations. We first obtain the existence of smooth solutions with small $L^2$-norm and essentially bounded densities. No smallness assumption is imposed on the $H^4$-norm of the initial data. Using a compactness argument, we further obtain the existence of weak solutions which may have discontinuities across some hypersurfaces in $\mathbb{R}^3$. We also provide a blow-up criterion of solutions in terms of the $L^\infty$-norm of density.

(1. 6) Equations (1.5)-(1.6) will be crucial in obtaining a priori bounds on the solutions, and the importance can be explained heuristically as follows: 1. In estimating higher-regularity norms (for example 1 0 R 3 t 3 |∇u| 4 dxdt) of the weak solutions, one cannot merely applying the embedding W 2,2 ֒→ W 1,4 as ∇u may be discontinuous across hypersrufaces of R 3 (see for example [6] for more details). With the help of F and ω, we are able to observe the following decomposition of ∆u: u j x k x k = ω j,k xj + (µ + λ) −1 F xj + (µ + λ) −1 (P − P (ρ)) xj . If we anticipate that F (·, t), ω(·, t) ∈ H 1 and P (·, t) − P (ρ) ∈ L 2 ∩ L ∞ , the term u j x k x k should then be in W −1,4 , and hence the desired bound for ∇u in L 4 follows by a Fourier-type multiplier theorem. 2. Another important application of the equations (1.5)-(1.6) can be revealed in studying the pointwise bounds on the density. With the help of the effective viscous flux F , we can rewrite equation (1.1) 1 as follows: where (x(t), t) is a particle path governed by u. Upon integrating with respect to time, we observe that the oscillation in density can be controlled by the time integral of −ρF (x(t), t). By utilizing the Poisson's equation (1.6) and the claimed a priori bounds on F , we are able to show that such time integral is bounded by the initial energy of the system which is taken to be small by our assumption. Hence the density remains bounded above in L ∞ as compare to itself initially.
(1. 8) It follows that µ λ > (q − 2) 2 4(q − 1) (1.9) for q = 6 and consequently for some q > 6, which we now fix. We also remark that the above conditions (1.7)-(1.9) as imposed on P , µ and λ are consistent with those used by Li-Matsumura [10] and Hoff [7] for compressible Navier-Stokes system. Condition (1.7) is considered to be more general than those used in Hoff [6] which includes the special case P (ρ) = Kρ γ for γ ≥ 1 and K > 0. The assumptions (1.8)-(1.9) are required for technical reasons in obtaining a priori estimates and will be particularly used in proving Lemma 2.5 (notice that (1.8)-(1.9) are consistent with those given in [6]).
Next we state the assumptions on the initial data (ρ 0 , u 0 , φ 0 ). We assume there is a positive number N , which may be arbitrarily large such that where q is defined in (1.9). From now on, for ρ 0 −ρ, u 0 , ∇φ 0 ∈ L 2 (R 3 ), we also write (1.11) for the sake of convenience without further referring.
Weak solutions to the system (1.1)-(1.2) can be defined as follows. Given T > 0, we say that (ρ −ρ, u, φ) is a weak solution of (1. ; and the following integral identities (1.12)-(1.14) hold for all t 1 , t 2 ∈ [0, T ] and C 1 test functions ϕ having uniformly bounded support in x for t ∈ [t 1 , t 2 ]: (1.14) We adopt the following usual notations for Hölder seminorms: for v : R 3 → R m and α ∈ (0, 1], v α = sup We denote the material derivative of a given function v byv = v t +∇v ·u, Finally if I ⊂ [0, ∞) is an interval, C 1 (I; X) will be the elements v ∈ C(I; X) such that the distribution derivative v t ∈ D ′ (R 3 × int I) is an element of C(I; X).
We make use of the following standard facts (refer to Ziemer [22] for details). First, given r ∈ [2,6] there is a constant C(r) such that for w ∈ H 1 (R 3 ), (1.15) Next, for any r ∈ (3, ∞) there is a constant C(r) such that for w ∈ W 1,r (R 3 ), where α = 1 − 3/r. If Γ is the fundamental solution of the Laplace operator on R 3 , then there is a constant C such that for any g ∈ L 2 (R 3 ) ∩ L 4 (R 3 ), The following are the main results of this paper. First of all, Theorem 1.1 shows that given T > 0, under a smallness assumption on the L 2 -norm of the initial data, the smooth classical solution to (1.1) exists on [0, T ].
Finally in Theorem 1.3, we obtain a blow-up criterion for the classical solutions to (1.1)-(1.2) for the isothermal case without any smallness assumption on the initial data. Theorem 1.3. Let the system parameters P , µ, λ be given and satisfy the conditions (1.8)-(1.9) and P (ρ) = Kρ, (1.30) where K > 0 is a given constant. Givenρ > 0 and (ρ 0 −ρ, u 0 , ∇φ 0 ) ∈ H 4 (R 3 ), assume that (ρ −ρ, u, φ) is the smooth classical solution on R 3 × [0, T ]. Let T * ≥ T be the maximal existence time of the solution. If T * < ∞, then we have (1.31) The rest of the paper is organised as follows. In Section 2, we derive a priori bounds for smooth, local-in-time solutions to (1.1)-(1.2) under the assumption that densities are non-negative and bounded. In Section 3 we derive pointwise bounds for the density, bounds which are independent both of time and of initial smoothness. This will then close the estimates as obtained in Section 2 to give an uncontingent estimate for the smooth solutions, thereby proving Theorem 1.1. In Section 4, together with the a priori bounds obtained in previous sections, we prove Theorem 1.2 by applying compactness arguments. Finally in Section 5, we give the blow-up criterion for solutions to (1.1)-(1.2) by obtaining estimates on solutions without smallness assumption on initial data.

Energy Estimates
In this section we derive a priori bounds for smooth, local-in-time solutions of (1.1)-(1.2) whose densities are non-negative and bounded. We first recall the following local-in-time existence theorem for the system (1.1)-(1.2) (see for example [9] and the references therein): Theorem 2.1. Givenρ > 0 and initial data (ρ 0 −ρ, u 0 , φ 0 ) ∈ H 4 (R 3 ), we can find T > 0 such that the classical solution (ρ −ρ, u, φ) to (1.1)-(1.2) exists on R 3 × [0, T ]. Moreover, (ρ −ρ, u, φ) satisfies Given T > 0, we now fixρ,ρ as described in Section 1 and a smooth classical 3) with initial data (ρ 0 −ρ, u 0 , ∇φ 0 ). With respect to (ρ −ρ, u, φ) , we then define functionals Φ(t) and H(t) for a given such solution by where σ(t) ≡ min{1, t}. We obtain a priori bounds for Φ(t) and H(t) under the assumptions that • the initial energy C 0 in (1.11) is small; • the density ρ remains bounded above and non-negative. The results can be summarised in the following theorem: then the following bound holds Unless otherwise specified, M will denote a generic positive constant which depends on T and the same quantities as the constant C T in the statement of Theorem 1.2 but independent the regularity of initial data. And for simplicity, we write P = P (ρ) andP = P (ρ), etc., without further referring.
We begin with the following L 2 -estimate on (ρ −ρ, u, φ) which is valid for all t ∈ [0, T ].
Proof. By direct computation, we readily have norm of (ρ −ρ) (see [6] for related discussion). On the other hand, using the equations (1.1) 1 and (1.1) 3 , we have and hence Therefore we obtain the following energy balance equation: (2.10) and the result (2.9) follows.
Next we prove the following lemma which gives some auxiliary bounds on φ. These bounds will be useful in later analysis.
where r > 3 and C(r) > 0 depends on r. Proof.
and (2.11) follows by (2.9). To prove (2.12), using (1.15) and (2.9), for r > 3, there exists α r ∈ (0, 1) and C(r) > 0 such that We prove the following auxiliary L q estimates on the velocity u which will be used for controlling the L 3 norm of ∇u. (2.14) Proof. The proof is similar to the one given in [7] page 323-324. From the momentum equation, we have Adding the equation |u| q (ρ t + div(ρu)) = 0 and integrating, we then obtain, for Using the estimate (2.12), for r = 4 the term t 0 R 3 |u| q ρ∇φ can be bounded by and using the hypothesis (1.9) on µ and λ, we can bound the integrand in the double integral on the left side of (2.15) from below as follows.
On the other hand, there exists some M > 0 such that for each ε > 0, the right side of (2.15) is bounded by Hence we conclude that for all ε > 0, By choosing ε sufficient small and applying Gronwall's inequality, (2.14) follows for all t ∈ [0, T ].
We recall the following bounds for u, ω and F in W 1,r which are required for the derivation of estimates for the auxiliary functionals H(t) and Φ(t). The proof can be found on page 505 in [10].
We are ready to prove some higher order estimates on u andu which are crucial in bounding the functional Φ(t) in terms of for some θ > 0.
Proof. The proof is similar to the one given in [6]. To prove (2.18), we first multiply the equation (1.1) 2 by σu and integrate to get For the term involving φ, using Hölder's inequality and the estimate (2.9), hence the term can be absorbed by the left side of (2.18), and hence (2.18) follows. Next, we apply the differential operator ∂ t + u · ∇ on the equation (1.1) 2 and make use of the transport theorem to obtain sup 0≤s≤t σ 3 Using the estimates (2.11), (2.12) and (2.13), we have can be readily bounded in terms of Φ. For the term M t 0 R 3 σ 3u · (u · ∇(ρ∇φ)) , we can bound it as follows. (2.21) In Lemma 2.8 listed below, we obtain the bound on H(t) in terms of Φ(t).
Proof. First we bound the term t 0 σ 3 ∇u(·, s) 4 L 4 as appeared in the definition of H(t). With the help of (2.16), We estimate Using the same method, the terms t 0 σ 3 ω(·, s) 4 L 4 ds can be bounded by Putting the estimates back into (2.23), we obtain the following bound on The first term on the right can be estimated by . Proceeding in the same way, the second term involving ω is also bounded by Hence we obtain It remains to estimate the summation term . The proof is similar to the one given in [6] page 239-241 and we just sketch here. We make use of the decomposition and write u as u = z + w so that Hence for each r ∈ (1, ∞), there is a constant M (r) such that for t > 0, Given j 1 , j 2 , j 3 , k 1 , k 2 , k 3 ∈ {1, 2, 3}, we have We readily have For A 3 , using (2.9) and (2.5), we can estimate it as follows.
0 . Substituting the above estimates back into (2.23), we obtain the bound (2.22) as required.
Proof of Theorem 2.2. Theorem 2.2 follows immediately from the bounds (2.21) and (2.22), and the fact that those functionals Φ(t) and H(t) are all continuous in time.

Pointwise bounds for the density and Proof of Theorem 1.1
In this section we derive pointwise bounds for the density ρ, bounds which are independent both of time and of initial smoothness. This will then close the estimates of Theorem 2.2 to give an uncontingent estimate for the functionals Φ(t) and H(t) defined in (2.4)-(2.5). The result is formulated as Theorem 3.1. Theorem 1.1 will then be proved using the a priori estimates derived from Theorem 3.1.
Proof. The proof consists of a maximum-principle argument applied along particle trajectories of u which is similar to the one given in [19] pg. 48-50 for the corresponding magnetohydrodynamics system as well as those in [17]- [18] with slight modifications. We choose a positive number b ′ satisfyinḡ for some positive τ . It then follows from Theorem 2.2 that where M is now fixed. We shall show that if C 0 is further restricted, then in fact 0 ≤ ρ < b ′ on R 3 × [0, τ ], and so by an open-closed argument that 0 ≤ ρ < b ′ on all of R 3 × [0, T ], we have Φ(t) + H(t) ≤ M C θ 0 as well. We will only give the proof for the upper bound, the proof of the lower bound is just similar.
Fix y ∈ R 3 and define the corresponding particle path x(t) by Suppose that there is a time t 1 ≤ τ such that ρ(x(t 1 ), t 1 ) = b ′ . We may take t 1 minimal and then choose t 0 < t 1 maximal such that ρ(x(t 0 ), t 0 ) =ρ. Thus We have from the definition (1.4) of F and the mass equation that Integrating from t 0 to t 1 and and abbreviating ρ(x(t), t) by ρ(t), etc., we then obtain for a constantM which depends on the same quantities as the M T from Theorem 3.1 and θ ′ > 0. If so, then from (3.4), where the last inequality holds because P is increasing and P (s) −P (s) is nonnegative on [t 0 , t 1 ]. But (3.6) cannot hold if C 0 is small depending onM , b ′ , andρ. Stipulating this smallness condition, we therefore conclude that there is no time t 1 such that ρ(t 1 ) = ρ(x(t 1 ), t 1 ) = b ′ . Since y ∈ R 3 was arbitrary, it follows that ρ < b ′ on R 3 × [0, τ ], as claimed. To prove (3.5), we rewrite the right hand side of (3.4) as a space-time integral. Let Γ be the fundamental solution of the Laplace operator in R 3 , then we apply (1.6) to obtain (3.7) Using (1.18), the first integral on the right of (3.7) is bounded in the same way as in Lemma 4.2 of Hoff [6] t1 t0 R 3 For the second integral on the right side of (3.7), we observe that, for all s ∈ [0, τ ], and (ρ −ρ, u, ∇φ)(·,t) ∈ H 4 (R 3 ). Since the system (1.1) is autonomous, we can therefore reapply Theorem 2.1 at the new initial timet to extend the solution (ρ −ρ, u, φ) eventually to all [0, T ] in finitely many steps by replacing δt with some smaller number δ T in (3.8). In other words, there exists δ T > 0 depending on T such that if the initial data (ρ 0 −ρ, u 0 , ∇φ 0 ) ∈ H 4 (R 3 ) is given satisfying (1.3) and (1.10)-(1.11) with C 0 < δ T , then (ρ −ρ, u, φ) can be extended to the whole time interval [0, T ]. It finishes the proof of Theorem 1.1.

Existence of weak solutions and Proof of Theorem 1.2
In this section, we prove Theorem 1.2 by obtaining weak solutions to the system (1.1)-(1.2) on R 3 × [0, T ] where T > 0 is any given time. To begin with, we let initial data (ρ 0 , u 0 , φ 0 ) be given satisfying the hypotheses (1.3) and (1.10)-(1.11) of Theorem 3.1, and we fix those constants δ T , M T and θ T defined in Theorems 3.1.
In view of Lemma 4.1, the desired weak solution (ρ, u, φ) will then be obtained by taking η → 0, with the use of the compactness provided by those bounds in (4.1)-(4.2) and (4.4). It can be summarised in the following lemma: There is a sequence η k → 0 and functions u, ρ and φ such that as k → ∞, ∇u η k (·, t), ∇ω η k (·, t) ⇀ ∇u(·, t), ∇ω(·, t) (4.8) weakly in L 2 (R 3 × [0, T ]); and ρ η k (·, t), ∆φ η k → ρ(·, t), ∆φ(·, t) (4.10) Proof. The uniform convergence (4.7) follows from Lemma 4.1 via a diagonal process, thus fixing the sequence {η k }. The weak-convergence statements in (4.8) and (4.9) then follow from the bound (4.1) and considerations based on the equality of weak-L 2 derivatives and distribution derivatives. The convergence of approximate densities in (4.10) for a further subsequence can be achieved by applying the argument given in Lions [12] pp. 21-23 and extended by Feireisl [5], pp. 63-64 and 118-127, and we omit the details here. We will prove Theorem 1.3 using a contradiction argument. Specifically, for the sake of contradiction, we assume that for some constantC > 0. Based on the assumption (5.4), we derive a priori estimates for the local smooth solution (ρ −ρ, u, φ) on [0, T ] with T ≤ T * . Those estimates are different from we did in Section 2 and Section 3 in the sense that there is no smallness assumption imposed on the initial data, hence a more delicate analysis is required in bounding the solution (ρ −ρ, u, φ).
To facilitate the proof, we introduce the following auxiliary functionals: We first recall the following lemma which gives estimates on the solutions of the Lamé operator µ∆ + (µ + λ)∇div. More detailed discussions can also be found in Sun-Wang-Zhang [20].
Proof. A proof can be found in [20] pg. 39 and we omit the details here.
Givenρ > 0 and initial data (ρ 0 −ρ, u 0 , ∇φ 0 ) ∈ H 4 (R 3 ), we define We begin to estimate the functionals Φ 1 , Φ 2 and Φ 3 under the assumption (5.4). Similar to the previous cases, M will denote a generic positive constant which further depends onC,C, T * and S 0 . We first have the following bounds based on the results we obtained in Section 2, namely for any 0 ≤ t ≤ T ≤ T * and r > 3, sup 0≤s≤t R 3 Next we are going to estimate Φ 1 which is given in the following lemma: Proof. Following the proof of Lemma 2.7, we have (5.16) Using (5.4) and (5.10), the first term on the right side of (5.16) can be estimated as follows. 1 , and the second term can be bounded by 3 . Applying the above bounds on (5.16), the result follows.
Before we estimate Φ 2 , we introduce the following decomposition on u. We write where u p and u s satisfy Using Lemma 5.2, the term u p can be bounded by We give the estimates for u s as follows.
We finally obtain the bound on Φ 3 in terms of Φ 2 .
Therefore Lemma 5.2 implies that Proof of Theorem 1.3. In view of the bounds (5.15), (5.25) and (5.27), we can conclude that for 0 ≤ t ≤ T ≤ T * , Together with the pointwise boundedness assumption (5.4) on ρ and apply the similar argument given in [16], we can show that for 0 ≤ t ≤ T ≤ T * , sup 0≤s≤t ||(ρ −ρ, u, ∇φ)(·, s)|| H 4 + for some M ′′ which depends on C 0 ,C, T * and the system parameters P , µ, λ and K. An open-and-closed argument on the time interval can then be applied which shows that the local solution (ρ −ρ, u, φ) can be extended beyond T * , which contradicts the maximality of T * . Therefore the assumption (5.4) does not hold and this completes the proof of Theorem 1.3.