Long time behavior of weak solutions to Navier-Stokes-Poisson system

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1 Long time behavior of weak solutions to Navier-Stokes-Poisson system Peter Bella Abstract. In (Disc. Cont. Dyn. Syst. 11 (24), 1, pp ), Ducomet et al. showed the existence of global weak solutions to the Navier Stokes Poisson system. We study the global behavior of such a solution. This is done by (1) proving uniqueness of a solution to the stationary system; (2) by showing convergence of a weak solution to the stationary solution. In (1) we consider only the case with repulsion. We prove our result in the case of a bounded domain with smooth boundary in R 3 and also in the case of the whole space R 3. Keywords. Compressible fluids, Navier-Stokes equation, Weak solutions, Poisson equation, Long time behavior. Mathematics Subject Classifications (2): 35B4, 35D99, 35Q3, 76N1 1. Introduction In [4], Ducomet et al. showed existence of a global weak solution to the Navier Stokes Poisson system for compressible fluid. In this work, we are interested in the global behavior of the weak solution; more precisely, we show convergence of the weak solution to the solution of the stationary problem. In general, the Navier Stokes system for compressible, isentropic gas has the form: ρ t + div (ρu) =, (1.1) (ρu) t + div (ρu u) + p = µ u + (λ + µ) (div u) + Gρ Φ, (1.2) where p is the pressure, Φ is an external force, G is a given constant, and the viscous coefficients µ and λ satisfy physical conditions µ >, λ µ. Depending on the sign of the constant G, the term Gρ Φ represents either attractive (G > e.g., gravitation) or repulsive (G < e.g., Coulomb force in the case of charged particles in plasma) interactions.

2 2 Peter Bella The pressure p should satisfy a constitutive law: p = p(ρ), p C 1 ([, )), p() =, 1 a ργ 1 p (ρ) aρ γ 1 for ρ and some a >, where the adiabatic index γ satisfies (1.3) γ > 3 2. This technical condition is very common for the Navier Stokes equations for a compressible fluid. Observe that monatomic gases, for which the pressure has the form p(ρ) = aρ 5/3, satisfy this condition. To close the system, we consider Poisson s equation for the potential Φ: Φ = ρ + g, (1.4) where g = g(x) is a given function in L 1 () L (). Equations (1.1), (1.2), and (1.4) form the Navier Stokes Poisson (NSP) system. Numerous questions for the NSP system, e.g., existence, long-time behavior, and stability of solutions, were studied over the last decade. In the case G >, Ducomet et al. [4] showed the existence of weak solutions, with Φ being a Newton potential defined on the whole R 3. Their proof works also in the case G <, providing the proof of existence of the weak solution in our setting. Later, Kobayashi and Suzuki [8] adopted their technique for the case when Φ satisfies Neumann boundary data on (i.e., Φ n = at ). If G <, existence of the weak solution in a bounded domain with Φ satisfying Dirichlet boundary condition was proved in [2]. All these existence results assume that the pressure satisfies condition similar to (1.3). For a different form of the pressure (p = aρlog d ρ), Tan and Zhang [12] proved an existence result in two dimensions using Orlicz-Sobolev function spaces. The long-time behavior of a solution to NSP was studied in several settings. In a series of papers, Makino et al. [1] considered the spherically symmetric case and proved stability of the stationary solution. A similar result, again assuming spherical symmetry, was obtained in [3]. For classical solutions, Li et al. [9] proved convergence of the solution and identified the optimal rate of this convergence. Their argument, based on semigroup theory, works only for initial data close to the stationary solution. Recently, Tan and Wang [11] considered a magnetohydrodynamic model with a Coulomb force. More precisely, they considered equations (1.1), (1.2), and (1.4) with G <, together with an equation for the magnetic field. In contrast with our setting, their potential Φ satisfies Neumann boundary data and the set is a bounded smooth domain in R 2 or R 3. A similar problem, but without magnetic field and with positive G, was considered in [7], where compactness of global trajectories was shown. At first glance, results [7, 11] look very similar to ours. But closer investigation reveals that new challenges arise in our setting because of the choice of the domain and the different conditions on Φ. More precisely, we consider the Newton potential Φ defined in the whole space R 3, decaying to

3 Long time behavior of weak solutions to NSP system 3 zero at infinity. Then the stationary problem, obtained as a limit when time goes to infinity, possesses non-trivial solutions, whereas in [7, 11] the only solutions to the stationary problem are trivial. Our analysis has two parts. First, we need to show convergence of a solution of the NSP system to some solution of the stationary problem. In this part, we follow the argument of Feireisl and Petzeltová [5]. A similar idea was used in [11]. Since in [5] the potential Φ is given, an additional idea is needed here to treat the coupling between density ρ and potential Φ. Furthermore, in the case = R 3 we need to control the behavior of the solution at infinity (Lemma 5.3). The second part of our analysis shows uniqueness of the weak solution of the stationary problem. First we prove regularity of such a solution and then we show its uniqueness. The proof of uniqueness is the only point in our argument where the assumption G < is needed. Since the solution of the stationary problem can have zero density in a large set (a process called cavitation), equation (2.4) for the density does not provide any information in this case. For this reason the stationary problem is a free boundary problem. As it is well known, the mathematical theory of free boundary problems is not yet complete and even some basic questions are not well understood. To avoid free boundary, some authors assume that the density is strictly positive in a great simplification. Unfortunately, cavities were observed in physical experiments and therefore assuming strictly positive density is very unnatural. For this reason, we do not assume this condition in our analysis. The paper is organized as follows. In Section 2 we define the notion of a finite energy weak solution and state the main results. Section 3 includes all estimates needed for the proof of convergence of the weak solution. In Section 4 we prove the uniqueness results for solutions of the stationary problem. Sections 5 and 6 contain the proof of convergence. 2. Weak formulation and the main results The equation of motion (1.2) should be satisfied in a domain R 3, where on the boundary we prescribe zero Dirichlet boundary condition for u: u =. The continuity equation (1.1) and the Poisson equation (1.4) hold in the whole R 3 assuming that u and ρ are extended by zero outside of the domain. If ρ,u were smooth, it would follow from equations (1.1) and (1.2) that the energy of the system is non-increasing, more precisely: d dt E(t) + µ u 2 + (λ + µ) div u 2 =, (2.1)

4 4 Peter Bella where E(t) = E[ρ,u](t) = 1 2 ρ(t,x) u(t,x) 2 + P(ρ(t,x)) 1 [ 2 Gρ(t,x) ρ(t,y) + 2g(y) R x y 3 ] dy dx, ρ p(z) P(ρ) = ρ z 2 dz. Under same smoothness asumptions it follows from the continuity equation (1.1) that b(ρ) t + div (b(ρ)u) + (b (ρ)ρ b(ρ))div u = (2.2) for any smooth function b. Solutions satisfying relation (2.2) are called renormalized [1]. Definition. Motivated by (2.1), (2.2), we define finite energy weak solution of the problem (1.1)-(1.4) in the set Q := (, ) as a triple of functions ρ,u,φ satisfying ρ ; ρ L loc (, ;L1 L γ ()),u L 2 loc (, ;[D1,2 ()]3 ); the energy inequality d dt E(t) + µ u 2 + (λ + µ) div u 2, (2.3) holds in D (, ) (energy E(t) was defined above); equation (1.1) is satisfied in D ((, ) R 3 ), where ρ and u are extended by zero outside of ; moreover, equation (1.1) is satisfied also in the sense of renormalized solutions, i.e. (2.2) holds in D (Q) for any b C 1 (R) such that b (z) for all z M = M(b); the momentum equation (1.2) is satisfied in D (Q) and the Poisson equation (1.4) holds a.e. in R 3 for any t (, ) assuming ρ is extended by zero outside of. The space D 1,2 (), used in the previous definition, is the closure of compactly supported smooth functions in with respect to the norm ( ) 1 v 2 2. In a similar manner, by D m,q () we denote homogeneous Sobolev spaces D m,q () := { u L 1 loc() : D l u L q (), l = m }. If is a bounded domain, these spaces coincide with usual Sobolev spaces W m,q () (for more details on homogeneous Sobolev Spaces see [6]).

5 Long time behavior of weak solutions to NSP system 5 The energy inequality (2.3) suggests, at least formally, that for some sequence of times t n we should have ρ(t n ) ρ s, ρu(t n ), Φ(t n ) Φ s, where ρ s,φ s is the solution of the stationary problem: p(ρ s ) = Gρ s Φ s in, (2.4) Φ s = ρ s + g a.e. in R 3. In the last equation ρ s and g are extended by outside of. Our main results show this fact for finite energy weak solutions in the repulsive case G <. Theorem 1. Let = R 3, G < and g. Then for any finite energy weak solution of the Navier Stokes Poisson system (1.1)-(1.4) we have where Φ s solves Φ s = g. ρ(t) in L γ (R 3 ), Φ(t) Φ s in D 2,γ (R 3 ), ess sup ρ(τ) u(τ) 2 dx as t, τ>t R 3 Theorem 2. Let R 3 be a bounded domain with the outer ball property and Lipschitz and connected boundary and let G <, g =. Then any finite energy weak solution of the Navier Stokes Poisson system (1.1)-(1.4) converges to a solution ρ s,φ s of the stationary problem, more precisely ρ(t) ρ s in L γ (), Φ(t) Φ s in W 2,γ (), ess sup ρ(τ) u(τ) 2 dx as t. τ>t Moreover, the limit ρ s,φ s is the unique solution of the stationary problem consistent with the conservation of mass. 3. Energy estimates and local convergence Lemma 3.1. Under the hypotheses of Theorem 1 or Theorem 2 let ρ,u,φ be a finite energy weak solution of (1.1)-(1.4). Then for every ǫ > there exists a constant C = C(ǫ, g L (R 3 )) such that for a.e. t > : 2g(y) + ρ(t,y) ) ρ(t, x) dy dx ǫ ρ(t) γ L ( ρ(t) R x y γ+c L 1 () + ρ(t) q L 1 (), 3 where q = 5γ 6 3γ 4 < 3.

6 6 Peter Bella Proof. We split the left hand side of the inequality into two integrals. The first one is easy to estimate: [ ] 2g(y) ρ(t, x) R x y dy dy dx 2 g L (R 3 ) ρ(t, x) 3 R x y dx 3 C 1 ρ(t) L 1 (). Since ρ L loc (, ;L1 L γ ()), we have ρ(t) L 6/5 () for a.e. t >. Then [ ] ρ(t,y) ρ(t, x) R x y dy dx ρ(t) 1 L 6/5 () x ρ(t) 3 L 6 (R 3 ) (3.1) C ρ(t) 2 L 6/5 (), where we used Young s inequality and Hardy Littlewood Sobolev Lemma. Finally, using interpolation and Young s inequality we have ρ(t) 2 L 6 5 (R 3 ) = ρ(t) 2 L 6 5 () ρ(t) 5γ 6 3(γ 1) L 1 () ρ(t) ǫ ρ(t) γ 5γ 6 L γ () + C 3γ 4 3 ρ(t) L 1 (). Since γ > 3 5γ 6 2, we get q = 3γ 4 < 3, what concludes the proof. γ 3(γ 1) L γ () Lemma 3.2 (Conservation of mass). Under the hypotheses of Theorem 1 or Theorem 2, the mass m[ρ(t)] is time invariant, i.e. m := ρ(t,x)dx = ρ(s,x)dx, for a.e. < s t. Furthermore, there exists a constant E such that ess sup ( ρ(t) + ) ρ(t)u(t) L2()3 + u(t) 2 L1 Lγ() L 2 () dt E. t>1 Proof. The proof of this Lemma is standard and can be found in [5]. Corollary 3.3. Under the hypotheses of Theorem 1 or Theorem 2 we have lim τ τ+2 τ 1 u 2 ρ L 2 () + 2 u L + ρ u 2 1 L 3γ γ+3 () L 1 L 6γ dt =. γ+6 () (3.2) Lemma 3.4 (Analogy of Lemma 4.1 in [5]). Let φ C () be such that Let b C 1 (R) satisfies where suppφ, φ 1, φ M in. b,b, b(z) = for z, zb (z) cz θ for z, { 1 < θ < min 4, 2 } 3 γ 1. 1

7 Long time behavior of weak solutions to NSP system 7 Then, under the hypotheses of Theorem 1 or Theorem 2, there exists Y = Y (b) such that τ+1 ( τ+2 ) lim sup τ p(ρ)b(ρ)φ 2 dxdt Y (b) lim sup τ p(ρ) φ dxdt + 1. τ Proof. The proof is the same as of Lemma 4.1 in [5]. The only difference is in the last but one estimate, which is: τ+2 ( i Φ)ρφψ(t τ)a i [φb(ρ)] dxdt Y 5 (b) ρ(t) L γ () Φ(t) L γ (R 3 ) τ 1 where we used τ 1 Φ W 2,γ (R 3 ) C ρ L γ (). Y 5 (b)c ρ(t) 2 L γ (), Corollary 3.5 (Corollary 4.1 in [5]). Under the hypotheses of Theorem 1 or Theorem 2 let { 1 < θ < min 4, 2 } 3 γ 1. Then, for every closed set K there exists a constant C = C(θ,K) such that τ+1 ρ γ+θ dxdt C(θ,K) for all τ > 1. τ K Now we consider a sequence of times τ n as in [5] (we can assume that τ n > 2) and define ρ n (t,x) := ρ(t + τ n,x) Φ n (t,x) := Φ(t + τ n,x) u n (t,x) := u(t + τ n,x) for t ( 1,2), x. Proposition 3.6 (Proposition 4.1 in [5]). Assuming the hypotheses of Theorem 1 or Theorem 2, every sequence τ n contains a subsequence such that weakly in L γ (( 1,2) ), ρ n = ρ(t + τ n ) ρ s strongly in L γ (( 1,2) K) for any compact set K. Moreover, ρ s is time-independent and ρ s dx ρ(t) = m. Proposition 3.7. Under the hypotheses of Theorem 1 or Theorem 2, any sequence τ n contains a subsequence such that Φ n Φs weakly* in L ( 1,2;D 2,γ ()),

8 8 Peter Bella limit function Φ s is time-independent and solves the stationary problem p(ρ s ) = Gρ s Φ s Proof. Since Φ(t) = ρ + g, we have in D (), Φ s = ρ s + g a.e. in R 3. Φ(t) D 2,γ (R 3 ) C 1( ρ L γ () + 1). Lemma 3.2 implies sup t>1 ρ(t) L γ () E and so Φ n is bounded in L ( 1,2;D 2,γ ()). Therefore we can find a weakly* converging subsequence (denoted again by Φ n ) and particularly Φ n Φs in L ( 1,2;D 2,γ ()), Φ n Φs in L ( 1,2;D 1,γ ()). (3.3) Now let φ D(( 1,2) ). Then there exists a compact set K such that suppφ ( 1,2) K and ρ n ρ s strongly in L γ (( 1,2) K) by Proposition 3.6. Testing equation (1.2) with φ we get: 2 1 ρ n u n φ t + ρ n u n u n φ + p(ρ n )div φdxdt = = 2 1 µ u n φ + (λ + µ)div u n div φ Gρ n Φ n φdxdt. Using Corollary 3.3, Proposition 3.6, and (3.3) we can pass to the limit and obtain 2 2 p(ρ s )div φdxdt = Gρ s Φ s φdxdt, and so 1 1 p(ρ s ) = Gρ s Φ s in D (( 1,2) ). Similarly, we multiply equation (1.4) with any test function φ D(( 1,2) ): 2 2 Φ n φdxdt = (ρ n + g)φdxdt. 1 1 Again, passing to the limit using (3.3) we get Φ s = ρ s + g, (3.4) in D (( 1,2) ). Since the right hand side in (3.4) is time-independent, so is the left hand side.

9 Long time behavior of weak solutions to NSP system 9 4. Uniqueness of the stationary solution Lemma 4.1 (Uniqueness of ρ s in R 3 ). Under the hypotheses of Theorem 1 there exists a unique weak solution ρ s L 1 (R 3 ) L γ (R 3 ) of the stationary problem p(ρ s ) = Gρ s Φ s, (4.1) Φ s = ρ s + g. (4.2) Proof. First we need to prove some regularity for ρ s. Since the right hand side of (4.2) belongs to L γ (R 3 ), we have Φ D 2,γ (R 3 ). Then the right hand side of (4.1) is in L q (R 3 ) for q = 3γ 6 γ ; so p(ρ s) D 1,q (R 3 ) L q (R 3 ), where 1/q = 1/q 1/3. Since the function p 1 (x) growths like x 1/γ, we get that ρ s L q γ (R 3 ), where q is strictly larger than 1. Using the bootstrap argument we get ρ s L 1 L ( R 3). Smoothness of the function p 1 implies that ρ s is continuous everywhere and differentiable in the set Thus, in the set +, we rewrite (4.1): + := { x R 3 : ρ s (x) > }. p (ρ s ) ρ s = G Φ s ρ s P(ρ s ) = G Φ s, where P(t) = t p (s) s ds for t > and P(t) = for t <. Therefore P(ρ s (x)) = GΦ(x) + C(x), x +, where C(x) is a constant function on every connected component of +, and so ρ s (x) = P 1 (GΦ s (x) + C(x)) for every x +. Since ρ s is a continuous function, the set + is open; we write it as a disjoint union of its connected components, i.e. + = i, i 1 where every i is connected and C(x) = C i > for all x i. Consider arbitrary i. By the continuity of ρ s we have GΦ s + C i = on i. If i is bounded, then, by the maximum principle for superharmonic function (recall that Φ s = ρ s + g and G < ), we obtain Φ s (x) C i x i G and so GΦ s (x) + C i inside i. Hence ρ s = in i, a contradiction.

10 1 Peter Bella Now let i be unbounded. Since lim x Φ s (x) =, there exists R > such that Φ s < Ci 2G outside of B(,R). Then for x i B(,R) C we have ( ) ρ s (x) = P 1 (GΦ s (x) + C i ) P 1 Ci δ > ; 2 so i B(,R) C and R 3 ρ s (x)dx =, a contradiction. Lemma 4.2 (Uniqueness of ρ s in a bounded domain). Under the hypotheses of Theorem 2 there exists a unique weak solution ρ s L 1 () L γ () of the stationary problem p(ρ s ) = Gρ s Φ s, Φ s = ρ s, where ρ s satisfies ρ s L1 () = m for a given m >. Proof. The first part of the proof (regularity of ρ s ) follows the proof of Lemma 4.1. Thus we have ρ s (x) = P 1 (GΦ s (x) + C i ) for x i. Fix one i. First we will prove that i. Since x i \ implies ρ s (x ) =, we only need to show max i ρ s >. Let us assume the contrary: for any x i we have ρ s (x ) = and Φ s (x ) = Ci G. By the definition, for any y i we have Φ s (y) < C i G. (4.3) Since Φ s in i, by the maximum principle we have min Φ s (x) = min Φ s (y), x i y i a contradiction with (4.3). Potential Φ s is a continuous function, which is non-negative and decays to at infinity, hence it achieves its maximum. Therefore we can define a set { } M := x R 3 : Φ s (x) = max Φ. R 3 and U := C {ρ s = }. We decompose int U = j 1V j V, where V is the only unbounded component and V j are remaining components. Recall that + := { x R 3 : ρ s (x) > }. We will show that + is connected and that +. Let us assume the contrary, particularly i for any i. We investigate where the maximum of Φ s could be. Here are the possible cases:

11 Long time behavior of weak solutions to NSP system 11 M V :. Since Φ s = in V and the maximum is achieved in V, we know that Φ s is constant in V. Hence, Φ s = in V and the maximum of Φ s over R 3 is. Hence =, a contradiction. M :. Let x M. Since M V =, we have Φ s (x ) > Φ s (x), x c V. Then using Hopf s Lemma we get that normal derivative of Φ s at the point x is strictly negative, and so Φ s can t have the maximum at x. M V j :. Let x M V j. Then x necessarily also belongs to some i and Φ s (x ) = Ci G. We know that i and that i. Take y i and y 1 \ i. Since is connected, there exists a curve Γ such that y = Γ() and y 1 = Γ(1). We find a point z = Γ(t) such that z i but Γ((t,t+ǫ)) i =. Since has Lipschitz boundary, there exists a sequence of points z n i \ converging to z. We know that Φ s (z n ) = Φ s (x ) and by continuity Φ s (z) = Φ s (x ). Since z, we showed that M, a contradiction with the previous case. M + :. Let x M i. Since G <, we know in particular that i M, a contradiction with the previous case. M V j :. Since Φ s = in V j and Φ s has the maximum in V j, Φ s has to be constant in V j and so the maximum is achieved also at the boundary of V j, a contradiction with the third case. We showed that is connected. Now we will prove that ρ s (x) = P 1 (GΦ s (x) + C), x. (4.4) Since this is true in +, it is enough to show that Φ s C/G in \ +. Let O be any component of \ +. Since +, we have O. Next, we know that Φ s + = C/G and Φ s in O, and so by the maximum principle we get Φ s C/G in O. It remains to show that for a given constant C there exists a unique solution ρ s,φ s of (4.4) and that for two solutions ρ s,1,ρ s,2 of (4.4) with constants C 1 > C 2 we have ρ s,1 > ρ s,2. To do the first, let us assume that ρ 1,Φ 1 and ρ 2,Φ 2 are two solutions of (4.4) with the same constant C. Then we have (Φ 1 Φ 2 ) = ( P 1 (GΦ 1 + C) P 1 (GΦ 2 + C) ) χ. Multiplying this equation by Φ 1 Φ 2, integrating over R 3 and using integration by parts we get R 3 (Φ 1 Φ 2 ) 2 + ( P 1 (GΦ 1 + C) P 1 (GΦ 2 + C) ) (Φ 1 Φ 2 ) χ =. Since P 1 is monotonically increasing and G <, the second term is less than or equal zero, and so Φ 1 = Φ 2. Now let C 1 > C 2 and let ρ 1,Φ 1 and ρ 2,Φ 2 be corresponding solutions. We define Φ := (GΦ 1 + C 1 ) (GΦ 2 + C 2 )

12 12 Peter Bella and := { x R 3 : Φ(x) < }. Since Φ 1 and Φ 2 go to as x goes to infinity, Φ goes to C 1 C 2 > and so the set is bounded. For x we have GΦ 1 (x)+c 1 < GΦ 2 (x)+c 2, and so ρ 1 (x) < ρ 2 (x). Hence Φ(x) = G( Φ 1 (x) Φ 2 (x)) = G(ρ 1 (x) ρ 2 (x)) > for x. If x \ the previous relation changes to equality. Then by the maximum principle the minimum of Φ over is attained at its boundary, where Φ =. Therefore Φ in, and so =. We showed that Φ in R 3 and so ρ 1 ρ 2. To prove that ρ 1 > ρ 2 we need to show that ρ 1 (x) > ρ 2 (x) for at least one x. If this is not the case (i.e. ρ 1 ρ 2 ), then Φ 1 Φ 2 and equation (4.4) can not be true with two different constants C 1 and C Proof of the strong convergence Lemma 5.1 (Proposition 6.1 in [5]). Let be a set with compact Lipschitz boundary and ρ,φ,u be a finite energy weak solution of NSP. Then for every ǫ > there exists an open neighborhood U (ǫ) of the boundary such that τ+1 lim sup p(ρ)dxdt ǫ. τ τ U (ǫ) Lemma 5.2 (Strong convergence of ρ n ). Let M be bounded. Then, under the hypotheses of Theorem 1 or Theorem 2, for every sequence τ n there exists a subsequence such that ρ n ρ s strongly in L γ ((,1) M). Proof. In the case = R 3 the Lemma follows directly from the Proposition 3.6. Let us therefore assume that. Fix ǫ >. Lemma 5.1 tells us that there exists a neighborhood U (ǫ) of the boundary such that τ+1 lim sup p(ρ)dxdt ǫ, τ and so using (1.3) we have lim sup τ τ τ+1 τ U (ǫ) U (ǫ) ρ(t,x) γ dxdt Cǫ. (5.1) Now let K be a compact set such that K U (ǫ) M. By Lemma 3.6 there exists a subsequence of τ n s.t. and so ρ n ρ s strongly in L γ ((,1) (K M)), lim ρ n L γ ((,1) (K M)) = ρ s L γ ((,1) (K M)). (5.2)

13 Long time behavior of weak solutions to NSP system 13 Using (5.1) and (5.2) we get lim sup ρ n Lγ ((,1) M) ρ s Lγ ((,1) M) + Cǫ1/γ, and finally lim sup ρ n Lγ ((,1) M) ρ s Lγ ((,1) M). (5.3) Since ρ n ρ s in L γ (), formula (5.3) holds with equality; Radon Riesz theorem then implies ρ n ρ s strongly in L γ ((,1) M)). Lemma 5.3 (About the travelling wave). Let R >. Then under the hypotheses of Theorem 1 every finite energy weak solution of the problem (1.1) (1.4) satisfies 1 lim sup sup ρ n (x,t) γ dxdt =. x R 3 B(x,R) Proof. Assume the contrary. Then there exists an ǫ >, a sequence of points {x n } n 1 R 3 and a subsequence of ρ n (denoted as ρ n ) such that 1 ρ n (x,t) γ dxdt ǫ. (5.4) B(x n,r) Now let us define shifted solutions : ρ n (t,x) := ρ n (t,x + x n ) and analogically ũ n and Φ n. Using Proposition 3.6 for K = B(,R) we get the strong convergence for the subsequence of ρ n to the stationary solution ρ s in L γ ((,1) B(,R)), i.e.: 1 1 ρ n (x,t) γ dxdt = ρ n γ dxdt, B(x n,r) a contradiction with (5.4). B(,R) Lemma 5.4 (Modification of Lemma 4.1 from [5]). Under the assumptions of Lemma 3.4 for = R 3 if 1 lim ρ n Φ n dxdt = (5.5) R 3 then 1 lim p(ρ n )b(ρ n )φ 2 dxdt =. τ R 3 Proof. The proof is the same as the proof in [5] but to this difference. The last but one estimate in the original proof reads: 1 1 ( i Φ n )ρ n φψ(t τ)a i [φb(ρ)] dxdt Y 5 (b) ρ n Φ n dxdt. R 3 R 3 Using (5.5) at this point we get our estimate.

14 14 Peter Bella Lemma 5.5. Under the assumptions of Theorem 1 the following holds: 1 lim sup ρ n γ dxdt =. R 3 Proof. Let R > and ǫ >. Then by Lemma 5.3 there exists n such that for all n n we have 1 ρ n γ dxdt ǫ. (5.6) sup x R 3 B(x,R) From this fact we would like to derive a similar estimates, but uniformly in time. Let x R 3 be arbitrary. Then from equation (5.6) we know that there exists a time t (depending on x ) such that ρ n (t,x) γ dx ǫ. B(x,R) Let φ be a smooth function such that φ = 1 on B(x,R 1), φ = on B(x,R) c, φ 1. Using equation (1.1) we get that for any t 1 (,1): φ(x)ρ n (x,t 1 )dx = φ(x)ρ n (x,t )dx + R 3 R 3 Estimate (3.2) and positivity of ρ n imply ρ n (x,t 1 )dx B(x,R 1) B(x,R) t1 for a sufficiently large n (independently of x ) and so ρ n (t,x)dx Cǫ 1/γ B(x,R 1) t R 3 ρ n u n φdxdt. ρ n (x,t )dx + ǫ for large n and any x,t. Since L γ (R 3 ) norm of ρ is bounded, we get by interpolation that ρ α n(t,x)dx Cǫ p B(x,R 1) for 1 α < γ with p = γ α γ(γ 1) >. Now we decompose function Φ n : Φ n (t) = ( ) 1 (ρ n + g)(t) = = ( ) 1 (ρ n χ B(x,R 1) )(t) + ( ) 1 (ρ n χ B(x,R 1) c)(t) + ( ) 1 (g) =: Φ 1 n(t) + Φ 2 n(t) + Ψ Using elliptic regularity result we get Φ 1 n (t,x) C ρn (t) L α (B(x,R)) Cǫp with any 3 2 < α < γ.

15 Long time behavior of weak solutions to NSP system 15 For the second part we have: Φ 2 n(t,x) = hence Φ 2 n (t,x) = B(x,R 1) c B(x,R 1) c ρ n (t,z) z x dz 1 R ρ(z) z x dz, B(x,R 1) c ρ n (t,z)dz m R 1. To simplify notation we define Φ n (t,x) := Φ 1 n(t,x) + Φ 2 n(t,x). Two previous estimates then imply Φ n (t,x) Cǫ p + m R 1. Sending first ǫ and then R we get that lim Φ n (t) = L (R 3 ) uniformly in t. From that, using ρ n (t) L 1 (R 3 ) = m, we obtain 1 lim ρ n (t,x)( Φ n (t,x))dxdt =. R 3 Hence 1 L2 lim Φ n = lim ρ n (t,x) Φ n (t,x)dxdt =. (5.7) (,1;L 2 (R 3 )) R 3 Now we would like to prove 1 lim ρ n Φ n dxdt R lim ρ n Φ n dxdt + lim ρ n Ψ dxdt =. (5.8) R 3 R 3 Since g L 1 (R 3 ) L (R 3 ) is independent of time (and so is Ψ), we get that 1 lim ρ n Ψ dxdt = R 3 as an easy consequence of Lemma 5.3. For γ 2 the function ρ n is bounded in L (,1;L 1 L γ (R 3 )), particularly ρ n is bounded in L 2 (,1;L 2 (R 3 )). Then using Hölder s inequality and (5.7) we get (5.8). Now let 3 2 < γ < 2. We want to show L1 lim Φ n =. (5.9) (,1;L γ (R 3 )) From the fact ρ n L (,1;L γ (R 3 )) C and from elliptic regularity results follows Φn C (5.1). (5.1) L (,1;L γ (R 3 ))

16 16 Peter Bella By interpolation we obtain: 1 p L2 Φ n dt C(1 α)p (5.1) Φ n, (5.11) (,1;L 2 (R 3 )) L γ (R 3 ) where 3 8γ 12 2 < γ < 2 implies α = 5γ 6 (,1) and p = 2 α > 2. Estimate (5.11), Hölder s inequality and (5.7) show (5.9). From that, using boundedness of ρ n in L (,1;L γ (R 3 )), we get (5.8) also for γ ( 3 2,2). Since we proved (5.8), Lemma 5.4 holds and so for φ 1 we have 1 lim p(ρ n )b(ρ n )dxdt =, (5.12) R 3 where b satisfies assumptions of Lemma 3.4. Let us now choose δ >. Then there exists b from Lemma 3.4 such that Then we have 1 1 ρ γ n = R 3 1 δ γ 1 ρ n +C R 3 Using (5.12) we have and hence for δ {ρ<δ} 1 b(z) = 1 for z δ. ρ γ n + lim sup 1 {ρ δ} ρ γ n R 3 p(ρ n )b(ρ n ) δ γ 1 m +C 1 1 lim R 3 ρ γ n dxdt δ γ 1, 1 R 3 p(ρ n )b(ρ n ). ρ γ n dxdt =. R 3 From Lemma 4.1 we see that the convergence is true not only for a chosen subsequence, but also for the whole sequence ρ n. 6. Proof of the main results Proof of Theorem 1 and Theorem 2. Since both Theorem 1 and Theorem 2 have almost identical proofs, we will prove them at once. In both cases, we will prove convergence of ρ(t) to ρ s and similar statement Φ(t) Φ s. Of course, in Theorem 1, the limiting ρ s and Φ s are equal zero (consequence of Lemma 4.1). We know (from Lemma 5.5 in the case of Theorem 1; using Lemma 5.2 and Lemma 4.2 in the case of Theorem 2) that for every sequence τ n the time shifts ρ n = ρ(t+τ n ) converge to the stationary state ρ s, more accurately and so ρ n ρ s strongly in L γ ((,1) ), (6.1) Φ n Φ s strongly in L γ (,1;D 2,γ ()). (6.2)

17 Long time behavior of weak solutions to NSP system 17 Energy inequality (2.3) implies convergence of the energy E(t) for t to some finite number Further, Corollary 3.3 says that lim τ E := ess lim E(t). t τ+1 τ 1 2 ρ u 2 dxdt =. Since ρ n ρ s in L q (,1;L 1 () L γ ()) for any 1 < q <, we get from (6.2) that τn+1 ρφ ρ s Φ s dxdt =. lim This inequality then implies τn+1 E = lim P(ρ) 1 τ 2 GρΦdxdt = τ n τ n P(ρ s ) 1 2 Gρ s Φ s dx = E[ρ s ], where we used (6.1) and P (t) Ct γ 1 for passing to the limit. We first finish the proof of Theorem 2. Using continuity equation (1.1) and Corollary 3.3 we have ρ(t) ρ s Φ(t) Φ s weakly in L γ () weakly in W 2,γ () as t. Since is a bounded domain, Φ(t) converge strongly to Φ s in L () by the compact embedding. Hence ρ(t)φ(t) ρ sφ s and (using P (t) = p (t) t >, i.e. convexity of P) E = P(ρ s ) 1 2 Gρ s Φ s dx liminf t ess lim sup t P(ρ(t)) 1 2 Gρ(t)Φ(t)dx limsup t 1 2 ρ(t) u(t) 2 + P(ρ(t)) 1 2 Gρ(t)Φ(t)dx = ess lim E(t) = E. t Hence ess sup ρ(τ) u(τ) 2 dx τ>t ρ(t) ρ s strongly in L γ (), and Φ(t) Φ s strongly in W 2,γ (). P(ρ(t)) 1 2 Gρ(t)Φ(t)dx To prove the case = R 3 (Theorem 1), we can t use compact imbedding as was used above. Instead, we will use the fact that the limiting ρ s is

18 18 Peter Bella identically zero, and so E =. Using we get liminf t ess lim sup t P(ρ(t)) 1 2 = ess lim t E(t) = E =. Therefore and G <, ρ, g, Φ Gρ(t)Φ(t)dx limsup P(ρ(t)) 1 t 2 Gρ(t)Φ(t)dx 1 2 ρ(t) u(t) 2 + P(ρ(t)) 1 2 Gρ(t)Φ(t)dx ess sup ρ(τ) u(τ) 2 dx τ>t R 3 ρ(t) strongly in L γ (R 3 ), Φ(t) Φ s strongly in D 2,γ (R 3 ). Acknowledgments This research was partially supported by Nečas Center for Mathematical Modeling, project LC652 financed by MSMT, and by NSF grant DMS The author also thanks Eduard Feireisl for suggesting the problem and stimulating discussions. References [1] R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989), no. 3, [2] Donatella Donatelli, Local and global existence for the coupled Navier-Stokes- Poisson problem, Quart. Appl. Math. 61 (23), no. 2, [3] B. Ducomet and A. Zlotnik, Stabilization and stability for the spherically symmetric Navier-Stokes-Poisson system, Appl. Math. Lett. 18 (25), no. 1, [4] Bernard Ducomet, Eduard Feireisl, Hana Petzeltová, and Ivan Straškraba, Global in time weak solutions for compressible barotropic self-gravitating fluids, Discrete Contin. Dyn. Syst. 11 (24), no. 1, [5] Eduard Feireisl and Hana Petzeltová, Large-time behaviour of solutions to the Navier-Stokes equations of compressible flow, Arch. Ration. Mech. Anal. 15 (1999), no. 1, [6] Giovanni P. Galdi, An introduction to the mathematical theory of the Navier- Stokes equations. Vol. I, Springer Tracts in Natural Philosophy, vol. 38, Springer-Verlag, New York, 1994, Linearized steady problems.

19 Long time behavior of weak solutions to NSP system 19 [7] Fei Jiang, Zhong Tan, and Qiaolian Yan, Asymptotic compactness of global trajectories generated by the Navier-Stokes-Poisson equations of a compressible fluid, NoDEA Nonlinear Differential Equations Appl. 16 (29), no. 3, [8] Takayuki Kobayashi and Takashi Suzuki, Weak solutions to the Navier-Stokes- Poisson equation, Adv. Math. Sci. Appl. 18 (28), no. 1, [9] Hai-Liang Li, A. Matsumura, and Guojing Zhang, Optimal decay rate of the compressible Navier-Stokes-Poisson system in R 3, 28. [1] Šárka Matušů-Nečasová, Mari Okada, and Tetu Makino, Free boundary problem for the equation of spherically symmetric motion of viscous gas. III, Japan J. Indust. Appl. Math. 14 (1997), no. 2, [11] Zhong Tan and Yanjin Wang, Global existence and large-time behavior of weak solutions to the compressible magnetohydrodynamic equations with Coulomb force, Nonlinear Anal. 71 (29), no. 11, [12] Yinghui Zhang and Zhong Tan, On the existence of solutions to the Navier- Stokes-Poisson equations of a two-dimensional compressible flow, Math. Methods Appl. Sci. 3 (27), no. 3, Peter Bella Courant Institute of Mathematical Sciences 251 Mercer St New York, NY 112 USA bella@cims.nyu.edu