On spherically symmetric motions of a viscous compressible barotropic and selfgravitating gas

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1 J. math. fluid. mech ), /99-, DOI.7/s9-3- c 29 Bikhäuse Velag Basel/Switzeland Jounal of Mathematical Fluid Mechanics On spheically symmetic motions of a viscous compessible baotopic and selfgavitating gas Benad Ducomet, Šáka Nečasová and Alexis Vasseu SUMMARY. We conside the Cauchy poblem fo the equations of spheically symmetic motions in R 3, of a selfgavitating baotopic gas, with possibly non monotone pessue law, in two diffeent situations: in the fist one we suppose that the viscosities µρ), and λρ) ae density-dependent and satisfy the Besch-Desjadins condition, in the second one we conside constant densities. In the two cases, we pove that the poblem admits a global weak solution, povided that the polytopic index γ satisfy γ >. Mathematics Subject Classification 2). 76N,35Q3. Keywods. spheically symmetic motion, selfgavitating gas, non monotone pessue law, density-dependent viscosities.. Intoduction We conside the Navie-Stokes-Poisson system in R 3 fo a compessible isentopic gas with density-dependent viscosities t ρ + divρ v) =, t ρ v) + divρ v v) div 2µρ)D v)) λρ)div v) + P ρ Φ =, Φ = 4πGρ..) Hee ρ is the density, v is the velocity, Φ is the Newtonian gavitational potential G > is the Newton constant), P is the pessue, D is the stain tenso with D v) = /2 v + t v).

2 2 Benad Ducomet, Šáka Nečasová and Alexis Vasseu JMFM The pessue P ρ) is elated to the density ρ by a geneal baotopic constitutive law see 4] fo motivations) satisfying { P C R + ), P ) =, a zγ b P z) az γ + b fo all z,.2) fo thee constants a >, b and γ >. Following Mellet and Vasseu 3], we suppose that µ and λ ae two viscosity coefficients satisfying µρ) = µ Ψρ), λρ) = 2µ ρψ ρ) Ψρ)),.3) whee Ψ is an inceasing function of ρ. The viscosity coefficients satisfy the constaints intoduced in 3] µ ρ) m, µ) λ ρ) m µ ρ), mµρ) 2µρ) + 3λρ) m µρ),.4) fo suitable positive constants µ, m, m and m. We also assume the uppe bound µρ) Cρ γ 2,.5) fo a positive constant C, the physical meaning of.5) follows fom the behavio of the tempeatue dependence of gaseous viscosity µ Cθ /2 and the baotopic equivalence θ Cρ γ see 3]). If γ 3, we also equie that lim inf ρ fo some ε >. We also define the auxiliay functions φ and Π µρ) >,.6) +ε ρ γ 3 ρφ ρ) = Ψ ρ), Πρ) := ρ ρ P z) z 2 dz. We want to solve the Cauchy poblem.) fo t > with initial conditions ρ t= = ρ x), ρ v t= = m x), on R 3..7) We focus on spheically symmetic solutions satisfying ρ x, t) = ρ x, t) and vx, t) = v x, t), and we study weak solutions fo the above poblem with x x

3 Vol ) on spheically symmetic motions 3 popeties ρ L, T ); L R 3 ) L γ R 3 ) ), ρ L, T ); H R 3 ) ), ρ v L, T ], L 2 R 3 )) ) 3, )) 9 9 µρ) D v) L 2, T ); W, loc R 3 ), λρ) div v L 2, T ); W, loc R ))) 3..8) We also assume the following conditions on the initial data ρ x) ρ x ) and m x) = m x ) x x : ρ on, m = on {x R 3 ρ x) = }, ρ L R 3 ) L γ R 3 ), ρ /2 µρ ) L 2 R 3 ), m 2 ρ L R 3 )..9) Fo any τ τ 2, the weak solution ρ, v) on R 3, T ] must satisfy, fo any test function ζ Cc R 3, T ]), the weak continuity equation τ 2 τ2 ρζ dx ρζ t + ρ v ζ) dx,.) R 3 R 3 τ = τ and fo any test function η Cc 2 R 3, T ]) ) 3 such that η, T ) =, the weak momentum equation T T m ηx, ) dx+ ρ v η t + v v : η) dx dt+ P ρ) div η dx dt R 3 R 3 R 3 T T + µρ) v ) η dx dt + 2µ ρ) ρ v j i ρ i η j dx dt R 3 R 3 T + µρ) v ) div η) dx dt R 3 T + 2 ρ µ T ρ)v i j ρ i ζ i dx dt + λρ) v ) div η) dx dt R 3 R 3 T T + ρ λρ) v ) ρ div η) dx dt = 4πGρ ρ η dx dt..) R 3 R 3 In this fomula, the five last tems in the left-hand side ae equested in ode to povide a igoous meaning to the viscous contibution in the equation of motion see 8]), and the ight hand side is a shotened notation fo T ) ρy, t) 4πGρ R 3 R x y dy η dx dt. 3 It is known afte 4] that the Cauchy poblem fo the system.)-.7) admits a enomalized) weak solution, when the viscosity coefficients ae positive constants, povided that the pessue P ρ) behaves like Cρ γ fo lage ρ, with

4 4 Benad Ducomet, Šáka Nečasová and Alexis Vasseu JMFM γ > 3/2. Let us obseve that the estiction on γ is the same as in the nongavitating case 7]. In 5] it was poved that the poblem.)-.7) admits also a weak solution when the viscosity coefficients ae density dependent functions elated by Besch- Desjadins elations ], povided that the pessue P ρ) behaves like Cρ γ fo lage ρ with a smalle γ > 4/3. In this last case, the estiction on γ was equested in ode to contol the gavitational tem. On the othe hand, in ecent woks, S. Jiang and P. Zhang ] in the enomalized case with constant viscosities), Z. Guo, Q. Jiu, Z. Xin 9] and T. Zhang, D. Fang 8] fo density dependent viscosities) poved that the Cauchy poblem fo the spheically symmetic vesion of the non-gavitating Navie-Stokes system also admits a weak solution, in the pue baotopic case P ρ) = Cρ γ when the condition γ > is achieved. In the pesent pape we show that the Cauchy poblem fo the spheically symmetic Navie-Stokes-Poisson system admits in both cases fo density dependent viscosities o in the enomalized case with constant viscosities) a weak solution, fo a geneal pessue law given by.2), without loss of γ i.e. with the optimal condition γ > ). In the density-dependent case Sections 2) we follow the stategy of Zhang and Fang 8] see also ]): fist conside the poblem.).7) when R 3 is eplaced by the ball B, R) then pass to the limit R. Moeove, as fa as the spheical symmety is concened with the classical divegence at the oigin, it is also equested to emove the oigin, and finally one is led to solve the poblem.).7) in the shell B ε,r := B, R)\B, ε) and pass to the limits ε, R ). In the enomalized case with constant viscosities Section 3) we adapt the poof of S. Jiang and P. Zhang ] to the Navie-Stokes-Poisson system with non-monotone pessue. 2. The density-dependent case The spheically symmetic vesion of.) eads ρ t + ρv) + 2ρv =, ρv t + vv ) = P + λ + 2µ) v + 2v )) 4µ v + ρf, 2.) in the domain, T ] R + fo the density ρ, t) and the velocity v, t). The Newton foce is now f, t) := Φ, t), whee the gavitational potential is easily computed as the suitable solution of the ode 2 2 Φ ) = 4πGρ.

5 Vol ) on spheically symmetic motions 5 One gets Φ, t) = with the kenel fo s, K, s) = 4πG fo s. s We conside Diichlet bounday conditions and initial conditions K, s)ρs, t)s 2 ds, 2.2) 2.3) v, t) =, fo t >, 2.4) ρ t= = ρ ), ρv t= = m ), on R ) Then ou main esult is the following Theoem 2.. Suppose that the initial data satisfy.9) and that T is an abitay positive numbe. Then fo any γ >, the poblem 2.)2.4)2.5) possesses a global weak solution satisfying.8) togethe with popeties.) and.). 2.. A pioi estimates fo appoximate solutions We follow the stategy of 8]: to get fist unifom bounds fo the density in an appoximate poblem depending on numbes < ε << and R >>, then pass to the limit ε, R ). So we conside the petubed poblem t ρ + divρ v) =, t ρ v) + divρ v v) div 2µ ε ρ)d v)) λ ε ρ)div v) + P ρ Φ =, Φ = 4πGρ, 2.6) in the tuncated domain B ε,r, T ], with the petubed viscosity coefficients µ ε ρ) = µρ) + 2 ερθ, λ ε ρ) = λρ) + θ ) ερ θ, 2.7) whee µ ε and λ ε have been adjusted in ode to satisfy elation.3) and θ 2/3, ). We want to solve 2.6) in B ε,r, T ], with Diichlet bounday conditions fo t >, and initial conditions ρ v Bε,R =, 2.8) ρ t= = ρ x), ρ v t= = m x), on B ε,r. 2.9)

6 6 Benad Ducomet, Šáka Nečasová and Alexis Vasseu JMFM The spheically symmetic vesion of 2.6) eads now ρ t + ρv) + 2ρv =, ρv t + vv ) = P + λ ε + 2µ ε ) v + 2v )) 4µ ε v ρf ɛ, in the domain ε, T ], with ε := ε, R), fo the tuncated Newton foce We supplement 2.) with and egulaized initial data f ε, t) := 4πG 2 ε ρs, t)s 2 ds. 2.) vε, t) = vr, t) =, fo t >, 2.) ρ t= = ρ,ε,r j δ, v t= = v,ε,r j δ, on ε, 2.2) whee j δ is a standad mollifie, and the tuncated data ae ρ ε) + ε fo, ε], ρ,ε,r ) = ρ ) + ε fo ε, R], ρ R) + ε fo R, ), and v,ε,r ) = fo, ε + 2δ], m ) fo ε + 2δ, R 2δ], ρ ) + ε fo R 2δ, ). We suppose now that ε R 3 and we denote by ρ ε,r,δ, v ε,r,δ ) the solution of the poblem 2.)2.)2.2). Finally in ode to extact convegent subsequences it is convenient to extend ρ ε,r,δ, v ε,r,δ ) to the whole space by setting ρ ε,r,δ, t) = ρ ε,r,δ, t) fo ε, R], 2.3) else, and ṽ ε,r,δ, t) = v ε,r,δ, t) fo ε, R], else, 2.4) and in ode to simplify the notations we denote in the following by ρ n, v n ) the appoximate solution ρ εn,r nn,δ n, ṽ εn,r n,δ n ), fo sequences ε n, R n, δ n, when n. In the same stoke, we wite n := ε n, R n ], µ n := µ εn ρ n ), λ n := λ εn ρ n ) and P n := P ρ n ). In the following, fo any measuable set ω := :=, ) and p, we denote by L p ω) the weighted Lebesgue space } L p ω) := {f L ploc ω) ; f) p 2 d <, ω

7 Vol ) on spheically symmetic motions 7 and we adopt the analogous notations H m ω) and W m,p ω) fo the coesponding weighted Sobolev spaces We begin with enegy-entopy estimates. Lemma 2.2. Thee exists a constant C independent on n such that, fo any t, T ]. ρ n 2 d C. 2.5) 2. ) 2 d + t 3. If 2 3λ n + 2µ n ) 2 ρ nv n ) 2 + Π n ρ n Φ n ] 2 3λ n + 2µ n 2 v n ) + 2µ n vn 2 d dτ C, 3λ n + µ n 2.6) 2 ρ v ) 2 + Πρ ) 2 ρ Φρ ) ] dx <, the following estimates hold ρ n v n L,T ;L 2 )) C, ρ n L,T ;L γ )) C, 2.7) µ n v n ) L 2,T ;L 2 )) C, µ n v n L 2,T ;L 2 )) C, 2.8) ρ L,T ;L )) C. 2.9) Poof.. Integating the fist equation 2.) and using bounday conditions gives 2.5). 2. Noting D t := t + v, using the identity ρ nd t φ 2 d = d dt ρ nφ 2 d valid fo any function φ smooth on, T ], multiplying the second equation 2.) by v n 2 and integating by pats, we get d dt 2 ρ nvn 2 + Π n ) 2 ρ nφ n 2 d = λ n + 2µ n ) 2 v n ) ] 2 2 4µ n vn) 2 2.2) Using the notations w = 2 v n, a = λ n + 2µ n and b = µ n one checks that the integand in the ight-hand side ewites λ n + 2µ n ) 2 v n ) 2 2 4µ n v 2 n) = 2 Obseving that the numbe ζ := 3a 4b 4a aw 2 8b ww + 2b 2 w2 = 2 6aζ3 4ζ) 3 aζw ζ ] aw 2 8b ww + 2b 2 is positive, we have w a 2ζ w 3 w2 ]. 33 4ζ) 3 2ζ whee all the tems in the ight hand side ae positive afte.4). Plugging into 2.2) and integating in t, we get 2.6). 3. Afte.2) one obseve that fo a C >. Πρ n ) Cρ γ n ρ n log ρ n ρ n ), ] d. ] 2 w,

8 8 Benad Ducomet, Šáka Nečasová and Alexis Vasseu JMFM Let us estimate the gavitational enegy. E n t) = ρ n Φρ n ) 2 d 2 = s 2 ρ n s, t)k, s) 2 ρ n, t) ds d+ 2 2 We get fist B = 2 ρ n, t) s 2 ρ n s, t) ds d ρ n, t) s 2 ρ n s, t) ds d+ 2 ρ n, t) 2 2 Now using Hölde inequality, we bound A as follows 2 ρ n, t) ) γ γ d s 2 ρ n s, t)k, s) 2 ρ n, t) ds d =: A+B. 2 ρ n, t) s 2 ρ n s, t)k, s) 2 ρ n, t) ds d ) γ γ sρ n s, t) ds d s 2 ρ n s, t) ds d 2 ρ n, t) 2 L ). γ ) s 2 ρ n s, t)k, s) ds) γ d. By convexity, the fist integal in the ight-hand side is bounded by ρ n, t) L ). Fo the second one, we note that the kenel K is homogeneous of degee and satisfies, fo γ > K, s) s γ ds <, so we can apply an inequality of Hady-Littlewood-Polya see 6] p. 27) γ ) s 2 ρ n s, t)k, s) ds) γ ) d C s 2γ ρ γ γ ns, t) ds C Finally, using 2.5), we get E n t) C + ρ γ n, t) L )). We conclude fom the pevious estimates that Πρ n ) ρ n Φρ n )] 2 d C s 2 ρ γ n ds ) s 2 ρ γ γ ns, t) ds. ) ] s 2 ρ n log ρ n ds s 2 ρ γ γ n ds. As the log is subdominant and γ >, we end with the lowe bound Πρ n ) ρ n Φρ n )] 2 d C s 2 ρ γ n ds C. 2.2) Plugging 2.2) into 2.6) and taking.4) into account gives immediately the inequalities 2.7), 2.8) and 2.9) by inspection. We have the following spheical vesion of the Besch-Desjadins entopy estimate see ])

9 Vol ) on spheically symmetic motions 9 Lemma 2.3. Thee exists a constant C independent on n such that, fo any t, T ]. ρ n v n + 2µ φρ n )) ) 2 2 d + 2 t +2µ P ρ n )Ψ ρ n ) ρ n) ) 2 t +2µ Π n 2 d ρ n 2 d dτ 2µ t ρ n Φ n 2 d Ψ Φ 2 d 2 v n ) ) 2 + 4vn 2 ] Ψn d dτ C, 2.22) with Ψ n := Ψρ n ). 2. If 2 ρ v + 2µ φρ )) ) 2 2 d <, the following estimates hold ρ n φ ρ n ) L,T ;L 2 )) C, 2.23) µ nρ n )ρ γ 2 n ρ n ) L 2,T ;L 2 )) C, 2.24) µ n ρ n ) v n L 2,T ;L 2 )) C. 2.25) Poof.. In the poof of this estimate, we omit the explicit n dependence. Fist multiplying the fist equation 2.) by φ and deivating with espect to, we get φ t + vφ + v φ + ρφ 2 2 v ) ] =. Multiplying by 2 ρφ, using the continuity equation and integating in, we get d ρφ 2 2 dt 2 d + ρφ 2 v 2 d + φ ρ 2 2 v ) ] Ψ 2 d =. 2.26) Fom the second equation 2.), we have ρv t + vv ) λ + 2µ) 2 2 v ) ] v 4µ + P + ρf =, which ewites, by using.3) ρv t +vv ) 2µ Ψ 2 2 v ) ] +2µ Ψ v 2µ ρψ Ψ) 2 2 v ) ] +P +ρf =. Multiplying by ρ Ψ 2 and integating, we obtain v t +vv )Ψ 2 d 2µ Ψ 2 2 v ) ] Ψ ρ 2 d+2µ Ψ ρψ ) 2 2 v ) ] Ψ ρ 2 d Ψ 2 +2µ ρ v 2 Ψ P d + 2 d Ψ Φ 2 d =. 2.27) ρ Multiplying 2.26) by 2µ and adding the esult to 2.27), we get v t + vv )Ψ 2 d + µ d 2ρφ 2 2 dt 2 d 2µ Ψ 2 2 v ) ] Ψ ρ 2 d

10 Benad Ducomet, Šáka Nečasová and Alexis Vasseu JMFM +2µ Checking the identity 2µ Ψ Ψ ρψ ) 2 2 v ) ] Ψ ρ 2 d + 2µ + P Ψ ρ2 ρ 2 d 2 2 v ) one gets finally v t + vv )Ψ 2 d + µ 2 ] Ψ ρ 2 d = 2µ +2µ Ψ 2 2 v ) The fist integal in 2.28) ewites t + vv )Ψ v 2 d = d vψ 2 d dt whee A ewites A = d vψ 2 d dt Ψ Φ 2 d =. Ψ 2 2 v ) ] Ψ 2 d Ψ ρψ ) 2 2 v ) ] ] Ψ 2 d, Ψ ρ 2 d d 2ρφ 2 dt 2 d + P Ψ ρ2 ρ 2 d Ψ Φ 2 d =. 2.28) v Ψ t ) 2 d + vv Ψ 2 d =: A, 2.29) 2 2 v ) ] 2 ρψ d 2 2 v) vρ Ψ d+ vv Ψ ρ 2 d =: U + U 2 + U 3 + U ) Computing the tem U 3, we obtain U 3 = vv Ψ ρ 2 d + 2 v 2 ) Ψ d, so we get A = d dt vψ 2 d 2 2 v ) ] 2 ρψ Ψ) d 2 2 v 2 +2v 2 )Ψ d. 2.3) Plugging into 2.29), we have vψ 2 d 2 2 v ) ] 2 ρψ Ψ) 2 d t + vv )Ψ v 2 d = d dt 2 v 2 + 2v 2 )Ψ d. 2.32) Fom 2.2) we have d dt 2 ρv2 + Π 2 ρφ so d dt ) 2 d = λ + 2µ) 2 v) ] 2 ] 2 4µv 2 ) d, ρ v2 2 2 d+2µ 2 2 v ) ] 2 ρψ Ψ) 2 d+ d Π 2 d d ρφ 2 d dt dt

11 Vol ) on spheically symmetic motions = 2µ 2 2 v ) ] 2 Ψ 2 d + 4µ v 2 ) Ψ d. 2.33) Now plugging 2.32) into 2.28) gives d vψ 2 d 2 dt 2 v ) ] 2 ρψ Ψ) 2 d 2 v 2 + 2v 2 )Ψ d + µ d 2ρφ 2 2 dt 2 d + P Ψ ρ2 ε ρ 2 d Ψ Φ 2 d =. 2.34) Multiplying this elation by 2µ gives d 2µ 2 dt ρφ 2 ) + 2µ vψ 2 d + 2µ P Ψ ρ2 ρ 2 d 2µ Ψ Φ 2 d 2µ 2 2 v ) ] 2 ρψ Ψ) 2 d = 2µ 2 v 2 + 2v 2 )Ψ d, 2.35) whee the integand in the last integal is positive. Finally, adding 2.35) to 2.33), integating in time and estoing the n dependence gives 2.22). 2. Obseving that afte Poisson equation Ψ Φ 2 d = ρψ 2 d, and using.5) togethe with 2.6) into 2.22), we get immediately the inequalities 2.23), 2.24) and 2.25) by inspection. Lemma 2.4. The sequence {ρ γ n} n N is bounded in L 5/3, T ; L 5/3 )). Poof. Afte 2.9) and 2.24), ρ γ/2 n is bounded in L 2, T ; H n )) and then ρ γ n is bounded in L 2, T ; L 3 n )). As, afte 2.9), ρ γ n is also bounded in L, T ; L n )), using Hölde inequality ρ γ n is also bounded in L 5/3, T ; L 5/3 n )), and then see 2.3) and 2.4)) in L 5/3, T ; L 5/3 )). Lemma 2.5. If ρ + v 2 ) log + v 2 ) L )), then the sequence {ρ n v 2 n log + v 2 n)} n N is bounded in L, T ; L )). Poof. Recalling the identity d ρ n F 2 d = ρ n F t + vf ) 2 d, dt n n and choosing F = 2 + v2 n) log + vn), 2 we get d dt n 2 ρ n+vn) 2 log+vn) 2 2 d = v n + log + v 2 n ) ) ρ n v n ) t + v n v n ) ] 2 d. n Then multiplying the momentum equation by v n + log + v 2 n ) ) 2 and integating by pats, we have d dt n 2 ρ n + vn) 2 log + vn) 2 2 d = v n + log + vn))p 2 ρ n )) 2 d n

12 2 Benad Ducomet, Šáka Nečasová and Alexis Vasseu JMFM + v n +log+vn) λ 2 n + 2µ n ) ] 2 2 v n ) n 2 d+ 4v n +log+v 2 v n n)µ n ) n 2 d 4πGv n + log + vn)ρ 2 n Φ n ) 2 d. n Using enegy estimates and.3), we get ] 2 d dt n 2 ρ n + vn) 2 log + vn) 2 2 d + m µ n n 2 2 v n ) 2 d v n +log+vn))p 2 ρ n )) 2 d n 4πGv n +log+vn)ρ 2 n Φ n ) 2 d =: I+J. n 2.36) Using.2) and Cauchy-Schwaz, the fist contibution in the ight-hand side of 2.36) is bounded as follows I = v n + log + vn))p 2 2 d µ n n 2 v n ) ] 2 d+ n 2 m and the last integal is bounded by n ρ n log + v 2 n ) ] 2 η +C + log + v n 2n)) P ρ n) 2 2 d, µ n ) η 2 2 d n ρ η 2 n µ n P 2 n ) 2 2 η n µ n 2 d fo any < η < 2. One checks that the fist integal is bounded afte pevious estimates, and the second one is also bounded povided that 2γ < 5γ 3 i.e. γ < 3. When γ > 3, the same esult holds if condition.6) is satisfied. So fo any γ > Fo the second contibution in 2.36), one has J C v n + log + vn) 2 ) ρ n, τ) n 2 η 2 I C. 2.37) ρ n s, τ) s 2 ds d C ρ n vnd, 2 n so using.3) J µ n vnd ) n Plugging 2.37) and 2.38) in 2.36), integating on, t) and using 2.6), we find that n 2 ρ n + vn) 2 log + vn) 2 2 d C, which ends the poof, by taking 2.3) and 2.4) into account. ] v n ) 2 d,

13 Vol ) on spheically symmetic motions Poof of Theoem 2. Following vebatim the stategy of 8], the poof of Theoem 2. is achieved in the same way as in 3] by showing fist the convegence of the density and the pessue, then impoving the convegence of ρ n v 2 n, poving the convegence of the momentum, showing the stong convegence of ρ n 2 v n ) and finally the convegence of diffusion tems. We only sketch the poof heeafte, only stessing the specific oles played by gavitation and pessue, and we efe to 8] fo complete details. ) Convegence of the density. Lemma 2.6. The sequence {ρ n } is bounded in L, T ; L p )) fo any p, and up to an extacted subsequence fo any q, 3). ρ n ρ stongly in C, T ; L q loc )), 2.39) Poof. Fom Lemma 2.3, one get sup t,t ] ρ n H n) C, so afte 2.3), the sequence { ρ n } is bounded in L, T ; L β )) fo β 2, 6]. So {ρ n } is bounded in L, T ; L 3 )) and {ρ n v n } is bounded in L, T ; L 3/2 )) afte Lemma 2.2. Using the continuity equation, { t ρn } ɛn K,Rn N is bounded in L, T ; W,3/2 K, N)), fo any K N 6. As ρ n ) = 2 ρ n ρ n ), we find that {ρ n ) } is bounded in L, T ; L 3/2 K, N)) fo ɛ n K and R n N. This implies, afte Aubin-Lions lemma that ρ n ρ stongly in C, T ; L 3/2 K, N)). Using finally the splitting ρ n ρ L,T ;L 3/2,N)) C K ρ n ρ L,T ;L 3, K )) + ρ n ρ L,T ;L 3/2 K,N)), we get the popety. 2) Convegence of the pessue and the gavitational foce. As a diect consequence of Lemma 2.4 and 2.6, we have Lemma 2.7. Fo γ > P ρ n ) conveges stongly to P ρ) in L loc, T ); L loc)), ρ n Φ n ) conveges stongly to ρφ in L loc, T ); L loc)). 3) Convegence of the momentum. Lemma 2.8. Up to extacted subsequences. ρ n v n m stongly in L loc, T ; L loc)) L 2, T ; L β loc )), fo any β < 3/2.

14 4 Benad Ducomet, Šáka Nečasová and Alexis Vasseu JMFM 2. ρn v n m ρ stongly in L 2 loc, T ; L 2 loc)), whee m ρ = when m =. In paticula m = a.e. on {ρ, t) = }, and thee exists a function v, t) such that m, t) = ρ, t)v, t). Poof. The second pat of the Lemma follows diectly fom 3] and we omit the poof. As { ρ n } is bounded in L, T ; L 2 L 6 )) and { ρ n v n } is bounded in L, T ; L 2 )), we get that {ρ n v n } is bounded in L, T ; L L 3/2 )). 2.4) Using the identity ρ n v n ) = 2 ρ n v n ρ n ) + ρ n ρn v n ) and Lemma 2.5, we see that { ρ n v n ) } ɛn K,Rn N is bounded in L, T ; L K, N)). In paticula the sequence {ρ n v n } ɛn K,Rn N is bounded in L 2, T ; W, K, N)). As {ρ n } ɛn K,Rn N is bounded in L, T ; L K, N)), we see that { t ρ n v n )} ɛn K,Rn N is bounded in L 2, T ; W 2,3/2 K, N)), so using the Aubin-Lions Lemma ρ n v n m stongly in L 2, T ; L β K, N)), fo any β, 3/2). Fom 2.4), we get also that fo any β, 3/2). ρ n v n m stongly in L 2, T ; L β, N)), 4) Convegence in the weak fomulation. In ode to complete the poof of Theoem 2., the last point is to check that the pai ρ, v) is actually a weak solution of 2.). In othe wods we must check that if ρ, v) is the limit of the sequence {ρ n }, {v n }) obtained in Lemma 2.6 and 2.8, then ρ and v satisfy elations.) and.). The fact that ρ, v) satisfy elation.) is aleady poved in 8] see Poposition 3.5). Let ψ C, T ] with ψ, t) = and ψ, T ) = be a test function such that Supp ψ, t), N]. Multiplying the second equation 2.) by 2 ψ and integating on, T ], we have t ρ n v n ψ t + ρ n vnψ 2 + P ρ n ) ψ + 2ψ )) 2 d dt

15 Vol ) on spheically symmetic motions 5 t T + + t t 2µρ n ) v n ) ψ + 2v ) nψ 2 d dt 2ɛ n ρ θ n λρ n ) + θɛ n P ρ n )) + ɛ n 2 v n ) ψ + v n ψ + v nψ v n ) + 2v n ) d dt ) ψ + 2ψ ρ,ɛn,r n, )v,ɛn,r n, )ψ, ) 2 d ) 2 d dt P ρn ) 2µρ n ) + λρ n ) + θɛ n ρ θ ) ] n vn ) ɛn, t) ɛ 2 n ψɛ n, t) dt = t ρ n Φρ n )) ψ 2 d dt, 2.4) fo any n such that ɛ n K and R n N. Following vebatim 8] one can pass to the limit n in all of the tems of the left-hand side in 2.4). Note in paticula that lim n T P ρn ) 2µρ n ) + λρ n ) + θɛ n ρ θ ) ] n vn ) ɛn, t) ɛ 2 n ψɛ n, t) dt =. Moeove, afte Lemma 2.7, one also gets that the same holds fo the gavitational tem in the ight hand side t t ρ n Φρ n )) ψ 2 d dt ρ Φρ)) ψ 2 d dt, when n, which ends the poof of Theoem The constant viscosities case Fo constant viscosities, the spheically symmetic vesion of.) eads ρ t + ρv) + 2ρv =, ρv t + vv ) = P + λ + 2µ) v + 2v )) 3.) ρφ, in the domain, T ], T ], ) fo the density ρ, t) and the velocity v, t). The gavitational potential is given as peviously by 2.2) and 2.3). We conside Diichlet bounday condition at the oigin and initial conditions v, t) =, fo t >, 3.2) ρ t= = ρ ), ρv t= = m ), on R ) We ecall that ρ, v) is a weak solution of 3.)3.2)3.3) if

16 6 Benad Ducomet, Šáka Nečasová and Alexis Vasseu JMFM. ρ a.e. and fo any T > see.8)) ρ L, T ; L γ )), ρv 2 L, T ; L )), v L, T ; L v )), L, T ; L )), ) ) ) ) ρ C, T ]; L γ 2γ γ+ loc ), ρv C, T ]; L w loc ), w 3.4) whee, ), and the index w means that the coesponding L p space is endowed with the weak topology. 2. Fo any t 2 t and any test function η CR ) see.) and.)) t 2 t2 ρη 2 d ρη t + ρvη ) 2 d =. 3.5) ρvη 2 d t 2 t2 t t t t ρvη t + ρv 2 η + P η + 2η )) t2 +λ + 2µ) v η + 2vη ) t 2 t2 +4πG η, t)ρ, t) t The following esult holds 2 d dt 2 d dt ρs, t) s 2 ds 2 d dt =. 3.6) Theoem 3.. Suppose that T is an abitay positive numbe and that the initial data satisfy and ρ L )) L γ )), 3.7) m L 2γ γ+ ), m 2 ρ L )). 3.8) Then fo any γ >, the poblem 3.)3.2)3.3) possesses a global weak solution satisfying 3.4) togethe with popeties 3.5) and 3.6). In the constant viscosities case, exta estimates on the density gadient using Besch-Desjadins inequality ae no moe available and one must ely on the oiginal Lions-Feieisl method 2] 8] 5] see 4]). Following ], we conside appoximate solutions in the spiit of Section 3 and pass to the limit. As in the density dependent case, we only concentate on the ole of gavitation and pessue, efeing to the pape of Jiang and Zhang ] fo futhe details.

17 Vol ) on spheically symmetic motions Appoximate poblem and pioi estimates Adapting the appoximating scheme of Hoff ] to the self-gavitating situation, we define in ɛ ɛ, ) the mollified data ρ ) ɛ ) := ) ] 2/γ ρ 2/γ j ɛ/2 ) + ɛe 2, 3.9) ρ ) ɛ m ) ɛ ) := m ) ] j ɛ/2 ), 3.) ρ 2/γ and ) m ) ɛ v ) ɛ ) := χ ɛ ), 3.) ρ ) ɛ whee ɛ >, j is the standad Fiedich mollifie and χ ɛ C ) is a cut-off such that if ɛ, χ ɛ ) = if 2ɛ. Then denoting by δ ɛ and δ ɛ,r the step functions if ɛ, if ɛ o R, δ ɛ ) =, δ ɛ,r ) = if > ɛ. if ɛ < < R, fo a R >, one checks that, unifomly in R {δ ɛ ρ ) ɛ ρ ɛe 2)} in L γ ), {δ ɛ ρ ) ɛ ρ ɛe 2)} in L ), and {δ ɛ,r ρ ) ɛ v ) ɛ m )} in L 2γ γ+ ), when ɛ, fo any γ >. We conside the appoximate poblem ρ ɛ ) t + ρ ɛ v) + 2ρ ɛv ɛ =, ρ ɛ v ɛ ) t + v ɛ v ɛ ) ) = P ρ ɛ ) + λ + 2µ) v ɛ ) + 2v )) ɛ ρ ɛ Φ ɛ ), 3.2) in the domain, T ] ɛ fo the density ρ ɛ and the velocity v ɛ. The appoximate gavitational potential is Φ ɛ, t) = with K is given by 2.3). We conside Diichlet bounday condition ɛ K, s)ρ ɛ s, t)s 2 ds, 3.3) v ɛ, ɛ) =, fo t >, 3.4)

18 8 Benad Ducomet, Šáka Nečasová and Alexis Vasseu JMFM and initial conditions Then we have ρ ɛ t= = ρ ) ɛ ), ρ ɛ v ɛ t= = m ) ɛ ), on ɛ. 3.5) Poposition 3.2. Suppose that ɛ and T ae abitay positive numbes and that the initial data satisfy ρ L )) L γ )), 3.6) and m L 2γ γ+ ), m 2 ρ L )). 3.7) Then fo any γ >, the exteio poblem 3.2)3.4)3.5) possesses a weak solution such that and ɛ ρ ɛ >, ρ ɛ C, T ; L γ loc ɛ)), v ɛ C, T ; L 2 )) C, T ; H )), 2 ρ ɛv ɛ ) 2 + ) 2 Πρ ɛ) 2 3λ + 2µ d + 2 whee C > does not depend on ɛ. t 2 v ɛ ) ) 2 + v 2 ] ɛ d dτ C, ɛ 3.8) The poof is a diect consequence of ]: the gavitational contibution is bounded a pioi due to the coe pesence, and the subdominent contibution in the pessue see.2)) is contolled as above by mass consevation. In ode to get pecompactness of the sequences {ρ ɛ }, {v ɛ }, as we have no moe diect infomation on the gadient of ρ ɛ, we need impoved L p L q ) estimates on ρ ɛ itself fo p, q lage enough, which is the matte of the following esult. Lemma Fo any test function φ C ) with φ) = O 3 ) as, thee exists a constant C independent on ɛ such that T ρ ɛ ) 2γ φ 4 d dt C, 3.9) ɛ 2. Let γ > 3/2. Fo any T > and fo any test function ψ C ) with φ and φ) = O 3 ) as, thee exists a constant C independent on ɛ such that T ρ ɛ ) γ+θ ψ d dt C, 3.2) ɛ fo any < θ < γ. Poof. In the poof of these estimates, we omit the explicit ɛ dependence.. Multiplying by φ and integating on, ) the momentum equation ρv) t ρv 2) + P = λ + 2µ) 2 2 v ) ) ρφ =, 3.2)

19 Vol ) on spheically symmetic motions 9 with Φ ρ) = 4πGρ 2 ɛ ρs, τ) s2 ds, we get φp = λ + 2µ) 2 2 v ) φ ρv2 φ + t ρvφ ds 2ρv 2 φ + ds + λ + 2µ) s 2 s s 2 v ) φ ds + s Multiplying this equality by ρ γ and obseving afte ] that P + ρv 2 )φ s ds ρφ s φ ds. 3.22) ρ γ ) t + vρ γ ) + 2γvργ = γ)v ρ γ, holds in D, T ) ɛ ) which implies, by multiplying by ρφv ds, that + ρ γ + ) ρφv ds t = ρ γ γ )v ρ γ + 2γvργ ) ρφv ds we get φp ρ γ = λ + 2µ) 2 2 v ) φργ + ρ γ ] γ )v ρ γ + 2γvργ +λ + 2µ)ρ γ t ) + vρ γ ρφv ds ] ρφv ds + v 2 ρ γ+ φ, ) ) ρφv ds + vρ γ ρφv ds t ρφv ds ρ γ P + ρv 2 )φ s ds + ρ γ s 2 s 2 v ) s φ s ds + ρ γ ρφ s φ ds. 2ρv 2 φ s Then multiplying by φ 3 and integating on, T ) ɛ, we obtain T T φp ρ γ φ 3 d dt = λ + 2µ) φ 3 2 ɛ ɛ 2 v ) φργ d dt T ) T ) + φ ρ 3 γ ρφv ds d dt + φ vρ 3 γ ρφv ds d dt ɛ t ɛ T ] + φ γ 3 )v ρ γ + 2γvργ ρφv ds d dt ɛ T T φ 3 ρv 2 φ s ds d dt + φ 3 ρ γ 2ρv 2 φ ds d dt ɛ ɛ s T +λ + 2µ) φ 3 ρ γ s 2 ɛ s 2 v ) φ s s ds d dt T T 9 φ 3 ρ γ P φ s ds d dt + φ 3 ρ γ ρφ s φ ds d dt =: J k. ɛ ɛ k= 3.23) ds

20 2 Benad Ducomet, Šáka Nečasová and Alexis Vasseu JMFM Afte ] one knows that fo any η > 7 T J k η φρ 2γ φ 4 d dt + C η. 3.24) ɛ k= Using.2) one gets fo a R > such that Supp φ, R] T R T R J 8 a φ 3 ρ γ ρ γ s 2 ds d dt + b φ 3 ρ γ ρs 2 ds d dt C. ɛ ɛ 3.25) In the same stoke T J 9 4πG φ 3 ρ γ ρφ s ɛ s 2 ρα, t) α 2 dα ds d dt ɛ η T φρ 2γ φ 4 d dt + T 2 ɛ 2η ρ L ) φ 2 ρ φ ds d dt ɛ s η T φρ 2γ φ 4 d dt + C η. 3.26) 2 ɛ Afte.2) the left-hand side of 3.23) can be estimated as follows T φp ρ γ φ 3 d dt T T ρ 2γ φ 4 d dt b ρ γ+ φ 4 d dt. ɛ aγ ɛ ɛ As x ηx γ + C η, we get T φp ρ γ φ 3 d dt ɛ aγ T ɛ ρ 2γ φ 4 d dt bη T ɛ ρ 2γ φ 4 d dt C η. Plugging this last inequality togethe with 3.25) and 3.26) into 3.23) and choosing η such that + b 2 )η = gives 3.9). 2aγ 2. The poof follows the same scheme as the pevious one. Let φ C ) be a test function such that φ on Supp ψ. Multiplying 3.22) by ψρ θ fo θ > and obseving afte ] that ρ θ ) + vρ θ) + 2θvρθ = θ)v t ρ θ, holds in D, T ) ɛ ) fo any θ >, we get ψp ρ θ = λ + 2µ) 2 2 v ) ψρθ + ρ θ ψ ] + θ )v ρ θ ψ + 2θvρθ ψ +λ + 2µ)ρ θ ψ ) ρφv ds + vρ θ ψ t ρφv ds ρ θ P +ρv 2 )φ s ds+ρ θ ψ s 2 s 2 v ) s φ s ds + ρ θ ψ ρφ s φ ds. ) ρφv ds 2ρv 2 φ s ds

21 Vol ) on spheically symmetic motions 2 Then integating on, T ) ɛ, we obtain T T ψp ρ θ d dt = λ + 2µ) 2 ɛ ɛ 2 v ) ψρθ d dt T ) T ) + ψ ρ θ ρφv ds d dt + ψ vρ θ ρφv ds d dt ɛ t ɛ T ] + ψ θ )v ρ θ + 2θvρθ ρφv ds d dt ɛ T T ψ ρv 2 φ s ds d dt + ψ ρ θ 2ρv 2 φ ds d dt ɛ ɛ s T +λ + 2µ) ψ ρ θ s 2 ɛ s 2 v ) s φ s ds d dt T T ψρ θ P φ s ds d dt + ψ ρ θ ρφ s φ ds d dt ɛ ɛ 9 =: I k. 3.27) k= Afte ] one knows that fo any η > 7 T I k η ψρ γ+θ d dt + C η. 3.28) ɛ k= Using.2) one gets fo a R > such that Supp φ, R] T R T R I 8 a ψρ γ ρ γ s 2 ds d dt + b ψρ γ ρs 2 ds d dt C. ɛ ɛ 3.29) In the same stoke T I 9 4πG ψ ρ θ ρφ s ɛ s 2 ρα, t) α 2 dα ds d dt ɛ η T φρ 2γ ψ d dt + T 2 ɛ 2η ρ L ) ψ ρ φ ds d dt ɛ s η T ψρ γ+θ d dt + C η. 3.3) 2 ɛ Afte.2) the left-hand side of 3.27) can be estimated as peviously T ψ P ρ θ d dt T T ψ ρ γ+θ d dt b ψ ρ θ+ d dt. ɛ aγ ɛ ɛ As x ηx γ + C η, we get T ψ P ρ θ d dt T T ψ ρ γ+θ d dt bη ψ ρ γ+θ d dt C η. ɛ aγ ɛ ɛ

22 22 Benad Ducomet, Šáka Nečasová and Alexis Vasseu JMFM Plugging this last inequality togethe with 3.29) and 3.3) into 3.27) and choosing η such that + b 2 )η = gives 3.2). 2aγ 3.2. Poof of Theoem 3. As in Section 2, we conside the extension to the pai ρ ɛ, v ɛ ) defined as ρ ɛ, t) = ρ ɛ, t) fo ɛ, ), 3.3) else, and ṽ ɛ, t) = v ɛ, t) fo ɛ, ), else, 3.32) and we denote by ρ ɛ, v ɛ ) this extension, by simplicity. Afte the enegy inequality, thee exist extacted sequences such that moeove afte enegy inequality ρ ɛ ρ weak in L, T ; L γ loc )), ρ ɛ ρ weak in L 2γ, T ; L 2γ loc )), v ɛ v weak in L 2, T ; H loc)), ρ L, T ; L γ )) L 2γ, T ; L 2γ loc )), v, v L 2, T ; L 2 )). In ode to complete the poof of Theoem 3., we have finally to pove that the couple ρ, v) satisfies the system 3.). Using Lemma 3.3 and poceeding exactly as Jiang and Zhang in ] one poves the following popeties of the weak limits. Fo any < θ < γ, one has ρ θ P ρ) ρ θ P ρ) + 3λ + 2µ)ρ θ v = w lim 2. Fo any < θ < such that 2 ɛ { θ + + 6θ + θ 2 ) γ } 3λ + 2µ)ρ θ ɛ v ɛ ), 3.33) ρ θ ρ θ L 2 θ, T ; L 2 θ, )), 3.34) fo any < θ < γ. 3. The limits ρ and v satisfy the enegy inequality 2 ρv2 + ) 2 Πρ) 2 3λ + 2µ t d + 2 v 2 + v 2] d dτ C, 3.35) 2 moeove ρ, ρ θ ) θ L, T ; L )). 3.36)

23 Vol ) on spheically symmetic motions 23 In paticula, passing to the limit into 3.2), we find that ρ, v) satisfies in D, T ) ) the system ρ t + ρv) + 2ρv =, ρv) t + ρv 2 + P ] + 2ρv2 = λ + 2µ) v + 2v )) ρφ. 3.37) As popety 3.34) excludes the concentation of mass at the oigin, the end of the poof is vebatim the same as that of Theoem. in ]. Acknowlegment Š. N. was suppoted by the Gant Agency of the Czech Republic n. 2/8/2 and by the Academy of Sciences of the Czech Republic, Institutional Reseach Plan N. AVZ953. The last pat of pape was done duing he stay in Cea, Apajon. She would like to thank fo hospitality of Pof. Ducomet and his collegues duing he stay thee. A.V. was patially suppoted by Nečas Cente fo Mathematicall Modelling LC652 financed by MŠMT. Refeences ] D. Besch, B. Desjadins. Some diffusive capillay models of Koteweg type. C. R. Acad. Sci. Pais, Section Mécanique 24; 332: ] D. Besch, B. Desjadins, D. Géad-Vaet. On compessible Navie-Stokes equations with density dependent viscosities in bounded domains. J. Math. Pues Appl. 27; 87: ] S. Chapman, T.G. Cowling. The mathematical theoy of non-unifom gases, Cambidge Univesity Pess, ] B. Ducomet, E. Feieisl, A. Petzeltová, I. Staškaba. Global in time weak solutions fo compessible baotopic self-gavitating fluid, Discete and Continuous Dynamical Systems 24; :3 3. 5] B. Ducomet, Š. Nečasová, A. Vasseu. On global motions of a compessible baotopic and self-gavitating gas with density-dependent viscosities, Submitted. 6] B. Ducomet, A. Zlotnik, Viscous compessible baotopic symmetic flows with fee bounday unde geneal mass foce, Pat I: Unifom-in-time bounds and stabilization, Mathematical Methods in the Applied Sciences 28 25) ] E. Feieisl. Compessible Navie-Stokes Equations with a Non-Monotone Pessue Law, Jounal of Diff. Equ. 22; 84: ] E. Feieisl. Viscous and/o heat conducting compessible fluids, in Handbook of mathematical fluid dynamics, Vol, S. Fiedlande and D. See Ed., Noth-Holland, 22. 9] Z. Guo, Q. Jiu, Z. Xin. Spheically symmetic isentopic compessible flows with density-dependent viscosity coefficients. SIAM J. Math. Anal. 28; 39: ] D. Hoff. Spheically symmetic solutions of the Navie-Stokes equations fo compessible, isothemal flow with lage discontinuous initial data. Indiana Univ. Math. J. 992; 4:

24 24 Benad Ducomet, Šáka Nečasová and Alexis Vasseu JMFM ] S. Jiang, P. Zhang. On spheically symmetic solutions of the compessible isentopic Navie-Stokes equations. Comm. Math. Phys. 2; 25: ] P.L. Lions. Mathematical topics in fluid mechanics, Vol 2: Compessible models, Oxfod Univesity Pess, ] A. Mellet, A. Vasseu. On the baotopic compessible Navie-Stokes equations. Comm. Patial Diffeential Equations 27; 32: ] A. Mellet, A. Vasseu. Existence and uniqueness of global stong solutions fo one-dimensional compessible Navie-Stokes equations. SIAM J. Math. Anal. 28; 39: ] A. Novotný, I. Staškaba. Intoduction to the mathematical theoy of compessible flow, Oxfod Univesity Pess, 24. 6] E. Stein. Singula integals and diffeentiability popeties of functions, Pinceton Univesity Pess, 97. 7] T. Zhang, D. Fang. Global behavio of spheically symmetic Navie-Stokes equations with density-dependent viscosity. J. Diffeential Equations 27; 236: ] T. Zhang, D. Fang. A note on spheically symmetic isentopic compessible flows with density-dependent viscosity coefficients. Nonlinea Analysis: Real Wold Applications in pess) Benad Ducomet CEA, DAM, DIF F-9297 Apajon, Fance benad.ducomet@cea.f Šáka Nečasová Mathematical Institute AS ČR Žitna 25, 5 67 Paha, Czech Republic matus@math.cas.cz Alexis Vasseu Depatment of Mathematics, Univesity of Texas at Austin Univesity Station C2 TX, USA vasseu@math.utexas.edu