ON THE GLOBAL REGULARITY OF SUB-CRITICAL EULER-POISSON EQUATIONS WITH PRESSURE

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1 ON THE GLOBAL REGULARITY OF SUB-CRITICAL EULER-POISSON EQUATIONS WITH PRESSURE EITAN TADMOR AND DONGMING WEI Abstract We prove that the one-dimensional Euler-Poisson system driven by the Poisson forcing together with the usual γ-law pressure, γ 1, admits global solutions for a large class of initial data Thus, the Poisson forcing regularizes the generic finite-time breakdown in the p-system Global regularity is shown to depend on whether or not the initial configuration of the Riemann invariants and density crosses an intrinsic critical threshold 1 Introduction It is well known that the systems of Euler equations for compressible flows can and will breakdown at a finite time even if the initial data are smooth A prototype eample for such systems is provided bythe system of isentropic gas dynamics ρ t +ρu) =0 11) ρu) t +ρu ) = p, where the pressure p = pρ) is given by the usual γ-law, pρ) =Aρ γ By using the method introduced in [La64] to deal with pairs of conservation laws, it can be shown that 11) will lose the C 1 -smoothness due to the appearance of shock discontinuities unless its two Riemann invariants are nondecreasing Thus, the finite time breakdown of 11) is generic in the sense that it holds for all but a small set of initial data On the other hand, if we replace the pressure by Poisson forcing, then we arrive at the system of Euler-Poisson equations ρ t +ρu) =0, 1) ρu) t +ρu ) = kρϕ k>0, subject to initial data u 0,ρ 0 > 0) Here ϕ = ϕρ) is the potential, which is dictated by the onedimensional) Poisson equation, ϕ = ρ In this case, there is a large set of initial configurations which yield global smooth solutions More precisely, [ELT01] have shown that 1) admits a global smooth solution if and only if 13) u 0 ) > kρ 0 ) Thus, following the terminology of [LT0], the curve u 0 + kρ 0 = 0 is a critical threshold in configuration space which separates between initial configurations leading to finite time breakdown and a large set of sub-critical initial configurations which yield global smooth solutions In particular, Date: October 4, Mathematics Subject Classification 35Q35, 35B30 Key words and phrases Euler-Poisson equations, Riemann Invariants, Critical thresholds, Global regularity Acknowledgment Research was supported in part by NSF grant and ONR grant N J

2 EITAN TADMOR AND DONGMING WEI 13) allows negative velocity gradients depending on the local amplitude of the density), which otherwise are ecluded in the case of inviscid Burgers equations, corresponding to k = 0 In this paper we turn our attention to the full Euler-Poisson equations driven by both pressure and Poisson forcing, ρ t +ρu) =0, 14) ρu) t +ρu ) = pρ) kρϕ, k > 0, ϕ = ρ These equations govern different phenomena, ranging from the largest scale of eg, the evolution gravitational collapse in stars, to applications in the smallest scale of eg, semi-conductors There is a considerable amount of literature available on the local and global behavior of Euler-Poisson and related problems Consult [Ma86] for local eistence in the small H s -neighborhood of a steady state of self-gravitating stars, [CW96] for global eistence of weak solutions with geometrical symmetry, [Gu98] for global eistence for 3-D irrotational flow, [MN95] for isentropic case, and [JR00] [PRV95] for isothermal case Consult [Pe90] [MP90] [Si85] [En96] [WC98], [BW98] and in particular, [En96] more about that below), for non-eistence results and singularity formation The question of global smoothness vs finite breakdown was studied in a recent series of works of Engelberg, Liu and Tadmor, in terms of a critical threshold phenomena for 1-D pressure-less Euler-Poisson equations, [ELT01] and -D restricted Euler-Poisson equations, [LT0, LT03] The natural question that arises in the present contet of full Euler-Poisson equations 14) is whether the pressure enforces a generic finite time breakdown or, whether the presence of Poisson forcing preserves global regularity for a large set of initial configurations We answer this question of competition between pressure and Poisson forcing, proving that the Euler-Poisson equations 14) with γ 1 admit global smooth solutions for a large set of sub-critical initial data such that u 0 ) > K 0 ρ0 )+ Aγ ρ 0) 15), γ 1 ρ 0 ) 3 γ Here, K 0 is a constant depending on k, γ and the initial data In the particular and important) case of isothermal equations, γ = 1, we have K 0 = k and 15) amounts to a sharp critical threshold, 16) u 0 ) kρ 0 )+ A ρ 0) ρ 0 ), γ =1 The inequalities 15),16) quantify the competition between the destabilizing pressure effects, as the range of sub-critical initial configurations shrinks with the growth of the amplitude of the pressure, A, while the stabilizing effect of the Poisson forcing increases the sub-critical range with a growing k In particular, 16) with A = 0 recovers the pressure-free critical threshold 13) Formation of singularities and global regularity of 14) were addressed earlier by Engelberg in [En96] His results show finite-time break-down if u 0 ) Aγ ρ 0 ) ρ 0 ) γ 3 is sufficiently negative at some point Our contribution here is to quantify the critical threshold behind this asymptotic statement To fully appreciate this quantified threshold, we turn to the converse statement in [En96, Theorem ]: it asserts the global regularity of 14) for a class of initial data such that u 0 ) Aγ ρ 0 ) ρ 0 ) γ 3 > 0 It is a non-generic class in the sense of requiring both Riemann invariants at t =0tobeglobally increasing) In fact, by 15) one has a negative threshold, K 0 ρ0, implying the eistence of a large class of sub-critical initial data with global regularity The paper is organized as follows In section, we reformulate the system 14) with its Riemann invariants as a preparation for the analysis carried out in sections 3 and 4 In section 3, we prove

3 ON THE GLOBAL REGULARITY OF SUB-CRITICAL EULER-POISSON EQUATIONS 3 our main results, providing sufficient conditions for large sets of sub-critical initial configurations which yield global smooth solution In section 4, we give eamples of finite time breakdown for supercritical initial data Combining our results in sections 3 and 4, they confirm the eistence of a critical threshold phenomena for the full Euler-Poisson equations 14) Riemann Invariants 1 The Euler-Poisson equations with γ-law pressure: γ > 1 We begin by rewriting the Euler-Poisson equations 14) as a first order quasilinear system 1) ρ + J ρ = 0, u u kϕ where the Jacobian J := u Aγρ γ t ρ has two different eigenvalues u λ := u Aγρ γ 1 <µ:= u + Aγρ γ 1 and let R and S denote the Riemann invariants of the corresponding Euler system 11) R := u Aγ γ 1 ρ γ 1 and S := u + Aγ ) γ 1 ρ γ 1 They satisfy the coupled system of equations, 3a) 3b) R t + λr = kϕ, S t + µs = kϕ, coupled through the Poisson equation φ = ρif we set r := R,s:= S then upon differentiation of 3) we get 4a) 4b) r t + λr + λ S rs + λ R r = kρ, s t + µs + µ S s + µ R rs = kρ Net, we observe that λ = R+S R+S 4 S R) and µ = 4 S R) Hence, epressed in terms of θ := γ 1, we have for γ 1, λ R = µ S = 1+θ and λ S = µ R = 1 θ, θ := γ 1 0, and the pair of equations 4) is recast into the form 5a) r + 1+θ r + 1 θ rs = kρ, 5b) s + 1+θ s + 1 θ rs = kρ Here and below {} := t + λ and {} := t + µ denote differentiation along the λ and µ particle paths, γ 1 + γ 1 Γ λ := {, t) ẋt) =λ ρ, t),u, t) )}, Γµ := {, t) ẋt) =µ ρ, t),u, t) )}

4 4 EITAN TADMOR AND DONGMING WEI To continue, we rewrite the equation for ρ as 6) ρ t + λρ )+ µ λ ρ + ρ s + r =0 Since s r = S R = Aγρ γ 3 ρ, it enables us to epress µ λ ρ = Aγρ γ 1 ρ = ρ s r, so that the ρ equation 6) can be written along the λ particle path as ρ + ρs = 0 Similarly, it can be written along the µ particle path as ρ + ρr = 0 Assembling the above equations together, we arrive at the following system governing r, s and ρ, r + 1+θ r + 1 θ rs = kρ, 7a) ρ + ρs =0, and 7b) s + 1+θ s + 1 θ rs = kρ, ρ + ρr =0 Finally, we use the integration factors 1/ ρ and r/ρ ρ in the first and second equations of each pair in 7), to conclude 8a) 8b) ) r ρ + 1+θ s ρ ) + 1+θ θ rs = k ρ, ρ ρ r θ rs = k ρ ρ ρ s The isothermal case γ =1 In this case, the two eigenvalues are λ = u A<µ= u + A with the corresponding Riemann invariants R = u A ln ρ and S = u + A ln ρ Their derivatives, r and s, satisfy the pair of equations, corresponding to 8a),8b) with θ =γ 1)/ =0, 9a) 9b) ) r ρ + 1 r = k ρ, ρ ) s ρ + 1 s = k ρ ρ 3 Global smooth solutions for sub critical initial data For the pressure-less Euler-Poisson equations 1), the evolution of u and ρ could be traced backwards along the same particle path to their initial data at t = 0 The scenario becomes more complicated with the additional pressure term, due to the coupling of r and s along different particle paths which are traced back to different neighborhoods of the initial line t = 0 This is the main obstacle in finding the sharp critical threshold of the full Euler-Poisson system 14) To this end, we will seek invariant regions for the coupled system, governing the Riemann invariants We begin this section with the following lemma Lemma 31 Given that the total charge E 0 := ρ 0)d <, then ρ, t) and u, t) remain uniformly bounded for all t>0

5 ON THE GLOBAL REGULARITY OF SUB-CRITICAL EULER-POISSON EQUATIONS 5 Proof Under the given condition, we can set eg, [ELT01, p 116]) ϕ, t)= 1 ) ρξ, t)dξ ρξ, t)dξ, which satisfies E 0 ϕ, t) E 0, for all t 0 and R Recall the transport equations 3a),3b) which govern the Riemann invariants along different characteristics R + kϕ = S + kϕ = 0 Since ϕ is bounded, these transport equations tell us that R and S remain uniformly bounded with at most a linear growth in time Indeed, for all M 1 we have 31) sup M { R, t), S, t) } C 0 + ke 0 t, C 0 := sup M+u t { R 0 ), S 0 ) } Take the sum and difference of S and R to find that u, t) and ρ, t) in ) remain bounded, 3) u := sup u, t) C 0 + ke 0 t, M sup M ρ, t) Const C 0 + ke 0 t) γ 1, γ > 1, ep ke 0 t), γ =1 We note in passing that the time growth asserted in 3) is probably not sharp; the estimate can be improved after taking into account the uniform bounds of R / ρ and S / ρ discussed in theorems 31 and 3 below Remark 31 According to Lemma 31, the only way that the full Euler-Poisson system 14) breaks down at a finite time is through the formation of shock discontinuities where u and/or ρ blow up, but neither will concentrate at any critical point This is in contrast to the breakdown of the pressure-less Euler-Poisson equations 1), where u, t) =ρ, t) simultaneously at the critical time 31 Critical Threshold for isothermal case: γ = 1 We begin with the isothermal case, γ = 1, which plays an important role in various applications Compared with the general case 8), the isothermal case becomes simpler due to the fact that θ = 0 decouples the dependence on r and s through the mied term θrs, which disappears from left hand side of 9) Here we prove the following sharp characterization of the critical threshold phenomena Theorem 31 Consider the isothermal Euler-Poisson system 14) with pressure forcing pρ) =Aρ, and subject to initial data u 0,ρ 0 > 0) with finite total charge, E 0 = ρ 0)d < The system admits a global smooth, C 1 -solution if and only if 33) u 0 ) kρ 0 )+ A ρ 0) ρ 0 ), R Remark 3 Epressed in terms of the Riemann invariants specified in, u ± Aρ /ρ, theorem 31 states that the isothermal Euler-Poisson equations admit global smooth solutions for sub-critical initial conditions, 34) Proof We define X := r ρ and Y := s 0 kρ 0 and r 0 kρ 0 s ρ Equations 9a),9b) then read

6 6 EITAN TADMOR AND DONGMING WEI 35a) 35b) It follows that and similarly, X Y ρ X = k X ), ρ Y = k Y ) > 0, X k, k ), =0, X = k, < 0, X > k, > 0, Y k, k ), =0, Y = k, < 0, Y > k Thus, starting with 34), X 0,Y 0 k, we find that X and Y remain bounded within the invariant region [ k, k], or otherwise, they are decreasing outside this interval We conclude that { } X,t),Y,t) ma k, X0 ),Y 0 ) Lemma 31 tells us that ρ is bounded The boundedness of X, Y and ρ imply that r = X ρ and s = Y ρ remain bounded for all t<, and hence the Euler-Poisson system 14) admits a global smooth C 1 -solution Conversely, suppose that there eists X 0 = X 0 ) < k We will show that this value will evolve along Γ λ 0, 0) such that X,t) will tend to at a finite time To this end, assume that Y is well behaved, ie, Y 0 ) { k so that Y,t) Y 1 := ma Y 0 ), } k for all t s otherwise, the finite time blow up of Y can be argued along the same lines) It follows that s = Y ρ Y 1 ρ and inserting this into ρ = ρs, we find ρ Y 1 ρ 3/ This yields the lower-bound ) ρ Y 1 t +/, ρ 0 and together with 35a), we conclude that X,t) satisfies the following Ricatti equation along the Γ λ -path, X 1 36) X Y 1 t +/ X, X 1 := X0 k)/x0 > 0 ρ 0 Integration of 36) yields 37) X,t) X 1 ln 1+ ρ 0 Y 1 t/ ) + Y 1 X 0 Thus, starting with X 0 < k<0 it follows that there eists a finite critical time t c > 0 such that Xt t c ) tends to The critical threshold condition 33) reflects the competition between the Poisson forcing and the pressure It yields global smooth solutions for a large set of initial configurations allowing negative velocity gradients In the particular case that there is no pressure, A = 0, 33) is reduced to the critical threshold condition of the pressure-less Euler-Poisson equations u 0 > kρ 0 ) of [ELT01] Y 1

7 ON THE GLOBAL REGULARITY OF SUB-CRITICAL EULER-POISSON EQUATIONS 7 3 Critical threshold for γ > 1 The equations for the Riemann invariants 8a), 8b) are coupled through the mied term, θrs/ We note in passing that it is possible to get rid of this mied term when integrating 7a), 7b) with the integration factors ρ γ 3)/4, and rρ γ 7)/4 3 γ)/4 in the first and second equations in each pair, yielding ) rρ θ θ sρ θ 1 r ρ θ 1 = kρ 1+θ, ) + 1+θ s ρ θ 1 = kρ 1+θ Nevertheless, it will prove useful to use the same integration factors, 1/ ρ and r/ρ ρ which led to 8) The main task is to identify the invariant region associated with 8), corresponding to the isothermal invariant region [ k, k] discussed in theorem 31 Theorem 3 Consider the Euler-Poisson system 14) with γ law pressure pρ) =Aρ γ, γ>1, subject to initial data u 0,ρ 0 > 0) with finite total charge, E 0 = ρ 0)d < Then, there eists a constant K 0 > 0 depending on k, γ and the initial conditions specified in 39b) below), such that the Euler-Poisson equations 14) admit a global smooth, C 1 -solution if, u 0 ) K 0 ρ0 )+ Aγ ρ 0) 38) ρ 0 ) 3 γ Before we turn to the proof of this theorem, several remarks are in order Remark 33 Epressed in terms of the Riemann invariants, r = u Aγρ 0 /ρ 3 γ)/ 0 and s = u + Aγρ 0 /ρ 3 γ)/ 0, the critical threshold 38) reads r 0 ) ρ0 ), s 0 ) 39a) ρ0 ) K 0 The constant K 0 is given by K 0 = θm 0 + θ M0 { } +8k1 + θ) k, r 0 ), M 0 = ma 1 + θ) ρ0 ), s 0 ) 39b) ρ0 ) We mention two simplifications which are summarized in the following two corollaries We first observe that if the initial configurations satisfy the upper-bound r 0 ),s 0 ) kρ 0 ) then 39b) yields M 0 = k k, hence K 0 =, and theorem 3 implies the following 1+θ Corollary 31 Consider the Euler-Poisson system 14) with γ law pressure pρ) =Aρ γ, γ>1, subject to initial data u 0,ρ 0 > 0) with finite total charge, E 0 = ρ 0)d < Then, the Euler- Poisson equations 14) admit a global smooth, C 1 -solution if for all R, u 0 ) kρ 0 ) Aγ ρ 0) 310) ρ 0 ) 3 γ The net result follows from the trivial inequality K 0 θm 0 θm 0 + 8k1 + θ) ) / 1 + θ) Corollary 3 Consider the Euler-Poisson system 14) with a γ-law pressure pρ) =Aρ γ, γ>1, subject to initial data u 0,ρ 0 > 0) with finite total charge, E 0 = ρ 0)d < Then, the Euler- Poisson equations 14) admit a global smooth, C 1 -solution, if for all R,

8 8 EITAN TADMOR AND DONGMING WEI 311) u 0 ) + kρ 0 ) γ ) γ 1 γ +1) ma { kρ0 ),u 0)+ Aγ ρ 0) ρ 0 ) 3 γ } + Aγ ρ 0) ρ 0 ) 3 γ Remark 34 We observe that as in the isothermal case, the critical threshold in its various versions 38),39), 310) and 311), allow a large set of initial configurations with negative velocity gradient, due to the competition between the stabilizing Poisson forcing kρφρ) and the destabilizing pressure Aρ γ ) In the etreme case that Poisson forcing is missing k =0, the breakdown of the system is generic unless u 0 is positive enough so that r 0,s 0 > 0) In the other etreme of a pressure-less Euler-Poisson, A =0,γ=1, the critical thresholds 38), 310) are reduced to u 0 ) > kρ 0 ), which coincides with the pressure-less critical threshold 13) found in [ELT01] Proof Epressed in terms of X := 31a) 31b) r ρ and Y := X = ρ Y = ρ k 1+θ s ρ, equations 8) read X + θ ) XY, k 1+θ Y + θ XY We seek an invariant region of the form [ K 0,M 0 ], with K 0,M 0 > 0 yet to be determined We begin by noticing that if X, Y M then 1 X + Y M, and recalling that θ 0, 31) then yield X ρ k 1+θ X + θ ) M, X > 0, Y ρ k 1+θ Y + θ ) M, Y > 0 This in turn implies that k + θm 313) X and Y are decreasing if X, Y > C +, C + = C + M) := 1+θ The solution of C + M) =M yields M = k Thus, X and Y are decreasing whenever X, Y > M = k, and we end up with the upper-bound { k, 314) X,t),Y,t) M 0, M 0 := ma X0 ),Y 0 )} In a similar manner, we study the lower bound of the invariant region By 314) and 31) yield X ρ k 1+θ X + θ ) 315a) M 0X, X < 0, Y ρ k 1+θ Y + θ ) 315b) M 0Y, Y < 0, which in turn, imply that 316a) X and Y are increasing if 0 X, Y > K 0, 1 We let Z + = ma{x, 0} and Z = min{z, 0} denote the positive and negative part of Z )

9 ON THE GLOBAL REGULARITY OF SUB-CRITICAL EULER-POISSON EQUATIONS 9 where K 0 is the smallest root of the quadratics on the right of 315), K 0 := θm 0 + θ M0 +8k1 + θ) 316b) 1 + θ) The critical threshold condition 38) tells us that at t =0,X 0,Y 0 K 0 and 316a) implies that X,t) and Y,t) remain above the same lower-bound, 38) As before, the bounds of X, Y and ρ imply that r = X ρ and s = Y ρ remain bounded, and hence the Euler-Poisson system 14) a global smooth, C 1 -solution 4 Finite time breakdown for super-critical initial data Consider the Euler-Poisson system 14) with a γ-law pressure, γ 1, and subject to initial data such that r 0 ),s 0 ) k Then, according to corollary 31, the following critical threshold is sufficient for the eistence of global smooth solutions, u 0 ) kρ 0 )+ Aγ ρ 0) ρ 0 ) 3 γ In this section we show that this critical threshold is also necessary for global regularity Theorem 41 Consider the Euler-Poisson system 14) with a γ-law pressure pρ) =Aρ γ, γ 1, subject to initial data u 0,ρ 0 > 0) The system loses the C 1 -smoothness if there eists an R such that u 0 ) < kρ 0 )+ Aγ ρ 0) 41) ρ 0 ) 3 γ Remark 41 Epressed in terms of the Riemann invariants, r = u Aγρ 0 /ρ 3 γ)/ 0 and s = u + Aγρ 0 /ρ 3 γ)/ 0, the condition 41) reads 4) R st r 0 ) < kρ 0 ), or s 0 ) < kρ 0 ) The lack of smoothness in this case was shown in theorem 31 for γ =1and is etended for γ>1 below Proof Recall equations 31) for X := 43a) 43b) X = ρ Y = ρ r ρ and Y := k 1+θ s ρ X + θ ) XY, k 1+θ Y + θ XY In the proof of theorem 3, we have shown that X and Y have an upper bound { k, 44) X,t),Y,t) M 0, M 0 := ma X0 ),Y 0 )} Suppose that there eists X 0 = X 0 ) < k We will show that this value will evolve along Γ λ 0, 0) such that X,t) will tend to at a finite time To this end, assume that Y is well behaved, ie, Y 0 ) k so that Y,t) M 0 for all t s otherwise, the finite time blow up of Y can be argued along the same lines) It follows that along Γ λ 0, 0) X = ρ k 1+θ X + θ ) XY < ρ k 1 ) 45) X )

10 10 EITAN TADMOR AND DONGMING WEI Following eactly what we have done in the proof of theorem 31, we obtain the inequality M 0 46) X,t) X 1 ln 1+ ρ 0 M 0 t/ ), + M 0 X 0 where X 1 := X0 k)/x 0 > 0 Thus, starting with X 0 < k<0 it follows that there eists a finite critical time t c > 0 such that Xt t c ) tends to We conclude with an eample for a finite time breakdown Eample: Suppose at t =0,u 0 ) = 0 and 1, < 0, ρ 0 ) = 1 ɛ, 0 ɛ, 1, >ɛ Thus Aγ 1 ) /ɛ, 0 <<ɛ, s 0 ) = ɛ 0, elsewhere If we choose ɛ small enough, then s 0 ) < kρ 0 ) for 0 <<ɛ According to theorem 41, the system 14) will break down in a finite time This eample shows that even if the fluid is near rest at t = 0, the pressure itself could still lead to collision References [BW98] M P Brenner, T P Witelski, On Spherically Symmetric Gravitational Collapse, J of Statistical Physics, Vol93,1998 [CW96] G -Q Chen, D Wang, Convergence of shock capturing scheme for the compressible Euler-Poisson equation, CommMathPhys ), [En96] S Engelberg, Formation of singularities in the Euler and Euler-Poisson equations, Physica D ), [ELT01] S Engelberg, H Liu and E Tadmor, Critical Thresholds in Euler-Poisson Equations, Indiana UnivMathJ 50, ) [Gu98] Y Guo, Smooth irrotational flows in the large to the Euler-Poisson system in R 3+1, CommMathPhys ), [JR00] S Junca, M Rascle, Relaation of the isothermal Euler-Poisson system to the drift-diffusion equations, QuartApplMath 58000), [La64] P D La, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J Math Physics 55), 1964), [LT0] H Liu, E Tadmor, Spectral dynamics of the velocity gradient field in restricted flows, Communications in Mathematical Physics 8 00), [LT03] H Liu, E Tadmor, Critical thresholds in D restricted Euler-Poisson equations SIAM J Appl Math ), no 6, [Ma86] T Makino, On a local eistence theorem for the evolution of gaseous stars, In: Patterns and WavesTNishida, MMinura, HFujii, eds), North-Holland/Kinokuniva,1986,pp [MN95] P Marcati, R Natalini, Weak solutions to a hydrodynamic model for semiconductors and relaation to the drift-diffusion equation, ArchRatMechAnal191995), [MP90] T Makino, B Perthame, Sur les solutions à symétrie sphérique de l équation d Euler-Poisson pour l évolution d étoiles gazeuses, Japan JApplMath71990), [Pe90] B Perthame, Noneistence of global solutions to the Euler-Poisson equations for repulsive forces, Japan JApplMath71990), [PRV95] F Poupaud, M Rascle, J-P Vila, Global solutions to the isothermal Euler-Poisson system with arbitrarily large data J Differential Equations ), no1, [Si85] T C Sideris, Formation of singularities in three-dimensional compressible fluids, CommunMathPhys101, )

11 ON THE GLOBAL REGULARITY OF SUB-CRITICAL EULER-POISSON EQUATIONS 11 [WC98] D Wang, G-Q Chen, Formation of singularities in compressible Euler-Poisson fluids with heat diffusion and damping relaation, JDiffEqs ),44-65 Eitan Tadmor) Department of Mathematics, Institute for Physical Science and Technology and Center of Scientific Computation And Mathematical Modeling CSCAMM) University of Maryland College Park, MD 074 USA address: tadmor@cscammumdedu URL: Dongming Wei) Department of Mathematics Center of Scientific Computation And Mathematical Modeling CSCAMM) University of Maryland College Park, MD 074 USA address: dwei@cscammumdedu URL: