Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 60, pp. 1–21. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2023.60 SPACE-TIME BEHAVIOR FOR RADIATIVE HYDRODYNAMICS MODEL WITH OR WITHOUT HEAT CONDUCTION MENGQIAN LIU, ZHIGANG WU Abstract. We consider space-time behaviors of smooth solutions for the ra- diative hydrodynamics system with or without heat conduction in the whole space R3 by using Green’s function method. This result exhibits the general- ized Huygens’ principle as the classical compressible Navier-Stokes equations [3, 26], which is different from the Hamer model for radiating gases in [36]. 1. Introduction In this paper, we are concerned with a radiation hydrodynamics model of the compressible Navier-Stokes equations coupled with an elliptic equation for radiative flux. Such model is used to describe the motion of viscous and heat-conducting flu- ids with radiative effects, and simulate supernova explosions, nonlinear stellar pul- sation and stellar winds in astrophysics. In particular, we shall study the equations of radiation hydrodynamics [9], ρt + div(ρu) = 0, (ρu)t + div(ρu⊗ u) +∇P = µdiv(∇u + (∇u)T ) + µ′∇ divu, (ρE)t + div(ρuE + uP ) + divq = κ∆θ + div(µ(∇u + (∇u)T ) · u + µ′ divu), −∇divq + q +∇(θ4) = 0. (1.1) Here ρ(x, t), u(x, t) ∈ R3, q(x, t) ∈ R3 and θ(x, t) are the fluid density, velocity, radiative heat flux and temperature, respectively. The total specific energy E = e+ 1 2 |u| 2 with the specific internal energy e. Without loss of generality, we consider the ideal polytropic gases for the system (1.1), that is, the pressure P = Rρθ and the specific internal energy e = Cvθ with the positive constants R and Cv. Positive constants µ and µ′ are coefficients of viscosity satisfying µ > 0 and 2µ+µ′ > 0, and κ denotes the coefficient of heat conduction. We consider two cases: κ > 0 (with heat conduction) and κ = 0 (without heat conduction). 2020 Mathematics Subject Classification. 35Q30, 35B40, 35Q35. Key words and phrases. Green’s function; radiative hydrodynamics model; space-time estimate. ©2023. This work is licensed under a CC BY 4.0 license. Submitted March 2, 2023. Published September 15, 2023. 1 2 M. LIU, Z. WU EJDE-2023/60 We would like to mention the Hamer model of radiating gases [10], vt + n∑ i=1 fi(v)xi + divq = 0, −∇divq + q +∇v = 0, (1.2) where v is a scalar unknown function. This simplified model provides a good ap- proximation to the radiation hydrodynamics model. There are many important efforts for the one dimensional case, see [28] for the time-asymptotic behavior of solutions with discontinuous initial data, [31] for the asymptotic stability of the rarefaction wave, [32] for the stability of traveling waves and L1-stability of con- stants, and [29] for the convergence rates toward the travelling waves. For the multi-dimensional case, [6] studied the relaxation limit of the initial value prob- lem, and [4, 7, 8] focused on decay rates to the planar rarefaction waves. Last but not least, we shall refer to [36], where pointwise space-time estimate for the global classical solution of the Cauchy problem in any dimension n ≥ 1 was given. In particular, they obtained the following estimate of v, |v(x, t)| ≤ C(1 + t)−n/2 ( (1 + t)−n/2(1 + |x|2 1 + t )−γ + (1 + |x− ct|2 1 + t )−γ ) , (1.3) with γ = min{r, 3n/4}. Here the vector c = (f ′1(0), · · · , f ′n(0)), and r is from the pointwise assumption on the initial data |v0| ≤ Cε0(1 + |x|2)−r with r > n 2 and suitably small constant ε0 > 0. Additionally, (1.3) immediately gives the Lp estimate with p ≥ 1. For small perturbation problems of the typical fluid models with suitable dis- sipation, L2-estimates can be derived by standard energy methods. However, the usual L2-estimates can only exhibit the dissipative properties. On the other hand, pointwise space-time estimates can provide more information of the solution, even exhibit the wave propagations. In this field, there are also a few results. The pio- neering works was Zeng [43] and Liu and Zeng [27] for one dimensional compressible fluid models. The isentropic compressible Navier-Stokes system in three dimensions was investigated by Hoff and Zumbrun [11, 12] and Liu and Wang [26] for the linear and nonlinear problems, respectively. In [26], they verified the generalized Huygens’ principle through detailed analysis on Green’s function and developing subtle non- linear estimates of the convolutions. A hyperbolic-parabolic system obeys the gen- eralized Huygens’ principle in [26] implies that its pointwise space-time description contains both a diffusion wave (D-wave): (1 + t)−3/2 ( 1 + |x|2 1+t )−3/2 and a Huygens’ wave (H-wave): (1 + t)−2 ( 1 + (|x|−ct)2 1+t )−3/2 in R3. Obviously, the L2-decay rates of these two diffusion waves are the same as the heat kernel. Recently, there are series of works on other compressible fluid models, for instance, the Navier-Stokes equa- tions in [1, 20, 25], the damped Euler equation in [38], the Navier-Stokes-Poisson equations in [37, 39, 40, 41] and references therein. We aim to obtain the pointwise description of the solution around the steady state for the Cauchy problem of (1.1) in this paper. The motivation is to find the difference between our pointwise result for the model (1.1) (κ ≥ 0) and the pointwise result (1.3) for the simplified model (1.2). Since the conservation of the system will be used in deriving the pointwise space-time behavior of the solution EJDE-2023/60 RADIATIVE HYDRODYNAMICS MODELS 3 to this nonlinear problem, the initial data for the system (1.1) is given as follows: (ρ,m, w)|t=0 = (ρ0,m0, w0)(x), x ∈ R3. (1.4) Now, we come back to some of results on the model (1.1) closely related to the topic in this paper, since there exactly are lots of literature devoted to the mathe- matical theory on radiative hydrodynamics. The system (1.1) can also be viewed as the Navier-Stokes-Fourier equations coupled with a parabolic equation with high order nonlinear term of the temperature θ. In one-dimensional case, the radiation takes a rather good effect on the system even when omitting the heat conduction and viscosity. In fact, the Cauchy problem of the system (1.1) guarantees a unique global smooth solution for the small perturbation. See Kawashima et al. [15, 16], Lin and Goudon [23] for global existence of classical solutions and Deng and Yang [2] for pointwise space-time behavior. In terms of the stability of three elementary waves, see Lattanzio et al. [18] and Lin et al. [24] for the shock wave, Lin et al. [22] for rarefaction wave and Wang and Xie [34] for viscous contact wave. He et al. [17] and Liao and Zhao [21] investigated the global existence and large-time behavior for the viscous radiative and reactive gas. For the stability of the composite of the elementary waves, we refer to Fan [5] about two viscous shock waves, Rohde et al. [30] about rarefaction and contact waves, and Xie [42] about viscous contact wave and rarefaction waves. Hong [13] gave the large-time behavior toward the combination of two rarefaction waves and viscous contact wave when there exist additional heat conduction and viscosity. We shall also refer to Li [19] for the for- mation of singularities of the large perturbation. In multi-dimensional case, Huang and Zhang [14] studied the asymptotic stability of planar rarefaction wave, Wan and Xu [33] considered radial symmetric classical solutions in an exterior domain and a bounded concentric annular domain. Finally, mostly close to our topic, Wang and Xie [35] obtained the existence of the classical global solution and the decay rate by energy method in H4-framework. Recently, the condition on the regularity of initial data was relaxed to H2-framework by Gong et al. in [9]. In deriving the pointwise description of the solution constructed in [35], the main difficulties include that deriving the representation of this Green’s matrix in Fourier space, dealing with the singularity from Riesz operator in the Green’s matrix for the pointwise estimates through finding suitable combinations for the singular components, and closing the ansatz for the nonlinear problem. In addition, compared with the non-isentropic Navier-Stokes equations in [3], the presence of nonlocal operator (1−∆)−1 arising from the relation of q and θ in (1.1)4 will bring some new differences when deriving the pointwise estimates for both the Green’s function and the nonlinear coupling. See the details in Section 2 and Section 3. Our main result can be stated as follows. Theorem 1.1. Assume that (ρ0 − ρ̄, ρ0u0, w0 − w̄) ∈ H5(R3), with ε0 := ‖(ρ0 − ρ̄, ρ0u0, w0 − w̄)‖H5(R3) small and the constants ρ̄ > 0 and w̄ > 0. Then there is a unique global classical solution (ρ, ρu, w) of the Cauchy problem (1.1)-(1.4). If further, the initial data satisfies the pointwise assumptions |∂αx (ρ0 − ρ̄, ρ0u0, w0 − w̄)| ≤ Cε0(1 + |x|2)−r1 , r1 > 21 10 , |α| ≤ 1. (1.5) 4 M. LIU, Z. WU EJDE-2023/60 Then for the base sound speed c := √ (1 + R Cv ) Rw̄Cv ρ̄ , it holds for |α| ≤ 1 that |∂αx (ρ− ρ̄, ρu)| ≤ C(1 + t)− 4+|α| 2 ( 1 + (|x| − ct)3 1 + t )−3/2 + C(1 + t)− 3+|α| 2 ( 1 + |x|2 1 + t )−3/2 , |∂αx (w − w̄)| ≤ C(1 + t)− 4+|α| 2 (( 1 + (|x| − ct)3 1 + t )−3/2 + ( 1 + |x|2 1 + t )−3/2) . (1.6) The base sound speed c = √ (1 + R Cv ) Rw̄Cv ρ̄ is the same as the non-isentropic Navier-Stokes systems in [3], which means that the radiation does not affect the propagation speed of the Huygens’ wave. Note that (1.6) exhibiting the generalized Huygens’ principle is completely different from (1.3) for the model (1.2). From the spectral analysis in Section 2, we know that the system is linear stable when κ > 0 and κ = 0, and these two cases are almost the same when analyzing Green’s function in different frequency parts. In other words, we can see that for radiative hydrodynamics, the radiation effect can do the same job as the heat conduction when deducing the space-time behavior of the solution. As a byproduct, one get Lp-decay estimate when |α| ≤ 1: ‖∂αx (ρ− ρ̄, ρu)(·, t)‖Lp(R3) ≤ C(1 + t)− ( 2− 5 2p ) − |α|2 , 1 < p ≤ 2, C(1 + t)− 3 2 (1− 1 p )− |α|2 , 2 ≤ p ≤ ∞, ‖∂αx (w − w̄)(·, t)‖Lp(R3) ≤ C(1 + t)−(2− 1 p )− |α|2 , 1 < p ≤ ∞. (1.7) This article is structured as follows. Section 2 shows the representation of Green’s function in Fourier space and giving the expansions of spectral and Green’s function in high-medium-low frequencies, and then establishes the pointwise estimate of Green’s function. The pointwise estimates of the solution for the nonlinear problem is given in Section 3. Some useful lemmas are stated in Appendix. 2. Green’s function In this section, we shall first derive the representation of Green’s function in Fourier space, and then establish pointwise estimates of Green’s function. 2.1. Linearization and Green’s function. Here we assume the steady state of the Cauchy problem (1.1)-(1.4) is (ρ̄, 0, w̄). For simplicity, we use (ρ,m, w) to denote the perturbation (ρ − ρ̄,m, w − w̄). Then, the system (1.1) can be rewritten as ρt + divm = 0, mt + R Cv ∇w − µ ρ̄ ∆m− (µ+ µ′) ρ̄ ∇divm = F1, wt + w̄ ρ̄ (1 + R Cv ) divm + κw̄ Cvρ̄2 ∆ρ+ 4w̄4 C4 v ρ̄ 5 ∆ 1−∆ ρ − κ Cvρ̄ ∆w − 4w̄3 C4 v ρ̄ 4 ∆ 1−∆ w = F2, (2.1) EJDE-2023/60 RADIATIVE HYDRODYNAMICS MODELS 5 where the nonlinear terms are F1 = −div m⊗m ρ+ ρ̄ + R 2Cv ∇ |m| 2 ρ+ ρ̄ − µ∆ ρm ρ̄(ρ+ ρ̄) − (µ+ µ′)∇ div ρm ρ̄(ρ+ ρ̄) , F2 = κ Cvρ̄2 ∆ (w̄ρ− ρ̄w)ρ ρ+ ρ̄ − κ 2Cv ∆ |m|2 (ρ+ ρ̄)2 + (1 + R Cv ) div (w̄ρ− ρ̄w)m ρ̄(ρ+ ρ̄) + R 2Cv div |m|2m (ρ+ ρ̄)2 + 1 C4 v ∆ 1−∆ 4w̄3(w̄ρ− ρ̄w)(ρ4 + 4ρ̄ρ3 + 6ρ̄2ρ2 + 4ρ̄3ρ) ρ̄5(ρ+ ρ̄)4 + 1 C4 v ∆ 1−∆ [w4 + 4w̄w3 + 6w̄2w2 (ρ+ ρ̄)4 − w̄4(ρ4 + 4ρ̄ρ3 + 6ρ̄2ρ2) ρ̄4(ρ+ ρ̄)4 ] + 1 C4 v ∆ 1−∆ [ |m|8 16(ρ+ ρ̄)8 − (w + w̄)|m|6 2(ρ+ ρ̄)7 + 3(w + w̄)2|m|4 2(ρ+ ρ̄)6 − 2(w + w̄)3|m|2 (ρ+ ρ̄)5 ] + div [ µ(∇( m ρ+ ρ̄ ) +∇( m ρ+ ρ̄ )T ) · m ρ+ ρ̄ + µ′ div( m ρ+ ρ̄ ) m ρ+ ρ̄ ] . (2.2) We consider Green’s function of the linearized system of (2.1) about the variables (ρ,m, w). The linearized equation with the initial condition can be written as ρt + divm = 0, mt + R Cv ∇w − µ ρ̄ ∆m− (µ+ µ′) ρ̄ ∇ divm = 0, wt + w̄ ρ̄ (1 + R Cv ) divm + κw̄ Cvρ̄2 ∆ρ+ 4w̄4 C4 v ρ̄ 5 ∆ 1−∆ ρ − κ Cvρ̄ ∆w − 4w̄3 C4 v ρ̄ 4 ∆ 1−∆ w = 0, (ρ,m, w)|t=0 = (ρ0,m0, w0). (2.3) Set U = (ρ,m, w)T , then ∂tU = AU, (2.4) where A is the differential operator corresponding to (2.3) and A =  0 −div 0 0 µ ρ̄∆ + µ+µ′ ρ̄ ∇div − R Cv ∇ − κw̄ Cv ρ̄2 ∆− 4w̄4 C4 v ρ̄ 5 ∆ 1−∆ − w̄ρ̄ (1 + R Cv ) div κ Cv ρ̄ ∆ + 4w̄3 C4 v ρ̄ 4 ∆ 1−∆  . In what follows, we use G(x, t) to denote the Green’s function of (2.3), which is defined as Gt = AG, G(x, 0) = δ(x)I. (2.5) Here δ(x) is the Dirac delta function and I is the 5× 5 identity matrix. 6 M. LIU, Z. WU EJDE-2023/60 2.2. Derivation of Green’s function Ĝ(ξ, t). In this subsection, we compute Green’s function for the linearized system of (2.1) with the initial data Û0 := (ρ̂0, m̂0, ŵ0)T = (ρ̂0, m̂0,1, m̂0,2, m̂0,3, ŵ0)T . (2.6) By employing the Fourier transform, it is obvious from (2.4) that ∂tÛ = A1Û , (2.7) where A1 =  0 −iξT 0 0 −µρ̄ |ξ| 2I − (µ+µ′) ρ̄ ξξT − R Cv iξ kw̄ Cv ρ̄2 |ξ|2 + 4w̄4|ξ|2 C4 v ρ̄ 5(1+|ξ|2) − w̄ρ̄ (1 + R Cv )iξT − κ Cv ρ̄ |ξ|2 − 4w̄3|ξ|2 C4 v ρ̄ 4(1+|ξ|2)  . After a direct computation, we can get the eigenvalues of A1: λ1, λ2, λ3,−µρ̄ ξ 2 (with multiplicity 2), and their corresponding right eigenvectors g1 iξ1 iξ2 iξ3 ν1  ,  g2 iξ1 iξ2 iξ3 ν2  ,  g3 iξ1 iξ2 iξ3 ν3  ,  0 ξ3 0 −ξ1 0  ,  0 0 ξ3 −ξ2 0  , (2.8) with g1 := |ξ|2 λ1 , g2 := |ξ|2 λ2 , g3 := |ξ|2 λ3 , ν1 := −Cv(2µ+ µ′) Rρ̄ |ξ|2 − Cv R λ1, ν2 := −Cv(2µ+ µ′) Rρ̄ |ξ|2 − Cv R λ2, ν3 := −Cv(2µ+ µ′) Rρ̄ |ξ|2 − Cv R λ3. Here λ1, λ2, and λ3 are the roots of the equation λ3 + ( κ Cvρ̄ + 4w̄3 C4 v ρ̄ 4(1 + |ξ|2) + 2µ+ µ′ ρ̄ )|ξ|2λ2 + [ κ(2µ+ µ′) Cvρ̄2 |ξ|4 + 4(2µ+ µ′)w̄3|ξ|4 C4 v ρ̄ 5(1 + |ξ|2) + (Cv +R)Rw̄ C2 v ρ̄ |ξ|2]λ+ κRw̄ C2 v ρ̄ 2 |ξ|4 + 4Rw̄4|ξ|4 C5 v ρ̄ 5(1 + |ξ|2) = 0. (2.9) Then the solution of the Cauchy problem (2.6)-(2.7) can be written as ρ̂ m̂1 m̂2 m̂3 ŵ  = A  0 ξ3 0 −ξ1 0 +B  0 0 ξ3 −ξ2 0   e− µ ρ̄ |ξ| 2t + C  g1 iξ1 iξ2 iξ3 ν1  eλ1t +D  g2 iξ1 iξ2 iξ3 ν2  eλ2t + E  g3 iξ1 iξ2 iξ3 ν3  eλ3t. (2.10) EJDE-2023/60 RADIATIVE HYDRODYNAMICS MODELS 7 By using the initial values, we have A  0 ξ3 0 −ξ1 0 +B  0 0 ξ3 −ξ2 0  =  0( I − ξξτ |ξ|2 ) m̂0,1( I − ξξτ |ξ|2 ) m̂0,2( I − ξξτ |ξ|2 ) m̂0,3 0  , (2.11) C g1 iξ ν1 +D g2 iξ ν2 + E g3 iξ ν3  =  ρ̂0 ξξT |ξ|2 m̂0 ŵ0  , (2.12) with C = C1ρ̂0 + C2 ξτ · m̂0i |ξ|2 + C3ŵ0, D = D1ρ̂0 +D2 ξτ · m̂0i |ξ|2 +D3ŵ0, E = E1ρ̂0 + E2 ξτ · m̂0i |ξ|2 + E3ŵ0, (2.13) and C1 = ν3 − ν2 Ω , C2 = ν3g2 − ν2g3 Ω , C3 = g2 − g3 Ω , D1 = ν1 − ν3 Ω , D2 = ν1g3 − ν3g1 Ω , D3 = g3 − g1 Ω , E1 = ν2 − ν1 Ω , E2 = ν2g1 − ν1g2 Ω , E3 = g1 − g2 Ω . (2.14) Here Ω = −ν3g2 + ν3g1 + ν1g2 + ν2g3 − ν2g1 − ν1g3. Therefore, the Fourier transform of Green’s function can be rewritten as Ĝ = ˆ̃Ge− µ ρ̄ |ξ| 2t + Ĝ1eλ1t + Ĝ2eλ2t + Ĝ3eλ3t, (2.15) with ˆ̃G =  0 01×3 0 03×1 I − ξξτ |ξ|2 03×1 0 01×3 0  , Ĝ1 = g1C1 g1C2 iξτ |ξ|2 g1C3 iξC1 −C2 ξξτ |ξ|2 iξC3 ν1C1 ν1C2 iξτ |ξ|2 ν1C3  , (2.16) Ĝ2 = g2D1 g2D2 · iξ τ |ξ|2 g2D3 iξD1 −D2 ξξτ |ξ|2 iξD3 ν2D1 ν2D2 · iξ τ |ξ|2 ν2D3  , Ĝ3 = g3E1 g3E2 iξτ |ξ|2 g3E3 iξE1 −E2 ξξτ |ξ|2 iξE3 ν3E1 ν3E2 iξτ |ξ|2 ν3E3  (2.17) 2.3. Pointwise estimates of Green’s function. In this section, we shall derive the pointwise estimate of Green’s function G(x, t) by using the expression of Ĝ(ξ, t) in (2.15)-(2.17) together with the high-low frequency decomposition. To this end, we divide the Green’s function as follows: G(x, t) := χ1(D)G(x, t) + χ2(D)G(x, t) + χ3(D)G(x, t), where χ1(ξ) = { 1, |ξ| < ε1, 0, |ξ| > 2ε1, χ3(ξ) = { 1, |ξ| > K + 1, 0, |ξ| < K, be the smooth cut-off functions with 2ε1 < K and χ2 = 1− χ1 − χ3. 8 M. LIU, Z. WU EJDE-2023/60 Low frequency part. Three roots of the characteristic equation (2.9) in low fre- quency can be estimated by using the implicit function theorem as in [20], and we omit the proof for simplicity. Lemma 2.1. For sufficiently small |ξ|, λ1 is real and λ2,3 are complex conjugate, and λ1 = − κ ρ̄ + 4w̄3 C3 v ρ̄ 4 Cv +R |ξ|2 + ∞∑ j=2 aj |ξ|2j ; (2.18) λ2,3 = − [2µ+ µ′ 2ρ̄ + R( κ 2Cv ρ̄ + 2w̄3 C4 v ρ̄ 4 ) Cv +R ] |ξ|2 + ∞∑ j=2 ā2j |ξ|2j ± i(c|ξ|+ ∞∑ j=2 ā2j−1|ξ|2j−1), (2.19) where the base sound speed c = √ (1 + R Cv ) Rw̄Cv ρ̄ , and the coefficients aj, ā2j and ā2j−1 are real. Subsequently, one has the following for the components in (2.14) after a direct computation. Lemma 2.2. For sufficiently small |ξ|, we have the following expansions: g1 = − Cv +R κ ρ̄ + 4w̄3 C3 v ρ̄ 4 + ∞∑ j=1 b2j |ξ|2j , ν1 = Cv R ( κ ρ̄ + 4w̄3 C3 v ρ̄ 4 Cv +R − 2µ+ µ′ ρ̄ )|ξ|2 + ∞∑ j=2 c2j |ξ|2j , g2,3 = − 2µ+µ′ 2ρ̄ + R( κ 2Cvρ̄ + 2w̄3 C4 vρ̄ 4 ) Cv+R c2 |ξ|2 + ∞∑ j=2 b̃2j |ξ|2j ∓ i( 1 c |ξ|+ ∞∑ j=2 b̃2j−1|ξ|2j−1), ν2,3 = [ κ 2ρ̄ + 2w̄3 C3 v ρ̄ 4 Cv +R − Cv(2µ+ µ′) 2Rρ̄ ]|ξ|2 + ∞∑ j=2 c̃2j |ξ|2j ∓ Cv R i(c|ξ|+ ∞∑ j=2 c̃2j−1|ξ|2j−1), Ω = −i2c R Cv(Cv +R) κ ρ̄ + 4w̄3 C3 v ρ̄ 4 |ξ|+ i ∞∑ j=2 ˘̆c|ξ|2j−1, C1 = − κ ρ̄ + 4w̄3 C3 v ρ̄ 4 Cv +R + . . . , C2 = R(κρ̄ + 4w̄3 C3 v ρ̄ 4 )2 c2Cv(Cv +R)2 |ξ|2 + · · · , C3 = R(κρ̄ + 4w̄3 C3 v ρ̄ 4 ) c2Cv(Cv +R) |ξ|2 + · · · , D1 = κ ρ̄ + 4w̄3 C3 v ρ̄ 4 2(Cv +R) + · · · , D2 = −1 2 + · · · , D3 = i 1 2c R Cv 1 |ξ| + · · · , E1 = κ ρ̄ + 4w̄3 C3 v ρ̄ 4 2(Cv +R) + · · · , E2 = −1 2 + · · · , E3 = −i 1 2c R Cv 1 |ξ| + · · · . Here and below, “ · · · ” denote the reminding terms, which do not affect the results. Lemma 2.2 and the expression of Ĝ, ˆ̃G, Ĝ1, Ĝ2, Ĝ3 immediately yield the following result. EJDE-2023/60 RADIATIVE HYDRODYNAMICS MODELS 9 Lemma 2.3. For sufficiently small |ξ|, we have the following: Ĝ11 = eλ1t − i κ 2ρ̄ + 2w̄3 C3 v ρ̄ 4 c(Cv +R) |ξ|(eλ2t − eλ3t) + · · · , Ĝ12 = −i R(κρ̄ + 4w̄3 C3 v ρ̄ 4 ) c2Cv(Cv +R) ξT eλ1t − 1 2c ξT |ξ| (eλ2t − eλ3t) + · · · , Ĝ13 = − R c2Cv eλ1t + R 2c2Cv (eλ2t + eλ3t) + · · · , Ĝ21 = −i (κρ̄ + 4w̄3 C3 v ρ̄ 4 ) Cv +R ξeλ1t + i (κρ̄ + 4w̄3 C3 v ρ̄ 4 ) 2(Cv +R) ξ(eλ2t + eλ3t) + · · · , Ĝ22 = (I − ξξT |ξ|2 )e− µ ρ̄ |ξ| 2t − R(κρ̄ + 4w̄3 C3 v ρ̄ 4 )2 c2Cv(Cv +R)2 ξξT eλ1t + 1 2 ξξT |ξ|2 (eλ2t + eλ3t) + · · · , Ĝ23 = i R(κρ̄ + 4w̄3 C3 v ρ̄ 4 ) c2Cv(Cv +R)2 ξeλ1t − R 2cCv ξ |ξ| (eλ2t − eλ3t) + · · · , Ĝ31 = Cv( κ ρ̄ + 4w̄3 C3 v ρ̄ 4 ) R(Cv +R) ( 2µ+ µ′ ρ̄ − κ ρ̄ + 4w̄3 C3 v ρ̄ 4 Cv +R )|ξ|2eλ1t − i cCv( κ ρ̄ + 4w̄3 C3 v ρ̄ 4 ) 2R(Cv +R) |ξ|(eλ2t − eλ3t) + · · · , Ĝ32 = i (κρ̄ + 4w̄3 C3 v ρ̄ 4 )2 c2(Cv +R)2 ( κ ρ̄ + 4w̄3 C3 v ρ̄ 4 Cv +R − 2µ+ µ′ ρ̄ )|ξ|2ξT eλ1t − cCv 2R ξT |ξ| (eλ2t − eλ3t) + · · · , Ĝ33 = κ ρ̄ + 4w̄3 C3 v ρ̄ 4 c2(Cv +R) ( κ ρ̄ + 4w̄3 C3 v ρ̄ 4 Cv +R − 2µ+ µ′ ρ̄ )|ξ|2eλ1t + 1 2 (eλ2t + eλ3t) + · · · . In fact, one can deal with the leading term in the low frequency part since the rest terms just have the faster temporal decay rate. Indeed, the Huygens’ wave is arising from the low frequency part based on the above frequency analysis. Although the low frequency part is basically the same as the non-isentropic Navier- Stokes equations in [3], for completeness, we still take a typical leading term in Ĝl22 for example. In particular, one has[1 2 (eλ2t + eλ3t) ξξT |ξ|2 + e− µ ρ̄ |ξ| 2t(I − ξξT |ξ|2 ) ] = cos(c|ξ|t)ξξ T |ξ|2 [ e−θ1|ξ| 2t+O(|ξ|4)t cos(|ξ|β(|ξ|2)t) ] − sin(|ξ|β(|ξ|2)t) |ξ| [ |ξ| sin(|ξ|β(|ξ|2)t) ]ξξT |ξ|2 e−θ1|ξ| 2t+O(|ξ|4)t + e− µ ρ̄ |ξ| 2t(I − ξξT |ξ|2 ) = (cos(c|ξ|t)− 1) ξξT |ξ|2 e−θ1|ξ| 2t︸ ︷︷ ︸ I1 + ξξT |ξ|2 ( e−θ1|ξ| 2t − e− µ ρ̄ |ξ| 2t ) ︸ ︷︷ ︸ I2 +e− µ ρ̄ |ξ| 2tI + cos(c|ξ|t)ξξ T |ξ|2 e−θ1|ξ| 2t(cos(|ξ|β(|ξ|2)t)− 1)︸ ︷︷ ︸ I3 10 M. LIU, Z. WU EJDE-2023/60 + cos(c|ξ|t)ξξ T |ξ|2 ( eO(|ξ|4)t − 1 ) cos(|ξ|β(|ξ|2)t)e−θ1|ξ| 2t︸ ︷︷ ︸ I4 − sin(|ξ|β(|ξ|2)t) |ξ| |ξ| sin(|ξ|β(|ξ|2)t) |ξ|2 ξξT e−θ1|ξ| 2t+O(|ξ|4)t︸ ︷︷ ︸ I5 , where β(·) is analytic and the constant θ1 > 0 based on Lemma 2.1. I1 and I2 are corresponding to Riesz waves I and II in [3, 20] for both the isentropic and non-isentropic compressible Navier-Stokes system. Their pointwise space-time descriptions are |I1| ≤ C ( (1 + t)−3/2 ( 1 + |x|2 1 + t )−3/2 + (1 + t)−2 ( 1 + (|x| − ct)2 1 + t )−N) , |I2| ≤ C(1 + t)−3/2 ( 1 + |x|2 1 + t )−N , (2.20) with an arbitrarily large integer N . Next we consider ξξT |ξ|2 e −θ1|ξ|2t(cos(|ξ|β(|ξ|2)t)− 1) in I3, ξξ T |ξ|2 ( eO(|ξ|4)t − 1 ) cos(|ξ|β(|ξ|2)t)e−θ1|ξ| 2t in I4 and [ |ξ| sin(|ξ|β(|ξ|2)t) ] ξξT |ξ|2 in I5. Because of their analyticities and the faster decay rates, after a direct com- putation for these rest terms as in [20, Lemma 5.4], the inverse Fourier transform of the first two terms can be bounded by C(1+t)−3/2 ( 1+ |x| 2 1+t )−N , and inverse Fourier transform of the last term can be bounded by C(1 + t)−2 ( 1 + |x|2 1+t )−N . Recall the standard convolution estimates in [3, 20]: If |∂αx f(x, t)| ≤ (1 + t)− |α|+n+k 2 ( 1 + |x|2 1 + t )−N , x ∈ Rn, for all N > 0, then |∂αxwt ∗x f(x, t)| ≤ C(1 + t)− |α|+n+k 2 (1 + t)− n−1 4 ( 1 + (|x| − ct)2 1 + t )−N , |∂αxw ∗x f(x, t)| ≤ C(1 + t)− |α|+n+k−1 2 (1 + t)− n−1 4 ( 1 + (|x| − ct)2 1 + t )−N . (2.21) Here wt = cos(c|ξ|t) and w = sin(c|ξ|t) |ξ| are the Fourier transform of wave operators. Hence, one has |F−1(I3, I4, I5)| ≤ C(1 + t)−2 ( 1 + (|x| − ct)2 1 + t )−N . (2.22) Obtaining estimates for the other entries in Green’s matrix in the low frequency part is much easier, and one can refer to [3, 20]. Proposition 2.4. The low frequency of Green’s function G(x, t) for the system (2.3)1,2,3 in 3 dimensional space has the following estimates for t > 0 and |α| ≥ 0: |∂αx (χ1(D)Gij)(x, t)| ≤ C(1 + t)− 4+|α| 2 ( 1 + (|x| − ct)2 1 + t )−N for (i, j) 6= (2, 2), (3, 1), (3, 3), EJDE-2023/60 RADIATIVE HYDRODYNAMICS MODELS 11 |∂αx (χ1(D)G22)(x, t)| ≤ C(1 + t)− 3+|α| 2 ( 1 + |x|2 1 + t )− 3+|α| 2 + C(1 + t)− 4+|α| 2 ( 1 + (|x| − ct)2 1 + t )−N , |∂αx (χ1(D)(G31, G33))(x, t)| ≤ C(1 + t)− 4+|α| 2 (( 1 + (|x| − ct)2 1 + t )−N + ( 1 + |x|2 1 + t )−N) . Note that the above estimates are crucial for us to show the different pointwise estimates between (ρ,m) and w in (1.6). High frequency part. In the same way, when |ξ| � 1 one can obtain the following expansions for λ1, λ2, λ3. Lemma 2.5. For |ξ| � 1, λ1, λ2 and λ3 are all real. Furthermore, when κ > 0, λ1 = − Rw̄ Cv(2µ+ µ′) + ∞∑ j=1 d1 j |ξ|−2j ; λ2 = − κ Cvρ̄ |ξ|2 + R2w̄ Cv[κ− Cv(2µ+ µ′)] − 4w̄3 C4 v ρ̄ 4 + ∞∑ j=1 d2 j |ξ|−2j ; λ3 = −2µ+ µ′ ρ̄ |ξ|2 + R Cv w̄[κ− (Cv +R)(2µ+ µ′)] (2µ+ µ′)[κ− Cv(2µ+ µ′)] + ∞∑ j=1 d3 j |ξ|−2j , where d1 j , d 2 j , d 3 j are real constants. When κ = 0, we have λ1 = 1 2 [ − 4w̄3 C4 v ρ̄ 4 − (Cv +R)Rw̄ C2 v (2µ+ µ′) + √( 4w̄3 C4 v ρ̄ 4 + (R− Cv)Rw̄ C2 v (2µ+ µ′) )2 + 4R3w̄2 C3 v (2µ+ µ′)2 ] + ∞∑ j=1 d4 j |ξ|−2j , λ2 = 1 2 [ − 4w̄3 C4 v ρ̄ 4 − (Cv +R)Rw̄ C2 v (2µ+ µ′) − √ ( 4w̄3 C4 v ρ̄ 4 + (R− Cv)Rw̄ C2 v (2µ+ µ′) )2 + 4R3w̄2 C3 v (2µ+ µ′)2 ] + ∞∑ j=1 d5 j |ξ|−2j , λ3 = −2µ+ µ′ ρ̄ |ξ|2 + Rw̄(Cv +R) C2 v (2µ+ µ′) + ∞∑ j=1 d6 j |ξ|−2j , where d4 j , d 5 j , d 6 j are real constants. As a result, one has the following result. Lemma 2.6. For sufficiently large |ξ|, when κ > 0, we have the following: g1 = −Cv(2µ+ µ′) Rw̄ |ξ|2 + ∞∑ j=0 g2j |ξ|−2j , ν1 = −Cv(2µ+ µ′) Rρ̄ |ξ|2 + w̄ 2µ+ µ′ + ∞∑ j=1 ν2j |ξ|−2j , 12 M. LIU, Z. WU EJDE-2023/60 g2 = −Cvρ̄ κ + ∞∑ j=1 g̃2j |ξ|−2j , ν2 = κ− Cv(2µ+ µ′) Rρ̄ |ξ|2 + ∞∑ j=0 ν̄2j |ξ|−2j , g3 = − ρ̄ 2µ+ µ′ + ∞∑ j=1 ˜̃g2j |ξ|−2j , ν3 = w̄[κ− (Cv +R)(2µ+ µ′)] (2µ+ µ′)[κ− Cv(2µ+ µ′)] + ∞∑ j=1 ¯̄ν2j |ξ|−2j , Ω = [κ− Cv(2µ+ µ′)]Cv(2µ+ µ′) R2ρ̄w̄ |ξ|4 + ∞∑ j=−1 Ω2j |ξ|−2j , which implies C1 = − Rw̄|ξ|−2 Cv(2µ+ µ′) + · · · , C2 = Rρ̄w̄|ξ|−2 Cv(2µ+ µ′)2 + · · · , C3 = R2ρ̄2w̄|ξ|−4 κCv(2µ+ µ′)2 + · · · , D1 = Rw̄|ξ|−2 Cv(2µ+ µ′)− κ + · · · , D2 = R2ρ̄w̄|ξ|−2 [κ− Cv(2µ+ µ′)]2 + · · · , D3 = Rρ̄|ξ|−2 κ− Cv(2µ+ µ′) + · · · , E1 = κRw̄|ξ|−2 Cv(2µ+ µ′)[κ− Cv(2µ+ µ′)] + · · · , E2 = −1 + · · · , E3 = Rρ̄|ξ|−2 Cv(2µ+ µ′)− κ + · · · . When κ = 0, denoting a1 = 1 2 [− 4w̄3 C4 v ρ̄ 4 − (Cv +R)Rw̄ C2 v (2µ+ µ′) + √ ( 4w̄3 C4 v ρ̄ 4 + (R− Cv)Rw̄ C2 v (2µ+ µ′) )2 + 4R3w̄2 C3 v (2µ+ µ′)2 ], a2 = 1 2 [− 4w̄3 C4 v ρ̄ 4 − (Cv +R)Rw̄ C2 v (2µ+ µ′) − √ ( 4w̄3 C4 v ρ̄ 4 + (R− Cv)Rw̄ C2 v (2µ+ µ′) )2 + 4R3w̄2 C3 v (2µ+ µ′)2 ], we obtain g1 = 1 a1 |ξ|2 + ∞∑ j=0 g2j |ξ|−2j , ν1 = −Cv(2µ+ µ′) Rρ̄ |ξ|2 − Cv R a1 + ∞∑ j=1 ν2j |ξ|−2j , g2 = 1 a2 |ξ|2 + ∞∑ j=0 g̃2j |ξ|−2j , ν2 = −Cv(2µ+ µ′) Rρ̄ |ξ|2 − Cv R a2 + ∞∑ j=1 ν̄2j |ξ|−2j , g3 = − ρ̄ 2µ+ µ′ + ∞∑ j=1 ˜̃g2j |ξ|−2j , ν3 = − (Cv +R)w̄ Cv(2µ+ µ′) + ∞∑ j=1 ¯̄ν2j |ξ|−2j , Ω = Cv(2µ+ µ′)(a2 − a1) Rρ̄a1a2 |ξ|4 + ∞∑ j=−1 Ω2j |ξ|−2j , which implies C1 = a1a2|ξ|−2 a2 − a1 + · · · , C2 = [ (Cv+R)w̄ Cv(2µ+µ′)a2 + Cv R ]|ξ|−2 Cv(2µ+µ′)(a1−a2) Rρ̄a1a2 + · · · , EJDE-2023/60 RADIATIVE HYDRODYNAMICS MODELS 13 C3 = Rρ̄a1|ξ|−2 Cv(2µ+ µ′)(a2 − a1) + · · · , D1 = a1a2|ξ|−2 a1 − a2 + · · · , D2 = [ (Cv+R)w̄ Cv(2µ+µ′)a1 + Cv R ]|ξ|−2 Cv(2µ+µ′)(a2−a1) Rρ̄a1a2 + · · · , D3 = Rρ̄a2|ξ|−2 Cv(2µ+ µ′)(a1 − a2) + · · · , E1 = − ρ̄a1a2|ξ|−4 2µ+ µ′ + · · · , E2 = −1 + · · · , E3 = Rρ̄|ξ|−2 Cv(2µ+ µ′) + · · · . Here all of the above coefficients in Sigma summation symbols are real constants. Lemma 2.7. For sufficiently large |ξ|, when κ > 0, it holds Ĝ11 = eλ1t + CvRρ̄w̄ κ[κ− Cv(2µ+ µ′)] 1 |ξ|2 eλ2t − κRρ̄w̄ Cv(2µ+ µ′)2[κ− Cv(2µ+ µ′)] 1 |ξ|2 eλ3t + · · · , Ĝ12 = − ρ̄ 2µ+ µ′ iξT |ξ|2 eλ1t − CvR 2ρ̄2w̄ κ[κ− Cv(2µ+ µ′)]2 iξT |ξ|4 eλ2t + ρ̄ 2µ+ µ′ iξT |ξ|2 eλ3t + · · · , Ĝ13 = − Rρ̄2 κ(2µ+ µ′) 1 |ξ|2 eλ1t − RCvρ̄ 2 κ[κ− Cv(2µ+ µ′) 1 |ξ|2 eλ2t + Rρ̄2 (2µ+ µ′)[κ− Cv(2µ+ µ′)] 1 |ξ|2 eλ3t + · · · , Ĝ21 = −Rw̄ Cv(2µ+ µ′) iξ |ξ|2 eλ1t − Rw̄ κ− Cv(2µ+ µ′) iξ |ξ|2 eλ2t + κRw̄ Cv(2µ+ µ′)[κ− Cv(2µ+ µ′)] iξ |ξ|2 eλ3t + · · · , Ĝ22 = (I − ξξT |ξ|2 )e− µ ρ̄ |ξ| 2t − Rρ̄w̄ Cv(2µ+ µ′)2 ξξT |ξ|4 eλ1t − R2ρ̄w̄ [κ− Cv(2µ+ µ′)]2 ξξT |ξ|4 eλ2t + ξξT |ξ|2 eλ3t + · · · , Ĝ23 = R2ρ̄2w̄ κCv(2µ+ µ′)2 iξ |ξ|4 eλ1t + Rρ̄ κ− Cv(2µ+ µ′) iξ |ξ|2 (eλ2t − eλ3t) + · · · , Ĝ31 = w̄ ρ̄ eλ1t + w̄ ρ̄ eλ2t − κRw̄2[κ− (Cv +R)(2µ+ µ′)] Cv(2µ+ µ′)2[κ− Cv(2µ+ µ′)]2 1 |ξ|2 eλ3t + · · · , Ĝ32 = − w̄ 2µ+ µ′ iξT |ξ|2 eλ1t + Rw̄ κ− Cv(2µ+ µ′) iξT |ξ|2 eλ2t + w̄[κ− (Cv +R)2µ+ µ′] (2µ+ µ′)[κ− Cv(2µ+ µ′)] iξT |ξ|2 eλ3t + · · · , Ĝ33 = − Rρ̄w̄ κ(2µ+ µ′) 1 |ξ|2 eλ1t + eλ2t + Rρ̄w̄[κ− (Cv +R)(2µ+ µ′)] (2µ+ µ′)[κ− Cv(2µ+ µ′)]2 1 |ξ|2 eλ3t + · · · . When κ = 0, we have Ĝ11 = a2 a2 − a1 eλ1t + a1 a1 − a2 eλ2t + ρ̄2a1a2 (2µ+ µ′)2 1 |ξ|4 eλ3t + · · · , 14 M. LIU, Z. WU EJDE-2023/60 Ĝ12 = [ (Cv+R)w̄ Cv(2µ+µ′)a2 + Cv R ] Cv(2µ+µ′)(a1−a2) Rρ̄a2 eλ1t + [ (Cv+R)w̄ Cv(2µ+µ′)a2 + Cv R ] Cv(2µ+µ′)(a2−a1) Rρ̄a1 eλ2t + ρ̄ 2µ+ µ′ iξT |ξ|2 eλ3t + · · · , Ĝ13 = Rρ̄ Cv(2µ+ µ′) 1 a2 − a1 eλ1t + Rρ̄ Cv(2µ+ µ′) 1 a1 − a2 eλ2t − Rρ̄2 Cv(2µ+ µ′)2 1 |ξ|2 eλ3t + · · · , Ĝ21 = a1a2 a2 − a1 iξ |ξ|2 eλ1t − a1a2 a2 − a1 iξ |ξ|2 eλ2t − ρ̄a1a2 2µ+ µ′ iξ |ξ|4 eλ3t + · · · , Ĝ22 = (I − ξξT |ξ|2 )e− µ ρ̄ |ξ| 2t + (Cv+R)w̄ Cv(2µ+µ′)a2 + Cv R Cv(2µ+µ′)(a2−a1) Rρ̄a1a2 ξξT |ξ|4 eλ1t − (Cv+R)w̄ Cv(2µ+µ′)a1 + Cv R Cv(2µ+µ′)(a2−a1) Rρ̄a1a2 ξξT |ξ|4 eλ2t + ξξT |ξ|2 eλ3t + · · · , Ĝ23 = Rρ̄a1 Cv(2µ+ µ′)(a2 − a1) iξ |ξ|2 eλ1t + Rρ̄a2 Cv(2µ+ µ′)(a1 − a2) iξ |ξ|2 eλ2t + Rρ̄ Cv(2µ+ µ′) iξ |ξ|2 eλ3t + · · · , Ĝ31 = −Cv(2µ+ µ′)a1a2 Rρ̄a2 − a1 eλ1t + Cv(2µ+ µ′)a1a2 Rρ̄a2 − a1 eλ2t − (Cv +R)w̄ρ̄a1a2 Cv(2µ+ µ′)2 1 |ξ|4 eλ3t + · · · , Ĝ32 = (Cv+R)w̄ Cv(2µ+µ′)a2 + Cv R (a2−a1) a1a2 iξT |ξ|2 eλ1t − (Cv+R)w̄ Cv(2µ+µ′)a1 + Cv R (a2−a1) a1a2 iξT |ξ|2 eλ2t + (Cv +R)w̄ Cv(2µ+ µ′) iξT |ξ|2 eλ3t + · · · , Ĝ33 = a1 a1 − a2 eλ1t − a2 a1 − a2 eλ2t − (Cv +R)Rw̄ρ̄ C2 v (2µ+ µ′)2 1 |ξ|2 eλ3t + · · · . From Lemmas 2.5–2.7, we can find that there basically exist two kinds of singular components in the high frequency part of Green’s function. One is like the heat kernel t−3/2e− |x|2 Ct arising from the term e−C|ξ| 2t in Fourier space. The second one is like a Dirac δ-function or some δ-like functions, which is rising from the term like |ξ|−βe−Ct in Fourier space and the integer β ≥ 1. Thus, from Lemma 4.2, we have the following description for the high frequency part. Proposition 2.8. There exists a constant C > 0 such that the high frequency part satisfies |∂αx (χ3(D)Gij −GS)(x, t)| ≤ Ce−t/C(1 + |x|2)−N , for all integer N > 0. Here the singular parts GS(x, t) satisfy GS(x, t) = e−t/C [ C1t − 3+|α| 2 e− |x|2 Ct + C2δ(x) ] . (2.23) Middle frequency part. The analysis for the middle frequency part is partially based on the idea in Li [20]. In particular, we derive the following estimates for the eigenvalues. EJDE-2023/60 RADIATIVE HYDRODYNAMICS MODELS 15 Lemma 2.9. When η < |ξ| ≤ K with two fixed positive constants η and K, there exists a constant b > 0 such that Re(λ1(|ξ|), λ2(|ξ|), λ3(|ξ|)) ≤ −b. (2.24) Proof. Note first that λ1 + λ2 + λ3 = −( κ Cvρ̄ + 4w̄3 C4 v ρ̄ 4(1 + |ξ|2) + 2µ+ µ′ ρ̄ )|ξ|2, λ1λ2 + λ1λ3 + λ2λ3 = κ(2µ+ µ′) Cvρ̄2 |ξ|4 + 4(2µ+ µ′)w̄3|ξ|4 C4 v ρ̄ 5(1 + |ξ|2) + (Cv +R)Rw̄ C2 v ρ̄ |ξ|2, λ1λ2λ3 = − κRw̄ C2 v ρ̄ 2 |ξ|4 − 4Rw̄4|ξ|4 C5 v ρ̄ 5(1 + |ξ|2) . (2.25) We will prove (2.24) by two steps: Step 1. Suppose there is a real root λ1 and λ1 > 0. Combining (2.25)1 and (2.25)2, one has λ2λ3 = κ(2µ+ µ′) Cvρ̄2 |ξ|4 + 4(2µ+ µ′)w̄3|ξ|4 C4 v ρ̄ 5(1 + |ξ|2) + (Cv +R)Rw̄ C2 v ρ̄ |ξ|2 + ( κ Cvρ̄ + 4w̄3 C4 v ρ̄ 4(1 + |ξ|2) + 2µ+ µ′ ρ̄ )|ξ|2λ1 + λ2 1. (2.26) On the other hand, by (2.25)3 we have λ2λ3 = −( κRw̄ C2 v ρ̄ 2 |ξ|4 + 4Rw̄4|ξ|4 C5 v ρ̄ 5(1 + |ξ|2) )/λ1. (2.27) This yields a contradiction since the signs of λ2λ3 in (2.26) and (2.27) are opposite. Accordingly, the assumption is not true. Step 2. Suppose that λ1 < 0 and there is a pair of conjugate imaginary roots λ2 and λ3 with λ2 + λ3 > 0. From (2.25)1, one has λ2 + λ3 = −( κ Cvρ̄ + 4w̄3 C4 v ρ̄ 4(1 + |ξ|2) + 2µ+ µ′ ρ̄ )|ξ|2 − λ1 > 0, (2.28) λ1 < −( κ Cvρ̄ + 4w̄3 C4 v ρ̄ 4(1 + |ξ|2) + 2µ+ µ′ ρ̄ )|ξ|2. (2.29) Then, combining (2.25)2, (2.25)3, and (2.29), a routine computation gives rise to λ2 + λ3 = λ1[κ(2µ+µ′) Cv ρ̄2 |ξ|4 + 4w̄3(2µ+µ′) C4 v ρ̄ 5(1+|ξ|2) |ξ| 4 + (Cv+R)Rw̄ C2 v ρ̄ |ξ|2] + κRw̄ C2 v ρ̄ 2 |ξ|4 + 4Rw̄4|ξ|4 C5 v ρ̄ 5(1+|ξ|2) λ2 1 ≤ ( − ( κ Cvρ̄ + 4w̄3 C4 v ρ̄ 4(1 + |ξ|2) + 2µ+ µ′ ρ̄ )|ξ|2[ κ(2µ+ µ′) Cvρ̄2 |ξ|4 + 4w̄3(2µ+ µ′) C4 v ρ̄ 5(1 + |ξ|2) |ξ|4 + R2w̄ C2 v ρ̄ |ξ|2]− Rw̄(2µ+ µ′) Cvρ̄2 |ξ|4 ) /λ2 1 < 0. Obviously, this contradicts (2.28). Therefore, λ2 + λ3 < 0, i.e., Re(λ2) < 0 and Re(λ3) < 0. This completes the proof. � 16 M. LIU, Z. WU EJDE-2023/60 Lemma 2.10. The Green function Ĝ(ξ, t) is analytic when |ξ|2 ≥ δ, where δ is any fixed positive constant. Proof. We shall only present the proof for Ĝ11(ξ, t), since the other entries in the Ĝ(ξ, t) can be treated similarly. First, from Lemmas 2.1 and 2.5, we can induce that (2.9) has not repeated roots. Then, we can see that Ĝ11(ξ, λ1, λ2, λ3, t) = Ĝ1,m 11 eλ1t + Ĝ2,m 11 eλ2t + Ĝ3,m 11 eλ3t = g1C1e λ1t + g2D1e λ2t + g3E1e λ3t = g1 ν3 − ν2 Ω eλ1t + g2 ν1 − ν3 Ω eλ2t + g3 ν2 − ν1 Ω eλ3t = g1Ω(ν3 − ν2)eλ1t + g2Ω(ν1 − ν3)eλ2t + g3Ω(ν2 − ν1)eλ3t Ω2 = A(ξ, λ1, λ2, λ3, t) (ν2g3 − ν3g2 + ν3g1 − ν1g3 + ν1g2 − ν2g1)2 , (2.30) where A(ξ, λ1, λ2, λ3, t) = g1(ν2g3 − ν3g2 + ν3g1 − ν1g3 + ν1g2 − ν2g1)(ν3 − ν2)eλ1t + g2(ν2g3 − ν3g2 + ν3g1 − ν1g3 + ν1g2 − ν2g1)(ν1 − ν3)eλ2t + g3(ν2g3 − ν3g2 + ν3g1 − ν1g3 + ν1g2 − ν2g1)(ν2 − ν1)eλ3t = g1(ν2g3 − ν3g2 + ν3g1 − ν1g3 + ν1g2 − ν2g1)(ν3 − ν2) ∞∑ n=1 (λ1t) n n! + g2(ν2g3 − ν3g2 + ν3g1 − ν1g3 + ν1g2 − ν2g1)(ν1 − ν3) ∞∑ n=1 (λ2t) n n! + g3(ν2g3 − ν3g2 + ν3g1 − ν1g3 + ν1g2 − ν2g1)(ν2 − ν1) ∞∑ n=1 (λ3t) n n! . We claim that gi and νi (i = 1, 2, 3) are symmetric about λi (i = 1, 2, 3). It is easy to obtain that the numerator is a symmetric power series in λi (i = 1, 2, 3). In fact, for instance, exchanging λ1 and λ2, the first term of A becomes the second term, the second term becomes the first term, and the last term is still itself. Thus, A(ξ, λ1, λ2, λ3, t) = A(ξ, λ2, λ1, λ3, t). Then, we have Ĝ11(ξ, λ1, λ2, λ3, t) = Ĝ11(ξ, λ2, λ1, λ3, t). It is well known that every symmetric polynomial can be written as a power sum of the elementary symmetric polynomials λ1 + λ2 + λ3, λ1λ2 + λ1λ3 + λ2λ3, and λ1λ2λ3. From (2.25), we know that the numerator can be written as a power series in |ξ|2 and therefore is entire in |ξ|2. Similarly, we notice that the denominator is a symmetric polynomial in λi (i = 1, 2, 3), so it can be written as an polynomial in |ξ|2. Therefore, Ĝ11(ξ, t) must be analytic when |ξ|2 ≥ δ > 0. Thus, we have completed the proof. � Lemmas 2.9 and 2.10 immediately yield the following result. EJDE-2023/60 RADIATIVE HYDRODYNAMICS MODELS 17 Proposition 2.11. There exists a constant b1 > 0 such that |∂αx (χ2(D)G(x, t))| ≤ Ce−b1t(1 + |x|2 1 + t )−N , where N can be arbitratily large. In summary, from Propositions 2.4–2.11, we can have the following pointwise description involving the Huygens’ wave and the diffusion wave. Theorem 2.12. The Green function G(x, t) for system (2.3)1,2,3 in the 3 dimen- sional space has the following estimates for t > 0: |∂αx (Gij −GS)(x, t))| ≤ C(1 + t)− 4+|α| 2 ( 1 + (|x| − ct)2 1 + t )−N for (i, j) 6= (2, 2), (3, 1), (3, 3), |∂αx (G22 −GS)(x, t) ) | ≤ C(1 + t)− 3+|α| 2 ( 1 + |x|2 1 + t )− 3+|α| 2 + C(1 + t)− 4+|α| 2 ( 1 + (|x| − ct)2 1 + t )−N , |∂αx ((G31 −GS , G33 −GS))(x, t))| ≤ C(1 + t)− 4+|α| 2 (( 1 + (|x| − ct)2 1 + t )−N + ( 1 + |x|2 1 + t )−N) . Here N > 0 is an arbitrary large constant and GS is defined in Proposition 2.8. 3. Pointwise estimates for the nonlinear system First, by using Duhamel’s principle, we represent the solution (ρ,m, w) for the nonlinear problem (2.1). ∂αx  ρ m w  = ∂αxG ∗x U0 + ∫ t 0 ∂αxG ∗x  0 F1 F2  (·, s)ds, (3.1) where the initial data U0 := (ρ0,m0, ω0)T , and the nonlinear terms F1, F2 are defined in (2.2). Initial propagation. Let (ρ̆, m̆, w̆) denote the linear part of the solution in (3.1). Theorem 2.12, the initial condition (1.5), and the representation (3.1) yield the linear estimates |∂αx (ρ̆, m̆)| ≤ 2Cε ( (1 + t)− 3+|α| 2 ( 1 + |x|2 1 + t )−3/2 + (1 + t)−2 ( 1 + (|x| − ct)2 1 + t )−3/2 ) , |∂αx w̆| ≤ 2Cε(1 + t)− 4+|α| 2 (( 1 + |x|2 1 + t )−3/2 + ( 1 + (|x| − ct)2 1 + t )−3/2 ) . (3.2) Here we have used the convolution estimate in Lemma 4.3 for the initial propaga- tion, and the different pointwise estimates for the first two rows and the last row of Green’s function in Theorem 2.12. 18 M. LIU, Z. WU EJDE-2023/60 Nonlinear Coupling. According to the above initial propagation, we should give the ansatz for the nonlinear problem when |α| ≤ 1, |∂αx (ρ,m)| ≤ 2Cε ( (1 + t)− 3+|α| 2 ( 1 + |x|2 1 + t )−3/2 + (1 + t)−2 ( 1 + (|x| − ct)2 1 + t )−3/2 ) , |∂αxw| ≤ 2Cε(1 + t)− 4+|α| 2 (( 1 + |x|2 1 + t )−3/2 + ( 1 + (|x| − ct)2 1 + t )−3/2 ) . (3.3) Now, we substitute (3.3) into the representation of the solution (ρ,m, w) in (3.1) to close the ansatz. To this end, we also split Green’s function into the regular term Gij − GS and the singular term GS . For the convolution between Gij − GS and the nonlinear terms, one can put all of the derivatives on Gij − GS and use the nonlinear convolution estimates in Lemma 4.4 to obtain the corresponding estimate as in the ansatz (3.2). We emphasize that although there exists a nonlocal operator ∆ 1−∆ in some nonlinear terms of F2, it actually does not affect the result. In fact, one can put this operator onto Green’s function by integration by parts, and it’s easy to see that it is harmless for the pointwise estimates of Green’s function in all of frequency parts. Next, we consider the nonlinear convolution between the singular part of Green’s function and the nonlinear term. We only take the nonlinear estimate of the mo- mentum m for example, since from this one can see why we need H5-framework for this quasi-linear problem. In fact, when estimating ∂kxm, one will encounter the term ∫ t 0 GS(·, t − τ) ∗x ∂kx [∂2 x(ρm)](·, τ)dτ . Noting that GS is like δ(x) with exponential decay rate, one has to put the derivative on the nonlinear term. That is, ∫ t 0 δ(·, t − τ) ∗x ∂kx [∂2 x(ρm)](·, τ)dτ . As a result, one can close the ansatz only when |k| ≥ 1. Indeed, when |k| = 1, we should use the pointwise information of ∂3 x(ρm), and hence it also requires ∂3 x(ρ,m) ∈ L∞(R3). This together with Sobolev inequality yields that we can close the ansatz in H5-framework. Finally, by using the smallness of ε and the continuity, one can close the ansatz (3.2) and hence proves Theorem 1.1. 4. Appendix Some useful lemmas are given here. The first one is used to derive the pointwise estimates of Green’s function in the low frequency. Lemma 4.1 ([38]). If there exists a constant C > 0 such that when |ξ| ≤ 1, f̂(ξ, t) satisfies |∂βξ (ξαf̂(ξ, t))| ≤ C(|ξ|(|α|−|β|)+ + |ξ||α|t|β|/2)(1 + (t|ξ|2))a exp(−b|ξ|2t), for some constant b > 0, each fixed integer a and any multi-indexes α, β with |β| ≤ 2N , then |∂αx f(x, t)| ≤ CN (1 + t)−(n+|α|) ( 1 + |x|2 1 + t )−N , (4.1) where N is a positive constant and can be arbitrarily large. The second lemma describes the singular part of the high frequency. Lemma 4.2 ([38]). If supp f̂(ξ) ⊂ OK =: {ξ, |ξ| ≥ K > 0}, and f̂(ξ) satisfies |∂βξ f̂(ξ)| ≤ C|ξ|−|β|−1 (or |Dβ ξ f̂(ξ)| ≤ C|ξ|−|β|), EJDE-2023/60 RADIATIVE HYDRODYNAMICS MODELS 19 then there exist distributions f1(x), f2(x) and a constant C0 such that f(x) = f1(x) + f2(x) + C0δ(x) (or f(x) = f1(x) + f2(x) + C0∂xδ(x)), where δ(x) is the Dirac function. Furthermore, for any |α| ≥ 0 and any positive integer N , we have |∂αx f1(x)| ≤ C(1 + |x|2)−N , ‖f2‖L1 ≤ C, supp f2(x) ⊂ {x; |x| < η0 � 1}. The next two lemmas are often used to deal with initial propagation and non- linear coupling, respectively. We also state several typical cases for completeness. Lemma 4.3 ([40]). There exists a constant C > 0 such that for n1, n2 > 3/2 and n3 = min{n1, n2}, we have∫ R3 ( 1 + |x− y|2 1 + t )−n1 ( 1 + |y|2 )−n2 dy ≤ C ( 1 + |x|2 1 + t )−n3 ; and for N ≥ r1 > 21.10, we have∫ R3 ( 1 + (|x− y| − ct)2 1 + t )−N( 1 + |y|2 )−r1 dy ≤ C ( 1 + (|x| − ct)2 1 + t )−3/2 . Lemma 4.4 ([25]). There exists a constant C > 0 such that∫ t 0 ∫ R3 (1 + t− s)−2 ( 1 + |x− y|2 1 + t− s )−2 (1 + s)−3 ( 1 + |y|2 1 + s )−3 dy ds ≤ C(1 + t)−2 ( 1 + |x|2 1 + t )−3/2 ,∫ t 0 ∫ R3 (1 + t− s)−2 ( 1 + |x− y|2 1 + t− s )−2 (1 + s)−4 ( 1 + (|y| − cs)2 1 + s )−3 dy ds ≤ C(1 + t)−2 (( 1 + |x|2 1 + t )−3/2 + ( 1 + (|x| − ct)2 1 + t )−3/2 ) ,∫ t 0 ∫ R3 (1 + t− s)−5/2 ( 1 + (|x− y| − c(t− s))2 1 + t− s )−N × (1 + s)−4 ( 1 + (|y| − cs)2 1 + s )−3 dy ds ≤ C(1 + t)−2 (( 1 + |x|2 1 + t )−3/2 + ( 1 + (|x| − ct)2 1 + t )−3/2 ) , where the constant N > 0 can be arbitrarily large. Acknowledgments. This research was supported by the National Natural Science Foundation of China (No. 11971100) and by the Natural Science Foundation of Shanghai (Grant No. 22ZR1402300). References [1] S. J. Deng, S. H. Yu; Green’s function and pointwise convergence for compressible Navier- Stokes equations, Quart. Appl. Math., 75 (3)(2017), 433-503. [2] S. J. Deng, X .F. 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Wang; Pointwise estimates for bipolar compressible Navier-Stokes-Poisson system in dimension three, Arch. Rational Mech. Anal., 226 (2)(2017), 587-638. [41] Z. G. Wu, W. K. Wang; Generalized Huygens’ principle for bipolar non-isentropic com- pressible Navier-Stokes-Poisson system in dimension three, J. Diff. Eqns., 269 (10)(2020), 7906-7930. [42] F. Xie; Nonlinear stability of combination of viscous contact wave with rarefaction waves for a 1D radiation hydrodynamics model, Discrete Contin. Dyn. Syst., 17 (3)(2012), 1075-1100. [43] Y. N. Zeng; L1 Asymptotic behavior of compressible isentropic viscous 1-D flow, Comm. Pure Appl. Math., 47 (1994), 1053-1082. Mengqian Liu Department of Mathematics, Donghua University, Shanghai, 201620, China Email address: hbulmq@163.com Zhigang Wu Department of Mathematics, Donghua University, Shanghai, 201620, China Email address: zgwu@dhu.edu.cn 1. Introduction 2. Green's function 2.1. Linearization and Green's function 2.2. Derivation of Green's function (,t) 2.3. Pointwise estimates of Green's function Low frequency part High frequency part Middle frequency part 3. Pointwise estimates for the nonlinear system Initial propagation Nonlinear Coupling 4. Appendix Acknowledgments References