GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR OF SPHERICALLY SYMMETRIC SOLUTIONS FOR THE MULTI-DIMENSIONAL INFRARELATIVISTIC MODEL

. In this paper, we establish the global existence, uniqueness and asymptotic behavior of spherically symmetric solutions for the multi-dimensional infrarelativistic model in H i × H i × H i × H i +1 ( i = 1 , 2 , 4).


1.
Introduction. As we know, the importance of thermal radiation in physical problems increases as the temperature is raised. Usually, the role of the radiation is one of transporting energy by radiative process at the moderate temperature, while the energy and momentum densities of the radiation field may become comparable to or even dominate the corresponding fluid quantities at the higher temperature. So the radiation field significantly affects the dynamics of the field. The theory of radiation hydrodynamics finds a wide range of applications, such as stellar atmospheres and envelopes, supernova explosions, stellar winds, physics of laser fusion, reentry of vehicles and many others. Therefore, the study of mathematical theory of radiation hydrodynamics is of great importance from both the mathematical theory and that of applications.
In this paper, we consider the motion of the compressible multi-dimensional viscous gas with radiation, which is a system of the Navier-Stokes equations coupled with a transport equation. We know that the energy in the radiation field to be carried by point, massless particles called photons, which are travelling at the speed c of light, characterized by their frequency ν, and their energy of each photon E = hν (where h is the Planck's constant), the momentum − → p = hν c − → Ω , where − → Ω is a unit vector and denotes the direction of travel of the photon (it requires two angular variables to specify − → Ω ). In a radiative transfer, it is conventional to introduce the specific radiative intensity I ≡ I(x, t, ν, − → Ω ) driven by the so-called radiative transfer integro-differential equation introduced and discussed by Chandrasekhar [3]. Meanwhile, we can derive global quantities by integrating with respect to the angular and frequency variables: the specific radiative energy density E R (x, t) per unit volume is then E R (x, t) = 1 c I(x, t, ν, − → Ω )d − → Ω dν, and the specific radiative Under the consideration of the three basic interactions between photons and matter, namely, absorption, scattering and emission, we find the transfer in the conventional form (see, e.g., [17,18,19] where I(ν, − → Ω ) ≡ I(x, t, − → Ω , ν), S n−1 is the unit ball in R n , S e (ν) ≡ S e (x, t, ν, ρ, θ), σ a (ν) ≡ σ a (x, t, ν, ρ, θ) and σ s (ν) ≡ σ s (x, t, ν, ρ, θ), respectively, denote the rate of energy emission due to spontaneous processes, the absorption coefficient and the scattering coefficient that also depend on the mass density ρ and the temperature θ of the matter. The scattering interaction serves to change the photon's characteristics ν and − → Ω to a new set of characteristics ν and − → Ω . The sign ν → ν stands for from ν to ν and − → Ω · − → Ω denotes the transfer from direction − → Ω to direction − → Ω as an argument of σ s (ν). Therefore, we can describe the scattering event by a probabilistic statement concerning this change as follows When the matter is in local thermodynamical equilibrium and radiation is present with coupling terms between matter and radiation, the coupled system can be read as (see, e.g., [18,19]) where ρ = ρ(x, t), U = U(x, t), θ = θ(x, t), e = e(x, t), Q = Q(x, t) stand for the density, the velocity, the absolute temperature, the internal energy and the heat flux, respectively, − → Π = −P (ρ, θ) − → I + − → π represents the material stress tensor for a Newtonian fluid with the viscous contribution − → π = 2µ − → D + λdiv U − → I with µ > 0 and nλ + 2µ ≥ 0, and the strain tensor the radiative energy source and the Planck's function B(ν, θ) describes the frequency-temperature black body distribution. The thermo-radiative flux Q satisfies the Fourier's law where κ(ρ, θ) is the heat conductivity coefficient. Now we would like to mention some results on this system. First, in the inviscid case, Lowrie, Morel and Hittinger [16], Buet and Després [2] have investigated the asymptotic regimes, and Dubroca and Feugeas [4], Lin [14] and Lin, Coulombel and Guodon [15] have considered the numerical aspects. Second, Zhang and Jiang [29] has given a proof of local-in-time existence and blow-up of solutions. Then Golse and Perthame [8] have investigated a simplified version of the system. Third, for the Cauchy problem in multi-dimension case, Li and Zhu [13] have investigated the blowup of smooth solutions under some sufficient conditions. For the one-dimensional initial-boundary value problem, Ducomet and Nečasová [5] have considered global existence and uniqueness of weak solutions in H 1 ×H 1 ×H 1 ×H 1 and Qin, Feng and Zhang [21] also proved the global existence and large-time behavior of solutions in H i ×H i 0 ×H i ×H i+1 (i = 1, 2) for the infrarelativistic model. Furthermore, Ducomet and Nečasová [6] obtained the asymptotic behavior of global strong solutions. In the pure scattering case, Ducomet and Nečasová [7] first investigated the asymptotic behavior of a motion of a viscous heat-conducting one-dimensional gas with radiation, and then Qin, Feng and Zhang [22] also proved the large-time behavior of solutions in H i × H i 0 × H i × H i−1 (i = 2, 3). Recently, Azevedo, Sauter and Thompson [1] also studied an approximation model of compressible radiative flow and established global classical solutions for this model in a slab under semi-reflexive boundary conditions using energy-entropy estimates and a homotopic version of the Leray-Schauder fixed point theorem together with classical Friedman-Schauder estimates for linear second order parabolic equations in boundary Hölder spaces. Qin and Zhang [24] obtained global existence and asymptotic behavior of cylindrically symmetric solutions for the 3D infrarelativistic model with radiation.
In this paper, we shall consider the radial solution on this model and establish the global existence, uniqueness and asymptotic behavior of spherically symmetric solutions for the compressible viscous gas with radiation. For spherically symmetric Navier-Stokes equations, we would like to refer to [9,10,11,12,28] and the references therein.
It is worth pointing out some difficulties encountered in this paper. The first difficulty encountered here is to establish the uniform point-wise upper bound of the specific volume v(x, t). To overcome it, we construct a key estimate (48) on the temperature θ 1+s (x, t) by Sobolev's and Hölder's inequalities. Then we can apply Gronwall's inequality to obtain the uniform-in-time upper bound of v(x, t). The second difficulty is to establish the H 1 estimate of the velocity u(x, t). To do this, motivated by [6], we introduce the auxiliary function F (ξ) (see Lemma 3.6) and adopt the similar technique to construct the corresponding estimate. The third difficulty is to deal with the radiation intensity I(x, t). From equation (24 4 ), we can obtain the expression (93) of the radiation density I by solving the ordinary differential equation. By virtue of (93), we mainly make full use of Lemmas 3.1-3.8 to establish some important estimates for I(x, t), such as Lemmas 3.9-3.10, 4.3 and 5.5. The last difficulty is to construct the estimates of the radiative energy source (S E ) R in equation (24 3 ). To overcome it, we mainly apply the relation between (S E ) R and I. By using the estimates on I, we can obtain the estimates on (S E ) R . In addition, in order to derive our desired results, we mainly use the embedding theorem and interpolation technique, and some idea from Qin [20], Qin and Huang [23] and Umehara and Tani [26,27]. Especially, compared with the 1D case, we have some essential new difficulties and techniques used in the proof. Firstly, the proof of (34) in Lemma 3.1 is obtained by the relation (38) and the integration by parts. Secondly, estimates on radiation term play a key role in our proofs. We construct these estimates by some new techniques, such as the Gronwall inequality in Lemma 3.9. Finally, high-order estimates can be established by some complicate calculations, such as (179), (219) and Lemma 5.5.
The notation in this paper will be as follows. Signs L p , 1 ≤ p ≤ +∞, and denote the usual Lebesgue spaces and Sobolev spaces on (0, L); · B denotes the norm in the space B, · := · L 2 .f (t) = L −1 L 0 f (x, t)dx. Letter C will denote the general constant but may be different, and letters C i (i = 1, 2, 4) will denote the universal constants depending on the norms of initial datum (v 0 , u 0 , θ 0 , I 0 ) in H i (see below the definitions of H i ) but being independent of t, respectively.
We organize our present paper as follows. In Section 2, we will induct the spherically symmetric infrarelativistic model and state our main results. Subsequently, we will complete the proofs of the global existence and asymptotic behavior to the generalized solutions in H i (i = 1, 2, 4) in Sections 3, 4 and 5, respectively.
Introduce the radiative energy the radiative flux and note pressure and energy of the matter have the thermodynamical relation Especially, if we assume that the fluid motion is small enough with respect to the velocity of light c so that we can drop all the 1 c factors in the previous formulation, then the system (17) can be rewritten as We assume that e, P, σ a , σ s , κ and B are C i+1 (i = 1, 2, 4) functions on 0 < v < +∞ and 0 ≤ θ < +∞ and for any v ≥ 0, we also suppose that the following growth conditions for any v ≥ v and θ ≥ 0: where constants s, q, α satisfy s ∈ [0, 1], q ≥ 1 + s, 0 ≤ α ≤ 2s + 1, the numbers c j (j = 1, · · · , 13) are positive constants and the nonnegative functions f, g, h, k, l, M are such that, for some γ > 0,

Remark 1. The Planck function in thermal equilibrium
characterizes the radiation, usually called blackbody radiation, emitted by a perfect radiator or black body, where h and k are the Planck and Boltzmann constants (see [17]). Obviously, this function satisfies our assumptions (A 8 ), (A 10 ) and (A 12 ).
We are now in a position to state our main theorems.
Theorem 2.3. Assume that the initial data (v 0 , u 0 , θ 0 , I 0 ) ∈ H 4 and the compatibility conditions hold. Then there exists a unique global solution (v(t), u(t), θ(t), where v, θ > 0 and r are also the same as those of Theorem 2.1. Moreover, we have, as t → +∞, In order to derive our results of time asymptotic behavior of the global solutions, we may use the following basic inequality (Lemma 2.4) in analysis by Shen and Zheng [25]. Lemma 2.4. Let T be given with 0 < T ≤ +∞. Suppose that y(t), h(t) are nonnegative continuous functions defined on [0, T ] and satisfy the following conditions: where A i (i = 1, 2, 3, 4) are given non-negative constants. Then for any r > 0 with 0 < r < T , the following estimates holds: where Proof. We can easily get (32) by integrating (24) 1 over Q t := (0, L) × (0, t) and noting the boundary conditions. Equation (24) 3 can be written as (36) We can deduce (33) from the maximal value principle and the positivity of θ 0 .
Multiplying (24) 2 by u, adding the resultant to (24) 3 , and then integrating it over Q t and using the boundary conditions, we deduce L 0 (e + 1 2 From the definitions of F R and (S E ) R , we can infer from (24) which, together with (19), leads to Thus, the contribution of the radiation term reads Combining (39) with (23) and assumptions ( Similarly to the proof of Lemma 2.1 in [21], estimate (35) can be shown, thus we omit it. The proof is now complete.
Remark 2. By Jensen's inequality, the mean value theorem and (35), we can know that there exists a point a(t) ∈ [0, L] and two positive constants α 1 , α 2 such that where α 1 , α 2 are two roots of the equation y − log y − 1 = C 1 . Let then, noting that r t = u, we can deduce from (24) 1 , by the mean value theorem, that there exists a point Proof. See, e.g., Lemma 4.1.8 in [20].
Proof. The main idea of the proof is similar to that of Lemma 5.2.4 in [23]. But the different key point here is that B(x, t) in the expression of v(x, t) depends not only on t but also on x. Thus we need a detailed analysis. It follows from Lemma 3.1 that which, together with the expression of D(x, t) in Lemma 3.2 and by Hölder's inequality, leads to (45) Noting that the assumptions (A 4 ), we deduce from Lemma 3.1 and Remark 2 that, for any 0 ≤ τ ≤ t, Now, by Hölder's inequality, we have for any x ∈ [0, L], v(x, τ ) and a(t) is defined in Remark 2. Then we have Thus it follows from Lemma 3.2, assumption (A 4 ), (46) and (48) that Applying Gronwall's inequality and Lemma 3.1, we can derive On the other hand, we also infer from (45) Thus we obtain the estimate (43). The proof is complete.
Proof. Estimate (51) has been obtained in Lemma 2.3 of [21]. The proofs of estimates (52)-(53) are similar to those of Lemma 2.4 in [21]. Thus we omit the details.
Lemma 3.5. Under the assumptions in Theorem 2.1, the following estimate holds for any t > 0, Proof. Multiplying (24) 4 by I and then integrating the resultant over [0, L] × S n−1 and using the boundary conditions, we obtain Noting the boundary condition (19), we know Integrating (55) with respect to ν over [0, +∞) and using Young's inequality, (56) and the assumption ( Similarly, we also infer from (55) by Young's inequality and the assumptions ( Therefore, we complete the proof of (54).
Obviously, we can obtain the following result from Lemmas 3.1 and 3.5.
Corollary 1. Under the assumptions in Theorem 2.1, the following estimate also holds for any t > 0, Lemma 3.6. Under the assumptions in Theorem 2.1, the following estimates hold for any t > 0, Proof. Here we adopt the technique from Lemma 7 in [6]. Noting the formula (40) in Remark 2, we can define the auxiliary function for any ξ > 0, Thus it follows from the assumption ( Noting that and using the assumptions (A 1 ) and (A 5 ) − (A 6 ), Lemma 3.3 and the Sobolev embedding theorem, we have for any ε > 0, Using the Cauchy-Schwarz inequality and the Sobolev embedding inequality, Lemma 3.1 and Lemma 3.5, and noting that the assumption α ≤ 1 + 2s, we can obtain
Now repeating the derivation of (47)-(48) and applying (57), we can conclude for Thus we readily obtain the next corollary.
Corollary 3. Under the assumptions in Theorem 2.1, the following estimate holds for any ( Proof. It follows from Corollary 2 and (53) and (57) that Lemma 3.7. Under the assumptions in Theorem 2.1, the following estimate holds for any t > 0, Proof. Multiplying (24) 2 by u t over Q t and using Young's inequality, we have for any ε > 0, which, by taking ε > 0 small enough, along with Lemmas 3.4 and 3.6, leads to (63).
Lemma 3.8. Under the assumptions in Theorem 2.1, the following estimates hold that for any t > 0, Proof. Let Then it is easy to verify that But we easily know from the assumptions ( We rewrite (24) 3 as Multiplying (67) by K t and integrating the resultant over Q t , we easily obtain Now we use Lemmas 3.1-3.7 to estimate each term in (68). Obviously, and applying Cauchy's inequality, we can deduce and applying Corollaries 2-3 and Lemma 3.6, we deduce for q 1 = max{ q 2 −2s−1, 0}, which, along with (71) and (72), gives It follows from Lemma 3.6 and Corollaries 2-3 that Now let us consider the various contributions in the second integral of (68). By Lemmas 3.1-3.7 and Corollaries 1 and 3, we have Noting the following facts and from equation (24) 3 , we can deduce Thus, by the Sobolev inequality, Lemmas 3.4 and 3.7, we can conclude By Lemma 3.4 and Corollary 3, using Cauchy's inequality, we have The last contribution in (68) can be estimated as follows, It follows from Corollaries 1-3 that Using the assumption (A 9 ), Cauchy's inequality and Corollary 1, we have Using the same technique, we also get Inserting the estimates (83)-(85) into (82), we get Inserting all previous estimates (69)-(71), (74)-(78), (80)-(81) and (86) into (68), we obtain with λ = max{q + s + 2, q + 2α − s, 3 2 (q − s) + α − s, 2q − s + 2, 3q−s+2 2 }. By Lemmas 3.1, 3.4, 3.6 and the Hölder inequality, there exists a point b(t) ∈ [0, L] such that for any t > 0, Thus we get sup 0≤τ ≤t Using assumptions on q, s and α, we easily know that λ < 2(q + s + 2). Thus, by Young's inequality and (87), it follows that Therefore, we complete the proof.
Lemma 3.9. The following estimates hold that for any t > 0, Proof. We consider the following integro-differential equation Using Young's inequality and Lemmas 3.5 and 3.8, we have for all ω ∈ (0, 1),
Lemma 3. 10. Under the assumptions in Theorem 2.1, the following estimates hold that for any t > 0,

YUMING QIN AND JIANLIN ZHANG
Proof. By Lemma 3.9, we have By virtue of the direct computation, we also have Using Lemma 3.9, we see that Similarly, Inserting (109)-(110) into (108), we obtain the desired estimate.
The next two lemmas are aimed at showing the asymptotic behavior of solutions to the problem (24) with the initial boundary conditions (18)- (20) in H 1 .
Lemma 3.12. Under the assumptions in Theorem 2.1, we have as t → +∞, Proof. Similarly to the proof of Lemma 3.4 in [21], we can also obtain (124). Here we omit it.
Till now we have completed the proof of Theorem 2.1.

Global existence and asymptotic behavior in
Proof. Estimate (125) has been obtained in Lemma 1.3.1 of [23]. Differentiating (24) 3 with respect to t and multiplying the result by θ t over [0, L], we have that for any ε > 0, d dt Integrating (127) with respect to t and using Lemmas 3.1-3.8 and Young's inequality, we derive for any ε > 0, It follows from Lemmas 3.7-3.9 that Inserting (129) into (128) and then taking ε > 0 small enough, we obtain Using the Gagliardo-Nirenberg interpolation inequality and Young's inequality, we derive from (24) 3 that Combining (130) and (131), we get (126).

Lemma 4.2.
Under the assumptions of Theorem 2.2, the following estimates hold that for any t > 0, Proof. Similarly to the proof of Lemma 1.3.2 in [23], we easily obtain (132). It follows from (24) 2 that Similarly, we can infer from (24) 3 that By the definition of (S E ) R and Lemma 3.9, it follows from that which, together with (134)-(135), implies (133).

Lemma 4.3.
Under the assumptions of Theorem 2.2, the following estimate holds for any t > 0, Proof. It follows from (24 4 ) and the definitions of I and I that Employing the Gagliardo-Nirenberg interpolation inequality and using Lemmas 3.9 and 4.2, we conclude and Similarly, we also infer that which, along with (138)-(140), leads to (137).
The next lemma concerns the asymptotic behavior of the global solution in H 2 .
Till now we have completed the proof of Theorem 2.2.

5.
Global existence and asymptotic behavior in H 4 . In this section, we shall prove Theorem 2.3, that is, the global existence and asymptotic behavior of solutions in H 4 to the problem (24) with the initial boundary conditions (18)- (20) under some relative assumptions.
Differentiating (24) 2 with respect to t twice, multiplying the resultant by u tt in L 2 (0, L), performing an integration by parts, and using Theorem 2.2 and the embedding theorem and the Young inequality, we can deduce Thus, by Theorem 2.2, which, along with (163) and (165) and Lemma 3.9, gives estimate (155).
In the same manner, differentiating (24) 3 with respect to t twice, multiplying the resultant by θ tt and performing an integration by parts over L 2 (0, L), and using the embedding theorem and the Young inequality, we have By virtue of Theorems 2.1-2.2 and Lemmas 4.1-4.3, and using the embedding theorem, we deduce that for any ε ∈ (0, 1), It follows from (174) by the Hölder inequality that Now we can estimate A 8 as By the induction of (129), we can easily obtain Using Hölder's inequality and the interpolation theorem, we can deduce from The- Differentiating (93) with respect to t twice and after the lengthy calculation, we can deduce Applying Gronwall's inequality to (180), we can obtain Inserting (178)-(179) and (181) into (177), we have Thus we infer from (168)-(176) and (182) that for any ε ∈ (0, 1) small enough, Therefore taking supremum in t on the left-hand side of (183), picking ε ∈ (0, 1) small enough, and using (165), we can derive estimate (156). The proof is thus complete.
Differentiating (24) 2 with respect to x three times, using Lemmas 5.1-5.2 and Theorem 2.2 and applying Poincaré's inequality, we deduce Thus, Using the same technique, we can deduce from (24) 3 that Hence, we complete the proofs of (193) and (194).
By Lemmas 5.1-155, we have proved the global existence of solutions in H 4 to the problem (24) in H 4 with arbitrary initial datum (u 0 , v 0 , θ 0 ) ∈ H 4 and the uniqueness of solution follows from that of solution in H 1 or in H 2 . Next, we shall show the asymptotic behavior of solutions.
Proof. Differentiating (24) 1 with respect to x three times and then multiplying the resultant by v xxx over L 2 (0, L), we have Applying Holder's inequality and the interpolation inequality, we can estimate each term in (246) as follows, Inserting (247)-(250) into (246), we have Analogously, we also deduce from equation (24) Naturally, we can also get the following estimates