### Journal Menu >>

 International Journal of Modern Nonlinear Theory and Application, 2013, 2, 186-193 http://dx.doi.org/10.4236/ijmnta.2013.23026 Published Online September 2013 (http://www.scirp.org/journal/ijmnta) Pointwise Estimates for Solutions to a System of Radiating Gas Shikuan Mao, Yongqin Liu School of Mathematics and Physics, North China Electric Power University, Beijing, China Email: shikuanmao@ncepu.edu.cn, yqliu2@ncepu.edu.cn Received August 22, 2013; revised September 8, 2013; accepted September 13, 2013 Copyright © 2013 Shikuan Mao, Yongqin Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT In this paper we focus on the initial value problem of a hyperbolic-elliptic coupled system in multi-dimensional space of a radiating gas. By using the method of Green function combined with Fourier analysis, we obtain the pointwise decay estimates of solutions to the problem. Keywords: Radiating Gas; Initial Value Problem; Pointwise Estimates 1. Introduction In this paper we consider the initial value problem  20div 0,,0,div0,, 0,,0 ,,ntnnuau q x tqq uxtuxu xx   (1.1) here is a constant vector, and are unknown functions of na1,,nqq,,xx,uuxt0.t,qxtnx11nn and Typically, represent the velocity and radiating heat flux of the gas respectively. ,uqThe system (1.1) is a simplified version of the model for the motion of radiating gas in n-dimensional space. More precisely, in a certain physical situation, the system (1.1) gives a good approximation to the following system describing the motion of radiating gas, which is a quite general model for compressible gas dynamics where heat radiative transfer phenomena are taken into account,  22412div 0,div 0,div 0,22div 0,tttuuuupIuueuepuqaqa           q(1.2) where ρ, u, p, e and θ are respectively the mass density, velocity, pressure, internal energy and absolute tempera-ture of the gas, while q is the radiative heat flux, and a1 and a2 are given positive constants depending on the gas itself. The first three equations are motivated by the usual Euler system, which describe the in-viscid flow of a compressible fluid and express conservation of mass, momentum and energy respectively. We refer to the book of Courant and Friedrichs [1] for a detailed derivation of several models in compressible gas-dynamics. The physical motivation of the fourth equation, which takes into account of heat radiation phenomena, is given in [2]. Moreover, the simplified model (1.1) was first recovered by Hamer (see [3]), and for the reduction of system (1.2) to system (1.1), see [2-4]. Concerning the investigation on the hyperbolic-elliptic coupled system in one-dimensional radiating gas, we refer to [5,6]. In the case of the muti-dimensional case, Francesco in [7] obtained the global well-posedness of the system (1.1) and analyzed the relaxation limits. Re-cently, in [8], Liu and Kawashima investigated the decay rate to diffusion wave for the initial value problem (1.1) in n(n ≥ 1)-dimensional space by using a time-weighted energy method. The rest of the paper is arranged as follows. Section 2 gives the full statement of our main theorem. In Section 3, we give estimates on the Green function by Fourier analysis which will be used in Section 5. Section 4 gives the global existence of solutions to the problem (2.3). In Section 5, we obtain the pointwise decay estimates of solutions. Before closing this section, we give some notations to be used below. Let f denote the Fourier transform of f defined by Copyright © 2013 SciRes. IJMNTA S. K. MAO, Y. Q. LIU 187  2·1ˆ:e2nnix d,fffxx and we de- note its inverse transform as 1.1ppnLL p is the usual Lebesgue space with the norm ·pL. Let be a nonnegative integer, sthen ssnHH denotes the Sobolev space of functions, equipped with the norm 2L12220:.sskxHLkff For 0,k111:;01,nnnkkkxixxikink  ksn.ik For non- negative integer , denotes the space of -times continuously differentiable functions on the interval k;CIHkI with values in the Sobolev space .nssHHFinally, in this paper, we denote every positive con-stant by the same symbol or c without confusion. [·] is the Gauss’s symbol. C2. Main Theorems and Proof For simplicity, without loss of generality, we choose 11,,22na in (1.1). That is, we will con- sider the following initial value problem:  10div 0,, 0,div0,, 0,,0 ,.inntxinnuuuq x tqq uxtuxu xx   (2.3) Our Main results are the following: Theorem 2.1. Let  3,1, 22,2,nsnn be an inte-ger. Assume that and put  10,sn nuH L100 0:.sHL Then there is a small positive con-stant such that if 00Eu u01,Eux then the problem (2.3) has a unique global solution with ,t210, ;,0, ;,sns nuC HuLH  12 10, ;0, ;.sn snqC HLH   Moreover, if 26,sn and for any multi-indexes  with 21,sn  there exists some constant 2nr such that 2001rxDu xCEx, then for any 24,sn the solution to Equation (2.3) has the following decay estimate,  220,11. We also have 1rnxxDuxt CEtt the following corollary by using Theorem 2.1. Corollary 2.2. Under the same assumptions in Theo-rem 2.1, the solution satisfies the following decay esti-mates:  42201,nkkxLutCE t with satisfying k042nks ;   214201,nkkxLqtCEt with satisfying k052nks . Remark. In Theorem 2.1, we do not need to assume that 10,nuL if The results in Corollary 2.2 is similar to those in [7]. 2.n3. The Global Existence of Solution This section is devoted to prove the global existence re-sult stated in Theorem 2.1. In [7], the global existence of solutions to the problem (2.3) is obtained, but for the completeness of this paper, here we give the sketch of the proof. Since a local existence result can be obtained by the standard method based on the successive approximation sequence, we omit its details and only derive the desired a priori estimates of solutions. From 112.32.3div 2.3, 2 we get that 110.iinnttx xiiu u uuuuu  (3.4) Now we make energy estimates by using (3.4) under the following a priori estimate: 1,xLut (3.5) here 11 is a given constant. Multiplying (3.4) by and integrating with respect to ux, by integration by parts we have that  1222d.dx22HLLututC ututtL (3.6) Multiplying 3.4lx by and integrating with respect to 1lxulx, by integration by parts we have that  11222dd,1llxxHLlxxLHut uttCutut l2. 1 (3.7) We add up (3.7) with 1ls and get that Copyright © 2013 SciRes. IJMNTA S. K. MAO, Y. Q. LIU 188  1212222dd.sssxxHHxxLHut uttCut ut  (3.8) Combining (3.6) with (3.8) we have that  11222dd.ssxHHxxLHut uttCut ut s (3.9) In view of (3.5), (3.9) yields that  122200d.ssstxHHHut uCu  (3.10) From 2 we have that thus (3.10) yields that 2.311,qu  1122200d.ssstxHHHqt qCu  (3.11) By the continuity argument, we have the following result. Theorem 3.1. Let 22ns be an integer. Assume that and put 0,snuH00:.sHEu Then there is a small positive constant 01 such that if 00,E then the problem (2.3) has a unique global solution with ux,t210, ;,0,;,sns nuC HuLH  12 10, ;0, ;.sn snqC HLH  4. Estimates on Green Function In order to study the problem (2.3), we start with the Green function (or the fundamental solution) to the linear problem corresponding to the Equation (3.4), which sat-isfies  0,, 0,,0 ,.nttnGGGx tGxx x  (4.12) By Fourier transform we get that, ˆˆˆ0,,0,ˆ,0 1,.nttnGGG tG (4.13) By direct calculation we have that 221ˆ,etGt. Let  131, ,1,,0,2 ,0,1,RR   be the smooth cut-off functions, where  and are any fixed positive numbers satisfying R01R. Set 2131,  and ˆˆ,,Gt Gt,1,2, 3.iii We are going to study ,iGxt, which is the inverse Fourier transform of ˆ,.iGt First we give a lemma which is important for us to make estimates on the low frequency part. ˆ,2NnCftLemma 4.1. If has compact sup-port in the variables , is a positive integer, and there exists a constant such that N0,bˆ,ft satis-fies  2||222,111e,kkkbttCttt 2ˆmDft for any multi-indexes , with 2N, then 2,,nkxNNDfxtCtB xt, where and are any fixed integers, mkx 0,,aa and 2,11.NNxBxt t maProof. If ,k by direct calculation we have that   21|| ||||222222ˆ,e ,d111ed1 .nnxkkknkmbtxD fxtCDftCtttCtt    t If ,k we also have  21222222ˆ,e,11 11ed1nnxkknkmbtxD fxtCDftCttttCt d.t  Let 0 when 21,xt and 2N when 21,xt we obtain from the above estimates that 221,min1,Nnk tDfxt Ctx. Since 22221,1 ,121,11,xtxxtxtt, we have Copyright © 2013 SciRes. IJMNTA S. K. MAO, Y. Q. LIU 189large such that 211m2212min 1,2,.11NNNNNtBxtxxt  . Since 2supp ;,G R we have that Thus we complete the proof. By using Lemma 4.1 we can get the following propo-sition about the estimates on 1,.GxtProposition 4.2. For sufficiently small , there exists constant such that 0C21,,nxNNDGxtCtBx t. Proof. For  being sufficiently small, by noticing that is a smooth function of ,ˆtG near 0 and using Taylor expansion we have that 22421,ˆeettO ttG . It yields that  222222ˆ,111e.tDGtCttt t Since  12121112,!ˆ,,!!DGtDDGt  and 1,  we have that  21||22222,111e.tDGtCttt t From Lemma 4.1 we obtain that 21,,nxNNDGxtCtBx t. Thus we complete the proof of Proposition 4.2. As for we have the following estimates. 2,GxtProposition 4.3. For fixed  and , there exist positive numbers and such that RmC22,e ,tmxNDGxtCBx t 2ˆ,etmGtC. (4.14) It yields that 2,etmxDG xtC. (4.15) Now we shall give an estimate on 2,xGxt by induction on . Assume that, if 1,l then  2ˆ,1etmDGt Ct , (4.16) which is true as 0 by (4.14) By using (4.13), we have the following problem for  222201,ˆˆˆ,,,11ˆ,0.tDG tDG tDG tDGt,,(4.17) By multiplying (4.17), whose variables are now changed to ,s by ˆ,Gts and integrating over the region 0,st, we have that  220ˆˆˆ,,, ,1tDG tG tsDGss  d. In view of (4.16) for 1,l it yields that   10ˆ,e1ed1ts sttmmRDGtCts Ct  e,m which shows that (4.16) is valid as .l This implies that, for 1,l  2·2{},ˆ,ede111e.nhxtixtmRtmxDG xtCD GtCtCt d  (4.18) By using (4.15) when 21xt and (4.18) with 2N when 21xt, as well as the fact that 22222,1 ,121,11,xtxxtxtt we get that . 22,e ,tmxNDGxtCBx t. Proof. For any fixed , we choose sufficiently mCopyright © 2013 SciRes. IJMNTA S. K. MAO, Y. Q. LIU 190 Thus we complete the proof of Proposition 4.3. Next we will come to consider First we give a lemmas which is useful in dealing with the high fre-quency part. 2,GxtLemma 4.4. If ˆsupp: ;,RfOR  and ˆf satisfies  1ˆˆ,,fCDfC 1,, then there exist distributions  12,fxfx and con-stant such that 0C,120fxf Cxxfx where x is the Dirac function. Furthermore, for positive integer 2,Nn211,NxDf xCx 122,supp;2 ,LfCfxxx0 with 0 being sufficiently small. The proof of Lemma 4.4 can be seen in [9]. Choose sufficiently large such that R221,21 if 1.R By Taylor expansion, we have that  24111()33ˆ,eOtGt . It is obvious that 23ˆ,etGtC. Since  24121211133ˆ,eOtDGtCDD , by direct calculation, we have that for 1, 123ˆ,etDGtC. By using Lemma 4.4 we have the following result. Proposition 4.5. For being sufficiently large, there exist distributions and con-stant such that R31Gx,,t32 ,,Gxt0C  2331320,e ,,tGxtGxt GxtCx, where x is the Dirac function. Furthermore, for positive integer 2Nn, the following estimates hold: 231 ,1,NxDG xtCx and 13232 0.,,supp.,;2 ,LGtCGtxx here 0 is sufficiently small. Combining Proposition 4.2, Proposition 4.3, and Proposition 4.5, we have the following theorem on the Green function. Theorem 4.6. For any multi-indexes , there exists a distribution 232 0,e ,t,KxtG xtCx such that the following estimate holds: 2,1 ,nxNDGKxtCtB xt, here, 2nN is an arbitrary positive integer. 5. Pointwise Estimates In this section, we focus on the pointwise estimates of solutions to the problem (2.3). By Duhamel principle, the solution to the Equation (3.4) with initial datum can be ex-pressed as following,  0,0uxu x 001,., 1:, ,.intxnuxtGtuxGtuuuxtuxtd Now we give a lemma which will be used in the fol-lowing analysis. Lemma 5.1. When 12,nnn2, and 31min ,,nnn2 we have that 13222211d111nnnnxy xyyCtt    . The proof of Lemma 5.1 can be seen in [9]. Since 12,,,:uxtG KxtKxtII, by using Lemma 5.1 and Theorem 4.6 with , we have that Nr2101,. nxrDICEtB xtNoticing that if 02,xy then 11211yCx2, we have that 22320 020e,e,.ntxxtrDIGxytCx yDu y yCEBx td Thus we obtain that  20,1,, 1.2nxrDu xtCEtnBxt s  (5.19) Next we come to make estimates on To this end, we will use the following lemma. ,.uxtLemma 5.2. Assume , then the following ine-qualities hold, 1n1) If 0,t, and 2At, then Copyright © 2013 SciRes. IJMNTA S. K. MAO, Y. Q. LIU Copyright © 2013 SciRes. IJMNTA 191221121111nnnnAAtt    Denote 12,0,,42,:1,,:sup, ,nnrxxTnsxttBx tMTDux x.  ;2 t, then 212 11nnAt. A2) If Now we come to make estimates to by using ,uxt Theorem 4.6 with and Lemma 5.2. We decompose Nr,xDu xt as following,    220;2 1220;2 12;2 122;2 120,1,,1,,d11,211,2iiiinxtnxxyxy itnxxyxy inttxyyxy inttxyyxy itRDu xtDGKxytuyyDGKxytuyyGKxytDu yyGKxytDu yyKddd,dd,dd131 3241425,1 ,dd:.inyxixytDuuyyIIIII  Next we estimate respectively by using Theorem 4.6. By using Lemma 5.2 (1), we have that 1, 2, 3, 4, 5iIi  122231 0;212220;2||221,11,1,dd,1,ddnnNryx ytnnNryx ynrtCM TtBxy tByyCM TtBxtByyCMTBx ttI  Now we estimate 32 .I in two cases. Case 1. 2xt. By using Lemma 5.2 (1), we have that   122232 0;2122220;2221,111,1,1,1 .tnnNryx yntnnNryx ynrICMTtBxytBy yCM TtBxy tBxtytCM TBxtt,dddd   Case 2. 2xt. By using Lemma 5.2 (2), we have that  122232 0;212220221,1 ,dd11,d,1.tnyx ynNrtnnnrrICMT tBxytBy yCM TtByCMTBxt t   Combining the two cases, we have that S. K. MAO, Y. Q. LIU 192 2232 ,1nrICMTBxt t. As for 41,I we also need to divide it into two cases. Case 1. 2xt. By using Lemma 5.2 (1), we have that    12241 ;222212222;222221,1,dd1(1)1,,dd1,1.nttNyx ynrnnttyx ynNrnrICMTtBxytBy yCM TtttBxt BytytCM TBxtt  Case 2. 2xt. By using Lemma 5.2 (2), we have that   12241 ;222212222;222211,dd1(1),dd,1.nttNyx ynrnnttyx yNnrICMTtBxytBy yCM TttBxytyCM TBxtt,  Combining the two cases, we have that 2241 ,1nrICMTBxtt. As for 42,I by direct calculation, we have that   12242 ;222222212;22221,1,dd1,(1 ),1.dd,nttNyx ynrnrnttNyx ynrICMTt BxytBy yCM TBxttBxytyCM TBxtt To estimate 5,I we will use the following result, which is obtained in [7]. Lemma 5.3. 1) If 3, 1,2, 2,2nsnn and  10,sn nuH L put 100 0:sHEu u,L then the following estimate holds:  4201,sknkkxHutCE t with 01 .ks2) If 2,2ns and put 0,2snuH n,00:,sHEu then the following estimate holds:  201skkkxHutCE t, with 0.ksWe estimate 5I as following,  253201201e1e1 ,d.iitntxxitntxxiICGtDuuCD uuxd Notice that consists of terms of 21inxxiDu12121 2,, 0,3.kkxxuukk kk Without loss of generality, we assume that then 12,kk1max,2.k Since 62ns and 4,2ns we have that   112,1nkkxuyMTBy,. By using Gagliardo-Nirenberg inequality and Lemma 5.3, we have that  222021, 2kkxLnuCE ks1.  Thus we have that   1212 20,,1 . nk kkkxxuyuy CEMT  Combined with Proposition 4.5 and the fact that 112211,11yxCtt if 12,xy it yields that  12250 020e1 d,1.tntnrICEMTCE MTBxtt Combining 313241 42,,,IIII and 5,I we have that  120,,,xhDuxtCEMTMTxt.(5.20) Proof of Theorem 2.1. In view of (5.19) and (5.20), we get that  1200 ,,,xhDu xtC EEM TM Txt . It yields that Copyright © 2013 SciRes. IJMNTA S. K. MAO, Y. Q. LIU Copyright © 2013 SciRes. IJMNTA 193 200 .MTCE EMTMT  Thus if 0 is suitably small, we obtain E0MTCE by the continuous dependence on the initial data. In view of Theorem 3.1, the proof of Theorem 2.1 is completed. 6. Acknowledgements The first author is partially supported by the National Natural Science Foundation of China (Grant No. 11201142) and by the Fundamental Research Funds for the Central Universities (Grant No. 11QL40). The second author is partially supported by the National Natural Science Foundation of China (Grant No. 11201144). REFERENCES [1] R. Courant and K. O. Friedrichs, “Supersonic Flow and ShockWaves,” Interscience Publishers, Inc., New York, 1948. [2] W. G. Vincenti and C. H. Kruger, “Introduction to Physi- cal Gas Dynamics,” Wiley, New York, 1965. [3] K. Hamer, “Nonlinear Effects on the Propogation of Sound Waves in a Radiating Gas,” Quarterly Journal of Mechanics & Applied Mathematics, Vol. 24, No. 2, 1971, pp. 155-168. http://dx.doi.org/10.1093/qjmam/24.2.155 [4] W. L. Gao and C. J. Zhu, “Asymptotic Decay toward the Planar Rarefaction Waves for a Model System of the Ra- diating Gas in Two Dimensions,” Mathematical Models and Methods in Applied Sciences, Vol. 18, No. 4, 2008, pp. 511-541. http://dx.doi.org/10.1142/S0218202508002760 [5] S. Kawashima and S. Nishibata, “Weak Solutions with a Shock to a Model System of the Radiating Gas,” Science Bulletin of Josai University, Vol. 5, Special Issue, 1998, pp. 119-130. [6] S. Kawashima and S. Nishibata, “Cauchy Problem for a Model System of the Radiating Gas: Weak Solution with a Jump and Classical Solutions,” Mathematical Models and Methods in Applied Sciences, Vol. 9, No. 1, 1999, pp. 69-91. http://dx.doi.org/10.1142/S0218202599000063 [7] M. D. Francesco, “Initial Value Problem and Relaxation Limits of the Hamer Model for Radiating Gases in Several Space Variables,” Nonlinear Differential Equations and Applications NoDEA, Vol. 13, No. 5-6, 2007, pp. 531-562. http://dx.doi.org/10.1007/s00030-006-4023-y [8] Y. Liu and S. Kawashima, “Asymptotic Behavior of So- lutions to a Model System of a Radiating Gas,” Commu- nications on Pure and Applied Analysis, Vol. 10, No. 1, 2011, pp. 209-223. http://dx.doi.org/10.3934/cpaa.2011.10.209 [9] W.-K. Wang and T. Yang, “The Pointwise Estimates of Solutions for Euler Equations with Damping in Multi- Dimensions,” Journal of Differential Equations, Vol. 173, No. 2, 2001, pp. 410-450. http://dx.doi.org/10.1006/jdeq.2000.3937