International Journal of Modern Nonlinear Theory and Application, 2013, 2, 186193 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 hyperbolicelliptic coupled system in multidimensional 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 2 0 div 0,,0, div0,, 0, ,0 ,, n t n n uau q x t qq uxt uxu xx (1.1) here is a constant vector, and are unknown functions of n a 1,, n qq ,,xx ,uuxt 0.t ,qxt n x 11n n and Typically, represent the velocity and radiating heat flux of the gas respectively. ,uq The system (1.1) is a simplified version of the model for the motion of radiating gas in ndimensional 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, 22 4 12 div 0, div 0, div 0, 22 div 0, t t t u uuupI uu euepu qaqa 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 inviscid 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 gasdynamics. 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 [24]. Concerning the investigation on the hyperbolicelliptic coupled system in onedimensional radiating gas, we refer to [5,6]. In the case of the mutidimensional case, Francesco in [7] obtained the global wellposedness 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 timeweighted 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 denote the Fourier transform of defined by C opyright © 2013 SciRes. IJMNTA
S. K. MAO, Y. Q. LIU 187 2 · 1 ˆ:e 2 n n ix d, f fxx and we de note its inverse transform as 1. 1 ppn LL p is the usual Lebesgue space with the norm · . Let be a nonnegative integer, s then sn HH denotes the Sobolev space of functions, equipped with the norm 2 L 1 2 2 2 0 :. s sk x HL k ff For 0,k 1 11 :;01, n n n k k k xi xx i kink ksn . i k For non negative integer , denotes the space of times continuously differentiable functions on the interval k ;CIH k 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. C 2. Main Theorems and Proof For simplicity, without loss of generality, we choose 11 ,, 22 n a in (1.1). That is, we will con sider the following initial value problem: 1 0 div 0,, 0, div0,, 0, ,0 ,. i nn tx i n n uuuq x t qq uxt uxu xx (2.3) Our Main results are the following: Theorem 2.1. Let 3,1, 22,2, n snn be an inte ger. Assume that and put 1 0, sn n uH L 1 00 0 :. s L Then there is a small positive con stant such that if 00 Eu u 01 ,E ux then the problem (2.3) has a unique global solution with ,t 21 0, ;,0, ;, sns n uC HuLH 12 1 0, ;0, ;. sn sn qC HLH Moreover, if 26,sn and for any multiindexes with 21,sn there exists some constant 2 n r such that 2 00 1 r x Du xCEx , then for any 24,sn the solution to Equation (2.3) has the following decay estimate, 2 2 0 ,11. We also have 1 r n x x Duxt 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: 42 201, nk k xL utCE t with satisfying k 04 2 n ks ; 2 1 42 01, nk k xL qtCEt with satisfying k05 2 n ks . Remark. In Theorem 2.1, we do not need to assume that 1 0, n uL 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 11 2.32.3div 2.3, 2 we get that 11 0. ii nn ttx x ii u u uuuuu (3.4) Now we make energy estimates by using (3.4) under the following a priori estimate: 1, xL ut (3.5) here 11 is a given constant. Multiplying (3.4) by and integrating with respect to u , by integration by parts we have that 12 22 d. dx2 2 LL ututC utut t L (3.6) Multiplying 3.4 l x by and integrating with respect to 1 l xul , by integration by parts we have that 1 1 22 2 d d ,1 ll xx HL l xx LH ut ut t Cutut 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 12 1 2 22 2 d d . ss s xx HH xx LH ut ut t Cut ut (3.8) Combining (3.6) with (3.8) we have that 1 1 22 2 d d . s s x HH xx LH ut ut t Cut ut s (3.9) In view of (3.5), (3.9) yields that 1 22 2 0 0d. ss t xH HH ut uCu (3.10) From 2 we have that thus (3.10) yields that 2.3 1 1,qu 11 22 2 0 0d. ss t xH HH qt qCu (3.11) By the continuity argument, we have the following result. Theorem 3.1. Let 2 2 n s be an integer. Assume that and put 0, sn uH00 :. H Eu 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 21 0, ;,0,;, sns n uC HuLH 12 1 0, ;0, ;. sn sn qC 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 ,. n tt n GGGx t Gxx x (4.12) By Fourier transform we get that, ˆˆˆ 0,,0, ˆ,0 1,. n tt n GGG t G (4.13) By direct calculation we have that 2 2 1 ˆ,e t Gt . Let 13 1, ,1,, 0,2 ,0,1, R R be the smooth cutoff functions, where and are any fixed positive numbers satisfying R 01R. Set 213 1, and ˆˆ ,,Gt Gt , 1,2, 3.i ii We are going to study , i Gxt, which is the inverse Fourier transform of ˆ,. i Gt First we give a lemma which is important for us to make estimates on the low frequency part. ˆ, 2 n Cft Lemma 4.1. If has compact sup port in the variables , is a positive integer, and there exists a constant such that N 0,b ˆ, t satis fies 2  2 22 , 11 1e , k kk bt t Cttt 2 ˆ m Df t for any multiindexes , with 2N , then 2 ,, nk xNN DfxtCtB xt , where and are any fixed integers, mk x 0,,aa and 2 ,1 1. N x Bxt t ma Proof. If ,k by direct calculation we have that 2 1   2 22 22 2 ˆ ,e ,d 11 1ed1 . n n x k kk nk mbt xD fxtCDft Ctt tCtt t If ,k we also have 2 1 2 22 22 2 ˆ ,e, 11 1 1ed1 n n x k k nk mbt xD fxtCDft Cttt tCt d . t Let 0 when 21, t and 2N when 21, t we obtain from the above estimates that 2 2 1 ,min1, N nk t Dfxt Ct x . Since 2 2 2 2 1,1 , 12 1,1 1, t x x t t t , we have Copyright © 2013 SciRes. IJMNTA
S. K. MAO, Y. Q. LIU 189 large such that 2 1 1m 22 12 min 1,2,. 11 NN N N N tBxt xx t . Since 2 supp ;,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,.Gxt Proposition 4.2. For sufficiently small , there exists constant such that 0C 2 1,, n xNN DGxtCtBx t . Proof. For being sufficiently small, by noticing that is a smooth function of , ˆt G near 0 and using Taylor expansion we have that 2 24 2 1 , ˆee ttO t t G . It yields that 2 2 22 22 ˆ, 11 1e. t DGt Ctt t t Since 12 12 1 1 12 , !ˆ,, !! DGt DGt and 1, we have that 2 1  2 22 22 , 11 1e. t DGt Ctt t t From Lemma 4.1 we obtain that 2 1,, n xNN DGxtCtBx t . Thus we complete the proof of Proposition 4.2. As for we have the following estimates. 2,Gxt Proposition 4.3. For fixed and , there exist positive numbers and such that R mC 2 2,e , t m xN DGxtCBx t 2 ˆ,e t m GtC . (4.14) It yields that 2,e t m x DG xtC . (4.15) Now we shall give an estimate on 2, Gxt by induction on . Assume that, if 1,l then 2 ˆ,1e t m DGt Ct , (4.16) which is true as 0 by (4.14) By using (4.13), we have the following problem for 22 22 0 1, ˆˆˆ ,,, 11 ˆ,0. t DG tDG tDG t DGt ,, (4.17) By multiplying (4.17), whose variables are now changed to , by ˆ,Gts and integrating over the region 0, t, we have that 2 2 0 ˆˆˆ ,,, , 1 t DG tG tsDGss d. In view of (4.16) for 1,l it yields that 1 0 ˆ,e1ed1 ts st tmm R DGtCts Ct e, m which shows that (4.16) is valid as .l This implies that, for 1,l 2 · 2 {} , ˆ,ed e11 1e. n h xt ix t m R t m xDG xt CD Gt Ct Ct d (4.18) By using (4.15) when 21 t and (4.18) with 2N when 21 t, as well as the fact that 2 2 2 2 2,1 , 12 1,1 1, t x x t t t we get that . 2 2,e , t m xN DGxtCBx t . Proof. For any fixed , we choose sufficiently m Copyright © 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,Gxt Lemma 4.4. If ˆ supp: ;, R OR and ˆ f satisfies 1 ˆˆ ,,fCDfC 1, , then there exist distributions 12 , xfx and con stant such that 0 C , 120 xf Cx xfx where is the Dirac function. Furthermore, for positive integer 2,Nn 2 11, x Df xCx 1 22 ,supp;2 , L fCfxxx 0 with 0 being sufficiently small. The proof of Lemma 4.4 can be seen in [9]. Choose sufficiently large such that R 2 2 1, 2 1 if 1.R By Taylor expansion, we have that 24 11 1() 33 ˆ,e Ot Gt . It is obvious that 2 3 ˆ,e t GtC . Since 24 12 12 11 1 33 ˆ,e Ot DGtCDD , by direct calculation, we have that for 1, 1 2 3 ˆ,e t DGtC . 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 R 31 Gx ,,t 32 ,,Gxt 0 C 2 331320 ,e ,, t GxtGxt GxtCx , where is the Dirac function. Furthermore, for positive integer 2Nn, the following estimates hold: 2 31 ,1, x DG xtCx and 1 3232 0 .,,supp.,;2 , L GtCGtxx 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 multiindexes , there exists a distribution 2 32 0 ,e , t , xtG xtCx such that the following estimate holds: 2 ,1 , n xN DGKxtCtB xt , here, 2 n N 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 :, ,. i n t x n uxtGtuxGtuu uxtuxt d Now we give a lemma which will be used in the fol lowing analysis. Lemma 5.1. When 12 ,nnn2, and 31 min ,,nnn2 we have that 13 2 22 2 11d1 11 n nn n xy x yyC tt . 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 2 10 1,. n xr DICEtB xt Noticing that if 0 2,xy then 11 2 11yCx 2 , we have that 2 2320 0 2 0 e, e,. n t xx t r DIGxytCx yDu y y CEBx t d Thus we obtain that 2 0 ,1 ,, 1. 2 n x r Du xtCEt n Bxt 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 2 t, then Copyright © 2013 SciRes. IJMNTA
S. K. MAO, Y. Q. LIU Copyright © 2013 SciRes. IJMNTA 191 22 1 121 111 nn n n AA tt Denote 1 2 ,0,, 4 2 ,:1,, :sup, , n n r x xT n s xttBx t MTDux x . ; 2 t, then 2 12 1 1 n nA t . 2) If Now we come to make estimates to by using ,uxt Theorem 4.6 with and Lemma 5.2. We decompose Nr , x Du xt as following, 2 2 0;2 1 2 2 0;2 1 2 ;2 1 2 2 ;2 1 2 0 , 1,, 1,,d 11, 2 11, 2 i i i i n x tn xx yxy i tn xx yxy i n t txy yxy i n t txy yxy i t R Du xt DGKxytuyy DGKxytuyy GKxytDu yy GKxytDu yy K dd d ,dd ,dd 1 31 3241425 ,1 ,dd :. i n yx i xytDuuyy IIIII Next we estimate respectively by using Theorem 4.6. By using Lemma 5.2 (1), we have that 1, 2, 3, 4, 5 i Ii 1 222 31 0;2 1 22 2 0;2  22 1,1 1,1,dd ,1 ,dd nn Nr yx y t nn Nr yx y n r t CM TtBxy tByy CM TtBxtByy CMTBx tt Now we estimate 32 . in two cases. Case 1. 2 t. By using Lemma 5.2 (1), we have that 1 222 32 0;2 12 22 2 0;2 22 1,1 1 1,1, 1 ,1 . tnn Nr yx y n t nn Nr yx y n r ICMTtBxytBy y CM TtBxy tBxty t CM TBxtt ,dd dd Case 2. 2 t. By using Lemma 5.2 (2), we have that 1 222 32 0;2 1 22 2 0 22 1 ,1 ,dd 11 ,d,1. tn yx y n Nr t nn n rr ICMT t BxytBy y CM Tt ByCMTBxt t Combining the two cases, we have that
S. K. MAO, Y. Q. LIU 192 22 32 ,1 n r ICMTBxt t. As for 41, we also need to divide it into two cases. Case 1. 2 t. By using Lemma 5.2 (1), we have that 1 22 41 ;2 2 2 2 1 2 22 2 ;2 2 2 22 1, 1,dd 1(1) 1 ,,dd 1 ,1. n t tN yx y n r n nt tyx y n Nr n r ICMTtBxyt By y CM Ttt t Bxt Byty t CM TBxtt Case 2. 2 t. By using Lemma 5.2 (2), we have that 1 22 41 ;2 2 2 2 1 2 22 2 ;2 2 22 1 1,dd 1(1) ,dd ,1. n t tN yx y n r n nt tyx y N n r ICMTtBxyt By y CM Ttt Bxyty CM TBxtt , Combining the two cases, we have that 22 41 ,1 n r ICMTBxtt . As for 42, by direct calculation, we have that 1 22 42 ;2 2 2 2 2 22 1 2 ;2 2 22 1, 1,dd 1, (1 ), 1. dd , n t tN yx y n r n r n t tN yx y n r ICMTt Bxyt By y CM TBxt tBxyty CM TBxtt To estimate 5, we will use the following result, which is obtained in [7]. Lemma 5.3. 1) If 3, 1, 2, 2, 2 n snn and 1 0, sn n uH L put 1 00 0 : s H Eu u, L then the following estimate holds: 42 01, sk nk k xH utCE t with 01 .ks 2) If 2, 2 n s and put 0,2 sn uH n, 00 :, H Eu then the following estimate holds: 2 01 sk k k xH utCE t , with 0.ks We estimate 5 as following, 2 532 01 2 01 e1 e1 ,d. i i tn t xx i tn t xx i ICGtDuu CD uux d Notice that consists of terms of 2 1i n xx i D u 12 121 2 ,, 0,3. kk xx uukk kk Without loss of generality, we assume that then 12 ,kk 1max,2.k Since 6 2 n s and 4, 2 n s we have that 1 12 ,1 nk k xuyMTBy,. By using GagliardoNirenberg inequality and Lemma 5.3, we have that 2 22 02 1, 2 k k xL n uCE ks 1. Thus we have that 12 12 2 0 ,,1 . nk k kk xx uyuy CEMT Combined with Proposition 4.5 and the fact that 11 22 11, 11 yx C tt if 1 2,xy it yields that 1 22 50 0 2 0 e1 d ,1. tn t n r ICEMT CE MTBxtt Combining 313241 42 ,,, III and 5, we have that 1 2 0, ,, xh DuxtCEMTMTxt . (5.20) Proof of Theorem 2.1. In view of (5.19) and (5.20), we get that 1 2 00 , ,, xh Du xtC EEM TM Txt . It yields that Copyright © 2013 SciRes. IJMNTA
S. K. MAO, Y. Q. LIU Copyright © 2013 SciRes. IJMNTA 193 2 00 .MTCE EMTMT Thus if 0 is suitably small, we obtain E 0 TCE 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. 155168. 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. 511541. 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. 119130. [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. 6991. 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. 56, 2007, pp. 531562. http://dx.doi.org/10.1007/s000300064023y [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. 209223. 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. 410450. http://dx.doi.org/10.1006/jdeq.2000.3937
