 Applied Mathematics, 2010, 1, 504-509 doi:10.4236/am.2010.16066 Published Online December 2010 (http://www.SciRP.org/journal/am) Copyright © 2010 SciRes. AM C 0Approximation on the Spatially Homogeneous Boltzmann Equation for Maxwellian Molecules* Minling Zheng School of Science, Huzhou Teacher College, Huzhou, China E-mail: mlzheng@yahoo.com.cn Received August 25, 2010; revised October 15, 2010; accepted October 19, 2010 Abstract In this paper we study the viscosity analysis of the spatially homogeneous Boltzmann equation for Maxwel-lian molecules. We first show that the global existence in time of the mild solution of the viscosity equation (,)tvfQf ff  . We then study the asymptotic behaviour of the mild solution as the coefficients 0, and an estimate on 0ff is derived. Keywords: Viscosity Boltzmann Equation, Mild Solution, Viscosity Approximation, Collision Kernel 1. Introduction In this paper we shall investigate the asymptotic properties of the solution of the viscosity Boltzmann equation for Maxwellian molecules ,tvfQf ff   in 30, R (1) as the viscosity coefficients 0. Here, (,)Qf f is the Boltzmann collision operator for Maxwellian molecules defined by its quadratic form  3**0*,2cos sinRQfgfg fgbddv where function b is nonnegative and continuous, and 1cos sin0,bL. Here the shorthand ,'fftv, **,fftv are used; *',vv are the post-collisional velocities corresponding to the pre-collisional velocities *,vv respectively, which submit to the elastic collision law ***',',vvvvvvvv  (2) where (,) denotes the scalar product. 2S the 2-D unit sphere and (')/|'|vv vv .  is the angle between *vv and , 0,. On physically, Q satisfies the symmetrization and translation invariance. For Maxwellian potential Q can be split into Q and _Q: ,,,QffQ ffQ ff  32 **,cosRSQffffb ddv  32**,,cosRSQff fLfLffb ddv The problem of viscosity approximation of the spa-tially homogeneous Boltzmann equation, namely wheth-er the solution of (1) converges to the solution of the equation ,tfQff in 30, R (3) as 0, is very interested for mathematical theory of Boltzmann equation as well as practical applications. We know that the energy of the solution of (1) is increasing with the time t due to the diffusion effect. We cannot expect that the so lu tion o f (1) approaches to the Maxwel-lian equilibrium in large time. This observation has re-cently been shown by Li-Matsumura . In early work of the authors an explicit estimate of ff in 1kL was derived which indicates also the dependence of time . It must be stressed this result excludes the case of Maxwellian molecules. Actually, the produce of mo-ments for cutoff potential is not valid for Maxwellian molecules. In this paper we shall study the viscosity ap-proximation for Maxwellian molecules. Our goal is to *This work was supported by Huzhou Natural Science Foundation(2008YZ06) and Innovation Team Foundation of Department of Educa-tion of Zhejiang Province (T200924). M. L. ZHENG Copyright © 2010 SciRes. AM 505study the existence and uniqueness of the global solution of the viscosity equation (1) in time, and to estimate ff explicitly in 0C-norm. The new tool is the Gagliardo-Nirenberg inequality. Let us mention some works about the spatially homo-geneous Boltzmann equation with cutoff potential, see [3-11] for example. For the Maxwellian molecules Mor-genstern first deduced the existence and uniqueness of the solution in 1L space . We also remark that the approximation with diffusion term in velocity variable was present in the work of DiPerna-Lions . Now we complement the equation (1) and (3 ) with the same initial condition: 300||(),ttffvvR . (4) In the sequel we always assume that 130() ()vLR . (5) It must be emphasized that the nonnegative hypothesis of ()vis not necessary in present paper. In the following we denote the mCnorm by ||m, and 3,0||max maxsupmjjjm jvRffandf Df  Here  is the multi-index. This paper is organized as follows. We introduce a mild solution to the Cauchy problem (1) and (4) in Sec-tion 2. We prove local existence of the mild solution by the contracted mapping principle. In Section 3, we pro-pose the global existence of the mild solution. Our main tool is the interpolation inequalities. Finally, we study the 2,pWestimate of f in Section 4 and deduce the following asymptotic expression 0ktffAe . 2. The Local Existence In this section we shall study the local existence of the solution of the C auchy problem (1), (4). Definition 1. Given 0. We call f is the mild solution to the Cauchy problem (1) and (4), if 1, 30, ;1pfCWRp  and satisfies 330,0,, ,,, 0tRRvtftvGtvdGtsvQf fsdsdt   (6) where 23/21||,exp,044tvGGtv ttt The following is the local existence theorem. Theorem 1. Given 0. Let 1, 3()pWR, 1p and satisfy (5). Then there exists 0T such that the Cauchy problem (1) and (4) has a unique mild solution ,0ftT . In order to prove Theorem 1, let us recall a well-known result which is often called convolution property. Proposition 2 ([10,14]). For any 1p, if 13 3(), ()pfLRgL R then there exist constant 0C dependent on bonly, such that 1,ppLLLQgfCg f The proof of Theorem 1 Consider the following space 1, 30, ;pfCTW R T is determined. Defined the mapping:ff by 0,0,**,,0ttv tsvvtftvGvGQffvdst(7) where *v denotes the convolution in variable v. By 13LR and the definition of the mild solu-tion (6), we have 1110,tLLLfQf fds Making use of the Prop. 2 and Gronwall’s lemma, we obtain the estimate of 1Lf. In terms of (7), we denote ,0ftvt by 1I and 2I. Obviously, 1, 1,1ppWWI By Prop. 2 and Young’s inequality, noting that 11tsLG, one obtain 1120,pppttsLL LLLIGQffdsCfft  Here the nonnegative constant C depends on b only. In what following, we denote C for various non-negative constants independent of  unless special statements. On the other hand, 120,ppttsLLLIGQffds 111220ptLLCtsffds 11122pLLCfft. (8) Therefore, M. L. ZHENG Copyright © 2010 SciRes. AM 506 1, 1,122ppWWICttf This follows the mapping  is closed. Let 12,ff, by the bilinearity of Q   112 2112,, ,Qf fQffQf ff  1221 12122,Qfff fLffffLf Thus  112 212,,ppLLQf fQffCff  12112 2,,pLCtt QffQf f  1,1212pWCTTff  (9) So, one deduces that the mapping  is locally Lip-schitz continuous. By choosing 0T suitably, such that 0tT , the Cauchy problem (1) and (4) exists a unique mild solution. 3. The Global Existence of the Wild Solution In order to prove the global existence of the mild solution above, it suffices to show that 1, ,0pWft  First, let us recall the N dimensional Gagliar-do-Nirenberg’s inequality: let 1,qr, j, m are integer s and 0jm . Suppose that ,1jam, (1a if /mjNr is a nonnegative integer). Then there exists a constant C dependent on q, r, j, m, a, N such that for any uD ()NR, 1qpraaLLLjmDu CDuu where 111jmaapm rNq  Lemma 3. Given 0. Let 1, 3()pWR, 1p and satisfy (5). Then 0, 0,Tt T , the solution of Cauchy problem (1) and ( 4) satisfies ,ppLLfCT Proof From (1), we have 311sgnppptLRdfff fdvpdt 31sgn ,pRfffQffdv  3221pRpffdv  3112sgn ,pRffQffdvII  (10) Obviously 10I. By Prop. 2 and Holder’s inequality  311,,ppppLLRfQffdvQff f  1ppppLLLCf fCf  (11) By these estimates above and Gronwall’s lemma, we derive ,ppLLfCT This finishes the proof of the lemma. Lemma 4. Given 0. Let 1, 3pWR, 1p and satisfy (5). Then 0, 0,Tt T , the solution of Cauchy problem (1) and ( 4) satisfies 1,,pp WLfCT . Proof In terms of (6), for any 0t 0,* *,.tttsftvGGQ ffds Therefore, 0**,tttsfGGQffds  (12) By Young’s inequality 10,ppqrtttsLLLL LfGGQffds  (13) where 1111pqr, 1,,pqr. Next we estimate qtsLGand ,LrQf f re-spectively. Noting that 21322,2zzeez R. Thus 2( )ts tsvGGts  231 122 2zts ze   131112222222etsCts  (14) Therefore 111 112qqqtsts tsLLLGGGCts  (15) M. L. ZHENG Copyright © 2010 SciRes. AM 507By Gagliardo-Nirenberg’s inequality,   ,, ,rp paLL LQffQf fQf f  1,paLQf f (16) where 1111,0 13aaarp p  By the translation invariance of Q, it is easily to show that ,,,Qf fQf fQff So making us of Prop. 2 again 11,.pppLLL LLQf fCffff (17) By (12), it gives 111110,tttsLLLLLfGGQffds  110tLLCfds  (18) that is 11,LLfCT (19) Plugging (19 ) i nt o ( 17 ), gi ves ,pppLLLQf fCff . (20) By (20) and (16), 1,pppraaLLLLQf fCfff  1pppaaaLLLCfff (21) Combining (15), (21) and (13), we deduce 1120ppppptaaaLLLLLfCtsfffds (22) According to the Lemma 3, one has  112200 .pp pttaLLLfCts dsCtsfds  (23) By Gronwall type inequality we obtain the desired re-sult. Next using the basic theory of parabolic equation and the a priori estimate above, we have the following theo-rem. Theorem 5. Gi ven 0. Let 1, 3pWR, 1p and satisfy (5). Then for any 0T the Cauchy problem (1) and (4) exists a unique mild solution f such that 1, 330, ;0,pfCTWR CTR 4. 2,pW Estimate and 0C Approximation In this section we shall make 2,pWestimate on f and deduce the explicit estimate on the viscosity approxima-tion. Theorem 6. Given 0, 2p. Then for any 2, 3pWR satisfying (5) the mild solution f of the Cauchy problem (1) and (4) belongs to 2,30, ;pCWR. Proof By the equation of (1), one has  ,ffQfft. (24) Thus  311sgnppptLRdfff fdvpdt  31sgn ,pRfffQffdv  12II (25) Next we estimate 1Iand 2I respectively.  311sgn pRIff fdv 3221pRpf fdv (26) and  312sgn ,pRIff Qffdv   321,pRpf fQffdv  322221,ppRpf fQfffdv  (27) By Young’s inequality, for any 0, 32221pRIpf fdv 3221,4pRpQf ffdv   (28) Employing Young’s inequality again, the second term of the above formulation can be estimated by  33121,24ppRRpppQf fdvfdvpp   (29) Taking 4, and plugging (28), (29) and (26) into (25), it gives M. L. ZHENG Copyright © 2010 SciRes. AM 508 3223114pppLRpdfffdvpdt  33211 2,ppRRpppQf fdvfdvpp   (30) By (20) and Gr onwall’s lemma, and the Schauder the- ory, we conclude the desired result. Now, we consider whether the mild solution of the Cauchy problem (1) and (4) converges to the solution of (3) and (4) in 0C-norm as 0. The following theorem is our main result. Theorem 7. Let 22,3()pCWR and satisfying (5), 2p. For any 0Tand 0, set f is the mild solution of the viscosity equation 330,0,|ttffQffinTRfinR   (31) and f is the solution of 330,0,|ttfQffinT RfinR  (32) Then,  0,, ktftft Ae  (33) where A and k are constants independent of . Furthermore, for any 0 0,ft f (34) if 10minlog ,tTkA  Proof By the theorem abov e and the result of spatially homogenous Boltzmann equation, we know that 2, 3,, pftvfWR. Let wf f, then ,,wfQffQfft  Therefore,  ||sgn sgn,,.wwf wtQf fQff (35) Noting that 0,,Qf fQff 0,,Qff fQf ff  110LLCf fff (36) By the estimate on 1Lf in Section 2, we know the estimate (36) is uniform in . Therefore 0,,Qf fQffCw (37) Next, we estimate 2f. Indeed, by (6) for any multi- indices , 2, we have  300,RDfGt Dvd 300,,,tRGtsDQ ffsvdsd  12II (38) noting that 3,1RGtvdv, 12I. Making use of Leibniz form ul a, for 2 ||2,,DQ ffQD fDf (39) Thus 11,12220tLWICfffds . (40) Noting that 1110,tLLLfQffds So, 2220tfCfdsCT  (41) By Gronwall’s inequality we can deduce the bound of 2f and the bound is independent of . Together these estimate with (35) we have 0203,,wfQffQfft  0CCw (42) Therefore, 0ktwt Ae This finishes the pr o of o f T h eor em 7. 5. Acknowledgements I am grateful to the reviewer for the meaningful sugges-tions. 6. References  H. L. Li and A. Matsumura, “Behaviour of the Fokker- Planck-Boltzmann Equation near a Maxwellian,” Archive for Rational Mechanics and Analysis, Vol. 189, No. 1, 2008, pp. 1-44.  M. Zheng and X. P. Yang, “Viscosity Analysis on the M. 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