华中科技大学 硕士学位论文 耗散Boltzmann方程的渐近性 姓名：肖志涛 申请学位级别：硕士 专业：应用数学 指导教师：张显文 20070528 ? Boltzmann? ? ? ? ? ?Boltzmann?ds?Boltzmann? ?pseudo-Maxwellian 11?Boltzmann ?d2? ? t ?Sobolev? ?Sobolev? ?L1?Sobolev? ?Boltzmann?ds? I Abstract Boltzmann equation is the practically mathematical model to describe the time and space evolution, which is an integro-diff erential equation that probability density satisfi es,and which provides a mathematical model for the statistical evolution of the moderaterly rarefi ed gas .The way is statistical average of microscopic state of gases that we have observated, and obtain macroscopic properity of the gases. First,we introduce Boltzmann equation and dsdistance simply,then our work fo- cus on dissipative Boltzmann equation in the pseudo-Maxwellian approximation11 ,we quanitfy the long-time behavior of a system of inelatic interactions,by mean of contractivity of a suitable distance d2in the set of probability measures.Boundedness of the moments,existence,uniqueness and regularity of a steay state are derived from this basic properties.The solutions of equation are proved to converge exponentially as t to this steady state in this distance.Futhermore,we prove a uniform bound in time on Sobolev norms of the solutions,provided the initial datum has a fi nite norm in the corresponding Sobolev space.Finally,these results are then combined, using inter- polation inequalities to obtain exponential convergence to the steady state in strong L1-norm,as well as various Sobolev norms. Key words: Boltzmann equationdsdistanceSteady stateExponential conver- gence II 独创性声明独创性声明 本人声明所呈交的学位论文是我个人在导师指导下进行的研究工作及取得 的研究成果。尽我所知，除文中已经标明引用的内容外，本论文不包含任何其他 个人或集体已经发表或撰写过的研究成果。对本文的研究做出贡献的个人和集 体，均已在文中以明确方式标明。本人完全意识到，本声明的法律结果由本人承 担。 学位论文作者签名： 日期： 年 月 日 学位论文版权使用授权书学位论文版权使用授权书 本学位论文作者完全了解学校有关保留、使用学位论文的规定，即：学校有 权保留并向国家有关部门或机构送交论文的复印件和电子版， 允许论文被查阅和 借阅。 本人授权华中科技大学可以将本学位论文的全部或部分内容编入有关数据 库进行检索，可以采用影印、缩印或扫描等复制手段保存和汇编本学位论文。 保密 ，在_年解密后适用本授权书。 本论文属于 不保密。 （请在以上方框内打“” ） 学位论文作者签名： 指导教师签名： 日期： 年 月 日 日期： 年 月 日 1? 1.1?Boltzmann? Boltzmann?L. Boltzmann(18441906)?1872?1? ? ?Boltzmann ? ? ? ? ?Boltzmann?E.Steban?Pethanme?2? ? ?(2 34). ?Boltzmann? f t + v xf = Q(f,f), f(v,0) = f0(v). (1.1) ?t (0,+),?x R3,?v R3. f(v,x,t)? ?x?v?t?Q? ? Q(f,f)(v) = Z R3 Z S2(Jf(v)f() f(v)f()B(|u|,)dnd. ?u = v ? ?J?(v,)?(v,)?Jacobi? (v,)?(v,)? ? v= v 1+ 2 (u n)n, = + 1+ 2 (u n)n. (1.2) ? ? (v ) n = (v,) n. 1 ?n?0 0?0= (2+ 3)/4 0? ?ds? ?ds? ? ? D= ? F P2(Rd); Z vidv = 0, Z |v|2dF(v) = d ? . ? ?ds?s = 2?ds? ? Prokhorovs?(F,G)? 0?U Rd? U= v Rd;d(v,U) 0? P M 2+(R d) = ? F D; Z |v|2+dF(v) M ? .(2.12) P M (Rd) = ? F D; Z (|v|2)dF(v) M ? .(2.13) ?r ?(r)/r . ?7,9. ?2.2?Fn D,F P2(Rd)? (i) Fn F,F D; (ii) (Fn,F) 0; (iii) (Fn,F) 0,F D; (iv) m 1,kFn Fk m 0,F D; (v) d2(Fn,F) 0. ?m1?m2? S P 2(Rd) ? ? 0? 0? ? F,G S? m1(F,G) m2(F,G) , 8 m2(F,G) m1(F,G) . ?2.3?M 0,? ?m 1,k k m ?d2?P M ? ? ?X0,Y0?X1,Y1? ?Fi(Gi)? Xi(Yi)? 0 0,f? d(t) dt = 1 2 4 B(t)1+ 2+ 2Fp(t), (3.1) ? t Z R3 f(v)vivjdv = (1 + )(3 ) 8 B 2 f (t) Z R3 f(v)vivjdv +2ij ?1 + 4 ?2 B (f(t)1+ 2 + 2ijFp f(t), (3.2) ?t 0? = 8F B(1 2) !2 +22p .(3.3) ?(2.1)?vivj? t Z R3 f(v)vivjdv= B 4 2 f (t) Z R3 Z R3 Z S2 f(v)f()v iv j vivjdndvd +Fp f(t) Z R3 vivjvfdv, ? v= 3 4 v + 1 + 4 + 1 + 4 |v |n, ? t Z R3 f(v)vivjdv= B 4 2 f (t) Z R3 Z R3 Z S2 f(v)f()(3 4 )2 1vivj +(1 + 4 )2(ij+ |v |2ninj 11 +|v |(inj+ jni) + (3 )(1 + ) 16 ) (vij+ ivj+ |v |(vinj+ vjni)dndvd +2ijFp f(t). ?(3.1),(3.2)?(3.1)? ?t 0,f? ?3.2?p = 1?(3.1)? ?z = e2Ft,? dz dt = 1 2 4 BeFtz1+ 2, ? z(t) = “ z 2 0 + 1 2 8 B F (eFt 1) #2 , ? (t) = e2Ft “ 2 0 + 1 2 8 B F (eFt 1) #2 .(3.4) ?