Content-Type: multipart/mixed; boundary="-------------9904231407150" This is a multi-part message in MIME format. ---------------9904231407150 Content-Type: text/plain; name="99-128.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-128.keywords" Boltzmann, granular media, kinetic equations ---------------9904231407150 Content-Type: application/postscript; name="boltzgran.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="boltzgran.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: 4inelasbol.dvi %%Pages: 25 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSCommandLine: dvips 4inelasbol.dvi -o boltzgran.ps %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 1999.04.16:1526 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N 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y(b)m(y)g(one)g(of)f(the)h (authors)g([6,)g(7)o(].)221 3598 y(Here,)50 b(the)d(main)d (di\013erence)j(is)e(the)i(absence)g(of)f(conserv)-5 b(ation)45 b(of)h(energy)h(at)e(the)i(lev)m(el)e(of)75 3718 y(the)i(collision)c(mec)m(hanism,)50 b(and)c(th)m(us,)52 b(at)46 b(the)h(lev)m(el)f(of)g(the)h(Boltzmann)e(equation.)85 b(In)47 b(fact,)75 3839 y(a)d(Maxw)m(ellian)e(distribution)g(can)i(not) g(b)s(e)g(a)g(solution)e(of)h(the)h(inelastic)f(Boltzmann)f(equation.) 75 3959 y(Ho)m(w)m(ev)m(er,)34 b(the)e(Dirac)d(delta)i(distribution)e (is)i(ob)m(viously)g(in)f(the)i(k)m(ernel)g(of)f(the)g(collision)e(op)s (erator.)75 4079 y(Therefore,)53 b(with)48 b(the)g(F)-8 b(ourier)47 b(metho)s(d,)k(w)m(e)e(can)f(describ)s(e)h(the)f(sp)s (ectrum)h(of)e(the)i(linearized)75 4200 y(homogeneous)33 b(equation)f(around)h(the)g(Dirac)e(delta)h(distribution.)221 4320 y(In)e(addition,)e(w)m(e)i(clarify)e(the)h(existence)i(of)e (self-similar)c(solutions)j(for)g(this)h(mo)s(del.)41 b(This)29 b(t)m(yp)s(e)75 4441 y(of)24 b(solutions)g(ha)m(v)m(e)i(b)s (een)g(considered)f(b)s(efore)g(for)g(the)g(inelastic)e(hard-sphere)j (Boltzmann)d(equation)75 4561 y([8,)35 b(14)o(].)50 b(They)36 b(ha)m(v)m(e)g(b)s(een)f(called)e(homogeneous)i(co)s(oling)d(states.)51 b(The)35 b(nearly)g(elastic)e(case)j(w)m(as)75 4681 y(studied)k(in)g ([14)o(].)66 b(In)41 b(the)f(presen)m(t)i(mo)s(del)c(w)m(e)j(state)g (and)f(pro)m(v)m(e)h(the)f(precise)h(conditions)e(under)75 4802 y(whic)m(h)33 b(these)h(self-similar)29 b(solutions)i(exist,)j (and)e(then)i(describ)s(e)f(the)g(large-time)d(b)s(eha)m(vior)i(of)h (the)75 4922 y(system.)43 b(These)31 b(results)e(are)g(accomplished)f (using)h(eigenfunction)f(expansion)h(of)g(the)g(solutions)f(in)75 5042 y(the)h(F)-8 b(ourier)28 b(transform)g(equation.)42 b(Finally)-8 b(,)27 b(w)m(e)j(study)g(the)f(momen)m(t)f(equations)h (for)f(this)h(system.)1926 5356 y(2)p eop %%Page: 3 3 3 2 bop 75 107 a Fn(W)-8 b(e)33 b(analyze)f(t)m(w)m(o)h(cases)h(under)g (whic)m(h)f(the)f(self-similar)d(solutions)i(are)i(asymptotically)d (relev)-5 b(an)m(t.)75 227 y(These)27 b(cases)g(are:)40 b(a\))25 b(nearly)g(elastic)g(particles)f(with)h(constan)m(t)h(or)f(v) -5 b(ariable)24 b(restitution)g(co)s(e\016cien)m(t)75 348 y(and)36 b(b\))g(small)e(temp)s(erature.)54 b(In)36 b(b)s(oth)g(cases,)i(the)f(self-similar)32 b(solution)i(is)i(near)g(a)g (Maxw)m(ellian)75 468 y(distribution.)52 b(Therefore,)38 b(w)m(e)f(can)f(mak)m(e)g(a)g(formal)e(transition)g(to)i(h)m(ydro)s (dynamics)g(in)g(the)g(spa-)75 588 y(tially)26 b(inhomogeneous)i (simpli\014ed)f(equation.)42 b(The)30 b(resulting)e(h)m(ydro)s(dynamic) h(description,)g(v)-5 b(alid)75 709 y(for)26 b(a\))h(and)f(b\),)j(is)d (the)h(usual)f(Euler)h(equations)g(for)f(gas-dynamics)g(with)h(a)f (dissipativ)m(e)g(term)h(in)f(the)75 829 y(temp)s(erature)g(equation)g (due)h(to)e(the)i(inelasticit)m(y)-8 b(.)39 b(Finally)-8 b(,)25 b(w)m(e)i(study)g(the)g(stabilit)m(y/instabilit)m(y)21 b(of)75 949 y(the)30 b(homogeneous)g(co)s(oling)e(state)i(in)g(the)g(h) m(ydro)s(dynamic)g(description.)42 b(In)30 b(fact,)h(in)e(the)h (constan)m(t)75 1070 y(restitution)41 b(co)s(e\016cien)m(t)i(case)g (the)f(homogeneous)g(co)s(oling)e(state)j(is)e(unstable.)73 b(In)42 b(the)h(v)-5 b(ariable)75 1190 y(case,)48 b(w)m(e)e(\014nd)e (that)g(the)h(system)g(is)f(linearly)e(stable)i(when)i(the)e (restitution)g(co)s(e\016cien)m(t)h(has)f(a)75 1311 y(deca)m(y)34 b(condition)d(of)h(the)h(same)g(order)f(of)g(the)h(collision)d (frequency)-8 b(.)75 1637 y Fo(2)161 b(Pseudo-Maxw)l(ellian)54 b(mo)t(del)g(for)g(dissipativ)l(e)317 1819 y(hard)g(spheres)75 2038 y Fn(In)39 b(this)g(section)g(w)m(e)h(in)m(tro)s(duce)f(the)h (basic)f(mo)s(del)e(w)m(e)j(will)d(treat)i(in)f(this)h(pap)s(er.)63 b(Assume)40 b(that)75 2159 y(w)m(e)i(are)e(studying)h(the)g(dynamics)f (of)g Fk(N)51 b Fn(p)s(erfect)41 b(spheres)i(of)d(diameter)f Fk(\033)45 b(>)d Fn(0)e(suc)m(h)i(that)e(they)75 2279 y(p)s(erform)k(inelastic)g(collisions.)79 b(If)45 b(\()p Fk(x;)17 b(v)t Fn(\))45 b(and)g(\()p Fk(x)31 b Fj(\000)g Fk(\033)t(n;)17 b(w)s Fn(\),)48 b(are)d(the)h(states)g(of)f(t)m(w)m(o)h (particles)75 2399 y(b)s(efore)40 b(collision,)e(where)j Fk(n)f Fj(2)g Fk(S)1344 2363 y Fi(2)1423 2399 y Fn(is)f(the)h(unit)f(v) m(ector)i(along)d(the)i(cen)m(ter)h(of)f(b)s(oth)f(spheres,)44 b(the)75 2520 y(p)s(ostcollisional)32 b(v)m(elo)s(cities)i(are)i(found) g(assuming)e(that)i(the)g(total)e(momen)m(tum)g(is)h(preserv)m(ed)k (but)75 2640 y(w)m(e)34 b(lo)s(ose)d(some)i(part)f(of)g(the)h(normal)e (relativ)m(e)h(v)m(elo)s(cit)m(y)-8 b(,)32 b(that)g(is,)1311 2814 y Fk(n)22 b Fj(\001)g Fn(\()p Fk(v)1530 2773 y Fh(0)1575 2814 y Fj(\000)h Fk(w)1748 2773 y Fh(0)1770 2814 y Fn(\))28 b(=)g Fj(\000)p Fk(e)p Fn(\(\()p Fk(v)e Fj(\000)d Fk(w)s Fn(\))e Fj(\001)h Fk(n)p Fn(\))75 2987 y(where)28 b(0)g Fk(<)f(e)h Fj(\024)g Fn(1)f(is)g(called)f(the)h(restitution)f(co)s (e\016cien)m(t.)42 b(Using)27 b(b)s(oth)g(information)d(w)m(e)k (construct)75 3108 y(the)33 b(p)s(ostcollisional)28 b(v)m(elo)s(cities) k(as)1499 3329 y Fk(v)1550 3288 y Fh(0)1656 3329 y Fn(=)1825 3262 y(1)p 1825 3306 49 4 v 1825 3397 a(2)1883 3329 y(\()p Fk(v)26 b Fn(+)c Fk(w)s Fn(\))g(+)2333 3262 y Fk(u)2389 3226 y Fh(0)p 2333 3306 79 4 v 2348 3397 a Fn(2)1477 3571 y Fk(w)1550 3530 y Fh(0)1656 3571 y Fn(=)1825 3504 y(1)p 1825 3548 49 4 v 1825 3639 a(2)1883 3571 y(\()p Fk(v)k Fn(+)c Fk(w)s Fn(\))g Fj(\000)2334 3504 y Fk(u)2390 3468 y Fh(0)p 2334 3548 79 4 v 2349 3639 a Fn(2)75 3776 y(where)34 b Fk(u)413 3740 y Fh(0)464 3776 y Fn(=)29 b Fk(u)21 b Fj(\000)i Fn(\(1)f(+)h Fk(e)p Fn(\)\()p Fk(u)f Fj(\001)g Fk(n)p Fn(\))p Fk(n)p Fn(,)33 b Fk(u)28 b Fn(=)g Fk(v)e Fj(\000)d Fk(w)36 b Fn(and)d Fk(u)2130 3740 y Fh(0)2181 3776 y Fn(=)28 b Fk(v)2336 3740 y Fh(0)2382 3776 y Fj(\000)22 b Fk(w)2554 3740 y Fh(0)2577 3776 y Fn(.)45 b(Let)33 b(us)g(denote)h(b)m(y)g Fk(v)3451 3740 y Fh(\003)3523 3776 y Fn(and)f Fk(w)3786 3740 y Fh(\003)75 3896 y Fn(the)e(precollisional)c(v)m(elo)s(cities)j(corresp)s(onding)h (to)g Fk(v)j Fn(and)d Fk(w)s Fn(.)42 b(Let)31 b(us)h(remark)e(that)h (the)g(co)s(e\016cien)m(t)75 4016 y(of)h(restitution)g(can)g(b)s(e)h(a) f(function)h(of)f(the)h(normal)d(relativ)m(e)i(v)m(elo)s(cit)m(y)g Fj(j)p Fn(\()p Fk(v)26 b Fj(\000)d Fk(w)s Fn(\))e Fj(\001)h Fk(n)p Fj(j)p Fn(.)221 4137 y(F)-8 b(ollo)m(wing)29 b(the)j(standard)g (pro)s(cedures)h(of)e(kinetic)h(theory)g([9];)g(w)m(e)g(deduce)i(that)d (the)h(equation)75 4257 y(for)g(the)h(probabilit)m(y)e(densit)m(y)i(of) f(particles)g(will)e(b)s(e)j(giv)m(en)g(b)m(y)1140 4407 y Fk(@)5 b(P)p 1140 4451 134 4 v 1149 4543 a(@)g(f)1306 4475 y Fn(+)22 b(\()p Fk(v)k Fj(\001)21 b(r)1647 4490 y Fg(x)1691 4475 y Fn(\))p Fk(P)41 b Fn(=)28 b(\()p Fk(N)k Fj(\000)23 b Fn(1\))p Fk(\033)2331 4434 y Fi(2)2370 4475 y Fk(Q)2447 4490 y Fg(B)2508 4475 y Fn(\()p Fk(P)s(;)17 b(P)d Fn(\))75 4693 y(where)589 4886 y Fk(Q)666 4901 y Fg(B)727 4886 y Fn(\()p Fk(P)s(;)j(P)d Fn(\))81 b(=)1230 4769 y Ff(Z)1276 4958 y Fe(I)-10 b(R)1333 4939 y(3)1388 4769 y Ff(Z)1434 4958 y Fg(S)1481 4939 y Fe(2)1519 4886 y Fn(\(\()p Fk(v)26 b Fj(\000)d Fk(w)s Fn(\))e Fj(\001)h Fk(n)p Fn(\))2046 4901 y Fi(+)2138 4790 y Ff(h)2187 4819 y Fn(1)p 2187 4863 49 4 v 2189 4955 a Fk(e)2246 4886 y(J)9 b(P)2386 4845 y Fi(\(2\))2480 4886 y Fn(\()p Fk(t;)17 b(v)t(;)g(v)2743 4845 y Fh(\003)2781 4886 y Fk(;)g(x)22 b Fn(+)g Fk(\033)t(n;)17 b(w)3234 4845 y Fh(\003)3273 4886 y Fn(\))1620 5091 y Fk(P)1697 5049 y Fi(\(2\))1791 5091 y Fn(\()p Fk(t;)g(x;)g(v)t(;)g(x)22 b Fj(\000)h Fk(\033)t(n;)17 b(w)s Fn(\))2551 4994 y Ff(i)2606 5091 y Fk(dw)i(dn)1926 5356 y Fn(3)p eop %%Page: 4 4 4 3 bop 75 107 a Fn(Here,)41 b Fk(J)49 b Fn(is)38 b(the)i(Jacobian)e (of)g(the)i(transformation)d(\()p Fk(v)t(;)17 b(w)s Fn(\))37 b(in)m(to)i(\()p Fk(v)2674 71 y Fh(\003)2713 107 y Fk(;)17 b(w)2830 71 y Fh(\003)2868 107 y Fn(\))39 b(and)g Fk(P)3218 71 y Fi(\(2\))3351 107 y Fn(is)f(the)i(join)m(t)75 227 y(probabilit)m(y)31 b(densit)m(y)i(of)f(t)m(w)m(o)h(particles.)221 348 y(As)d(w)m(e)g(said)f(in)g(the)g(in)m(tro)s(duction)f(w)m(e)j(are)e (going)f(to)h(study)h(the)g(homogeneous)f(case)h(of)f(simpli-)75 468 y(\014ed)i(Boltzmann)f(and)h(Boltzmann-Ensk)m(og)f(equations)h(for) f(dissipativ)m(e)h(hard)f(spheres)j(dynamics.)75 588 y(Therefore,)27 b(w)m(e)d(ha)m(v)m(e)h(to)f(explain)e(the)j(relation)c (b)s(et)m(w)m(een)26 b Fk(P)2280 552 y Fi(\(2\))2397 588 y Fn(and)e Fk(P)14 b Fn(.)40 b(In)24 b(the)g(classical)e(Boltzmann) 75 709 y(equation)32 b(this)h(relation)d(is)i(giv)m(en)h(b)m(y)h(the)f (molecular)d(c)m(haos)j(h)m(yp)s(othesis)h([17],)e(that)h(is,)1151 902 y Fk(P)1228 861 y Fi(\(2\))1322 902 y Fn(\()p Fk(t;)17 b(x;)g(v)t(;)g(y)t(;)g(w)s Fn(\))25 b(=)j Fk(P)14 b Fn(\()p Fk(t;)j(x;)g(v)t Fn(\))p Fk(P)d Fn(\()p Fk(t;)j(y)t(;)g(w)s Fn(\))872 b(\(2.1\))75 1095 y(for)31 b(an)m(y)-8 b(,)33 b Fk(t)28 b Fj(\025)g Fn(0,)k Fk(x;)17 b(v)t(;)g(y)t(;)g(w)29 b Fj(2)f Fn(I)-20 b(R)1272 1053 y Fi(3)1312 1095 y Fn(.)43 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Fn(where)284 2221 y Fk(H)8 b Fn(\()p Fk(k)s(;)17 b(v)t(;)g(w)s Fn(\))25 b(=)j Fj(\000)p Fk(iv)1022 2075 y Ff( )1098 2153 y Fk(k)p 1098 2197 55 4 v 1101 2289 a Fn(2)1184 2221 y(+)1292 2153 y(1)22 b Fj(\000)g Fk(e)p 1292 2197 216 4 v 1375 2289 a Fn(4)1518 2221 y Fk(k)j Fn(+)1702 2153 y(1)d(+)g Fk(e)p 1702 2197 215 4 v 1784 2289 a Fn(4)1926 2221 y Fj(j)p Fk(k)s Fj(j)p Fk(n)2094 2075 y Ff(!)2181 2221 y Fn(+)g Fk(iw)2402 2075 y Ff( )2467 2221 y Fj(\000)2554 2153 y Fk(k)p 2554 2197 55 4 v 2557 2289 a Fn(2)2641 2221 y(+)2749 2153 y(1)g Fj(\000)h Fk(e)p 2749 2197 216 4 v 2832 2289 a Fn(4)2975 2221 y Fk(k)i Fn(+)3159 2153 y(1)d(+)g Fk(e)p 3159 2197 215 4 v 3242 2289 a Fn(4)3383 2221 y Fj(j)p Fk(k)s Fj(j)p Fk(n)3551 2075 y Ff(!)75 2497 y Fn(Therefore,)536 2756 y Fj(L)605 2771 y Fi(+)664 2756 y Fn(\()p Fk(';)17 b(')p Fn(\))27 b(=)1082 2689 y(1)p 1052 2733 108 4 v 1052 2825 a(4)p Fk(\031)1187 2639 y Ff(Z)1233 2828 y Fg(S)1280 2809 y Fe(2)1335 2756 y Fk(')1416 2635 y Ff(\022)1486 2689 y Fn(3)22 b Fj(\000)h Fk(e)p 1486 2733 216 4 v 1570 2825 a Fn(4)1712 2756 y Fk(k)i Fn(+)1896 2689 y(1)d(+)g Fk(e)p 1896 2733 215 4 v 1979 2825 a Fn(4)2120 2756 y Fj(j)p Fk(k)s Fj(j)p Fk(n)2288 2635 y Ff(\023)2366 2756 y Fk(')2447 2635 y Ff(\022)2517 2689 y Fn(1)g(+)g Fk(e)p 2517 2733 V 2600 2825 a Fn(4)2741 2756 y(\()p Fk(k)k Fj(\000)c(j)p Fk(k)s Fj(j)p Fk(n)p Fn(\))3161 2635 y Ff(\023)3255 2756 y Fk(dn)75 3008 y Fn(Hence,)34 b(w)m(e)g(ha)m(v)m(e)g(pro)m(v)m(ed)g(the)f(follo)m (wing)75 3236 y Fd(Lemma)k(3.1)49 b Fc(The)34 b(F)-7 b(ourier)34 b(tr)-5 b(ansform)35 b(e)-5 b(quation)34 b(c)-5 b(orr)g(esp)g(onding)33 b(to)j Fn(\(3.2\))e Fc(is)h(given)f(by) 897 3439 y Fk(@)5 b(')p 897 3483 121 4 v 911 3574 a(@)g(t)1056 3506 y Fn(=)1199 3439 y(1)p 1169 3483 108 4 v 1169 3574 a(4)p Fk(\031)1304 3389 y Ff(Z)1350 3577 y Fg(S)1397 3558 y Fe(2)1452 3506 y Fj(f)p Fk(')p Fn(\()p Fk(t;)17 b(k)1734 3521 y Fi(+)1792 3506 y Fn(\))p Fk(')p Fn(\()p Fk(t;)g(k)2062 3521 y Fh(\000)2121 3506 y Fn(\))22 b Fj(\000)g Fk(')p 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4446 y Fn(exp)q(\()p Fj(\000)p Fk(i)p Fn(\()p Fk(k)25 b Fj(\001)d Fk(v)t Fn(\)\))p Fk(f)2348 4461 y Fi(0)2387 4446 y Fn(\()p Fk(v)t Fn(\))17 b Fk(dv)75 4713 y Fd(Remark)37 b(3.2)49 b Fn(Assuming)c(that)g Fk(f)11 b Fn(\()p Fk(v)t Fn(\))49 b Fj(\024)h Fk(A)17 b Fn(exp)q(\()p Fj(\000)p Fk(\013)q Fj(j)p Fk(v)t Fj(j)2263 4676 y Fi(2)2301 4713 y Fn(\),)49 b(w)m(e)d(can)g(use)g(the)g(complex)f (F)-8 b(ourier)75 4833 y(v)j(ariable)47 b Fk(k)57 b Fn(=)e Fk(i)732 4807 y Fn(~)729 4833 y Fk(k)d Fn(\(t)m(w)m(o-side)c(Laplace)g (transform\))g(and)g(obtain)f(the)i(same)f(equation)h(for)f(the)75 4953 y(p)s(ositiv)m(e)32 b(function)48 b(~)-65 b Fk(')p Fn(\()923 4927 y(~)920 4953 y Fk(k)s Fn(\))28 b(=)f Fk(')p Fn(\()p Fk(i)1281 4927 y Fn(~)1278 4953 y Fk(k)s Fn(\).)1926 5356 y(8)p eop %%Page: 9 9 9 8 bop 221 107 a Fn(W)-8 b(e)28 b(see)g(from)e(Lemma)g(3.1)h(that)g (our)g(simpli\014cation)d(of)j(the)h(mo)s(del)d(leads)i(to)g(a)g(v)m (ery)i(simpli\014ed)75 227 y(homogeneous)34 b(equation)f(that)h(in)f(F) -8 b(ourier)32 b(transform)h(can)h(b)s(e)g(completely)f(describ)s(ed.) 48 b(The)34 b(rest)75 348 y(of)e(this)g(section)h(is)f(dev)m(oted)i(to) e(review)h(v)m(ery)h(quic)m(kly)f(the)g(existence)h(and)f(uniqueness)h (theory)f(for)75 468 y(the)g(F)-8 b(ourier)31 b(transform)h(equation)g (\(3.4\).)221 588 y(The)44 b(existence)g(theorems)f(for)f(the)h (classical)e(elastic)g(case)j(\()p Fk(e)g Fn(=)h(1\))d(w)m(ere)i(pro)m (v)m(ed)g(b)m(y)f(Mor-)75 709 y(genstern,)e(Wild)d(and)g(others)i (\(see)f([6])g(and)g(references)i(therein\).)61 b(These)41 b(results)e(can)g(b)s(e)g(easily)75 829 y(generalized)32 b(to)g(the)h(inelastic)e(case.)45 b(Consider)33 b(a)f(c)m(hange)h(in)f (time)f(v)-5 b(ariable)31 b(of)h(the)h(t)m(yp)s(e)948 1038 y Fk(\034)39 b Fn(=)28 b(1)22 b Fj(\000)g Fn(exp)q(\()p Fj(\000)p Fk(t)p Fn(\))33 b Fk(;)212 b(')p Fn(\()p Fk(r)m(;)17 b(k)s Fn(\))28 b(=)f(exp)q(\()p Fj(\000)p Fk(t)p Fn(\)\010\()p Fk(\034)6 b(;)17 b(k)s Fn(\))75 1247 y(then)33 b(\(3.4\))f(leads)h(to) 1627 1349 y Fk(@)5 b Fn(\010)p 1627 1394 128 4 v 1635 1485 a Fk(@)g(\034)1792 1417 y Fn(=)27 b Fj(L)1964 1432 y Fi(+)2023 1417 y Fn(\(\010)p Fk(;)17 b Fn(\010\))75 1617 y(with)32 b(\010\()p Fk(k)s(;)17 b Fn(0\))28 b(=)f Fk(')785 1632 y Fi(0)825 1617 y Fn(\()p Fk(k)s Fn(\).)43 b(W)-8 b(e)33 b(lo)s(ok)e(for)h(the)h(solution)e(in)h(p)s(o)m(w)m(er)i (series)f(expansion)g(of)f(the)h(t)m(yp)s(e)1490 1884 y(\010\()p Fk(\034)6 b(;)17 b(k)s Fn(\))27 b(=)1944 1776 y Fh(1)1920 1801 y Ff(X)1913 1982 y Fg(n)p Fi(=0)2063 1884 y Fn(\010)2133 1899 y Fg(n)2180 1884 y Fn(\()p Fk(k)s Fn(\))p Fk(\034)2363 1843 y Fg(n)75 2156 y Fn(and)33 b(w)m(e)g(obtain)f(a)g(simple)f(recurren)m(t)j(sequence)h(of)d (equalities)1077 2365 y(\010)1147 2380 y Fi(0)1270 2365 y Fn(=)83 b Fk(')1493 2380 y Fi(0)980 2563 y Fn(\010)1050 2578 y Fg(n)p Fi(+1)1270 2563 y Fn(=)1528 2495 y(1)p 1439 2540 228 4 v 1439 2631 a Fk(n)22 b Fn(+)g(1)1736 2455 y Fg(n)1697 2480 y Ff(X)1693 2664 y Fg(k)r Fi(=0)1838 2563 y Fj(L)1907 2578 y Fi(+)1966 2563 y Fn(\(\010)2074 2578 y Fg(k)2117 2563 y Fk(;)17 b Fn(\010)2231 2578 y Fg(n)p Fh(\000)p Fg(k)2371 2563 y Fn(\))33 b Fk(;)212 b(n)28 b Fj(\025)g Fn(0)703 b(\(3.5\))75 2844 y(Noting)32 b(that)h Fj(j)p Fk(')702 2859 y Fi(0)741 2844 y Fj(j)c(\024)g Fn(1,)k(w)m(e)i(obtain)d Fj(j)p Fn(\010)1560 2859 y Fg(n)1607 2844 y Fj(j)c(\024)i Fn(1)j(for)f(an)m(y)i Fk(n)29 b Fj(\025)h Fn(0.)45 b(Then)34 b(the)g(series)g(\(3.5\))f(con)m(v)m (erges)75 2965 y(uniformly)38 b(on)i Fk(\034)51 b Fj(2)41 b Fn([0)p Fk(;)17 b Fn(1\).)65 b(Using)39 b(the)i(same)e(ideas)h(as)g (in)g([6)o(])g(section)g(13,)i(w)m(e)f(can)f(deduce)i(the)75 3085 y(follo)m(wing)30 b(result.)221 3205 y(Let)e(us)h(de\014ne)f Fk(f)11 b Fn(\()p Fk(t;)17 b(v)t Fn(\))27 b(to)h(b)s(e)g(a)f(solution)f (of)h(\(3.2\))h(if)e(its)h(F)-8 b(ourier)27 b(transform)f(is)i(a)f(c)m (haracteristic)75 3326 y(function)32 b(for)g(an)m(y)h Fk(t)28 b Fj(\025)g Fn(0)33 b(and)f(solv)m(es)i(\(3.4\).)75 3541 y Fd(Prop)s(osition)i(3.3)49 b Fc(Pr)-5 b(oblem)34 b Fn(\(3.4\))g Fc(with)h Fk(f)1749 3556 y Fi(0)1816 3541 y Fj(2)28 b Fk(L)1976 3505 y Fi(1)2016 3541 y Fn(\(I)-20 b(R)2141 3500 y Fi(3)2180 3541 y Fn(\))637 3780 y Fk(f)685 3795 y Fi(0)752 3780 y Fj(\025)29 b Fn(0)34 b Fk(;)1184 3663 y Ff(Z)1230 3852 y Fe(I)-10 b(R)1286 3833 y(3)1342 3780 y Fk(f)1390 3795 y Fi(0)1429 3780 y Fn(\()p Fk(v)t Fn(\))17 b Fk(dv)31 b Fn(=)c(1)35 b Fk(;)2132 3663 y Ff(Z)2178 3852 y Fe(I)-10 b(R)2234 3833 y(3)2289 3780 y Fj(j)p Fk(v)t Fj(j)2396 3739 y Fi(2)2435 3780 y Fk(f)2483 3795 y Fi(0)2522 3780 y Fn(\()p Fk(v)t Fn(\))17 b Fk(dv)31 b Fn(=)c(3)p Fk(\022)2992 3795 y Fi(0)3060 3780 y Fk(<)g Fj(1)75 4023 y Fc(has)k(a)h(unique)f(classic)-5 b(al)31 b(solution)g Fk(')p Fn(\()p Fk(t;)17 b(k)s Fn(\))32 b Fc(for)f Fk(t)d(>)g Fn(0)p Fc(.)43 b(Mor)-5 b(e)g(over,)32 b(ther)-5 b(e)32 b(exists)f(a)h(unique)f(function)75 4144 y Fk(f)11 b Fn(\()p Fk(t;)17 b(v)t Fn(\))27 b Fj(\025)h Fn(0)35 b Fc(such)f(that)554 4383 y Fk(f)39 b Fj(2)28 b Fk(L)801 4342 y Fh(1)876 4383 y Fn(\(0)p Fk(;)17 b Fj(1)p Fn(;)g Fk(L)1217 4342 y Fi(1)1255 4383 y Fn(\(I)-20 b(R)1380 4341 y Fi(3)1420 4383 y Fn(\)\))34 b Fk(;)216 b(')p Fn(\()p Fk(t;)17 b(v)t Fn(\))27 b(=)2174 4266 y Ff(Z)2220 4454 y Fe(I)-10 b(R)2276 4435 y(3)2332 4383 y Fk(f)11 b Fn(\()p Fk(t;)17 b(v)t Fn(\))g(exp)o(\()p Fj(\000)p Fk(i)p Fn(\()p Fk(k)26 b Fj(\001)c Fk(v)t Fn(\)\))17 b Fk(dv)t Fn(;)75 4626 y Fc(the)35 b(function)f Fk(f)11 b Fn(\()p Fk(t;)17 b(v)t Fn(\))35 b Fc(de\014nes)e(ther)-5 b(efor)g(e)35 b(a)g(unique)f(solution)h(of)g Fn(\(3.2\))f Fc(satisfying)g(the)h(initial)g(c)-5 b(on-)75 4746 y(dition)34 b Fk(f)11 b Fn(\(0)p Fk(;)17 b(v)t Fn(\))27 b(=)g Fk(f)813 4761 y Fi(0)853 4746 y Fn(\()p Fk(v)t Fn(\))p Fc(.)75 4962 y Fd(Remark)37 b(3.4)49 b Fn(The)37 b(uniqueness)g(is)e(related)h (to)f(the)h(explicit)e(form)h(that)g(w)m(e)i(ha)m(v)m(e)g(for)e Fk(')p Fn(\()p Fk(t;)17 b(k)s Fn(\))36 b(in)75 5082 y(terms)d(of)f(a)g (series.)44 b(More)33 b(details)e(can)i(b)s(e)g(found)g(in)f(section)g (13)g(of)g([6].)1926 5356 y(9)p eop %%Page: 10 10 10 9 bop 75 107 a Fo(4)161 b(Linearized)54 b(equation)75 326 y Fn(First)32 b(of)g(all,)e(let)i(us)h(write)g Fj(L)1171 341 y Fi(+)1229 326 y Fn(\()p Fk(';)17 b(')p Fn(\))32 b(in)g(a)g(di\013eren)m(t)h(w)m(a)m(y)-8 b(.)45 b(Equation)32 b(\(3.4\))g(can)h(b)s(e)g(written)f(as)1186 598 y Fj(L)1255 613 y Fi(+)1313 598 y Fn(\()p Fk(';)17 b(')p Fn(\))27 b(=)1731 530 y(1)p 1702 574 108 4 v 1702 666 a(4)p Fk(\031)1836 480 y Ff(Z)1882 669 y Fg(S)1929 650 y Fe(2)1984 598 y Fk(F)2077 451 y Ff( )2153 530 y Fj(j)p Fk(k)s Fj(j)p Fk(n)22 b Fj(\000)h Fk(k)p 2153 574 344 4 v 2301 666 a Fn(2)2507 451 y Ff(!)2606 598 y Fk(dn)75 864 y Fn(where)1028 1004 y Fk(F)14 b Fn(\()p Fk(u)p Fn(\))27 b(=)h Fk(')p Fn(\()p Fj(\000)p Fk(z)t(u)p Fn(\))p Fk(')p Fn(\()p Fk(k)d Fn(+)d Fk(z)t(u)p Fn(\))66 b(and)f Fk(z)32 b Fn(=)2588 936 y(1)22 b(+)g Fk(e)p 2588 980 215 4 v 2670 1072 a Fn(2)2844 1004 y Fk(:)75 1199 y Fn(No)m(w)33 b(w)m(e)h(use)f(the)g (iden)m(tit)m(y)661 1398 y(1)p 631 1442 108 4 v 631 1533 a(4)p Fk(\031)766 1348 y Ff(Z)812 1537 y Fg(S)859 1518 y Fe(2)914 1465 y Fk(F)1007 1319 y Ff( )1083 1398 y Fj(j)p Fk(k)s Fj(j)p Fk(n)22 b Fj(\000)g Fk(k)p 1083 1442 344 4 v 1230 1533 a Fn(2)1436 1319 y Ff(!)1535 1465 y Fk(dn)28 b Fn(=)1869 1398 y(1)p 1785 1442 218 4 v 1785 1533 a(2)p Fk(\031)t Fj(j)p Fk(k)s Fj(j)2029 1348 y Ff(Z)2075 1537 y Fe(I)-10 b(R)2131 1518 y(3)2187 1465 y Fk(\016)t Fn(\(2)p Fk(\030)26 b Fj(\001)c Fk(k)j Fn(+)d Fj(j)p Fk(\030)5 b Fj(j)2718 1424 y Fi(2)2756 1465 y Fn(\))p Fk(F)2887 1319 y Ff( )2964 1398 y Fk(\030)p 2963 1442 49 4 v 2963 1533 a Fn(2)3022 1319 y Ff(!)3121 1465 y Fk(d\030)36 b(:)346 b Fn(\(4.1\))75 1737 y(Equation)32 b(\(4.1\))g(can)h(b)s(e)g(v) m(eri\014ed)g(b)m(y)h(writing)d(the)i(righ)m(t)f(hand)h(side)f(in)g(p)s (olar)f(co)s(ordinates)h(as)777 1891 y Ff(Z)823 2080 y Fe(I)-10 b(R)880 2061 y(3)935 2008 y Fk(\016)t Fn(\(2)p Fk(\030)26 b Fj(\001)c Fk(k)j Fn(+)d Fj(j)p Fk(\030)5 b Fj(j)1466 1967 y Fi(2)1504 2008 y Fn(\))p Fk(F)1636 1862 y Ff( )1712 1941 y Fk(\030)p 1711 1985 V 1711 2077 a Fn(2)1770 1862 y Ff(!)1869 2008 y Fk(d\030)32 b Fn(=)2108 1941 y Fj(j)p Fk(k)s Fj(j)p 2108 1985 110 4 v 2139 2077 a Fn(2)2244 1891 y Ff(Z)2290 2080 y Fg(S)2337 2061 y Fe(2)2392 2008 y Fk(F)2486 1862 y Ff( )2561 1941 y Fj(j)p Fk(k)s Fj(j)p Fk(n)22 b Fj(\000)h Fk(k)p 2561 1985 344 4 v 2709 2077 a Fn(2)2915 1862 y Ff(!)3014 2008 y Fk(dn)75 2275 y Fn(Mean)m(while,)33 b(directly)f(b)m(y)h(the)g(c)m(hange)h(of)e (v)-5 b(ariables)31 b Fk(\030)h Fn(=)c Fk(r)s(n)k Fn(w)m(e)i(ha)m(v)m (e)409 2424 y Ff(Z)455 2612 y Fe(I)-10 b(R)511 2593 y(3)567 2541 y Fk(\016)t Fn(\(2)p Fk(\030)26 b Fj(\001)c Fk(k)j Fn(+)d Fj(j)p Fk(\030)5 b Fj(j)1098 2500 y Fi(2)1136 2541 y Fn(\))p Fk(F)1267 2395 y Ff( )1344 2473 y Fk(\030)p 1343 2518 49 4 v 1343 2609 a Fn(2)1402 2395 y Ff(!)1501 2541 y Fk(d\030)87 b Fn(=)1841 2424 y Ff(Z)1887 2612 y Fg(S)1934 2593 y Fe(2)1989 2424 y Ff(Z)2072 2450 y Fh(1)2035 2612 y Fi(0)2163 2541 y Fk(\016)t Fn(\()p Fk(r)s Fn(\()p Fk(r)25 b Fn(+)d(2)p Fk(k)j Fj(\001)c Fk(n)p Fn(\)\))p Fk(F)2902 2420 y Ff(\022)2973 2473 y Fk(r)s(n)p 2973 2518 105 4 v 3001 2609 a Fn(2)3088 2420 y Ff(\023)3165 2541 y Fk(r)3212 2500 y Fi(2)3268 2541 y Fk(dr)e(dn)1682 2791 y Fn(=)83 b(2)1907 2674 y Ff(Z)1952 2862 y Fg(S)1999 2843 y Fe(2)2054 2791 y Fj(j)p Fn(\()p Fk(k)25 b Fj(\001)d Fk(n)p Fn(\))2342 2806 y Fh(\000)2401 2791 y Fj(j)p Fk(F)14 b Fn(\()p Fj(\000)p Fn(\()p Fk(k)25 b Fj(\001)d Fk(n)p Fn(\))p Fk(n)p Fn(\))17 b Fk(dn)1682 3013 y Fn(=)1841 2895 y Ff(Z)1887 3084 y Fg(S)1934 3065 y Fe(2)1989 3013 y Fj(j)p Fk(k)25 b Fj(\001)d Fk(n)p Fj(j)p Fk(F)14 b Fn(\()p Fj(\000)p Fn(\()p Fk(k)25 b Fj(\001)d Fk(n)p Fn(\))p Fk(n)p Fn(\))17 b Fk(dn)75 3251 y Fn(Therefore,)725 3449 y(1)p 696 3494 108 4 v 696 3585 a(4)p Fk(\031)830 3400 y Ff(Z)876 3588 y Fg(S)923 3569 y Fe(2)978 3517 y Fk(F)1071 3371 y Ff( )1147 3449 y Fj(j)p Fk(k)s Fj(j)p Fk(n)22 b Fj(\000)g Fk(k)p 1147 3494 344 4 v 1294 3585 a Fn(2)1500 3371 y Ff(!)1599 3517 y Fk(dn)28 b Fn(=)1934 3449 y(1)p 1849 3494 218 4 v 1849 3585 a(2)p Fk(\031)t Fj(j)p Fk(k)s Fj(j)2093 3400 y Ff(Z)2139 3588 y Fg(S)2186 3569 y Fe(2)2241 3517 y Fj(j)p Fk(k)d Fj(\001)d Fk(n)p Fj(j)p Fk(F)14 b Fn(\()p Fj(\000)p Fn(\()p Fk(k)25 b Fj(\001)d Fk(n)p Fn(\))p Fk(n)p Fn(\))17 b Fk(dn)32 b(:)75 3789 y Fn(Using)g(this)g(form)m(ula)f(in)h(\(3.4\),)g(w)m(e)i(ha)m(v)m (e)g(pro)m(v)m(ed)g(the)f(follo)m(wing)c(iden)m(tit)m(y)-8 b(.)75 4005 y Fd(Lemma)37 b(4.1)49 b Fc(Equation)35 b Fn(\(3.4\))f Fc(c)-5 b(an)35 b(b)-5 b(e)34 b(written)h(as)258 4196 y Fk(@)5 b(')p 258 4240 121 4 v 272 4332 a(@)g(t)417 4264 y Fn(=)560 4196 y(1)p 530 4240 108 4 v 530 4332 a(2)p Fk(\031)664 4146 y Ff(Z)710 4335 y Fg(S)757 4316 y Fe(2)812 4167 y Ff(n)868 4264 y Fk(')p Fn(\()p Fk(z)t Fn(\()p Fk(k)26 b Fj(\001)21 b Fk(n)p Fn(\))p Fk(n)p Fn(\))p Fk(')p Fn(\()p Fk(k)26 b Fj(\000)c Fk(z)t Fn(\()p Fk(k)k Fj(\001)c Fk(n)p Fn(\))p Fk(n)p Fn(\))h Fj(\000)f Fk(')p Fn(\(0\))p Fk(')p Fn(\()p Fk(k)s Fn(\))2564 4167 y Ff(o)2619 4164 y(\014)2619 4214 y(\014)2619 4264 y(\014)2657 4196 y Fk(k)j Fj(\001)d Fk(n)p 2657 4240 185 4 v 2694 4332 a Fj(j)p Fk(k)s Fj(j)2851 4164 y Ff(\014)2851 4214 y(\014)2851 4264 y(\014)16 b Fk(dn)28 b Fn(=)f Fj(L)p Fn(\()p Fk(';)17 b(')p Fn(\))172 b(\(4.2\))75 4536 y Fc(wher)-5 b(e)34 b Fk(z)f Fn(=)541 4497 y Fi(1+)p Fg(e)p 541 4513 123 4 v 585 4571 a Fi(2)674 4536 y Fc(.)221 4752 y Fn(Using)g(previous)g(form)m(ula)d(for)i Fj(L)p Fn(\()p Fk(';)17 b(')p Fn(\))32 b(w)m(e)h(\014nd)h(out)e(the)h(follo)m (wing)d(prop)s(ert)m(y)883 4962 y Fj(L)p Fn(\(exp\()p Fk(i)p Fn(\()p Fk(k)c Fj(\001)c Fk(u)p Fn(\)\))p Fk(;)17 b Fn(exp\()p Fk(i)p Fn(\()p Fk(k)25 b Fj(\001)d Fk(u)p Fn(\)\)\))27 b(=)g(0)636 b(for)32 b(an)m(y)h Fk(u)28 b Fj(2)g Fn(I)-20 b(R)3516 4920 y Fi(3)427 5107 y Fj(L)p Fn(\(exp)q(\()p Fk(i)p Fn(\()p Fk(k)25 b Fj(\001)d Fk(u)p Fn(\)\))p Fk(';)17 b Fn(exp\()p Fk(i)p Fn(\()p Fk(k)26 b Fj(\001)21 b Fk(u)p Fn(\)\))p Fk(')p Fn(\))27 b(=)h(exp)q(\()p Fk(i)p Fn(\()p Fk(k)d Fj(\001)d Fk(u)p Fn(\)\))p Fj(L)p Fn(\()p Fk(';)17 b(')p Fn(\))179 b(for)32 b(an)m(y)h Fk(u)28 b Fj(2)g Fn(I)-20 b(R)3516 5065 y Fi(3)3624 5107 y Fn(\(4.3\))1901 5356 y(10)p eop %%Page: 11 11 11 10 bop 75 107 a Fn(Let)22 b(us)h(study)h(equation)e(\(4.2\))f(near)i (the)f(\014xed)i(p)s(oin)m(t)d Fk(')2121 122 y Fh(\003)2188 107 y Fn(=)27 b(1.)40 b(The)23 b(b)s(eha)m(vior)f(of)g(the)h(solutions) e(near)75 227 y(an)m(y)30 b(other)f(exp)q(\()p Fk(i)p Fn(\()p Fk(k)18 b Fj(\001)d Fk(u)p Fn(\)\))28 b(for)h(an)m(y)h Fk(u)d Fj(2)h Fn(I)-20 b(R)1628 185 y Fi(3)1668 227 y Fn(,)30 b Fj(j)p Fk(u)p Fj(j)c(6)p Fn(=)i(0,)h(is)g(quite)g(similar)d (to)i(the)i(case)g Fj(j)p Fk(u)p Fj(j)c Fn(=)i(0,)i(due)f(to)75 348 y(prop)s(ert)m(y)k(\(4.3\).)42 b(In)32 b(fact,)g(\(4.3\))g(is)f (the)h(equiv)-5 b(alen)m(t)32 b(of)f(the)h(translational)d(in)m(v)-5 b(ariance)31 b(of)g(equation)75 468 y(\(2.6\).)45 b(Let)33 b(us)h(remark)f(that)g(linearizing)d(the)k(F)-8 b(ourier)32 b(transformed)h(equation)g(around)g(the)h(state)75 588 y Fk(')139 603 y Fh(\003)206 588 y Fn(=)28 b(1)k(is)g(lik)m(e)g (linearizing)d(our)k(original)c(equation)j(around)h(the)g(Delta)e (Dirac)g(distribution.)221 709 y(Th)m(us,)k(let)c(us)j(consider)f(p)s (erturbations)f(of)g Fk(')1927 724 y Fh(\003)1994 709 y Fn(=)27 b(1)33 b(in)f(equation)g(\(4.2\))g(of)g(the)h(form)1205 929 y Fk(')p Fn(\()p Fk(t;)17 b(k)s Fn(\))27 b(=)h(1)22 b(+)g Fk( )t Fn(\()p Fk(t;)17 b(k)s Fn(\))32 b(;)212 b Fj(k)p Fk( )t Fj(k)27 b(\034)g Fn(1)75 1149 y(with)32 b Fj(k)22 b(\001)g(k)32 b Fn(is)g(the)h(con)m(tin)m(uous)h(norm.)42 b(W)-8 b(e)33 b(obtain)f(the)h(linearized)e(equation)343 1356 y Fk(@)5 b( )p 343 1400 124 4 v 359 1492 a(@)g(t)505 1424 y Fn(=)28 b Fj(L)p Fk( )86 b Fn(=)1026 1356 y(1)p 996 1400 108 4 v 996 1492 a(2)p Fk(\031)1130 1307 y Ff(Z)1177 1495 y Fg(S)1224 1476 y Fe(2)1288 1356 y Fj(j)p Fk(k)25 b Fj(\001)d Fk(n)p Fj(j)p 1288 1400 240 4 v 1354 1492 a(j)p Fk(k)s Fj(j)1538 1327 y Ff(h)1577 1424 y Fk( )t Fn(\()p Fk(z)t Fn(\()p Fk(k)k Fj(\001)c Fk(n)p Fn(\))p Fk(n)p Fn(\))g(+)g Fk( )t Fn(\()p Fk(k)k Fj(\000)c Fk(z)t Fn(\()p Fk(k)k Fj(\001)c Fk(n)p Fn(\))p Fk(n)p Fn(\))g Fj(\000)h Fk( )t Fn(\(0\))f Fj(\000)g Fk( )t Fn(\()p Fk(k)s Fn(\))3527 1327 y Ff(i)75 1694 y Fn(W)-8 b(e)33 b(can)g(pro)m(v)m(e)h(the)f(follo)m(wing)d(result.)75 1922 y Fd(Theorem)37 b(4.2)49 b Fc(Polynomial)35 b(eigenfunctions)g (and)h(c)-5 b(orr)g(esp)g(onding)34 b(eigenvalues)h(of)h Fj(L)g Fc(ar)-5 b(e)36 b(given)75 2043 y(by)909 2070 y Ff(8)909 2145 y(>)909 2170 y(>)909 2195 y(<)909 2344 y(>)909 2369 y(>)909 2394 y(:)1024 2158 y Fj(L)p Fk(')1157 2173 y Fg(nl)q(m)1288 2158 y Fn(\()p Fk(k)s Fn(\))28 b(=)f Fj(\000)p Fk(\025)1683 2173 y Fg(nl)1752 2158 y Fk(')1816 2173 y Fg(nl)q(m)1948 2158 y Fn(\()p Fk(k)s Fn(\))1024 2294 y Fk(')1088 2309 y Fg(nl)q(m)1219 2294 y Fn(\()p Fk(k)s Fn(\))h(=)g Fj(j)p Fk(k)s Fj(j)1591 2258 y Fi(2)p Fg(n)p Fi(+)p Fg(l)1749 2294 y Fk(Y)1806 2309 y Fg(l)q(m)1911 2198 y Ff(\020)1990 2255 y Fg(k)p 1970 2271 78 4 v 1970 2328 a Fh(j)p Fg(k)r Fh(j)2058 2198 y Ff(\021)1024 2429 y Fk(n)g Fn(=)g(0)p Fk(;)17 b Fn(1)p Fk(;)g(:)g(:)g(:)50 b Fn(;)116 b Fk(l)30 b Fn(=)d(0)p Fk(;)17 b Fn(1)p Fk(;)g(:)g(:)g(:)50 b Fn(;)117 b Fk(m)28 b Fn(=)f Fj(\000)p Fk(l)r(;)17 b(:)g(:)g(:)g(;)g(l)75 2606 y Fc(wher)-5 b(e)34 b Fk(Y)407 2621 y Fg(l)q(m)495 2606 y Fn(\()p Fk(u)p Fn(\))h Fc(ar)-5 b(e)34 b(the)h(spheric)-5 b(al)34 b(harmonics)f(with)717 2872 y Fk(\025)774 2887 y Fg(nl)926 2872 y Fn(=)83 b(1)22 b(+)g Fk(\016)1297 2887 y Fg(n)p Fi(0)1379 2872 y Fk(\016)1422 2887 y Fg(l)q Fi(0)1506 2872 y Fj(\000)h Fn(2)1655 2751 y Ff(\032)1717 2872 y Fk(z)1766 2831 y Fi(2)p Fg(n)p Fi(+)p Fg(l)1942 2755 y Ff(Z)2025 2782 y Fi(1)1988 2944 y(0)2081 2872 y Fk(\013)2144 2831 y Fi(2)p Fg(n)p Fi(+)p Fg(l)q Fi(+1)2393 2872 y Fk(P)2456 2887 y Fg(l)2482 2872 y Fn(\()p Fk(\013)q Fn(\))17 b Fk(d\013)22 b Fn(+)1284 3130 y(+)1519 3062 y(1)p 1370 3107 346 4 v 1370 3198 a Fk(z)t Fn(\(2)h Fj(\000)f Fk(z)t Fn(\))1743 3013 y Ff(Z)1826 3039 y Fi(1)1789 3201 y(1)p Fh(\000)p Fg(z)1935 3130 y Fk(\013)1998 3089 y Fi(2)p Fg(n)p Fi(+)p Fg(l)q Fi(+1)2247 3130 y Fk(P)2310 3145 y Fg(l)2353 2984 y Ff( )2428 3062 y Fn(1)g Fj(\000)h Fk(z)k Fn(+)22 b Fk(\013)2832 3026 y Fi(2)p 2428 3107 443 4 v 2470 3198 a Fn(\(2)g Fj(\000)h Fk(z)t Fn(\))p Fk(\013)2881 2984 y Ff(!)2980 3130 y Fk(d\013)3094 3009 y Ff(\033)3155 3130 y Fk(;)75 3407 y Fc(wher)-5 b(e)34 b Fk(P)413 3422 y Fg(l)474 3407 y Fc(ar)-5 b(e)35 b(the)f(L)-5 b(e)g(gendr)g(e)35 b(p)-5 b(olynomials.)43 b(Mor)-5 b(e)g(over,)34 b Fk(\025)2293 3422 y Fg(nl)2390 3407 y Fk(>)27 b Fn(0)35 b Fc(exc)-5 b(ept)34 b(two)h(values)1663 3627 y Fk(\025)1720 3642 y Fi(00)1822 3627 y Fn(=)28 b Fk(\025)1983 3642 y Fi(01)2085 3627 y Fn(=)f(0)1387 b(\(4.4\))75 3847 y Fc(In)34 b(p)-5 b(articular,)822 4127 y Fk(\025)879 4142 y Fg(n)p Fi(0)1045 4127 y Fn(=)83 b(1)21 b(+)h Fk(\016)1415 4142 y Fg(n)p Fi(0)1520 4127 y Fj(\000)1719 4060 y Fn(1)p 1630 4104 228 4 v 1630 4196 a Fk(n)g Fn(+)g(1)1883 3981 y Ff(")1932 4127 y Fk(z)1981 4086 y Fi(2)p Fg(n)2086 4127 y Fn(+)2194 4060 y(1)g Fj(\000)g Fn(\(1)g Fj(\000)h Fk(z)t Fn(\))2660 4024 y Fi(2\()p Fg(n)p Fi(+1\))p 2194 4104 695 4 v 2288 4196 a Fn(1)f Fj(\000)g Fn(\(1)g Fj(\000)h Fk(z)t Fn(\))2754 4167 y Fi(2)2898 3981 y Ff(#)2974 4127 y Fn(=)1045 4409 y(=)83 b(1)21 b(+)h Fk(\016)1415 4424 y Fg(n)p Fi(0)1520 4409 y Fj(\000)1719 4342 y Fn(1)p 1630 4386 228 4 v 1630 4478 a Fk(n)g Fn(+)g(1)1883 4263 y Ff(")1932 4409 y Fk(z)1981 4368 y Fi(2)p Fg(n)2086 4409 y Fn(+)2227 4302 y Fg(n)2188 4326 y Ff(X)2184 4511 y Fg(k)r Fi(=0)2313 4409 y Fn(\(1)f Fj(\000)i Fk(z)t Fn(\))2608 4368 y Fi(2)p Fg(k)2687 4263 y Ff(#)2752 4409 y Fk(:)221 4710 y Fn(W)-8 b(e)34 b(refer)f(to)g([6)o(,)g(7])g(for)g (all)d(the)k(details)e(of)g(the)h(deriv)-5 b(ation)32 b(since)h(the)g(pro)s(of)g(is)f(iden)m(tical)f(with)75 4831 y(minor)g(c)m(hanges.)45 b(Also)32 b(w)m(e)h(refer)g(to)f([16])h (for)f(prop)s(erties)g(of)g(the)h(spherical)f(harmonics.)1901 5356 y(11)p eop %%Page: 12 12 12 11 bop 75 107 a Fd(Remark)37 b(4.3)49 b Fn(As)d(a)f(consequence)j (when)e Fk(e)f Fn(is)g(constan)m(t,)k(0)g Fk(<)g(e)h(<)f Fn(1)c(the)g(sp)s(ectrum)h(of)f(the)75 227 y(linearized)d(op)s(erator)h (consists)i(of)e(t)m(w)m(o)i(zero)f(eigen)m(v)-5 b(alues)44 b(\(4.4\))f(that)g(corresp)s(onds)j(to)d(the)h(t)m(w)m(o)75 348 y(conserv)-5 b(ation)26 b(la)m(ws)g(\(mass)g(and)h(momen)m(tum\))e (and)h(a)g(sequence)j(of)c(isolate)g(p)s(oin)m(ts)h(on)g(the)h(negativ) m(e)75 468 y(part)34 b(of)f(the)h(real)f(axis.)48 b(It)34 b(is)f(clear)g(that)h Fk(\025)1711 483 y Fg(nl)1810 468 y Fj(!)29 b Fn(1)34 b(as)g(\(2)p Fk(n)23 b Fn(+)g Fk(l)r Fn(\))30 b Fj(!)f(1)p Fn(.)48 b(Note)34 b(that)f(the)i(sp)s(ectrum)75 588 y(is)d(giv)m(en)h(b)m(y)g(the)g(v)-5 b(alues)33 b Fj(\000)p Fk(\025)1156 603 y Fg(nl)1225 588 y Fn(.)221 709 y(In)g(particular,)452 989 y Fk(\025)509 1004 y Fg(n)p Fi(1)619 989 y Fn(=)28 b(1)22 b Fj(\000)903 922 y Fn(\(1)g Fj(\000)h Fk(a)p Fn(\))1201 886 y Fi(2)p Fg(n)p Fi(+1)p 903 966 471 4 v 1025 1058 a Fk(n)f Fn(+)g(2)1405 989 y Fj(\000)1601 922 y Fn(1)p 1515 966 221 4 v 1515 1058 a(1)g(+)g Fk(a)1762 843 y Ff(\()1926 922 y Fk(a)p 1838 966 228 4 v 1838 1058 a(n)h Fn(+)f(1)2086 922 y(1)f Fj(\000)i Fk(a)2307 886 y Fi(2\()p Fg(n)p Fi(+1\))p 2086 966 450 4 v 2179 1058 a Fn(1)f Fj(\000)h Fk(a)2401 1029 y Fi(2)2567 989 y Fn(+)2764 922 y(1)p 2675 966 228 4 v 2675 1058 a Fk(n)f Fn(+)g(2)2922 922 y(1)g Fj(\000)g Fk(a)3143 886 y Fi(2\()p Fg(n)p Fi(+2\))p 2922 966 450 4 v 3016 1058 a Fn(1)g Fj(\000)g Fk(a)3237 1029 y Fi(2)3381 843 y Ff(\))75 1272 y Fk(n)51 b Fj(\025)g Fn(0,)e(where)f Fk(a)i Fn(=)h(1)31 b Fj(\000)h Fk(z)t Fn(.)85 b(Th)m(us,)51 b Fk(\025)1659 1287 y Fi(01)1784 1272 y Fn(=)g(0)45 b(for)h(an)m(y)h(v) -5 b(alue)45 b(0)51 b Fj(\024)g Fk(a)g Fj(\024)g Fn(1)p Fk(=)p Fn(2.)83 b(Finally)-8 b(,)47 b(w)m(e)75 1392 y(reco)m(v)m(er)28 b(in)e(the)h(case)g Fk(z)32 b Fn(=)c(1)e(w)m(ell-kno)m(wn)g(form)m(ula) f(for)h(the)h(eigen)m(v)-5 b(alues)26 b(of)g(the)h(linearized)d (classical)75 1512 y(Boltzmann)31 b(equation)i(for)f(Maxw)m (ell-molecules)f(giv)m(en)h(b)m(y)1124 1773 y Fk(\025)1181 1788 y Fg(nl)1278 1773 y Fn(=)27 b(1)22 b(+)g Fk(\016)1593 1788 y Fg(n)p Fi(0)1676 1773 y Fk(\016)1719 1788 y Fg(l)q Fi(0)1802 1773 y Fj(\000)h Fn(4)1968 1656 y Ff(Z)2050 1683 y Fi(1)2013 1845 y(0)2106 1773 y Fk(\013)2169 1732 y Fi(2)p Fg(n)p Fi(+)p Fg(l)q Fi(+1)2418 1773 y Fk(P)2481 1788 y Fg(l)2507 1773 y Fn(\()p Fk(\013)q Fn(\))17 b Fk(d\013)75 2028 y Fn(The)34 b(main)c(di\013erence)k(from)d(the)i (elastic)f(case)h(\()p Fk(z)f Fn(=)c(1\))k(is)g(that)h(w)m(e)g(obtain)f (a)g(small)e(eigen)m(v)-5 b(alue)1646 2248 y Fk(\025)1703 2263 y Fi(10)1805 2248 y Fn(=)27 b Fk(z)t Fn(\(1)c Fj(\000)g Fk(z)t Fn(\))75 2468 y(as)33 b(a)f(result)h(of)f(p)s(erturbation)f(of)i (the)g(v)-5 b(alue)32 b Fk(\025)1819 2483 y Fi(10)1921 2468 y Fn(=)27 b(0)33 b(for)f Fk(z)g Fn(=)c(1,)k(that)g(is,)h(the)g (elastic)e(case.)75 2800 y Fo(5)161 b(Isotropic)53 b(equation)h(and)f (its)h(self-similar)i(solutions)75 3019 y Fn(Let)38 b(us)g(come)f(bac)m (k)i(for)e(a)g(while)f(to)i(the)f(original)e(Boltzmann)h(equation)h (\(2.6\).)58 b(W)-8 b(e)38 b(can)f(try)h(to)75 3140 y(lo)s(ok)31 b(for)h(solutions)g(of)g(self-similar)d(t)m(yp)s(e,)k(that)g(is,)1188 3360 y Fk(f)11 b Fn(\()p Fk(t;)17 b(v)t Fn(\))27 b(=)g Fk(\032\022)s Fn(\()p Fk(t)p Fn(\))1792 3319 y Fh(\000)p Fi(3)p Fg(=)p Fi(2)1958 3360 y Fk(g)t Fn(\()p Fk(\022)s Fn(\()p Fk(t)p Fn(\))2206 3319 y Fh(\000)p Fi(1)p Fg(=)p Fi(2)2370 3360 y Fn(\()p Fk(v)f Fj(\000)d Fk(u)p Fn(\)\))911 b(\(5.1\))75 3580 y(where)35 b Fk(\032;)17 b(u;)g(\022)s Fn(\()p Fk(t)p Fn(\))34 b(are)g(the)g(mass,)h(momen)m(tum)d(and)i(temp) s(erature)g(of)g Fk(f)11 b Fn(.)47 b(These)36 b(states)f(are)f(called) 75 3700 y(homogeneous)f(co)s(oling)d(states.)221 3821 y(Therefore,)k Fk(g)i Fn(m)m(ust)d(satisfy)865 3948 y Ff(Z)911 4136 y Fe(I)-10 b(R)967 4118 y(3)1023 4065 y Fk(g)19 b(dw)30 b Fn(=)e(1)k(;)1566 3948 y Ff(Z)1612 4136 y Fe(I)-10 b(R)1669 4118 y(3)1724 4065 y Fk(w)s(g)19 b(dw)30 b Fn(=)e(0)k(;)2340 3948 y Ff(Z)2386 4136 y Fe(I)-10 b(R)2443 4118 y(3)2498 4065 y Fj(j)p Fk(w)s Fj(j)2627 4024 y Fi(2)2665 4065 y Fk(g)20 b(dw)30 b Fn(=)d(1)75 4319 y(Computing)k(the)i(deriv)-5 b(ativ)m(e)33 b(of)f Fk(\022)s Fn(\()p Fk(t)p Fn(\))g(w)m(e)i(ha)m(v)m(e)771 4531 y Fk(d)p 753 4575 86 4 v 753 4667 a(dt)866 4481 y Ff(Z)912 4670 y Fe(I)-10 b(R)968 4651 y(3)1024 4598 y Fj(j)p Fk(v)25 b Fj(\000)e Fk(u)p Fj(j)1308 4557 y Fi(2)1346 4598 y Fk(f)28 b(dv)j Fn(=)c(3)p Fk(\032\022)1801 4557 y Fh(0)1825 4598 y Fn(\()p Fk(t)p Fn(\))h(=)f Fj(\000)2168 4573 y Fn(~)2144 4598 y Fk(B)2234 4531 y Fn(1)22 b Fj(\000)g Fk(e)2449 4495 y Fi(2)p 2234 4575 256 4 v 2337 4667 a Fn(4)2499 4598 y Fk(\032)2549 4557 y Fi(2)2589 4598 y Fk(V)f Fn(\()p Fk(g)t(;)c(g)t Fn(\))p Fk(\022)s Fn(\()p Fk(t)p Fn(\))3048 4557 y Fi(3)p Fg(=)p Fi(2)75 4847 y Fn(where)1140 4967 y Fk(V)k Fn(\()p Fk(g)t(;)c(g)t Fn(\))26 b(=)1570 4850 y Ff(Z)1616 5039 y Fe(I)-10 b(R)1672 5020 y(6)1728 4967 y Fj(j)p Fk(v)25 b Fj(\000)e Fk(w)s Fj(j)2029 4926 y Fi(2)2067 4967 y Fk(g)t Fn(\()p Fk(v)t Fn(\))p Fk(g)t Fn(\()p Fk(w)s Fn(\))17 b Fk(dv)g(dw)35 b(:)1901 5356 y Fn(12)p eop %%Page: 13 13 13 12 bop 75 107 a Fn(Simplifying)29 b(w)m(e)k(deduce)1214 380 y(3)p Fk(\022)1311 339 y Fh(0)1334 380 y Fn(\()p Fk(t)p Fn(\))28 b(=)f Fj(\000)1677 355 y Fn(~)1653 380 y Fk(B)6 b(\032)1793 313 y Fn(1)22 b Fj(\000)h Fk(e)2009 277 y Fi(2)p 1793 357 256 4 v 1897 449 a Fn(4)2059 380 y Fk(V)e Fn(\()p Fk(g)t(;)c(g)t Fn(\))p Fk(\022)s Fn(\()p Fk(t)p Fn(\))2518 339 y Fi(3)p Fg(=)p Fi(2)2659 380 y Fk(:)938 b Fn(\(5.2\))75 631 y(Substituting)32 b(\(5.1\))g(in)m(to)g (\(5.2\))g(w)m(e)h(deduce)h(the)f(follo)m(wing)d(equation)j(for)f Fk(g)t Fn(:)963 896 y Fj(\000)1050 828 y Fn(3)p 1050 873 49 4 v 1050 964 a(2)1109 896 y Fk(\032\022)1207 855 y Fh(0)1231 896 y Fn(\()p Fk(t)p Fn(\))p Fk(\022)s Fn(\()p Fk(t)p Fn(\))1501 855 y Fh(\000)p Fi(5)p Fg(=)p Fi(2)1698 896 y Fn(div)1831 911 y Fg(w)1888 896 y Fn(\()p Fk(w)s(g)t Fn(\))26 b(=)i Fk(\032)2268 855 y Fi(2)2331 871 y Fn(~)2307 896 y Fk(B)5 b(\022)s Fn(\()p Fk(t)p Fn(\))2545 855 y Fh(\000)p Fi(1)2662 871 y Fn(~)2640 896 y Fk(Q)p Fn(\()p Fk(g)t(;)17 b(g)t Fn(\))685 b(\(5.3\))75 1141 y(where)877 1274 y(~)855 1299 y Fk(Q)p Fn(\()p Fk(g)t(;)17 b(g)t Fn(\))26 b(=)1283 1182 y Ff(Z)1330 1371 y Fe(I)-10 b(R)1386 1352 y(3)1441 1153 y Ff(")1635 1232 y Fn(1)p 1500 1276 320 4 v 1500 1367 a Fk(e)1545 1339 y Fi(2)1584 1367 y Fn(\()p Fk(\022)s Fn(\()p Fk(t)p Fn(\)\))1829 1299 y Fk(g)t Fn(\()p Fk(w)1991 1258 y Fh(\003)2029 1299 y Fn(\))p Fk(g)t Fn(\()p Fk(v)2207 1258 y Fh(\003)2246 1299 y Fn(\))22 b Fj(\000)g Fk(g)t Fn(\()p Fk(v)t Fn(\))p Fk(g)t Fn(\()p Fk(w)s Fn(\))2783 1153 y Ff(#)2863 1299 y Fk(dw)34 b(:)75 1536 y Fn(Using)e(that)h(the)g(second)g(momen)m(t)f(of)g Fk(g)k Fn(is)c(1)g(w)m(e)i(ha)m(v)m(e)g Fk(V)21 b Fn(\()p Fk(g)t(;)c(g)t Fn(\))26 b(=)i(2)k(and)h(then)1393 1815 y Fk(\022)1441 1774 y Fh(0)1465 1815 y Fn(\()p Fk(t)p Fn(\))27 b(=)h Fj(\000)p Fn(2)1856 1790 y(~)1833 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3714 a Fi(2)1112 3679 y Fj(j)p Fk(k)s Fj(j)1222 3643 y Fi(2)1294 3679 y Fk(;)719 b(\020)35 b Fn(=)2231 3640 y Fg(k)r Fh(\001)p Fg(n)p 2231 3656 101 4 v 2243 3714 a Fh(j)p Fg(k)r Fh(j)2375 3679 y Fk(;)212 b(z)32 b Fn(=)2805 3640 y Fi(1+)p Fg(e)p 2805 3656 123 4 v 2848 3714 a Fi(2)2970 3679 y Fk(;)542 3830 y(k)25 b Fj(\001)d Fk(k)719 3845 y Fh(\000)805 3830 y Fn(=)919 3791 y Fg(z)p 919 3807 36 4 v 919 3864 a Fi(2)964 3830 y Fn(\(1)g Fj(\000)h Fk(\020)8 b Fn(\))p Fj(j)p Fk(k)s Fj(j)1372 3794 y Fi(2)1605 3830 y Fn(and)278 b Fj(j)p Fk(k)2119 3845 y Fi(+)2178 3830 y Fj(j)2206 3794 y Fi(2)2272 3830 y Fn(=)28 b Fj(j)p Fk(k)s Fj(j)2486 3794 y Fi(2)2541 3734 y Ff(h)2581 3830 y Fn(1)22 b Fj(\000)2761 3787 y Fi(1)p Fh(\000)p Fg(\020)p 2761 3807 126 4 v 2806 3864 a Fi(2)2897 3830 y Fk(z)t Fn(\(2)g Fj(\000)h Fk(z)t Fn(\))3242 3734 y Ff(i)3331 3830 y Fk(:)75 4062 y Fn(No)m(w,)45 b(w)m(e)d(can)g(ev)-5 b(aluate)42 b(equation)f(\(3.4\))g(in)g(spherical)g(co)s(ordinates)h 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Fk(k)s Fj(j)3411 4356 y Fi(2)p 3301 4436 149 4 v 3351 4528 a Fn(2)3460 4313 y Ff(!\))3626 4459 y Fk(d\020)75 4769 y Fn(where)34 b Fk(\014)f Fn(=)28 b Fk(z)t Fn(\(2)22 b Fj(\000)h Fk(z)t Fn(\).)44 b(Fixing)31 b Fk(\021)g Fn(=)1463 4723 y Fh(j)p Fg(k)r Fh(j)1542 4699 y Fe(2)p 1463 4746 113 4 v 1501 4803 a Fi(2)1618 4769 y Fn(and)h Fk(s)c Fn(=)1994 4726 y Fi(1)p Fh(\000)p Fg(\020)p 1994 4746 126 4 v 2040 4803 a Fi(2)2163 4769 y Fn(w)m(e)33 b(ha)m(v)m(e)948 4980 y Fk(@)5 b(')p 948 5025 121 4 v 962 5116 a(@)g(t)1106 5048 y Fn(=)1210 4931 y Ff(Z)1293 4957 y Fi(1)1256 5119 y(0)1349 4952 y Ff(n)1404 5048 y Fk(')p Fn(\()p Fk(z)1555 5007 y Fi(2)1595 5048 y Fk(s\021)t Fn(\))p Fk(')p Fn(\(\(1)21 b Fj(\000)i Fk(\014)6 b(s)p Fn(\))p Fk(\021)t Fn(\))21 b Fj(\000)i Fk(')p Fn(\(0\))p Fk(')p Fn(\()p Fk(\021)t Fn(\))2778 4952 y Ff(o)2866 5048 y Fk(ds)661 b Fn(\(5.5\))1901 5356 y(13)p eop %%Page: 14 14 14 13 bop 75 109 a Fn(where)34 b Fk(\014)f Fn(=)28 b(1)22 b Fj(\000)729 63 y Fi(\(1)p Fh(\000)p Fg(e)p Fi(\))906 39 y Fe(2)p 729 86 213 4 v 818 144 a Fi(4)951 109 y Fn(.)44 b(No)m(w,)33 b(the)g(temp)s(erature)g Fk(\022)s Fn(\()p Fk(t)p Fn(\))f(is)h(giv)m(en)f(b)m(y)1530 351 y Fk(\022)s Fn(\()p Fk(t)p Fn(\))c(=)f Fj(\000)p Fk(')1961 310 y Fh(0)1961 376 y Fg(\021)2004 351 y Fn(\()p Fk(t;)17 b(\021)t Fn(\))2211 251 y Ff(\014)2211 301 y(\014)2211 351 y(\014)2238 405 y Fg(\021)r Fi(=0)75 601 y Fn(pro)m(vided)33 b Fk(')p Fn(\()p Fk(t;)17 b Fn(0\))27 b(=)h(1.)221 721 y(So)40 b(if)e Fk(\014)45 b Fn(=)39 b(constan)m(t)h(in)f(\(5.5\),)i(then)f(the) f(equation)h(is)e(in)m(v)-5 b(arian)m(t)38 b(under)j(dilations)c Fk(\021)43 b Fj(!)c Fk(\013)q(\021)75 841 y Fn(with)c Fk(\013)e Fj(2)g Fn(I)-20 b(R)582 800 y Fi(+)641 841 y Fn(.)52 b(Therefore,)37 b(similar)32 b(to)j(the)h(elastic)e(case)j Fk(\014)h Fn(=)32 b(1)j([6,)h(7)o(],)h(w)m(e)f(can)g(pro)m(v)m(e)g (that)g(the)75 962 y(equation)c(\(5.5\))g(admits)g(self-similar)c (solutions)1455 1182 y Fk(')p Fn(\()p Fk(t;)17 b(\021)t Fn(\))27 b(=)44 b(~)-65 b Fk(')p Fn(\(exp)q(\()p Fj(\000)p Fk(\026t)p Fn(\))p Fk(\021)t Fn(\))75 1402 y(with)32 b Fk(\026)c Fj(2)g Fn(I)-20 b(R)33 b(where)49 b(~)-65 b Fk(')33 b Fn(satis\014es)g(the)g(equation)801 1666 y Fk(\026\021)922 1598 y(d)16 b Fn(~)-65 b Fk(')p 922 1642 115 4 v 928 1734 a(d\021)1068 1666 y Fn(+)1166 1549 y Ff(Z)1249 1575 y Fi(1)1212 1737 y(0)1305 1569 y Ff(n)1376 1666 y Fn(~)h Fk(')p Fn(\()p Fk(z)1512 1625 y Fi(2)1552 1666 y Fk(s\021)t Fn(\))16 b(~)-65 b Fk(')o Fn(\(\(1)22 b Fj(\000)h Fk(\014)6 b(s)p Fn(\))p Fk(\021)t Fn(\))21 b Fj(\000)38 b Fn(~)-64 b Fk(')o Fn(\(0\))16 b(~)-65 b Fk(')p Fn(\()p Fk(\021)t Fn(\))2734 1569 y Ff(o)2822 1666 y Fk(ds)27 b Fn(=)h(0)221 1931 y(Assuming)k(that)1213 2103 y(~)-65 b Fk(')p Fn(\()p Fk(\021)t Fn(\))27 b(=)1551 1995 y Fh(1)1526 2020 y Ff(X)1520 2201 y Fg(n)p Fi(=0)1679 2036 y Fn(\()p Fj(\000)p Fn(1\))1881 1999 y Fg(n)p 1679 2080 250 4 v 1761 2171 a Fk(n)p Fn(!)1954 2103 y(~)-64 b Fk(')2003 2118 y Fg(n)2049 2103 y Fk(\021)2101 2062 y Fg(n)2181 2103 y 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Ff(\))3624 4875 y Fn(\(6.4\))1901 5356 y(17)p eop %%Page: 18 18 18 17 bop 75 107 a Fn(pro)m(vided)33 b(there)g(are)g(no)f(resonances,)j (i.e.,)971 393 y Fk(\025)1028 408 y Fg(m)1122 393 y Fj(6)p Fn(=)1226 285 y Fg(m)p Fh(\000)p Fi(1)1242 310 y Ff(X)1238 495 y Fg(k)r Fi(=1)1395 393 y Fk(r)1439 408 y Fg(k)1482 393 y Fk(\025)1539 408 y Fg(k)1614 393 y Fn(with)1852 285 y Fg(m)p Fh(\000)p Fi(1)1869 310 y Ff(X)1864 495 y Fg(k)r Fi(=1)2022 393 y Fk(k)s(r)2120 408 y Fg(k)2190 393 y Fn(=)28 b Fk(m)k Fn(and)h Fk(r)2645 408 y Fg(k)2715 393 y Fj(\025)c Fn(0)j Fk(:)75 686 y Fn(Otherwise)j(w)m(e)g(obtain)e (additional)e(terms)j(in)g(\(6.3\).)47 b(The)35 b(resonances)h(are)f (absen)m(t)g(in)e(the)i(elastic)75 806 y(case)e Fk(e)28 b Fn(=)g(1)k([6])h(and)f(they)i(app)s(ear)e(only)g(for)g(sp)s(ecial)g (v)-5 b(alues)32 b(of)h Fk(e)p Fn(.)221 926 y(Finally)-8 b(,)33 b(let)h(us)i(clarify)d(that)h(the)i(self-similar)30 b(solution)k(can)h(dominate)e(in)h(the)h(limit)d Fk(t)f Fj(!)g(1)75 1047 y Fn(when)39 b 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Fk(e)1106 4546 y Fi(2)1146 4582 y Fn(\))p Fk(t)28 b Fn(=)f(constan)m(t)q(.)1901 5356 y(18)p eop %%Page: 19 19 19 18 bop 75 107 a Fb(6.2)135 b(Non-constan)l(t)46 b(restitution)g(co)t (e\016cien)l(t)75 292 y Fn(Let)34 b(us)h(no)m(w)f(assume)h(that)e Fk(e)p Fn(\()p Fk(\022)s Fn(\))i(and)f(that)f(the)i(b)s(eha)m(vior)e (of)h Fk(e)p Fn(\()p Fk(\022)s Fn(\))g(for)f(small)f(energy)j (collisions)c(is)75 412 y(more)37 b(elastic.)57 b(Consider)38 b(that)f Fk(e)p Fn(\()p Fk(\022)s Fn(\))g(is)g(a)g(decreasing)h(con)m (tin)m(uous)g(function)f(of)g Fk(\022)k Fn(that)c(tends)h(to)75 532 y(some)32 b(0)c Fk(<)f(e)544 547 y Fi(0)612 532 y Fk(<)g Fn(1)33 b(as)f Fk(\022)f Fj(!)c(1)33 b Fn(and)f(that)h Fk(e)p Fn(\(0\))27 b(=)h(1.)43 b(W)-8 b(e)33 b(will)d(assume)j(that) 1566 790 y(lim)1564 850 y Fg(\022)r Fh(!)p Fi(0)1854 722 y Fk(e)p Fn(\()p Fk(\022)s Fn(\))p 1731 767 416 4 v 1731 858 a(1)22 b Fj(\000)g Fk(C)7 b(\022)2026 829 y Fg(\013=)p Fi(2)2184 790 y Fn(=)28 b(1)1287 b(\(6.5\))75 1032 y(with)32 b(0)c 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Fn(exp)2275 3782 y Ff( )2341 3928 y Fj(\000)2460 3860 y(j)p Fk(v)t Fj(j)2567 3824 y Fi(2)p 2428 3904 209 4 v 2428 3996 a Fn(2)p Fk(\022)s Fn(\()p Fk(t)p Fn(\))2646 3782 y Ff(!)75 4161 y Fk(e)120 4176 y Fi(0)187 4161 y Fj(!)g Fn(1,)k Fk(t)c Fj(!)f(1)p Fn(,)32 b(\(1)22 b Fj(\000)h Fk(e)1026 4125 y Fi(2)1026 4186 y(0)1066 4161 y Fn(\))p Fk(t)k Fn(=)33 b(constan)m(t.)75 4491 y Fo(7)161 b(Euler)54 b(equations.)71 b(Hydro)t(dynamic)54 b(limit.)75 4710 y Fn(On)48 b(the)h(basis)f(of)g(the)g(previous)h(considerations,)j(w)m (e)d(assume)g(the)g(Maxw)m(ellian)e(form)g(of)h(the)75 4831 y(distribution)31 b(function,)h(pro)m(vided)h(that)f(the)h(mean)f (free)h(path)1815 5072 y Fk(\017)28 b Fn(=)2011 5005 y(1)p 1996 5049 80 4 v 1996 5140 a Fk(B)1901 5356 y Fn(19)p eop %%Page: 20 20 20 19 bop 75 107 a Fn(is)35 b(small)d(enough.)52 b(Then,)37 b(w)m(e)f(obtain)e(formally)f(the)i(follo)m(wing)e(dissipativ)m(e)h (Euler)h(equations)g(for)75 227 y(densit)m(y)f Fk(\032)p Fn(\()p Fk(t;)17 b(x)p Fn(\),)33 b(bulk)f(v)m(elo)s(cit)m(y)h Fk(u)p Fn(\()p Fk(t;)17 b(x)p Fn(\))32 b(and)h(temp)s(erature)f Fk(\022)s Fn(\()p Fk(t;)17 b(x)p Fn(\))33 b(as)1557 423 y Fk(@)5 b(\032)p 1557 467 108 4 v 1565 559 a(@)g(t)1696 490 y Fn(+)22 b(div\()p Fk(\032u)p Fn(\))60 b(=)g(0)1379 727 y Fk(@)5 b(u)p 1379 772 113 4 v 1389 863 a(@)g(t)1524 795 y Fn(+)22 b(\()p Fk(u)f Fj(\001)h(r)p Fn(\))p Fk(u)g Fn(+)2094 727 y(1)p 2094 772 51 4 v 2094 863 a Fk(\032)2154 795 y Fj(r)p Fk(p)60 b Fn(=)g(0)881 1061 y Fk(@)5 b(\022)p 881 1105 106 4 v 887 1197 a(@)g(t)1018 1129 y Fn(+)22 b(\()p Fk(u)g Fj(\001)f(r)p Fn(\))p Fk(\022)26 b Fn(+)1581 1061 y(2)p 1581 1105 49 4 v 1581 1197 a(3)1639 1129 y Fk(\022)s Fn(div)p Fk(u)60 b Fn(=)g Fj(\000)2159 1061 y Fn(1)p 2159 1105 V 2164 1197 a Fk(\017)2228 1061 y Fn(1)22 b Fj(\000)h Fk(e)p Fn(\()p Fk(\022)s Fn(\))2568 1025 y Fi(2)p 2228 1105 380 4 v 2394 1197 a Fn(4)2617 1129 y Fk(\032G)p Fn(\()p Fk(\032)p Fn(\))p Fk(\022)2918 1087 y Fi(3)p Fg(=)p Fi(2)3624 800 y Fn(\(7.1\))75 1373 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b(of)h(the)h(constan)m(t)g Fk(S)52 b Fn(in)45 b(our)i(appro)m(ximation)d(\(2.4\))h(is)h(c)m (hosen.)86 b(Th)m(us,)75 2437 y(the)39 b(material)d(of)i(this)g (section)g(relates)h(not)f(only)g(to)g(our)h(mo)s(del,)f(but)h(to)f (the)h(dissipativ)m(e)f(Euler)75 2557 y(equations)33 b(deriv)m(ed)g(directly)f(from)g(the)h(equation)f(\(2.3\).)75 2779 y Fd(Remark)37 b(7.2)49 b Fn(The)31 b(equations)f(\(7.1\))g(with)f (constan)m(t)i Fk(e)g Fn(w)m(ere)g(brie\015y)f(considered)h(b)s(efore)f (b)m(y)h(sev-)75 2899 y(eral)22 b(authors)i(\(see)g([1,)f(8]\).)40 b(Our)24 b(aim)d(is)i(to)g(stress)i(some)e(qualitativ)m(e)e (di\013erences)k(b)s(et)m(w)m(een)g(constan)m(t)75 3019 y(and)33 b(temp)s(erature)f(dep)s(enden)m(t)j(restitution)c(co)s (e\016cien)m(ts)221 3240 y(Th)m(us,)45 b(w)m(e)c(consider)g(separately) g(t)m(w)m(o)g(cases.)69 b(F)-8 b(or)40 b(brevit)m(y)i(w)m(e)f(refer)g (b)s(elo)m(w)g(to)f(our)g(sp)s(eci\014c)75 3361 y(mo)s(del,)29 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1827 3713 a(3)1886 3644 y Fk(t)32 b(u)2009 3659 y Fg(x)2113 3644 y Fn(=)60 b(0)3624 3388 y(\(8.2\))75 3844 y(F)-8 b(rom)31 b(the)i(last)f(t)m(w)m(o)h(equations)g(w)m(e)h(ha) m(v)m(e)1349 4036 y(3)p 1349 4080 V 1349 4171 a(5)1424 4007 y Ff(\020)1473 4103 y Fk(t)1508 4062 y Fh(\000)p Fi(1)1603 4103 y Fk( )1666 4118 y Fg(t)1696 4007 y Ff(\021)1746 4157 y Fg(t)1836 4103 y Fn(=)60 b Fk(u)2028 4118 y Fg(xt)2157 4103 y Fn(=)2320 4036 y Fk(\017)p 2303 4080 75 4 v 2303 4171 a(t)2338 4142 y Fi(2)2387 4103 y Fk( )2450 4118 y Fg(xx)2534 4103 y Fk(:)1063 b Fn(\(8.3\))75 4354 y(Next,)34 b(w)m(e)f(searc)m(h)h(for)e(solutions)g(of)g(\(8.3\))g(of)g(the)h(form) 1366 4574 y Fk( )t Fn(\()p Fk(t;)17 b(x)p Fn(\))60 b(=)h Fk(y)t Fn(\()p Fk(t)p Fn(\))17 b(exp)o(\()p Fk(i)p Fn(\()p Fk(k)26 b Fj(\001)c Fk(x)p Fn(\)\))75 4794 y(then,)33 b Fk(y)t Fn(\()p Fk(t)p Fn(\))f(m)m(ust)h(satisfy)f(the)h(equation)1479 4991 y(3)p 1479 5035 49 4 v 1479 5127 a(5)1555 4962 y Ff(\020)1604 5058 y Fk(t)1639 5017 y Fh(\000)p Fi(1)1734 5058 y Fk(y)1782 5073 y Fg(t)1811 4962 y Ff(\021)1861 5112 y Fg(t)1951 5058 y Fn(=)60 b Fj(\000)2192 4991 y Fk(\017)p 2174 5035 75 4 v 2174 5127 a(t)2209 5098 y Fi(2)2259 5058 y Fk(k)2313 5017 y Fi(2)2352 5058 y Fk(y)t(:)1901 5356 y Fn(23)p eop %%Page: 24 24 24 23 bop 75 107 a Fn(or)32 b(equiv)-5 b(alen)m(tly)d(,)1511 266 y Fk(ty)1598 225 y Fh(00)1662 266 y Fj(\000)22 b Fk(y)1813 225 y Fh(0)1896 266 y Fn(=)60 b Fj(\000)2119 199 y Fn(5)p 2119 243 49 4 v 2119 334 a(3)2178 266 y Fk(\017k)2271 225 y Fi(2)2311 266 y Fk(y)t(:)1234 b Fn(\(8.4\))75 485 y(A)36 b(long)f(time)g(asymptotics)g(is)h(giv)m(en)g(b)m(y)h(the)g (function)e Fk(y)2272 500 y Fh(1)2380 485 y Fn(=)f(exp)q(\()p Fk(\013)2740 405 y Fj(p)p 2822 405 36 4 v 2822 485 a Fk(t)p Fn(\),)j(where)h Fk(\013)e Fn(is)g(found)g(b)m(y)75 606 y(plugging)31 b(in)m(to)g(\(8.4\))h(and)h(matc)m(hing)f(it,)f (obtaining)1680 817 y Fk(\013)1743 781 y Fi(2)p 1680 862 103 4 v 1707 953 a Fn(4)1820 885 y(=)d Fj(\000)2011 817 y Fn(5)p 2011 862 49 4 v 2011 953 a(3)2070 885 y Fk(\017k)2163 844 y Fi(2)2203 885 y Fk(:)75 1136 y Fn(Therefore,)34 b(the)f(linear)e(w)m(a)m(v)m(es)k(solutions)c(of)h(system)i(\(8.3\))e (are)g(asymptotically)f(giv)m(en)h(b)m(y)1300 1368 y Fk( )t Fn(\()p Fk(t;)17 b(x)p Fn(\))27 b(=)h(exp)1874 1272 y Ff(\020)1923 1368 y Fk(ik)s Fn(\()p Fk(x)23 b Fj(\000)g Fn(2)p Fk(\030)2318 1383 y Fg(o)2355 1283 y Fj(p)p 2438 1283 75 4 v 85 x Fk(\017t)q Fn(\))2551 1272 y Ff(\021)75 1633 y Fn(where)47 b Fk(\030)413 1648 y Fg(o)501 1633 y Fn(=)628 1533 y Ff(q)p 711 1533 147 4 v 100 x Fn(5)p Fk(=)p Fn(3)e(on)h(the)g(bac)m(kground)h(of)f(the)g (spatially)e(homogeneous)i(solution,)i(that)d(is,)75 1762 y(substituting)f Fk( )49 b Fn(in)44 b(\(8.2\))h(and)g(solving)e (for)i Fk(\032)g Fn(and)g Fk(u)p Fn(.)80 b(Th)m(us,)49 b(in)44 b(this)h(case)h(w)m(e)g(obtain)e(linear)75 1882 y(stabilit)m(y)c(of)g(the)i(homogeneous)g(solutions)e(whic)m(h)i(is)f (another)g(di\013erence)h(with)f(resp)s(ect)i(to)e(the)75 2003 y(constan)m(t)33 b(case.)221 2123 y(Finally)-8 b(,)30 b(w)m(e)k(remark)e(that)g(the)h(asymptotic)f(phase)i(v)m(elo)s(cit)m(y) e(of)g(the)h(w)m(a)m(v)m(es)i(is)d(giv)m(en)h(b)m(y)1184 2418 y Fk(C)7 b Fn(\()p Fk(t)p Fn(\))28 b(=)f(2)p Fk(\030)1595 2433 y Fg(o)1660 2351 y Fk(d)p 1643 2395 86 4 v 1643 2487 a(dt)1739 2333 y Fj(p)p 1822 2333 75 4 v 85 x Fk(\017t)h Fn(=)2027 2257 y Ff(s)p 2110 2257 129 4 v 2120 2351 a Fn(5)p 2120 2395 49 4 v 2120 2487 a(3)2189 2351 y Fk(\017)p 2189 2395 40 4 v 2191 2487 a(t)2266 2418 y Fn(=)2370 2257 y Ff(s)p 2453 2257 264 4 v 2463 2351 a Fn(5)p 2463 2395 49 4 v 2463 2487 a(3)2521 2418 y Fk(\022)2566 2433 y Fg(o)2605 2418 y Fn(\()p Fk(t)p Fn(\))75 2680 y(and)33 b(coincides)f(with)g(the)h(time)e(dep)s(enden)m(t)k(sp)s(eed)f(of)e (sound)h(in)f(the)h(usual)f(Euler)h(gas.)75 3020 y Fd(Ac)m(kno)m (wledgmen)m(ts)221 3140 y Fn(The)f(\014rst)f(author)g(thanks)h(TICAM)g (for)e(kind)h(hospitalit)m(y)e(and)i(\014nancial)e(supp)s(ort)i(during) f(his)75 3261 y(visit)21 b(in)h(No)m(v)m(em)m(b)s(er-Decem)m(b)s(er)h (1998.)39 b(The)23 b(second)h(author)e(is)g(supp)s(orted)h(b)m(y)g(the) g(DGESIC)f(MEC-)75 3381 y(Spain)28 b(P)m(erfeccionamien)m(to)g(de)g(Do) s(ctores)g(en)h(el)f(Extranjero)h(fello)m(wship.)41 b(P)m(artially)26 b(supp)s(orted)j(b)m(y)75 3502 y(Russian)47 b(Basic)g(Researc)m(h)i(F) -8 b(oundation)46 b(\(gran)m(t)h(Boltzmann\),)j(TMR)d(Europ)s(ean)h (Union)f(ERB)75 3622 y(FMBX-CT97-0157)31 b(and)i(DGES)f(\(Spain\))g (PB95-1203.)75 4075 y Fo(References)124 4294 y Fn([1])49 b(Bellomo,)30 b(N.,)k(Esteban,)h(M.,)f(Lac)m(ho)m(wicz,)g(M.,)g Fc(Nonline)-5 b(ar)35 b(kinetic)g(e)-5 b(quations)35 b(with)g(dissip)-5 b(a-)276 4415 y(tive)34 b(c)-5 b(ol)5 b(lisions)p Fn(,)31 b(Appl.)i(Math.)g(Letter,)g Fd(8)p Fn(,)f(47-52)f(\(1995\).)124 4618 y([2])49 b(Benedetto,)36 b(D.,)e(Caglioti,)e(E.,)j Fc(The)h(c)-5 b(ol)5 b(lapse)35 b(phenomenon)g(in)h(one)g(dimensional)f(inelastic)276 4738 y(p)-5 b(oint)34 b(p)-5 b(article)35 b(system)p Fn(,to)d(app)s(ear)g(in)g(Ph)m(ysica)i(D.)124 4942 y([3])49 b(Benedetto,)30 b(D.,)f(Caglioti,)e(E.,)j(Carrillo,)d(J.)i(A.,)h (Pulviren)m(ti,)e(M.,)i Fc(A)i(non-maxwel)5 b(lian)29 b(ste)-5 b(ady)276 5062 y(distribution)32 b(for)g(one-dimensional)e(gr) -5 b(anular)33 b(me)-5 b(dia)p Fn(,)29 b(J.)i(Stat.)f(Ph)m(ys.,)i Fd(91)p Fn(,)f(979-990)d(\(1998\).)1901 5356 y(24)p eop %%Page: 25 25 25 24 bop 124 107 a Fn([4])49 b(Benedetto,)30 b(D.,)f(Caglioti,)d(E.,)k (Pulviren)m(ti,)e(M.,)i Fc(A)i(Kinetic)e(Equation)h(for)g(Gr)-5 b(anular)31 b(Me)-5 b(dia)p Fn(,)276 227 y(Math.)32 b(Mo)s(d.)h(and)g (Num.)f(An.,)h Fd(31)p Fn(,)g(615-641)d(\(1997\).)124 431 y([5])49 b(Bizon,)23 b(C.,)i(Shattuc)m(k,)h(Swift,)e(J.)f(B.,)h(M.) f(D.,)h(McCormic)m(k,)h(W.)d(D.,)j(Swinney)-8 b(,)25 b(H.)d(L.,)j Fc(Patterns)276 551 y(in)40 b(3D)g(vertic)-5 b(al)5 b(ly)40 b(oscil)5 b(late)-5 b(d)40 b(gr)-5 b(anular)40 b(layers:)56 b(simulation)39 b(and)h(exp)-5 b(eriment)p Fn(,)40 b(Ph)m(ys.)g(Rev.)276 671 y(Letters,)33 b Fd(80)p Fn(,)g(57-60)e(\(1998\).)124 875 y([6])49 b(Bob)m(ylev,)31 b(A.)f(V.,)h Fc(Exact)h(solutions)h(of)f(the)g(nonline)-5 b(ar)32 b(Boltzmann)f(e)-5 b(quation)32 b(and)g(the)h(the)-5 b(ory)276 995 y(of)34 b(r)-5 b(elaxation)34 b(of)h(a)f(maxwel)5 b(lian)34 b(gas)p Fn(,)e(T)-8 b(eoret.)33 b(Mat.)g(Fiz.,)e Fd(60)p Fn(,)i(280-310)e(\(1984\).)124 1199 y([7])49 b(Bob)m(ylev,)39 b(A.)e(V.,)i Fc(The)f(the)-5 b(ory)39 b(of)g(the)g(nonline)-5 b(ar)38 b(sp)-5 b(atial)5 b(ly)39 b(uniform)g(Boltzmann)e(e)-5 b(quation)276 1319 y(for)34 b(Maxwel)5 b(l)34 b(mole)-5 b(cules)p Fn(,)32 b(So)m(v.)h(Sci.)g(Rev.)g (C.)g(Math.)g(Ph)m(ys.,)h Fd(7)p Fn(,)f(111-233)e(\(1988\).)124 1522 y([8])49 b(Brey)-8 b(,)31 b(J.)g(J.,)g(Duft)m(y)-8 b(,)31 b(J.)g(W.,)g(San)m(tos,)h(A.,)f Fc(Dissip)-5 b(ative)32 b(dynamics)g(for)h(har)-5 b(d)32 b(spher)-5 b(es)p Fn(,)30 b(J.)h(Stat.)276 1643 y(Ph)m(ys.,)j Fd(87)p Fn(,)f(1051-1068)d (\(1997\).)124 1846 y([9])49 b(Cercignani,)27 b(C.,)j(Illner,)d(R.,)i (Pulviren)m(ti,)e(M.,)i Fc(\\The)h(mathematic)-5 b(al)29 b(the)-5 b(ory)31 b(of)f(dilute)g(gases")p Fn(,)276 1967 y(Springer)i(series)h(in)f(Applied)g(Mathematical)e(Sciences,)k(v)m(ol) e(106,)g(Springer-V)-8 b(erlag)31 b(1994.)75 2170 y([10])49 b(Constan)m(tin,)40 b(P)-8 b(.,)40 b(Grossman,)f(E.,)h(Mungan,)h(M.,)f (,)g Fc(Inelastic)f(c)-5 b(ol)5 b(lisions)39 b(of)g(thr)-5 b(e)g(e)41 b(p)-5 b(articles)276 2290 y(on)34 b(a)h(line)f(as)h(a)f (two)h(dimensional)e(bil)5 b(liar)-5 b(d)p Fn(,)32 b(Ph)m(ysica)h(D,)f Fd(83)p Fn(,)h(409-420)e(\(1995\).)75 2494 y([11])49 b(Esteban,)k(M.)c(J.,)j(P)m(erthame,)h(B.,)f Fc(On)d(the)g(mo)-5 b(di\014e)g(d)48 b(Ensko)-5 b(g)49 b(e)-5 b(quation)49 b(for)g(elastic)g(and)276 2614 y(inelastic)39 b(c)-5 b(ol)5 b(lisions.)38 b(Mo)-5 b(dels)40 b(with)g(spin)p Fn(,)f(Ann.)g(Inst.)g(H.)f(P)m(oincar)g(Anal.)g(Non)g(Linaire,)g Fd(8)p Fn(,)276 2734 y(289-308)30 b(\(1991\).)75 2938 y([12])49 b(Gidasp)s(o)m(w,)c(D.)f Fc(Multiphase)h(Flow)f(and)g (Fluidization,)j(Continuum)d(and)h(Kinetic)f(The)-5 b(ory)276 3058 y(Descriptions)p Fn(,)31 b(Academic)h(Press,)i(1994.)75 3262 y([13])49 b(Goldman,)26 b(D.,)h(Shattuc)m(k,)j(M.)e(D.,)f(Bizon,)h (C.,)h(McCormic)m(k,)g(W.)e(D.,)h(Swift,)f(J.)h(B.,)g(Swinney)-8 b(,)276 3382 y(H.)31 b(L.,)h Fc(A)n(bsenc)-5 b(e)33 b(of)g(inelastic)g (c)-5 b(ol)5 b(lapse)33 b(in)g(a)h(r)-5 b(e)g(alistic)34 b(thr)-5 b(e)g(e)33 b(b)-5 b(al)5 b(l)34 b(mo)-5 b(del)p Fn(,)30 b(Ph)m(ys.)k(Rev.)e(E,)g Fd(57)p Fn(,)276 3502 y(4831-4833)e(\(1998\).)75 3706 y([14])49 b(Goldsh)m(tein,)32 b(A.,)h(Shapiro,)g(M.,)g Fc(Me)-5 b(chanics)35 b(of)g(c)-5 b(ol)5 b(lisional)33 b(motion)i(of)g(gr)-5 b(anular)35 b(materials.)276 3826 y(Part)g(I.)f(Gener)-5 b(al)35 b(hydr)-5 b(o)g(dynamic)34 b(e)-5 b(quations)p Fn(,)32 b(J.)g(Fluid)f(Mec)m(h.,)j Fd(282)p Fn(,)f(75-114)e(\(1995\).)75 4030 y([15])49 b(Jenkins,)39 b(J.)f(T.,)i(Ric)m(hman,)e(M.)g(W.,)h Fc(Gr)-5 b(ad's)39 b(13-moment)f(system)i(for)f(a)h(dense)e(gas)h(of)g (in-)276 4150 y(elastic)34 b(spher)-5 b(es)p Fn(,)32 b(Arc)m(h.)h(Rat.)f(Mec)m(h.)i(Anal.,)e Fd(87)p Fn(,)h(355-377)e (\(1985\).)75 4353 y([16])49 b(Mathews,)34 b(J.,)f(W)-8 b(alk)m(er,)32 b(R.,)h Fc(Mathematic)-5 b(al)34 b(metho)-5 b(ds)34 b(of)h(physics)p Fn(,)d(Addison-W)-8 b(esley)33 b(1970.)75 4557 y([17])49 b(Sela,)31 b(N.,)i(Goldhirsc)m(h,)e(I.,)i Fc(Hydr)-5 b(o)g(dynamic)34 b(e)-5 b(quations)34 b(for)g(r)-5 b(apid)34 b(\015ows)h(of)f(smo)-5 b(oth)34 b(inelastic)276 4677 y(spher)-5 b(es,)33 b(to)i(Burnett)h(or)-5 b(der)p Fn(,)32 b(J.)h(Fluid)d(Mec)m(h.,)35 b Fd(361)p Fn(,)d(41-74)g (\(1998\).)1901 5356 y(25)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9904231407150--