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)659 4886 y Fk(@)5 b( )p 659 4930 124 4 v 675 5022 a(@)g(t)810 4953 y Fn(\()p Fk(x)903 4968 y Fh(2)943 4953 y Fk(;)17 b(x)1042 4968 y Fh(3)1081 4953 y Fk(;)g(t)p Fn(\))p Fk(;)g Fn(0)p Fk(;)g Fn(0)1384 4807 y Fm(!)1487 4953 y Fj(\001)39 b Fk(\027)51 b Fn(=)1773 4807 y Fm( )1848 4886 y Fk(@)5 b( )p 1848 4930 V 1864 5022 a(@)g(t)1982 4807 y Fm(!)33 b(,)2204 4765 y(v)2204 4812 y(u)2204 4862 y(u)2204 4911 y(t)p 2292 4765 1041 4 v 42 x Fn(1)38 b(+)2494 4807 y Fm( )2583 4886 y Fk(@)5 b( )p 2569 4930 152 4 v 2569 5022 a(@)g(x)2680 5037 y Fh(2)2731 4807 y Fm(!)2797 4830 y Fh(2)2875 4953 y Fn(+)2990 4807 y Fm( )3079 4886 y Fk(@)g( )p 3065 4930 V 3065 5022 a(@)g(x)3176 5037 y Fh(3)3227 4807 y Fm(!)3293 4830 y Fh(2)3349 4953 y Fk(:)187 b Fn(\(22\))p eop %%Page: 6 6 6 5 bop 0 100 a Fq(On)38 b(the)f(Collapse)g(of)i(Tubes)e(Carried)h(by)f (3D)h(Incompressible)e(Flo)n(ws)p 0 113 3268 4 v 420 w Fn(6)0 407 y(Similarly)-8 b(,)30 b(subtracting)k(the)h(t)m(w)m(o)f 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b(b)p Fn(],)35 b(and)g(let)0 4672 y Fk(t)35 4687 y Fh(0)102 4672 y Fj(2)28 b Fn(\(0)p Fk(;)17 b(T)d Fn(\))32 b(b)s(e)h(a)f(time)f (to)i(b)s(e)f(pic)m(k)m(ed)i(b)s(elo)m(w.)43 b(W)-8 b(e)33 b(de\014ne)1208 5100 y Fk(B)5 b Fn(\()p Fk(t)p Fn(\))28 b(=)g Fk(b)22 b Fj(\000)1727 4967 y Fl(T)1693 4983 y Fm(Z)1703 5222 y Fl(t)1809 5100 y Fn(max)1829 5160 y Fl(x)p Fp(2)p Fl(Q)2024 5100 y Fj(j)p Fk(u)p Fn(\()p Fk(x;)17 b(\034)11 b Fn(\))p Fj(j)17 b Fk(d\034)27 b(;)1035 b Fn(\(30\))p eop %%Page: 9 9 9 8 bop 0 100 a Fq(On)38 b(the)f(Collapse)g(of)i(Tubes)e(Carried)h(by)f (3D)h(Incompressible)e(Flo)n(ws)p 0 113 3268 4 v 420 w Fn(9)0 407 y(and)1207 603 y Fk(A)p Fn(\()p Fk(t)p Fn(\))28 b(=)g Fk(a)22 b Fn(+)1728 469 y Fl(T)1694 485 y Fm(Z)1704 725 y Fl(t)1810 603 y Fn(max)1830 662 y Fl(x)p Fp(2)p Fl(Q)2025 603 y Fj(j)p Fk(u)p Fn(\()p Fk(x;)17 b(\034)11 b Fn(\))p Fj(j)17 b Fk(d\034)27 b(:)1034 b Fn(\(31\))0 1019 y(W)-8 b(e)24 b(are)g(assuming)g(that)f Fk(u)p Fn(\()p Fk(x;)17 b(\034)11 b Fn(\))24 b(is)g(con)m(tin)m(uous)h(on)f Fk(Q)5 b Fj(\002)g Fn([0)p Fk(;)17 b(T)d Fn(\),)25 b(and)f(that)2775 932 y Fl(T)2764 948 y Fm(R)2766 1097 y Fh(0)2836 1019 y Fn(max)2856 1079 y Fl(x)p Fp(2)p Fl(Q)3034 1019 y Fj(j)p Fk(u)p Fn(\()p Fk(x;)17 b(\034)11 b Fn(\))p Fj(j)p Fk(d\034)39 b(<)27 b Fj(1)p Fn(.)0 1193 y(It)33 b(follo)m(ws)e(that)h Fk(A)p Fn(\()p Fk(t)p Fn(\))p Fk(;)17 b(B)5 b Fn(\()p Fk(t)p Fn(\))33 b(are)g Fk(C)1328 1157 y Fh(1)1399 1193 y Fn(functions)g(on)f([0)p Fk(;)17 b(T)d Fn(\),)32 b(and)h(that)1118 1575 y Fk(a)28 b Fj(\024)g Fk(A)p Fn(\()p Fk(t)p Fn(\))g Fk(<)f(B)5 b Fn(\()p Fk(t)p Fn(\))28 b Fj(\024)g Fk(b)33 b Fn(for)f Fk(t)c Fj(2)g Fn([)p Fk(t)2382 1590 y Fh(0)2422 1575 y Fk(;)17 b(T)d Fn(\))j Fk(;)944 b Fn(\(32\))0 1861 y(pro)m(vided)33 b(w)m(e)g(pic)m(k)f Fk(t)784 1876 y Fh(0)856 1861 y Fn(close)g(enough)g(to)g Fk(T)14 b Fn(.)43 b(W)-8 b(e)33 b(pic)m(k)f Fk(t)2092 1876 y Fh(0)2164 1861 y Fn(so)g(that)g(\(32\))g(holds.)43 b(Th)m(us,)33 b(\012)3336 1876 y Fl(t)3366 1861 y Fn(\()p Fk(J)3458 1876 y Fl(t)3488 1861 y Fn(\))28 b Fj(\032)g Fk(Q)0 2011 y Fn(for)k Fk(t)c Fj(2)g Fn([)p Fk(t)368 2026 y Fh(0)408 2011 y Fk(;)17 b(T)d Fn(\).)43 b(Immediately)30 b(from)i(\(30\),)g (\(31\),)g(w)m(e)h(obtain)581 2393 y Fk(B)660 2351 y Fp(0)684 2393 y Fn(\()p Fk(t)p Fn(\))27 b(=)44 b Fj(\000)p Fk(A)1092 2351 y Fp(0)1116 2393 y Fn(\()p Fk(t)p Fn(\))28 b(=)g(max)1378 2452 y Fl(x)p Fp(2)p Fl(Q)1573 2393 y Fj(j)p Fk(u)p Fn(\()p Fk(x;)17 b(t)p Fn(\))p Fj(j)44 b(\025)301 b Fn(max)2061 2456 y Fl(x)p Fp(2)p Fh(\012)2199 2464 y Fb(t)2226 2456 y Fh(\()p Fl(A)p Fh(\()p Fl(t)p Fh(\)\))14 b Fp([)e Fh(\012)2536 2464 y Fb(t)2564 2456 y Fh(\()p Fl(B)s Fh(\()p Fl(t)p Fh(\)\))2794 2393 y Fj(j)p Fk(u)2878 2408 y Fh(3)2917 2393 y Fn(\()p Fk(x;)17 b(t)p Fn(\))p Fj(j)408 b Fn(\(33\))0 2701 y(\(recall)31 b Fk(u)c Fn(=)h(\()p Fk(u)579 2716 y Fh(1)618 2701 y Fk(;)17 b(u)718 2716 y Fh(2)756 2701 y Fk(;)g(u)856 2716 y Fh(3)895 2701 y Fn(\)\).)0 2934 y(>F)-8 b(rom)31 b(\(29\))h(and)h(\(33\))f(w)m(e)h (see)h(at)e(once)h(that)1200 3248 y Fk(d)p 1183 3292 86 4 v 1183 3384 a(dt)1295 3316 y Fn(V)-8 b(ol)16 b(\012)1523 3331 y Fl(t)1553 3316 y Fn(\()p Fk(J)1645 3331 y Fl(t)1674 3316 y Fn(\))44 b Fj(\025)h Fn(0)32 b(for)g Fk(t)c Fj(2)g Fn([)p Fk(t)2327 3331 y Fh(0)2367 3316 y Fk(;)17 b(T)d Fn(\))j Fk(:)999 b Fn(\(34\))0 3602 y(On)33 b(the)g(other)f(hand,)h (\(4\))f(and)h(\(32\))f(sho)m(w)i(that)e(V)-8 b(ol)16 b(\012)2090 3617 y Fl(t)2115 3626 y Fd(0)2154 3602 y Fn(\()p Fk(J)2246 3617 y Fl(t)2271 3626 y Fd(0)2310 3602 y Fn(\))45 b Fk(>)f Fn(0.)0 3834 y(Consequen)m(tly)-8 b(,)690 4186 y(lim)17 b(inf)724 4245 y Fl(t)p Fp(!)p Fl(T)10 b Fp(\000)993 4186 y Fn(V)-8 b(ol)16 b(\012)1221 4201 y Fl(t)1295 4186 y Fj(\025)45 b Fn(lim)17 b(inf)1451 4245 y Fl(t)p Fp(!)p Fl(T)10 b Fp(\000)1720 4186 y Fn(V)-8 b(ol)16 b(\012)1948 4201 y Fl(t)1978 4186 y Fn(\()p Fk(J)2070 4201 y Fl(t)2099 4186 y Fn(\))44 b Fj(\025)h Fn(V)-8 b(ol)16 b(\012)2531 4201 y Fl(t)2556 4210 y Fd(0)2595 4186 y Fn(\()p Fk(J)2687 4201 y Fl(t)2712 4210 y Fd(0)2751 4186 y Fn(\))45 b Fk(>)f Fn(0)17 b Fk(:)0 4518 y Fn(The)34 b(pro)s(of)d(of)h(our)h(theorem)f(is)g(complete.)2047 b Fa(\004)p eop %%Page: 10 10 10 9 bop 0 100 a Fq(On)38 b(the)f(Collapse)g(of)i(Tubes)e(Carried)h(by) f(3D)h(Incompressible)e(Flo)n(ws)p 0 113 3268 4 v 371 w Fn(10)0 407 y Fo(5)215 b Fr(References)119 696 y Fn(1.)49 b(J.)29 b(Beale,)g(T.)g(Kato,)f(and)h(A.)g(Ma)5 b(jda,)29 b(\\Remarks)g(on)f(the)h(breakdo)m(wn)h(of)e(smo)s(oth)g(solutions)244 846 y(for)k(the)h(3D)f(Euler)g(equations,")h Fg(Comm.)44 b(Math.)h(Phys.)p Fn(,)32 b Fc(94)p Fn(,)h(pp.61-64,)f(\(1984\).)119 1078 y(2.)49 b(P)-8 b(.)29 b(Constan)m(tin,)h(C.)f(F)-8 b(e\013erman,)29 b(and)g(A.)g(Ma)5 b(jda,)30 b(\\Geometric)d(constrain) m(ts)i(on)g(p)s(oten)m(tially)244 1228 y(singular)41 b(solutions)g(for)h(the)h(3D)f(Euler)g(equations,")j Fg(Commun.)71 b(Part.)i(Di\013.)e(Eq.)p Fn(,)45 b Fc(21)p Fn(,)244 1377 y(pp.559-571,)31 b(\(1996\).)119 1610 y(3.)49 b(D.)38 b(Cordoba)g(and)g(C.)h(F)-8 b(e\013erman,)39 b(\\Scalars)f(con)m(v)m(ected)j(b)m(y)e(a)f(2D)f(incompressible)g (\015o)m(w,")244 1759 y(\(to)32 b(app)s(ear\).)119 1992 y(4.)49 b(D.)32 b(Cordoba)i(and)f(C.)g(F)-8 b(e\013erman,)33 b(\\Beha)m(vior)g(of)f(sev)m(eral)i(2D)e(\015uid)h(equations)g(in)f (singular)244 2141 y(scenarios,")h(submitted)f(to)g Fg(Pr)-5 b(o)g(c.)45 b(Nat.)g(A)-5 b(c)g(ad.)45 b(Sci.)p Fn(,)32 b(U.S.A.)0 5186 y(Jan)m(uary)h(29,)f(2001)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF