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(incompressible)g(v)m(elo)s(cit)m(y)h(\014eld)f(u\(x,t\))h(as)g(ab)s(o) m(v)m(e)h(with)324 4227 y(v)m(orticit)m(y)32 b(denoted)i(b)m(y)f Fk(!)t Fn(.)43 b(The)33 b(2D)f(Euler)h(equation)f(ma)m(y)g(b)s(e)h (written)f(in)g(the)h(form)1745 4447 y(\()p Fk(@)1834 4462 y Fg(t)1886 4447 y Fn(+)22 b Fk(u)g Ff(\001)g(r)2195 4462 y Fg(x)2239 4447 y Fn(\))16 b Fk(!)31 b Fn(=)d(0)899 b(\(2\))1182 4592 y Fk(u)27 b Fn(=)h Ff(r)1452 4551 y Fi(?)1452 4617 y Fg(x)1511 4592 y Fk( )69 b(and)c( )31 b Fn(=)d(\()p Ff(\0004)2270 4607 y Fg(x)2314 4592 y Fn(\))2352 4551 y Fi(\000)p Fj(1)2446 4592 y Fk(!)t(;)324 4812 y Fn(with)k(u\(x,0\))h(equal)f(to)h(a)f(giv)m(en)h(smo)s(oth)e(div)m (ergence)j(free)f Fk(u)2613 4827 y Fj(0)2652 4812 y Fn(\()p Fk(x)p Fn(\).)1918 5251 y(3)p eop %%Page: 4 4 4 3 bop 324 548 a Fo(3)161 b(Results)324 767 y Fn(Asssume)33 b(that)e(q)i(=)e(q\(x,t\))i(is)e(a)g(solution)g(to)g(\(1\))g(or)h (\(2\),)g(and)f(that)h(a)f(lev)m(el)h(curv)m(e)h(of)324 887 y(q)g(can)g(b)s(e)f(parameterized)g(b)m(y)1228 1218 y Fk(x)1283 1233 y Fj(2)1351 1218 y Fn(=)27 b Fk(\036)1512 1233 y Fg(\032)1552 1218 y Fn(\()p Fk(x)1645 1233 y Fj(1)1685 1218 y Fk(;)17 b(t)p Fn(\))65 b Fk(f)11 b(or)67 b(x)2139 1233 y Fj(1)2207 1218 y Ff(2)28 b Fn([)p Fk(a;)17 b(b)p Fn(])946 b(\(3\))324 1429 y(with)32 b Fk(\036)604 1444 y Fg(\032)672 1429 y Ff(2)c Fk(C)843 1392 y Fj(1)882 1429 y Fn(\([)p Fk(a;)17 b(b)p Fn(])23 b Ff(\\)f Fn([0)p Fk(;)17 b(T)1412 1392 y Fi(\003)1451 1429 y Fn(\)\),)33 b(in)e(the)i(sense)i(that)990 1639 y Fk(q)t Fn(\()p Fk(x)1130 1654 y Fj(1)1170 1639 y Fk(;)17 b(\036)1272 1654 y Fg(\032)1312 1639 y Fn(\()p Fk(x)1405 1654 y Fj(1)1444 1639 y Fk(;)g(t)p Fn(\))p Fk(;)g(t)p Fn(\))28 b(=)f Fk(G)p Fn(\()p Fk(\032)p Fn(\))66 b Fk(f)11 b(or)67 b(x)2350 1654 y Fj(1)2417 1639 y Ff(2)28 b Fn([)p Fk(a;)17 b(b)p Fn(])p Fk(;)709 b Fn(\(4\))324 1849 y(and)32 b(for)h(certain)f Fk(\032)g Fn(to)h(b)s(e)g(sp)s(eci\014ed)g(b)s(elo)m(w.)470 1970 y(The)h(stream)e(function)g Fk( )k Fn(satis\014es)1648 2180 y Ff(r)1731 2139 y Fi(?)1790 2180 y Fk( )c Fn(=)c Fk(u:)1365 b Fn(\(5\))470 2391 y(F)-8 b(rom)31 b(\(3\))h(and)h(\(4\),)f (w)m(e)i(ha)m(v)m(e)1484 2583 y Fk(@)5 b(q)p 1460 2627 152 4 v 1460 2719 a(@)g(x)1571 2734 y Fj(1)1644 2650 y Fn(+)1776 2583 y Fk(@)g(q)p 1752 2627 V 1752 2719 a(@)g(x)1863 2734 y Fj(2)1924 2583 y Fk(@)g(\036)2038 2598 y Fg(\032)p 1924 2627 155 4 v 1926 2719 a Fk(@)g(x)2037 2734 y Fj(1)2117 2650 y Fn(=)27 b(0)1168 b(\(6\))1484 2991 y Fk(@)5 b(q)p 1484 3036 104 4 v 1490 3127 a(@)g(t)1620 3059 y Fn(+)1752 2991 y Fk(@)g(q)p 1728 3036 152 4 v 1728 3127 a(@)g(x)1839 3142 y Fj(2)1900 2991 y Fk(@)g(\036)2014 3006 y Fg(\032)p 1900 3036 155 4 v 1931 3127 a Fk(@)g(t)2093 3059 y Fn(=)27 b(0)1192 b(\(7\))470 3315 y(By)33 b(\(1\),)f(\(2\),)h(\(5\),)f(\(6\))g (and)h(\(7\))f(w)m(e)i(obtain)951 3543 y Fk(@)5 b(\036)1065 3558 y Fg(\032)p 951 3588 V 982 3679 a Fk(@)g(t)1199 3610 y Fn(=)82 b Ff(\000)1475 3493 y Fg(@)t(q)p 1475 3514 75 4 v 1479 3572 a(@)t(t)p 1444 3588 136 4 v 1475 3646 a(@)t(q)p 1454 3667 116 4 v 1454 3724 a(@)t(x)1535 3733 y Fd(2)1618 3610 y Fn(=)1731 3528 y Fk(<)28 b Ff(\000)1935 3484 y Fg(@)t( )p 1922 3505 V 1922 3562 a(@)t(x)2003 3571 y Fd(2)2048 3528 y Fk(;)2114 3484 y Fg(@)t( )p 2101 3505 V 2101 3562 a(@)t(x)2182 3571 y Fd(1)2254 3528 y Fk(>)g Ff(\001)f Fk(<)2547 3484 y Fg(@)t(q)p 2527 3505 V 2527 3562 a(@)t(x)2608 3571 y Fd(1)2652 3528 y Fk(;)2726 3484 y Fg(@)t(q)p 2706 3505 V 2706 3562 a(@)t(x)2787 3571 y Fd(2)2859 3528 y Fk(>)p 1731 3588 1204 4 v 2296 3646 a Fg(@)t(q)p 2275 3667 116 4 v 2275 3724 a(@)t(x)2356 3733 y Fd(2)1199 3939 y Fn(=)82 b Fk(<)28 b Ff(\000)1562 3871 y Fk(@)5 b( )p 1548 3916 152 4 v 1548 4007 a(@)g(x)1659 4022 y Fj(2)1710 3939 y Fk(;)1778 3871 y(@)g( )p 1764 3916 V 1764 4007 a(@)g(x)1875 4022 y Fj(1)1953 3939 y Fk(>)28 b Ff(\001)f Fk(<)2256 3812 y Fg(@)t(q)p 2236 3833 116 4 v 2236 3891 a(@)t(x)2317 3900 y Fd(1)p 2226 3916 136 4 v 2256 3974 a Fg(@)t(q)p 2236 3995 116 4 v 2236 4052 a(@)t(x)2317 4061 y Fd(2)2371 3939 y Fk(;)17 b Fn(1)27 b Fk(>)1199 4231 y Fn(=)82 b Fk(<)28 b Ff(\000)1562 4164 y Fk(@)5 b( )p 1548 4208 152 4 v 1548 4299 a(@)g(x)1659 4314 y Fj(2)1710 4231 y Fk(;)1778 4164 y(@)g( )p 1764 4208 V 1764 4299 a(@)g(x)1875 4314 y Fj(1)1953 4231 y Fk(>)28 b Ff(\001)f Fk(<)h Ff(\000)2303 4164 y Fk(@)5 b(\036)2417 4179 y Fg(\032)p 2303 4208 155 4 v 2305 4299 a Fk(@)g(x)2416 4314 y Fj(1)2468 4231 y Fk(;)17 b Fn(1)27 b Fk(>)324 4481 y Fn(Next)740 4648 y Fk(@)p 692 4693 152 4 v 692 4784 a(@)5 b(x)803 4799 y Fj(1)871 4716 y Fn(\()p Fk( )t Fn(\()p Fk(x)1069 4731 y Fj(1)1108 4716 y Fk(;)17 b(\036)1210 4731 y Fg(\032)1250 4716 y Fn(\()p Fk(x)1343 4731 y Fj(1)1383 4716 y Fk(;)g(t)p Fn(\))p Fk(;)g(t)p Fn(\)\))83 b(=)1920 4648 y Fk(@)5 b( )p 1906 4693 V 1906 4784 a(@)g(x)2017 4799 y Fj(1)2090 4716 y Fn(+)2212 4648 y Fk(@)g( )p 2198 4693 V 2198 4784 a(@)g(x)2309 4799 y Fj(2)2370 4648 y Fk(@)g(\036)2484 4663 y Fg(\032)p 2370 4693 155 4 v 2372 4784 a Fk(@)g(x)2483 4799 y Fj(1)1738 4969 y Fn(=)82 b Fk(<)28 b Ff(\000)2101 4901 y Fk(@)5 b( )p 2087 4946 152 4 v 2087 5037 a(@)g(x)2198 5052 y Fj(2)2249 4969 y Fk(;)2317 4901 y(@)g( )p 2303 4946 V 2303 5037 a(@)g(x)2414 5052 y Fj(1)2492 4969 y Fk(>)28 b Ff(\001)f Fk(<)h Ff(\000)2842 4901 y Fk(@)5 b(\036)2956 4916 y Fg(\032)p 2842 4946 155 4 v 2844 5037 a Fk(@)g(x)2955 5052 y Fj(1)3007 4969 y Fk(;)17 b Fn(1)27 b Fk(>)1918 5251 y Fn(4)p eop %%Page: 5 5 5 4 bop 470 548 a Fn(Therefore)1230 710 y Fk(@)5 b(\036)1344 725 y Fg(\032)p 1230 754 155 4 v 1262 845 a Fk(@)g(t)1423 777 y Fn(=)1584 710 y Fk(@)p 1537 754 152 4 v 1537 845 a(@)g(x)1648 860 y Fj(1)1715 777 y Fn(\()p Fk( )t Fn(\()p Fk(x)1913 792 y Fj(1)1953 777 y Fk(;)17 b(\036)2055 792 y Fg(\032)2094 777 y Fn(\()p Fk(x)2187 792 y Fj(1)2227 777 y Fk(;)g(t)p Fn(\))p Fk(;)g(t)p Fn(\)\))938 b(\(8\))470 1022 y(With)29 b(this)g(form)m(ula)e(w)m(e)j(can)f(write)g(a)g (explicit)f(equation)h(for)g(the)g(c)m(hange)h(of)f(time)324 1142 y(of)j(the)h(area)f(b)s(et)m(w)m(een)j(t)m(w)m(o)e(\014xed)h(p)s (oin)m(ts)e(a,)g(b)h(and)g(t)m(w)m(o)g(lev)m(el)f(curv)m(es)j(\()p Fk(\036)3090 1157 y Fg(\032)3126 1166 y Fd(1)3164 1142 y Fk(;)17 b(\036)3266 1157 y Fg(\032)3302 1166 y Fd(2)3341 1142 y Fn(\);)1140 1351 y Fk(d)p 1122 1395 86 4 v 1122 1487 a(dt)1235 1278 y Fh(\022)1308 1283 y(Z)1408 1309 y Fg(b)1363 1508 y(a)1442 1418 y Fn([)p Fk(\036)1527 1433 y Fg(\032)1563 1442 y Fd(2)1602 1418 y Fn(\()p Fk(x)1695 1433 y Fj(1)1734 1418 y Fk(;)g(t)p Fn(\))22 b Ff(\000)h Fk(\036)2031 1433 y Fg(\032)2067 1442 y Fd(1)2106 1418 y Fn(\()p Fk(x)2199 1433 y Fj(1)2238 1418 y Fk(;)17 b(t)p Fn(\)])p Fk(dx)2488 1433 y Fj(1)2528 1278 y Fh(\023)1101 1621 y Fn(=)28 b Fk( )t Fn(\()p Fk(b;)17 b(\036)1453 1636 y Fg(\032)1489 1645 y Fd(2)1528 1621 y Fn(\()p Fk(b;)g(t)p Fn(\))p Fk(;)g(t)p Fn(\))22 b Ff(\000)h Fk( )t Fn(\()p Fk(a;)17 b(\036)2221 1636 y Fg(\032)2257 1645 y Fd(2)2295 1621 y Fn(\()p Fk(a;)g(t)p Fn(\))p Fk(;)g(t)p Fn(\))1129 1766 y(+)p Fk( )t Fn(\()p Fk(a;)g(\036)1463 1781 y Fg(\032)1499 1790 y Fd(1)1537 1766 y Fn(\()p Fk(a;)g(t)p Fn(\))p Fk(;)g(t)p Fn(\))22 b Ff(\000)h Fk( )t Fn(\()p Fk(b;)17 b(\036)2230 1781 y Fg(\032)2266 1790 y Fd(1)2305 1766 y Fn(\()p Fk(b;)g(t)p Fn(\))p Fk(;)g(t)p Fn(\))819 b(\(9\))470 1971 y(Assume)42 b(that)g(t)m(w)m(o)g(lev)m(el)f(curv)m(es)i Fk(\036)1850 1986 y Fg(\032)1886 1995 y Fd(1)1966 1971 y Fn(and)f Fk(\036)2223 1986 y Fg(\032)2259 1995 y Fd(2)2339 1971 y Fn(collapse)e(when)j(t)e(tends)h(to)f Fk(T)3522 1935 y Fi(\003)324 2091 y Fn(uniformly)30 b(in)i Fk(a)c Ff(\024)g Fk(x)1121 2106 y Fj(1)1189 2091 y Ff(\024)g Fk(b)33 b Fn(i.e.)698 2359 y Fk(\036)756 2374 y Fg(\032)792 2383 y Fd(2)830 2359 y Fn(\()p Fk(x)923 2374 y Fj(1)963 2359 y Fk(;)17 b(t)p Fn(\))22 b Ff(\000)h Fk(\036)1260 2374 y Fg(\032)1296 2383 y Fd(1)1334 2359 y Fn(\()p Fk(x)1427 2374 y Fj(1)1467 2359 y Fk(;)17 b(t)p Fn(\))28 b Ff(\030)1810 2291 y Fn(1)p 1727 2336 215 4 v 1727 2427 a Fk(b)22 b Ff(\000)h Fk(a)1968 2223 y Fh(Z)2067 2249 y Fg(b)2023 2449 y(a)2102 2359 y Fn([)p Fk(\036)2187 2374 y Fg(\032)2223 2383 y Fd(2)2261 2359 y Fn(\()p Fk(x)2354 2374 y Fj(1)2394 2359 y Fk(;)17 b(t)p Fn(\))22 b Ff(\000)h Fk(\036)2691 2374 y Fg(\032)2727 2383 y Fd(1)2765 2359 y Fn(\()p Fk(x)2858 2374 y Fj(1)2898 2359 y Fk(;)17 b(t)p Fn(\)])p Fk(dx)3148 2374 y Fj(1)324 2610 y Fn(In)36 b(other)g(w)m(ords;)i(the)e(distance)g (b)s(et)m(w)m(een)i(t)m(w)m(o)e(lev)m(el)g(sets)h(are)e(comparable)g (for)g Fk(a)e Ff(\024)324 2731 y Fk(x)379 2746 y Fj(1)446 2731 y Ff(\024)28 b Fk(b)p Fn(.)470 2851 y(Let)1258 2971 y Fk(\016)t Fn(\()p Fk(x)1398 2986 y Fj(1)1438 2971 y Fk(;)17 b(t)p Fn(\))27 b(=)h Ff(j)p Fk(\036)1772 2986 y Fg(\032)1808 2995 y Fd(2)1846 2971 y Fn(\()p Fk(x)1939 2986 y Fj(1)1979 2971 y Fk(;)17 b(t)p Fn(\))22 b Ff(\000)h Fk(\036)2276 2986 y Fg(\032)2312 2995 y Fd(1)2350 2971 y Fn(\()p Fk(x)2443 2986 y Fj(1)2483 2971 y Fk(;)17 b(t)p Fn(\))p Ff(j)324 3139 y Fn(b)s(e)33 b(the)g(thic)m(kness)h(of)e(the)h (fron)m(t.)470 3260 y(W)-8 b(e)30 b(de\014ne)g(semi-uniform)d(collapse) h(on)h(a)g(curv)m(e)i(if)d(\(3\))h(and)h(\(4\))e(holds)h(and)h(there) 324 3380 y(exists)j(a)f(constan)m(t)i Fk(c)p Fn(,)e(indep)s(enden)m(t)i (of)e(t,)h(suc)m(h)h(that)1338 3585 y Fk(min\016)t Fn(\()p Fk(x)1654 3600 y Fj(1)1695 3585 y Fk(;)17 b(t)p Fn(\))27 b Ff(\025)i Fk(c)22 b Ff(\001)g Fk(max\016)t Fn(\()p Fk(x)2390 3600 y Fj(1)2430 3585 y Fk(;)17 b(t)p Fn(\))324 3789 y(for)32 b Fk(a)c Ff(\024)g Fk(x)712 3804 y Fj(1)779 3789 y Ff(\024)h Fk(b)p Fn(,)k(and)f(for)g(all)f Fk(t)d Ff(2)g Fn([0)p Fk(;)17 b(T)1808 3753 y Fi(\003)1847 3789 y Fn(\).)470 3910 y(W)-8 b(e)33 b(call)e(the)i(length)f(b-a)g(of)g(the) h(in)m(terv)-5 b(al)31 b([a,b])i(the)g(length)f(of)g(the)h(fron)m(t.) 470 4030 y(No)m(w)g(w)m(e)h(can)f(state)g(the)g(follo)m(wing)d(theorem) 324 4220 y Fc(Theorem)37 b(1.)49 b Fb(F)-7 b(or)44 b(a)i(QG)f(solution) g(with)h(a)f(semi-uniform)f(fr)-5 b(ont,)48 b(the)e(thickness)324 4341 y Fk(\016)t Fn(\()p Fk(t)p Fn(\))35 b Fb(satis\014es)1099 4557 y Fk(\016)t Fn(\()p Fk(t)p Fn(\))27 b Fk(>)h(e)1433 4516 y Fi(\000)p Fg(e)1521 4492 y Fa(At)p Fd(+)p Fa(B)1764 4557 y Fk(f)11 b(or)72 b(al)r(l)g(t)28 b Ff(2)g Fn([0)p Fk(;)17 b(T)2517 4516 y Fi(\003)2556 4557 y Fn(\))p Fk(:)324 4761 y Fb(Her)-5 b(e,)30 b(the)f(c)-5 b(onstants)29 b(A)h(and)e(B)h (may)g(b)-5 b(e)29 b(taken)g(to)g(dep)-5 b(end)28 b(only)h(on)g(the)g (length)g(of)g(the)324 4882 y(fr)-5 b(ont,)35 b(the)g(semi-uniformity)f (c)-5 b(onstant,)35 b(the)g(initial)f(thickness)g Fk(\016)t Fn(\(0\))p Fb(,)h(and)f(the)h(norm)324 5002 y(of)f(the)h(initial)g (datum)g Fk(\022)1231 5017 y Fj(0)1270 5002 y Fn(\()p Fk(x)p Fn(\))h Fb(in)e Fk(L)1622 4966 y Fj(1)1684 5002 y Ff(\\)23 b Fk(L)1839 4966 y Fi(1)1914 5002 y Fb(.)1918 5251 y Fn(5)p eop %%Page: 6 6 6 5 bop 470 548 a Fn(Pro)s(of:)43 b(F)-8 b(rom)31 b(\(9\))h(w)m(e)i(ha) m(v)m(e)470 881 y Ff(j)525 814 y Fk(d)p 508 858 86 4 v 508 950 a(dt)603 881 y(A)p Fn(\()p Fk(t)p Fn(\))p Ff(j)28 b Fk(<)1025 814 y(C)p 956 858 215 4 v 956 950 a(b)23 b Ff(\000)g Fk(a)1181 881 y(sup)1332 896 y Fg(a)p Fi(\024)p Fg(x)1464 905 y Fd(1)1498 896 y Fi(\024)p Fg(b)1587 881 y Ff(j)p Fk( )t Fn(\()p Fk(x)1775 896 y Fj(1)1815 881 y Fk(;)17 b(\036)1917 896 y Fg(\032)1953 905 y Fd(2)1991 881 y Fn(\()p Fk(x)2084 896 y Fj(1)2124 881 y Fk(;)g(t)p Fn(\))p Fk(;)g(t)p Fn(\))22 b Ff(\000)g Fk( )t Fn(\()p Fk(x)2639 896 y Fj(1)2679 881 y Fk(;)17 b(\036)2781 896 y Fg(\032)2817 905 y Fd(2)2855 881 y Fn(\()p Fk(x)2948 896 y Fj(1)2988 881 y Fk(;)g(t)p Fn(\))p Fk(;)g(t)p Fn(\))p Ff(j)138 b Fn(\(10\))324 1104 y(where)975 1335 y Fk(A)p Fn(\()p Fk(t)p Fn(\))28 b(=)1383 1268 y(1)p 1300 1312 V 1300 1404 a Fk(b)23 b Ff(\000)g Fk(a)1541 1200 y Fh(Z)1641 1226 y Fg(b)1597 1425 y(a)1675 1335 y Fn([)p Fk(\036)1760 1350 y Fg(\032)1796 1359 y Fd(2)1835 1335 y Fn(\()p Fk(x)1928 1350 y Fj(1)1968 1335 y Fk(;)17 b(t)p Fn(\))83 b Ff(\000)g Fk(\036)2386 1350 y Fg(\032)2422 1359 y Fd(1)2461 1335 y Fn(\()p Fk(x)2554 1350 y Fj(1)2594 1335 y Fk(;)17 b(t)p Fn(\)])p Fk(dx)2844 1350 y Fj(1)2883 1335 y Fk(;)324 1571 y Fn(and)32 b(C)h(is)g(determined)f(b)m(y)h(the)g(semi-uniformit)m (y)d(constan)m(t)j(c.)470 1691 y(The)h(estimate)e(of)g(the)h (di\013erence)g(of)f(the)i(v)-5 b(alue)32 b(of)g(the)h(stream)f (function)h(at)f(t)m(w)m(o)324 1812 y(di\013eren)m(t)27 b(p)s(oin)m(ts)g(that)h(are)f(close)g(to)g(eac)m(h)i(other)e(is)g (obtained)g(b)m(y)h(writing)e(the)i(stream)324 1932 y(function)k(as)h (follo)m(ws;)1256 2170 y Fk( )t Fn(\()p Fk(x;)17 b(t)p Fn(\))28 b(=)f Ff(\000)1758 2034 y Fh(Z)1814 2260 y Fj(\012)1895 2102 y Fk(\022)s Fn(\()p Fk(x)c Fn(+)f Fk(y)t(;)17 b(t)p Fn(\))p 1895 2147 430 4 v 2057 2238 a Ff(j)p Fk(y)t Ff(j)2335 2170 y Fk(dy)t(;)324 2436 y Fn(and)32 b(this)h(is)f(b)s(ecause)i Fk( )d Fn(=)d(\()p Ff(\0004)1565 2451 y Fg(x)1609 2436 y Fn(\))1647 2399 y Fi(\000)1712 2372 y Fd(1)p 1712 2384 31 3 v 1712 2425 a(2)1756 2436 y Fk(\022)s Fn(.)470 2556 y(Therefore)726 2768 y Fk( )t Fn(\()p Fk(z)876 2783 y Fj(1)916 2768 y Fk(;)17 b(t)p Fn(\))22 b Ff(\000)g Fk( )t Fn(\()p Fk(z)1304 2783 y Fj(2)1344 2768 y Fk(;)17 b(t)p Fn(\))84 b(=)1705 2632 y Fh(Z)1760 2858 y Fj(\012)1832 2768 y Fk(\022)s Fn(\()p Fk(y)t Fn(\)\()2187 2701 y(1)p 2056 2745 314 4 v 2056 2836 a Ff(j)p Fk(y)24 b Ff(\000)e Fk(z)2300 2851 y Fj(1)2340 2836 y Ff(j)2400 2768 y(\000)2642 2701 y Fn(1)p 2509 2745 V 2509 2836 a Ff(j)p Fk(y)j Ff(\000)e Fk(z)2755 2851 y Fj(2)2795 2836 y Ff(j)2832 2768 y Fn(\))p Fk(dy)1545 3030 y Fn(=)1705 2894 y Fh(Z)1760 3120 y Fi(j)p Fg(y)r Fi(\000)p Fg(z)1905 3129 y Fd(1)1939 3120 y Fi(j\024)p Fj(2)p Fg(\034)2109 3030 y Fn(+)2202 2894 y Fh(Z)2256 3120 y Fj(2)p Fg(\034)8 b(<)p Fi(j)p Fg(y)r Fi(\000)p Fg(z)2530 3129 y Fd(2)2565 3120 y Fi(j\024)p Fg(k)2699 3030 y Fn(+)2792 2894 y Fh(Z)2846 3120 y Fg(k)r(<)p Fi(j)p Fg(y)r Fi(\000)p Fg(z)3085 3129 y Fd(1)3119 3120 y Fi(j)1544 3238 y Ff(\021)84 b Fk(I)1748 3253 y Fj(1)1809 3238 y Fn(+)22 b Fk(I)1950 3253 y Fj(2)2012 3238 y Fn(+)g Fk(I)2153 3253 y Fj(3)2192 3238 y Fk(:)324 3426 y Fn(where)34 b Fk(\034)39 b Fn(=)27 b Ff(j)p Fk(z)863 3441 y Fj(1)925 3426 y Ff(\000)22 b Fk(z)1069 3441 y Fj(2)1109 3426 y Ff(j)p Fn(.)470 3547 y(F)-8 b(urthermore)902 3759 y Ff(j)p Fk(I)973 3774 y Fj(1)1013 3759 y Ff(j)82 b(\024)i(jj)p Fk(\022)s Ff(jj)1444 3774 y Fg(L)1492 3755 y Fe(1)1582 3759 y Ff(\001)1632 3623 y Fh(Z)1687 3849 y Fi(j)p Fg(y)r Fi(\000)p Fg(z)1832 3858 y Fd(1)1866 3849 y Fi(j\024)p Fj(2)p Fg(\034)2019 3759 y Fn(\()2199 3691 y(1)p 2067 3736 V 2067 3827 a Ff(j)p Fk(y)25 b Ff(\000)e Fk(z)2313 3842 y Fj(1)2352 3827 y Ff(j)2412 3759 y Fn(+)2652 3691 y(1)p 2520 3736 V 2520 3827 a Ff(j)p Fk(y)i Ff(\000)e Fk(z)2766 3842 y Fj(2)2806 3827 y Ff(j)2843 3759 y Fn(\))p Fk(dy)1123 3966 y Ff(\024)84 b Fk(C)7 b(\034)470 4155 y Fn(W)-8 b(e)36 b(de\014ne)g(s)g(to)f(b)s(e)g(a)g(p)s(oin)m(t)f(in)h (the)g(line)f(b)s(et)m(w)m(een)k Fk(z)2496 4170 y Fj(1)2571 4155 y Fn(and)d Fk(z)2808 4170 y Fj(2)2848 4155 y Fn(,)h(then)f Ff(j)p Fk(y)27 b Ff(\000)e Fk(z)3385 4170 y Fj(1)3424 4155 y Ff(j)32 b(\024)324 4275 y Fn(2)p Ff(j)p Fk(y)25 b Ff(\000)d Fk(s)p Ff(j)32 b Fn(and)h Fk(I)912 4290 y Fj(2)984 4275 y Fn(can)g(b)s(e)g(estimated)e(b)m(y)983 4512 y Ff(j)p Fk(I)1054 4527 y Fj(2)1093 4512 y Ff(j)82 b(\024)i Fk(C)7 b(\034)34 b Ff(\001)1566 4377 y Fh(Z)1622 4602 y Fj(2)p Fg(\034)8 b(<)p Fi(j)p Fg(y)r Fi(\000)p Fg(z)1896 4611 y Fd(1)1930 4602 y Fi(j\024)p Fg(k)2064 4512 y Fk(max)2255 4527 y Fg(s)2293 4512 y Ff(jr)p Fn(\()2564 4445 y(1)p 2452 4489 275 4 v 2452 4581 a Ff(j)p Fk(y)24 b Ff(\000)f Fk(s)p Ff(j)2735 4512 y Fn(\))p Ff(j)p Fk(dy)1203 4788 y Ff(\024)84 b Fk(C)7 b(\034)34 b Ff(\001)1566 4652 y Fh(Z)1622 4878 y Fj(2)p Fg(\034)8 b(<)p Fi(j)p Fg(y)r Fi(\000)p Fg(z)1896 4887 y Fd(1)1930 4878 y Fi(j\024)p Fg(k)2064 4788 y Fk(max)2255 4803 y Fg(s)2435 4720 y Fn(1)p 2303 4765 314 4 v 2303 4856 a Ff(j)p Fk(y)24 b Ff(\000)f Fk(s)p Ff(j)2577 4827 y Fj(2)2626 4788 y Fk(dy)1203 5002 y Ff(\024)84 b Fk(C)7 b(\034)34 b Ff(\001)21 b(j)c Fn(log)f Fk(\034)11 b Ff(j)1918 5251 y Fn(6)p eop %%Page: 7 7 7 6 bop 470 548 a Fn(W)-8 b(e)33 b(use)h(the)f(conserv)-5 b(ation)32 b(of)g(energy)i(to)e(estimate)g Fk(I)2512 563 y Fj(3)2584 548 y Fn(b)m(y)1623 752 y Ff(j)p Fk(I)1694 767 y Fj(3)1733 752 y Ff(j)c(\024)g Fk(C)h Ff(\001)22 b Fk(\034)470 956 y Fn(Finally)-8 b(,)30 b(b)m(y)j(c)m(ho)s(osing)g Fk(\034)39 b Fn(=)27 b Ff(j)p Fk(z)1610 971 y Fj(1)1672 956 y Ff(\000)c Fk(z)1817 971 y Fj(2)1856 956 y Ff(j)32 b Fn(w)m(e)i(obtain)907 1160 y Ff(j)p Fk( )t Fn(\()p Fk(z)1085 1175 y Fj(1)1125 1160 y Fk(;)17 b(t)p Fn(\))22 b Ff(\000)g Fk( )t Fn(\()p Fk(z)1513 1175 y Fj(2)1553 1160 y Fk(;)17 b(t)p Fn(\))p Ff(j)27 b(\024)h Fk(M)10 b Ff(j)p Fk(z)2007 1175 y Fj(1)2070 1160 y Ff(\000)22 b Fk(z)2214 1175 y Fj(2)2254 1160 y Ff(jj)p Fk(l)r(og)t Ff(j)p Fk(z)2512 1175 y Fj(1)2572 1160 y Ff(\000)h Fk(z)2717 1175 y Fj(2)2757 1160 y Ff(jj)575 b Fn(\(11\))324 1364 y(where)30 b(M)f(is)f(a)h(constan)m(t)g(that)g(dep)s(end)h(on)f(the)g (initial)c(data)j Fk(\022)2661 1379 y Fj(0)2701 1364 y Fn(.)42 b(\(See)30 b(details)e(in)g([3].\))470 1485 y(Then)34 b(w)m(e)f(ha)m(v)m(e)324 1713 y Ff(j)379 1646 y Fk(d)p 362 1691 86 4 v 362 1782 a(dt)457 1713 y(A)p Fn(\()p Fk(t)p Fn(\))p Ff(j)83 b(\024)978 1646 y Fk(M)p 923 1691 215 4 v 923 1782 a(b)22 b Ff(\000)h Fk(a)1147 1713 y(sup)1298 1728 y Fg(a)p Fi(\024)p Fg(x)1430 1737 y Fd(1)1464 1728 y Fi(\024)p Fg(b)1553 1713 y Ff(j)p Fk(\036)1639 1728 y Fg(\032)1675 1737 y Fd(2)1714 1713 y Fn(\()p Fk(x)1807 1728 y Fj(1)1847 1713 y Fk(;)17 b(t)p Fn(\))22 b Ff(\000)g Fk(\036)2143 1728 y Fg(\032)2179 1737 y Fd(1)2218 1713 y Fn(\()p Fk(x)2311 1728 y Fj(1)2351 1713 y Fk(;)17 b(t)p Fn(\))p Ff(jj)p Fk(l)r(og)t Ff(j)p Fk(\036)2739 1728 y Fg(\032)2775 1737 y Fd(2)2812 1713 y Fn(\()p Fk(x)2905 1728 y Fj(1)2944 1713 y Fk(;)g(t)p Fn(\))22 b Ff(\000)h Fk(\036)3241 1728 y Fg(\032)3277 1737 y Fd(1)3316 1713 y Fn(\()p Fk(x)3409 1728 y Fj(1)3448 1713 y Fk(;)17 b(t)p Fn(\))p Ff(jj)752 1959 y(\024)955 1891 y Fk(C)29 b Ff(\001)22 b Fk(M)p 923 1936 318 4 v 923 2027 a Ff(\001)p Fn(\()p Fk(b)g Ff(\000)h Fk(a)p Fn(\))1251 1959 y Ff(j)p Fk(A)p Fn(\()p Fk(t)p Fn(\))p Ff(jj)p Fk(l)r(og)t(A)p Fn(\()p Fk(t)p Fn(\))p Ff(j)324 2213 y Fn(and)32 b(therefore)1385 2448 y Fk(A)p Fn(\()p Fk(t)p Fn(\))c Fk(>>)f(A)p Fn(\(0\))p Fk(e)2019 2407 y Fi(\000)p Fg(e)2140 2339 y Fa(C)s Fe(\001)p Fa(M)p 2117 2347 176 3 v 2117 2387 a Fe(\001)p Fd(\()p Fa(b)p Fe(\000)p Fa(a)p Fd(\))2302 2359 y Fa(t)324 2652 y Fc(Theorem)37 b(2.)49 b Fb(F)-7 b(or)34 b(a)h(2D)g(Euler)h(solution)f(with)g(a)g (semi-uniform)f(fr)-5 b(ont,)36 b(the)f(thick-)324 2773 y(ness)f Fk(\016)t Fn(\()p Fk(t)p Fn(\))h Fb(satis\014es)1085 2977 y Fk(\016)t Fn(\()p Fk(t)p Fn(\))28 b Fk(>)f(e)1419 2936 y Fi(\000)p Fj([)p Fg(At)p Fj(+)p Fg(B)s Fj(])1777 2977 y Fk(f)11 b(or)72 b(al)r(l)h(t)28 b Ff(2)g Fn([0)p Fk(;)17 b(T)2531 2936 y Fi(\003)2569 2977 y Fn(\))p Fk(:)470 3181 y Fn(Here,)33 b(the)f(constan)m(ts)h(A)e(and)h(B)g(ma)m(y)f(b)s(e) h(tak)m(en)g(to)g(dep)s(end)g(only)g(on)f(the)h(length)324 3301 y(of)e(the)h(fron)m(t,)f(the)h(semi-uniformit)m(y)d(constan)m(t,)j (the)g(initial)c(thic)m(kness)32 b Fk(\016)t Fn(\(0\),)f(and)f(the)324 3422 y(norm)h(of)i(the)g(initial)28 b(v)m(orticit)m(y)33 b(in)e Fk(L)1709 3385 y Fj(1)1771 3422 y Ff(\\)23 b Fk(L)1926 3385 y Fi(1)2001 3422 y Fn(.)470 3542 y(The)48 b(pro)s(of)e(theorem)h (2)f(is)h(similar)c(to)k(theorem)g(1)f(with)h(the)g(di\013erence)h (that)324 3662 y(instead)32 b(of)g(the)h(estimate)f(\(11\),)g(w)m(e)i (ha)m(v)m(e)1172 3867 y Ff(j)p Fk( )t Fn(\()p Fk(z)1350 3882 y Fj(1)1390 3867 y Fk(;)17 b(t)p Fn(\))22 b Ff(\000)g Fk( )t Fn(\()p Fk(z)1778 3882 y Fj(2)1818 3867 y Fk(;)17 b(t)p Fn(\))p Ff(j)27 b(\024)i Fk(M)10 b Ff(j)p Fk(z)2273 3882 y Fj(1)2335 3867 y Ff(\000)22 b Fk(z)2479 3882 y Fj(2)2519 3867 y Ff(j)324 4071 y Fn(where)29 b(M)g(is)e(a)h(constan)m (t)h(that)f(dep)s(end)h(on)f(the)h(initial)24 b(data)k Fk(u)2665 4086 y Fj(0)2704 4071 y Fn(.)41 b(\(See)29 b(details)e(in)h([3].\))470 4191 y(Similar)45 b(estimates)i(can)i(b)s (e)f(obtain)f(for)h(2D)f(ideal)f(Magneto-h)m(ydro)s(dynamics)324 4311 y(\(MHD\))27 b(Equation,)i(with)e(the)i(extra)f(assumption)f(that) 2454 4231 y Fh(R)2521 4257 y Fg(T)2572 4234 y Fe(\003)2501 4346 y Fj(0)2628 4311 y Ff(j)p Fk(u)p Ff(j)2740 4326 y Fg(L)2788 4307 y Fe(1)2856 4311 y Fn(\()p Fk(s)p Fn(\))p Fk(ds)h Fn(is)f(b)s(ounded)324 4432 y(up)37 b(to)f(the)h(time)e(of)i (the)g(blo)m(w-up.)55 b(This)37 b(estimates)f(are)h(consequence)j(of)c (applying)324 4552 y(the)45 b(Mean)g(v)-5 b(alue)43 b(theorem)h(in)g (\(10\).)78 b(Nev)m(ertheless)47 b(in)c(the)i(case)g(of)f(MHD)g(these) 324 4673 y(estimates)32 b(impro)m(v)m(e)g(the)h(results)g(obtain)f(in)g ([6].)324 4862 y Fc(Ac)m(kno)m(wledgmen)m(ts)k(1.)49 b Fb(This)40 b(work)g(was)g(initial)5 b(ly)40 b(supp)-5 b(orte)g(d)40 b(by)h(the)g(A)n(meric)-5 b(an)324 4983 y(Institute)35 b(of)g(Mathematics.)1918 5251 y Fn(7)p eop %%Page: 8 8 8 7 bop 324 548 a Fo(References)373 767 y Fn([1])48 b(P)-8 b(.)36 b(Constan)m(tin,)g(A.)f(J.)g(Ma)5 b(jda,)36 b(and)f(E.)h(T)-8 b(abak.)51 b(F)-8 b(ormation)33 b(of)h(strong)h(fron)m(ts)524 887 y(in)28 b(the)h(2-D)e(quasigeostrophic)i(thermal)e(activ)m(e)h (scalar.)37 b Fb(Nonline)-5 b(arity)p Fn(,)29 b(7:1495{)524 1008 y(1533,)j(1994.)373 1211 y([2])48 b(P)-8 b(.)39 b(Constan)m(tin,)i(Q.)d(Nie)g(and)h(N.)g(Sc)m(horghofer.)62 b(Nonsingular)37 b(surface-quasi-)524 1332 y(geostrophic)c(\015o)m(w)44 b Fb(Phys.)34 b(L)-5 b(ett.)36 b(A)p Fn(,)d(24:168-172.)373 1535 y([3])48 b(D.)37 b(Cordoba.)55 b(Nonexistence)39 b(of)d(simple)f(h)m(yp)s(erb)s(olic)h(blo)m(w-up)h(for)f(the)h(quasi-) 524 1655 y(geostrophic)c(equation.)43 b Fb(A)n(nn.)34 b(of)h(Math.)p Fn(,)e(148\(3\),)e(1998.)373 1859 y([4])48 b(D.)29 b(Cordoba)g(and)g(C.)g(F)-8 b(e\013erman.)37 b(Beha)m(vior)29 b(of)g(sev)m(eral)g(2D)g(\015uid)f(equations)h(in)524 1979 y(singular)i(scenarios.)44 b Fb(submitte)-5 b(d)35 b(to)g(Pr)-5 b(o)g(c.)35 b(Natl.)g(A)-5 b(c)g(ad.)35 b(Sci.)f(USA)373 2183 y Fn([5])48 b(D.)26 b(Cordoba)f(and)h(C.)g(F)-8 b(e\013erman.)32 b(Scalars)25 b(con)m(v)m(ected)j(b)m(y)f(a)e(2D)g (incompressible)524 2303 y(\015o)m(w.)44 b Fb(pr)-5 b(eprint)373 2506 y Fn([6])48 b(D.)30 b(Cordoba)g(and)g(C.)g(Marliani.)37 b(Ev)m(olution)29 b(of)g(curren)m(t)i(sheets)h(and)e(regularit)m(y)524 2627 y(of)j(ideal)e(incompressible)h(magnetic)g(\015uids)h(in)f(2D.)45 b Fb(Comm.)34 b(Pur)-5 b(e)35 b(Appl.Math)p Fn(,)524 2747 y(53\(4\):512-524,)30 b(2000.)373 2950 y([7])48 b(A.)27 b(Ma)5 b(jda)27 b(and)g(E.)g(T)-8 b(abak.)34 b(A)27 b(t)m(w)m(o-dimensional)d(mo)s(del)h(for)h(quasi-geostrophic)524 3071 y(\015o)m(w:)74 b(comparison)46 b(with)h(the)h(t)m(w)m (o-dimensional)d(Euler)i(\015o)m(w.)88 b Fb(Physic)-5 b(a)48 b(D)g Fn(,)524 3191 y(98:515-522,)31 b(1996)373 3395 y([8])48 b(K.)36 b(Ohkitani)f(and)h(M.)h(Y)-8 b(amada.)52 b(In)m(viscid)37 b(and)f(in)m(viscid-limit)31 b(b)s(eha)m(vior)36 b(of)g(a)524 3515 y(surface)e(quasi-geostrophic)e(\015o)m(w.)44 b Fb(Phys.)35 b(Fluids)p Fn(,)d(9:876-882.)1918 5251 y(8)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF