Content-Type: multipart/mixed; boundary="-------------0503171240560" This is a multi-part message in MIME format. ---------------0503171240560 Content-Type: text/plain; name="05-110.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-110.comments" MCS sections: 76D05 35Q30 76D17 76Exx ---------------0503171240560 Content-Type: text/plain; name="05-110.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-110.keywords" Navier-Stokes equations, Burgers vortices, stability, large-Reynolds-number asymptotics ---------------0503171240560 Content-Type: application/postscript; name="burg2D.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="burg2D.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.92b Copyright 2002 Radical Eye Software %%Title: sub_burg2D.dvi %%Pages: 17 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentFonts: CMR17 CMR12 CMBX10 CMR10 CMBX12 CMMI12 CMSY10 CMR8 %%+ CMEX10 CMMI8 CMBX8 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All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSAM10 %!PS-AdobeFont-1.1: MSAM10 2.1 %%CreationDate: 1993 Sep 17 09:05:00 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /square put readonly def /FontBBox{8 -463 1331 1003}readonly def /UniqueID 5031981 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMTI12 %!PS-AdobeFont-1.1: CMTI12 1.0 %%CreationDate: 1991 Aug 18 21:06:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-36 -251 1103 750}readonly def /UniqueID 5000829 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 525003F3DBE5BF07B2E83E66B7F97DDD7CE0EEB75A78BD9227BF359D002B6ADB 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX8 %!PS-AdobeFont-1.1: CMBX8 1.0 %%CreationDate: 1991 Aug 20 16:36:07 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-59 -250 1235 750}readonly def /UniqueID 5000766 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueID 5087383 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 5250011D19E9366EB6FD153D3A100CAA6212E3D5D93990737F8D326D347B7EDC 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 12 /vextendsingle put dup 13 /vextenddouble put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 26 /braceleftbigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 82 /integraltext put dup 90 /integraldisplay put dup 110 /braceleftBig put dup 111 /bracerightBig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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%%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F00F963068B8B731A88D7740B0DDAED1B3F82 7DB9DFB4372D3935C286E39EE7AC9FB6A9B5CE4D2FAE1BC0E55AE02BFC464378 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1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueID 5000794 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF4E9D2405B169CD5365D6ECED5D768D66D6C 68618B8C482B341F8CA38E9BB9BAFCFAAD9C2F3FD033B62690986ED43D9C9361 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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Fm(k)2425 1284 y Fi(L)2473 1265 y Fg(2)2512 1267 y Fm(k)p Fn(f)11 b Fm(k)2671 1284 y Fi(L)2719 1265 y Fg(2)2801 1267 y Fm(\024)45 b Fn(C)2993 1282 y Fk(1)3032 1267 y Fm(k)p Fn(')p Fm(k)3196 1284 y Fi(L)3244 1265 y Fg(2)3282 1267 y Fm(k)p Fn(f)11 b Fm(k)3441 1284 y Fi(L)3489 1265 y Fg(2)3560 1267 y Fn(;)0 1447 y Fr(whic)m(h)34 b(pro)m(v)m(es)g(\(18\).)146 1593 y(W)-8 b(e)39 b(no)m(w)g(return)g(to)g(the)g(op)s(erator)f Fm(L)f Fr(:)h Fn(D)s Fr(\()p Fm(L)p Fr(\))f Fm(!)h Fn(X)46 b Fr(de\014ned)40 b(b)m(y)f Fn(D)s Fr(\()p Fm(L)p Fr(\))e(=)h Fn(G)3174 1557 y Fk(1)p Fi(=)p Fk(2)3284 1593 y Fn(D)s Fr(\()p Fn(L)p Fr(\),)i Fn(X)46 b Fr(=)0 1713 y Fn(G)77 1677 y Fk(1)p Fi(=)p Fk(2)187 1713 y Fn(H)8 b Fr(,)34 b(and)g Fm(\000L)48 b Fr(=)e Fn(G)921 1677 y Fk(1)p Fi(=)p Fk(2)1031 1713 y Fn(LG)1174 1677 y Ff(\000)p Fk(1)p Fi(=)p Fk(2)1340 1713 y Fr(.)i(By)35 b(construction,)g Fm(L)f Fr(is)h(selfadjoin)m(t)g(in)f Fn(X)8 b Fr(,)35 b Fm(\000L)47 b(\025)g Fr(1)p Fn(=)p Fr(2,)34 b(and)0 1833 y(\()p 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Fr(+)p Fm(1)32 b Fn(;)212 b( )2132 1597 y Ff(\000)2191 1582 y Fr(\()p Fn(r)s Fr(\))44 b Fm(\030)h Fn(r)2527 1540 y Fk(2)2664 1582 y Fr(as)32 b Fn(r)f Fm(!)c Fr(0)32 b Fn(:)0 1828 y Fr(W)-8 b(e)33 b(observ)m(e)h(that)f(the)g(\\p)s(oten)m(tial")f(4)p Fn(=r)1549 1792 y Fk(2)1610 1828 y Fm(\000)23 b Fn(h)p Fr(\()p Fn(r)s Fr(\))32 b(in)h(\(28\))f(is)h(strictly)g(p)s(ositiv)m (e,)h(b)s(ecause)1010 2084 y(inf)1007 2144 y Fi(r)r(>)p Fk(0)1131 1974 y Fj(\020)1219 2017 y Fr(4)p 1201 2061 V 1201 2152 a Fn(r)1248 2124 y Fk(2)1319 2084 y Fm(\000)22 b Fn(h)p Fr(\()p Fn(r)s Fr(\))1597 1974 y Fj(\021)1701 2084 y Fr(=)48 b(inf)1821 2144 y Fi(z)s(>)p Fk(0)1947 1974 y Fj(\020)2017 2017 y Fr(1)p 2016 2061 50 4 v 2016 2152 a Fn(z)2098 2084 y Fm(\000)2311 2017 y Fn(z)p 2208 2061 256 4 v 2208 2152 a(e)2253 2124 y Fi(z)2315 2152 y Fm(\000)23 b Fr(1)2473 1974 y Fj(\021)2560 2084 y Fn(>)28 b Fr(0)k Fn(:)0 2335 y Fr(By)e(the)h(Maxim)m(um)h(Principle)f([15],)f (it)g(follo)m(ws)h(that)f Fn( )2085 2299 y Ff(0)2081 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Fr(\()p Fn(r)s Fr(\))2095 3705 y Fj(Z)2194 3731 y Ff(1)2150 3930 y Fi(r)2330 3773 y Fn(z)2379 3737 y Fk(3)p 2296 3818 V 2296 3909 a Fr(4)p Fn(w)2415 3924 y Fk(0)2463 3841 y Fn( )2526 3856 y Fk(+)2586 3841 y Fr(\()p Fn(z)t Fr(\))p Fn(h)p Fr(\()p Fn(z)t Fr(\))17 b(d)p Fn(z)38 b(;)115 b(r)30 b(>)d Fr(0)33 b Fn(:)132 b Fr(\(30\))0 4099 y(It)33 b(is)g(clear)g(that)f(\012)c(:)g Fl(R)885 4114 y Fk(+)971 4099 y Fm(!)g Fl(R)k Fr(is)h(a)f(smo)s(oth)h (function)g(satisfying)178 4367 y(\012\()p Fn(r)s Fr(\))45 b Fm(\030)538 4226 y Fj(\032)654 4306 y Fr(\012)724 4321 y Fk(+)783 4306 y Fn(r)830 4269 y Fk(2)1007 4306 y Fn(as)83 b(r)31 b Fm(!)c Fr(0)32 b Fn(;)654 4426 y Fr(\012)724 4441 y Ff(\000)783 4426 y Fn(r)830 4390 y Ff(\000)p Fk(2)1007 4426 y Fn(as)83 b(r)31 b Fm(!)c Fr(+)p Fm(1)32 b Fn(;)1887 4367 y Fr(where)99 b(\012)2304 4382 y Ff(\006)2408 4367 y Fr(=)44 b Fm(\000)2622 4231 y Fj(Z)2722 4257 y Ff(1)2678 4457 y Fk(0)2857 4299 y Fn(z)2906 4263 y Fk(3)p 2823 4344 V 2823 4435 a Fr(4)p Fn(w)2942 4450 y Fk(0)2991 4367 y Fn( )3054 4382 y Ff(\006)3113 4367 y Fr(\()p Fn(z)t Fr(\))p Fn(h)p Fr(\()p Fn(z)t Fr(\))17 b(d)p Fn(z)39 b(:)0 4636 y Fr(\(The)48 b(v)-5 b(alues)47 b(\012)627 4651 y Fk(+)739 4636 y Fm(\031)52 b(\000)p Fr(0)p Fn(:)p Fr(38)47 b(and)g(\012)1440 4651 y Ff(\000)1551 4636 y Fm(\031)53 b(\000)p Fr(17)p Fn(:)p Fr(5)46 b(w)m(ere)i(found)f(n)m (umerically)i(in)e([11].\))86 b(Similar)0 4757 y(estimates)32 b(hold)f(for)f(all)h(deriv)-5 b(ativ)m(es.)45 b(Going)29 b(bac)m(k)j(to)e(\(26\),)h(w)m(e)h(see)f(that)g Fn(!)g Fr(:)c Fl(R)3043 4772 y Fk(+)3130 4757 y Fm(!)g Fl(R)j Fr(is)h(smo)s(oth)0 4877 y(and)41 b(rapidly)i(decreasing)f(at)g (in\014nit)m(y)-8 b(.)71 b(Moreo)m(v)m(er)43 b Fn(!)t Fr(\(0\))e(=)i(0,)g Fn(!)t Fr(\()p Fn(r)s Fr(\))f Fn(<)g Fr(0)g(for)e(all)i Fn(r)j(>)e Fr(0,)g(and)e(it)0 4997 y(is)g(easy)g(to)e(v)m(erify)j(that)d(the)i(T)-8 b(a)m(ylor)40 b(expansion)i(of)d Fn(!)t Fr(\()p Fn(r)s Fr(\))g(at)h Fn(r)j Fr(=)d(0)g(con)m(tains)h Fe(even)e Fr(p)s(o)m(w)m(ers)i(of)f Fn(r)0 5118 y Fr(only)-8 b(.)61 b(Th)m(us)40 b(the)e(function)h(the)g (function)f Fn(w)1706 5133 y Ff(1)1818 5118 y Fr(:)g Fl(R)1967 5082 y Fk(2)2043 5118 y Fm(!)f Fl(R)h Fr(de\014ned)h(b)m(y)h (\(23\))d(is)i(smo)s(oth,)h(rapidly)0 5238 y(decreasing)34 b(at)e(in\014nit)m(y)-8 b(,)34 b(and)f(satis\014es)h(\003)p Fn(w)1633 5253 y Ff(1)1735 5238 y Fr(=)28 b Fm(M)p Fn(G)k Fr(b)m(y)h(construction.)947 b Fd(\003)146 5431 y Fr(As)47 b(a)f(consequence)k(w)m(e)d(can)g(pro)m(v)m(e)g(that)g(the)f(function)h Fn(w)2463 5446 y Fi(\013)2559 5431 y Fr(de\014ned)g(in)g(\(13\))f(is)h (uniformly)0 5551 y(b)s(ounded)33 b(in)g Fn(Y)21 b Fr(:)1865 5800 y(9)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 0 100 a Fl(Corollary)38 b(3.3)49 b Fe(Ther)-5 b(e)34 b(exists)h Fn(K)1320 115 y Fk(3)1387 100 y Fn(>)27 b Fr(0)35 b Fe(such)g(that)g Fm(k)p Fn(w)2115 115 y Fi(\013)2164 100 y Fm(k)2214 115 y Fi(Y)2302 100 y Fm(\024)28 b Fn(K)2490 115 y Fk(3)2530 100 y Fm(j)p Fn(\013)q Fm(j)p Fn(=)p Fr(\(1+)n Fm(j)p Fn(\013)q Fm(j)p Fr(\))34 b Fe(for)g(al)5 b(l)35 b Fn(\013)28 b Fm(2)g Fl(R)p Fe(.)0 353 y Fl(Pro)s(of:)42 b Fr(F)-8 b(rom)36 b(\(13\))g(and)g(\(15\))g(w)m(e)i(kno)m(w)f(that)g Fm(k)p Fn(w)1960 368 y Fi(\013)2009 353 y Fm(k)2059 368 y Fi(Y)2153 353 y Fm(\024)e Fn(K)2348 368 y Fk(2)2388 353 y Fm(j)p Fn(\013)q Fm(jk)p Fn(G)p Fm(k)2684 368 y Fi(Y)2743 353 y Fr(.)55 b(On)37 b(the)g(other)f(hand,)i(in)0 473 y(view)c(of)e(Prop)s (osition)h(3.1,)f(w)m(e)i(ha)m(v)m(e)g(\()p Fm(L)21 b(\000)i Fn(\013)q Fr(\003\))p Fn(w)1881 488 y Fi(\013)1957 473 y Fr(=)28 b Fm(\000)p Fn(\013)q Fm(M)p Fn(G)f Fr(=)h(\()p Fm(L)22 b(\000)g Fn(\013)q Fr(\003\))p Fn(w)2996 488 y Ff(1)3092 473 y Fm(\000)h(L)p Fn(w)3331 488 y Ff(1)3405 473 y Fr(,)33 b(hence)1233 693 y Fn(w)1303 708 y Fi(\013)1396 693 y Fr(=)45 b Fn(w)1587 708 y Ff(1)1683 693 y Fm(\000)23 b Fr(\()p Fm(L)f(\000)g Fn(\013)q Fr(\003\))2180 652 y Ff(\000)p Fk(1)2274 693 y Fm(L)p Fn(w)2413 708 y Ff(1)2520 693 y Fn(:)1059 b Fr(\(31\))0 913 y(Using)34 b(\(14\),)f(w)m(e)i(infer) f(that)f Fm(k)p Fn(w)1212 928 y Fi(\013)1261 913 y Fm(k)1311 928 y Fi(Y)1401 913 y Fm(\024)d(k)p Fn(w)1628 928 y Ff(1)1702 913 y Fm(k)1752 928 y Fi(Y)1835 913 y Fr(+)23 b Fn(K)2017 928 y Fk(1)2057 913 y Fm(kL)p Fn(w)2246 928 y Ff(1)2320 913 y Fm(k)2370 928 y Fi(X)2437 913 y Fr(.)46 b(Com)m(bining)35 b(b)s(oth)e(estimates)i(w)m(e)0 1034 y(obtain)d(the)h(desired)i (result.)2634 b Fd(\003)146 1229 y Fr(A)33 b(more)g(detailed)g (analysis)h(rev)m(eals)h(that)d Fn(w)1833 1244 y Ff(1)1940 1229 y Fr(is)h(indeed)h(the)f(limit)g(of)f Fn(w)2929 1244 y Fi(\013)3011 1229 y Fr(as)h Fm(j)p Fn(\013)q Fm(j)26 b(!)i(1)p Fr(:)0 1482 y Fl(Prop)s(osition)38 b(3.4)49 b Fe(Ther)-5 b(e)47 b(exists)h Fn(K)1454 1497 y Fk(4)1546 1482 y Fn(>)53 b Fr(0)48 b Fe(such)g(that)g Fm(k)p Fn(w)2339 1497 y Fi(\013)2421 1482 y Fm(\000)32 b Fn(w)2600 1497 y Ff(1)2675 1482 y Fm(k)2725 1497 y Fi(Y)2838 1482 y Fm(\024)53 b Fn(K)3051 1497 y Fk(4)3090 1482 y Fn(=)p Fr(\(1+)p Fm(j)p Fn(\013)q Fm(j)p Fr(\))47 b Fe(for)h(al)5 b(l)0 1602 y Fn(\013)28 b Fm(2)g Fl(R)p Fe(.)0 1856 y Fl(Pro)s(of:)51 b Fr(By)45 b(\(31\))e(and)i(\(14\),)h(w)m(e)g(ha)m(v)m (e)g Fm(k)p Fn(w)1698 1871 y Fi(\013)1777 1856 y Fm(\000)30 b Fn(w)1954 1871 y Ff(1)2029 1856 y Fm(k)2079 1871 y Fi(Y)2187 1856 y Fm(\024)48 b Fn(K)2395 1871 y Fk(1)2435 1856 y Fm(kL)p Fn(w)2624 1871 y Ff(1)2698 1856 y Fm(k)2748 1871 y Fi(X)2815 1856 y Fr(.)79 b(On)44 b(the)h(other)f(hand,)0 1976 y(pro)s(ceeding)37 b(exactly)h(as)f(in)f(the)h(pro)s(of)f(of)g (Prop)s(osition)h(3.1,)g(it)f(is)h(straigh)m(tforw)m(ard)g(to)f(sho)m (w)i(that)0 2096 y(there)30 b(exists)i Fn(z)558 2111 y Ff(1)661 2096 y Fm(2)c(S)7 b Fr(\()p Fl(R)944 2060 y Fk(2)983 2096 y Fr(\))30 b(suc)m(h)h(that)f(\003)p Fn(z)1590 2111 y Ff(1)1692 2096 y Fr(=)e Fm(L)p Fn(w)1935 2111 y Ff(1)2039 2096 y Fr(\(the)i(details)g(are)g(left)g(to)f(the)h (reader\).)43 b(Then,)0 2217 y(using)33 b(\(31\),)f(w)m(e)i(\014nd)f (for)f(all)h Fn(\013)28 b Fm(6)p Fr(=)f(0)1042 2481 y Fn(w)1112 2496 y Fi(\013)1206 2481 y Fr(=)44 b Fn(w)1396 2496 y Ff(1)1493 2481 y Fr(+)1608 2414 y(1)p 1601 2458 63 4 v 1601 2549 a Fn(\013)1673 2481 y(z)1718 2496 y Ff(1)1815 2481 y Fm(\000)1932 2414 y Fr(1)p 1925 2458 V 1925 2549 a Fn(\013)1998 2481 y Fr(\()p Fm(L)21 b(\000)i Fn(\013)q Fr(\003\))2395 2440 y Ff(\000)p Fk(1)2489 2481 y Fm(L)p Fn(z)2603 2496 y Ff(1)2710 2481 y Fn(;)0 2736 y Fr(hence)36 b Fm(k)p Fn(w)393 2751 y Fi(\013)465 2736 y Fm(\000)24 b Fn(w)636 2751 y Ff(1)710 2736 y Fm(k)760 2751 y Fi(Y)851 2736 y Fm(\024)31 b(j)p Fn(\013)q Fm(j)1078 2700 y Ff(\000)p Fk(1)1171 2736 y Fr(\()p Fm(k)p Fn(z)1304 2751 y Ff(1)1379 2736 y Fm(k)1429 2751 y Fi(Y)1513 2736 y Fr(+)23 b Fn(K)1695 2751 y Fk(1)1734 2736 y Fm(kL)p Fn(z)1898 2751 y Ff(1)1973 2736 y Fm(k)2023 2751 y Fi(X)2090 2736 y Fr(\).)48 b(Com)m(bining)36 b(b)s(oth)e(estimates)h(w)m(e)h (obtain)0 2857 y(the)d(desired)h(result.)2938 b Fd(\003)0 3214 y Fo(4)161 b(Existence)52 b(of)i(asymmetric)g(v)l(ortices)0 3458 y Fr(As)35 b(is)f(explained)i(in)f(the)f(in)m(tro)s(duction,)i(w)m (e)f(shall)g(pro)m(v)m(e)g(the)g(existence)i(of)d(a)f(stationary)i (solution)0 3579 y(of)d(\(6\))g(b)m(y)i(solving)f(Eq.\(12\),)g(namely) 959 3826 y Fn(w)47 b Fr(=)d Fn(\025w)1323 3841 y Fi(\013)1394 3826 y Fr(+)22 b(\()p Fm(L)g(\000)g Fn(\013)q Fr(\003\))1889 3785 y Ff(\000)p Fk(1)1983 3716 y Fj(\020)2043 3826 y Fl(v)h Fm(\001)f(r)p Fn(w)j Fm(\000)d Fn(\025)p Fm(M)p Fn(w)2702 3716 y Fj(\021)2793 3826 y Fn(;)786 b Fr(\(32\))0 4074 y(where)38 b Fn(w)356 4089 y Fi(\013)441 4074 y Fr(is)f(de\014ned)h(in)f(\(13\))f(and)g(\(as)h(usual\))g Fl(v)h Fr(denotes)g(the)e(v)m(elo)s(cit)m(y)j(\014eld)e(obtained)g (from)f Fn(w)0 4194 y Fr(b)m(y)k(the)f(Biot-Sa)m(v)-5 b(art)38 b(la)m(w)h(\(3\).)62 b(T)-8 b(o)38 b(b)s(ound)h(the)g (nonlinear)h(term)f(in)g(\(32\),)h(w)m(e)f(use)h(the)f(follo)m(wing)0 4314 y(bilinear)33 b(estimate:)0 4568 y Fl(Lemma)39 b(4.1)49 b Fe(Ther)-5 b(e)42 b(exists)h Fn(K)1235 4583 y Fk(5)1317 4568 y Fn(>)g Fr(0)g Fe(such)g(that,)i(if)e Fn(w)s(;)36 b Fr(~)-68 b Fn(w)44 b Fm(2)f Fn(Y)65 b Fe(and)42 b(if)h Fl(v)i Fe(is)d(the)h(velo)-5 b(city)43 b(\014eld)0 4688 y(obtaine)-5 b(d)34 b(fr)-5 b(om)34 b Fn(w)k Fe(by)d(the)g(Biot-Savart) f(law,)g(then)h Fl(v)24 b Fm(\001)d(r)f Fr(~)-69 b Fn(w)30 b Fm(2)f Fn(X)42 b Fe(and)1254 4908 y Fm(k)p Fl(v)24 b Fm(\001)e(r)e Fr(~)-69 b Fn(w)r Fm(k)1642 4923 y Fi(X)1754 4908 y Fm(\024)44 b Fn(K)1958 4923 y Fk(5)1998 4908 y Fm(k)p Fn(w)s Fm(k)2171 4923 y Fi(Y)2231 4908 y Fm(k)19 b Fr(~)-68 b Fn(w)r Fm(k)2403 4923 y Fi(Y)2498 4908 y Fn(:)0 5161 y Fl(Pro)s(of:)34 b Fr(W)-8 b(e)30 b(\014rst)g(observ)m(e)i (that)d Fn(w)h Fm(2)e Fn(L)1513 5125 y Fi(p)1553 5161 y Fr(\()p Fl(R)1675 5125 y Fk(2)1714 5161 y Fr(\))i(for)f(all)h Fn(p)d Fm(2)h Fr([1)p Fn(;)17 b Fr(+)p Fm(1)p Fr(\),)30 b(and)g(that)f Fm(k)p Fn(w)s Fm(k)3190 5176 y Fi(L)3238 5157 y Fb(p)3305 5161 y Fm(\024)f Fn(C)3480 5176 y Fi(p)3520 5161 y Fm(k)p Fn(w)s Fm(k)3693 5176 y Fi(Y)3752 5161 y Fr(.)0 5282 y(Indeed)37 b(this)g(is)f(pro)m(v)m(ed)h(in)f(\(20\))g (if)f Fn(p)f Fm(2)f Fr([1)p Fn(;)17 b Fr(2],)36 b(and)g(if)g Fn(p)d Fm(\025)h Fr(2)h(this)i(follo)m(ws)g(from)e(the)h(em)m(b)s (eddings)0 5402 y Fn(Y)58 b(,)-17 b Fm(!)36 b Fn(H)350 5366 y Fk(1)389 5402 y Fr(\()p Fl(R)511 5366 y Fk(2)550 5402 y Fr(\))g Fn(,)-17 b Fm(!)37 b Fn(L)837 5366 y Fi(p)877 5402 y Fr(\()p Fl(R)999 5366 y Fk(2)1038 5402 y Fr(\).)59 b(Next,)39 b(b)m(y)g(the)f(Calder\023)-49 b(on-Zygm)m(und)39 b(inequalit)m(y)h([20)o(],)g(the)e(v)m(elo)s(cit)m(y)1841 5800 y(10)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 0 100 a Fr(\014eld)32 b Fl(v)h Fr(asso)s(ciated)g(to)e Fn(w)j Fr(satis\014es)f Fm(kr)p Fl(v)r Fm(k)1590 115 y Fi(L)1638 96 y Fb(p)1705 100 y Fm(\024)28 b Fn(C)1880 115 y Fi(p)1920 100 y Fm(k)p Fn(w)s Fm(k)2093 115 y Fi(L)2141 96 y Fb(p)2211 100 y Fr(for)j(1)d Fn(<)f(p)h(<)f Fm(1)p Fr(.)43 b(Using)32 b(the)g(Gagliardo-)0 220 y(Niren)m(b)s(erg)h(inequalit)m(y)i([12],)d(w)m(e)i(deduce)g(that)f Fl(v)c Fm(2)f Fn(L)2045 184 y Ff(1)2120 220 y Fr(\()p Fl(R)2242 184 y Fk(2)2281 220 y Fr(\))k(and)h(that)627 436 y Fm(k)p Fl(v)r Fm(k)788 451 y Fi(L)836 432 y Fc(1)949 436 y Fm(\024)44 b Fn(C)7 b Fm(kr)p Fl(v)r Fm(k)1391 385 y Fk(1)p Fi(=)p Fk(2)1391 468 y Fi(L)1439 450 y Fg(3)1501 436 y Fm(k)p Fl(v)r Fm(k)1662 385 y Fk(1)p Fi(=)p Fk(2)1662 468 y Fi(L)1710 450 y Fg(6)1815 436 y Fm(\024)45 b Fn(C)7 b Fm(k)p Fn(w)s Fm(k)2187 385 y Fk(1)p Fi(=)p Fk(2)2187 468 y Fi(L)2235 450 y Fg(3)2296 436 y Fm(k)p Fn(w)s Fm(k)2469 385 y Fk(1)p Fi(=)p Fk(2)2469 474 y Fi(L)2517 455 y Fg(3)p Fb(=)p Fg(2)2661 436 y Fm(\024)45 b Fn(C)7 b Fm(k)p Fn(w)s Fm(k)3033 451 y Fi(Y)3125 436 y Fn(;)0 638 y Fr(where)39 b(w)m(e)g(also)g(used)g(the)f(Hardy-Littlew)m(o)s(o)s(d-Sob)s(olev)h (inequalit)m(y)h(\(21\))d(with)i Fn(q)i Fr(=)c(6,)i Fn(p)e Fr(=)g(3)p Fn(=)p Fr(2.)0 758 y(W)-8 b(e)33 b(conclude)h(that)e Fl(v)24 b Fm(\001)e(r)e Fr(~)-69 b Fn(w)30 b Fm(2)e Fn(X)41 b Fr(and)895 961 y Fm(k)p Fl(v)24 b Fm(\001)e(r)e Fr(~)-69 b Fn(w)r Fm(k)1283 976 y Fi(X)1395 961 y Fm(\024)44 b(k)p Fl(v)r Fm(k)1677 976 y Fi(L)1725 957 y Fc(1)1794 961 y Fm(kr)20 b Fr(~)-69 b Fn(w)r Fm(k)2049 976 y Fi(X)2161 961 y Fm(\024)44 b Fn(C)7 b Fm(k)p Fn(w)s Fm(k)2532 976 y Fi(Y)2592 961 y Fm(k)20 b Fr(~)-69 b Fn(w)r Fm(k)2764 976 y Fi(Y)2857 961 y Fn(;)0 1163 y Fr(whic)m(h)34 b(is)f(the)g (desired)h(result.)2561 b Fd(\003)146 1353 y Fr(W)-8 b(e)33 b(can)g(no)m(w)g(pro)m(v)m(e)h(the)f(main)g(result)h(of)e(this)h (section:)0 1586 y Fl(Prop)s(osition)38 b(4.2)49 b Fe(Cho)-5 b(ose)33 b Fn(\025)1188 1601 y Fk(0)1255 1586 y Fm(2)28 b Fr(\(0)p Fn(;)17 b Fr(1\))34 b Fe(such)h(that)1014 1788 y Fr(2)p Fn(K)1146 1803 y Fk(2)1185 1788 y Fn(\025)1242 1803 y Fk(0)1326 1788 y Fn(<)44 b Fr(1)35 b Fn(;)116 b Fe(and)99 b Fr(16)p Fn(K)2108 1803 y Fk(1)2147 1788 y Fn(K)2230 1803 y Fk(3)2270 1788 y Fn(K)2353 1803 y Fk(5)2392 1788 y Fn(\025)2449 1803 y Fk(0)2533 1788 y Fm(\024)44 b Fr(1)35 b Fn(:)841 b Fr(\(33\))0 1990 y Fe(Then,)42 b(for)f(al)5 b(l)41 b Fn(\025)e Fm(2)h Fr([0)p Fn(;)17 b(\025)979 2005 y Fk(0)1018 1990 y Fr(])41 b Fe(and)f(al)5 b(l)41 b Fn(\013)f Fm(2)g Fl(R)p Fe(,)i(e)-5 b(quation)41 b(\(32\))g(has)f(a)h(unique)h(solution)e Fn(w)3446 1954 y Fi(\013;\025)3595 1990 y Fm(2)g Fn(Y)0 2110 y Fe(such)50 b(that)g Fm(k)p Fn(w)574 2074 y Fi(\013;\025)683 2110 y Fm(k)733 2125 y Fi(Y)849 2110 y Fm(\024)56 b Fr(\(4)p Fn(K)1152 2125 y Fk(1)1191 2110 y Fn(K)1274 2125 y Fk(5)1314 2110 y Fr(\))1352 2074 y Ff(\000)p Fk(1)1446 2110 y Fe(.)89 b(This)49 b(solution)h(dep)-5 b(ends)49 b(smo)-5 b(othly)49 b(on)g(the)h(p)-5 b(ar)g(ameters)0 2231 y Fr(\()p Fn(\013)q(;)17 b(\025)p Fr(\))27 b Fm(2)h Fl(R)22 b Fm(\002)g Fr([0)p Fn(;)17 b(\025)743 2246 y Fk(0)782 2231 y Fr(])35 b Fe(and)f (satis\014es)h Fm(k)p Fn(w)1523 2195 y Fi(\013;\025)1632 2231 y Fm(k)1682 2246 y Fi(Y)1770 2231 y Fm(\024)28 b Fr(4)p Fn(K)2007 2246 y Fk(3)2047 2231 y Fn(\025)p Fm(j)p Fn(\013)q Fm(j)p Fn(=)p Fr(\(1+)n Fm(j)p Fn(\013)q Fm(j)p Fr(\))p Fe(.)0 2464 y Fl(Pro)s(of:)37 b Fr(Fix)c Fn(\025)28 b Fm(2)g Fr([0)p Fn(;)17 b(\025)871 2479 y Fk(0)910 2464 y Fr(],)32 b Fn(\013)d Fm(2)f Fl(R)p Fr(,)k(and)h(c)m(ho)s(ose)g Fn(r)d(>)e Fr(0)k(suc)m(h)i(that)825 2721 y(4)p Fn(K)964 2680 y Fi(\013)957 2745 y Fk(3)1014 2721 y Fn(\025)44 b Fm(\024)h Fn(r)i Fm(\024)1582 2653 y Fr(1)p 1459 2698 294 4 v 1459 2789 a(4)p Fn(K)1591 2804 y Fk(1)1631 2789 y Fn(K)1714 2804 y Fk(5)1796 2721 y Fn(;)114 b Fr(where)99 b Fn(K)2374 2680 y Fi(\013)2367 2745 y Fk(3)2468 2721 y Fr(=)2621 2653 y Fn(K)2704 2668 y Fk(3)2743 2653 y Fm(j)p Fn(\013)q Fm(j)p 2598 2698 287 4 v 2598 2789 a Fr(1)22 b(+)g Fm(j)p Fn(\013)q Fm(j)2927 2721 y Fn(:)652 b Fr(\(34\))0 2987 y(Let)33 b Fn(F)252 2950 y Fi(\013;\025)389 2987 y Fr(:)28 b Fn(Y)49 b Fm(!)27 b Fn(Y)54 b Fr(b)s(e)33 b(the)g(quadratic)g(map)g(de\014ned)h(b)m(y)604 3211 y Fn(F)681 3169 y Fi(\013;\025)791 3211 y Fr(\()p Fn(w)s Fr(\))44 b(=)g Fn(\025w)1231 3226 y Fi(\013)1302 3211 y Fr(+)22 b(\()p Fm(L)g(\000)h Fn(\013)q Fr(\003\))1798 3169 y Ff(\000)p Fk(1)1891 3100 y Fj(\020)1951 3211 y Fl(v)h Fm(\001)e(r)p Fn(w)i Fm(\000)f Fn(\025)p Fm(M)p Fn(w)2611 3100 y Fj(\021)2702 3211 y Fn(;)114 b(w)30 b Fm(2)e Fn(Y)54 b(:)0 3435 y Fr(Using)33 b(Prop)s(osition)g(2.1,)f (Corollary)h(3.3)f(and)h(Lemma)g(4.1,)f(w)m(e)i(\014nd)f(for)f(all)h Fn(w)d Fm(2)e Fn(Y)21 b Fr(:)866 3637 y Fm(k)p Fn(F)993 3596 y Fi(\013;\025)1103 3637 y Fr(\()p Fn(w)s Fr(\))p Fm(k)1302 3652 y Fi(Y)1406 3637 y Fm(\024)44 b Fn(K)1617 3596 y Fi(\013)1610 3661 y Fk(3)1667 3637 y Fn(\025)22 b Fr(+)g Fn(K)1927 3652 y Fk(1)1967 3637 y Fn(K)2050 3652 y Fk(5)2089 3637 y Fm(k)p Fn(w)s Fm(k)2262 3596 y Fk(2)2262 3661 y Fi(Y)2344 3637 y Fr(+)g Fn(K)2525 3652 y Fk(2)2564 3637 y Fn(\025)p Fm(k)p Fn(w)s Fm(k)2794 3652 y Fi(Y)2887 3637 y Fn(:)692 b Fr(\(35\))0 3839 y(Similarly)-8 b(,)34 b(for)e(all)g Fn(w)s(;)k Fr(~)-68 b Fn(w)29 b Fm(2)f Fn(Y)22 b Fr(:)425 4063 y Fm(k)p Fn(F)552 4022 y Fi(\013;\025)661 4063 y Fr(\()p Fn(w)s Fr(\))g Fm(\000)h Fn(F)1009 4022 y Fi(\013;\025)1118 4063 y Fr(\()d(~)-69 b Fn(w)s Fr(\))p Fm(k)1317 4078 y Fi(Y)1422 4063 y Fm(\024)1543 3953 y Fj(\020)1603 4063 y Fn(K)1686 4078 y Fk(1)1725 4063 y Fn(K)1808 4078 y Fk(5)1848 4063 y Fr(\()p Fm(k)p Fn(w)s Fm(k)2059 4078 y Fi(Y)2141 4063 y Fr(+)22 b Fm(k)d Fr(~)-68 b Fn(w)r Fm(k)2411 4078 y Fi(Y)2471 4063 y Fr(\))22 b(+)g Fn(K)2712 4078 y Fk(2)2752 4063 y Fn(\025)2809 3953 y Fj(\021)2868 4063 y Fm(k)p Fn(w)j Fm(\000)42 b Fr(~)-69 b Fn(w)s Fm(k)3235 4078 y Fi(Y)3328 4063 y Fn(:)251 b Fr(\(36\))0 4300 y(Let)38 b Fn(B)254 4315 y Fi(r)329 4300 y Fr(=)e Fm(f)p Fn(w)j Fm(2)e Fn(Y)h Fm(j)17 b(k)p Fn(w)s Fm(k)1016 4315 y Fi(Y)1112 4300 y Fm(\024)37 b Fn(r)s Fm(g)p Fr(.)58 b(It)38 b(follo)m(ws)h(from)e(\(33\),)i(\(34\),)g (and)e(\(35\))h(that)f Fn(F)3259 4264 y Fi(\013;\025)3407 4300 y Fr(maps)h Fn(B)3741 4315 y Fi(r)0 4421 y Fr(in)m(to)32 b(itself,)h(b)s(ecause)h Fn(K)911 4385 y Fi(\013)904 4445 y Fk(3)960 4421 y Fn(\025)22 b Fr(+)f Fn(K)1219 4436 y Fk(1)1258 4421 y Fn(K)1341 4436 y Fk(5)1381 4421 y Fn(r)1428 4385 y Fk(2)1488 4421 y Fr(+)g Fn(K)1668 4436 y Fk(2)1707 4421 y Fn(\025r)31 b Fm(\024)d Fn(r)s(=)p Fr(4)20 b(+)h Fn(r)s(=)p Fr(4)g(+)g Fn(r)s(=)p Fr(2)27 b(=)g Fn(r)s Fr(.)43 b(Similarly)33 b(\(36\))f(implies)0 4541 y(for)g(all)h Fn(w)s(;)j Fr(~)-68 b Fn(w)29 b Fm(2)f Fn(B)669 4556 y Fi(r)707 4541 y Fr(:)401 4743 y Fm(k)p Fn(F)528 4702 y Fi(\013;\025)638 4743 y Fr(\()p Fn(w)s Fr(\))22 b Fm(\000)g Fn(F)985 4702 y Fi(\013;\025)1095 4743 y Fr(\()e(~)-69 b Fn(w)s Fr(\))p Fm(k)1294 4758 y Fi(Y)1398 4743 y Fm(\024)45 b Fn(\024)p Fm(k)p Fn(w)25 b Fm(\000)42 b Fr(~)-69 b Fn(w)s Fm(k)1943 4758 y Fi(Y)2036 4743 y Fn(;)114 b Fr(where)99 b Fn(\024)44 b Fr(=)2754 4704 y Fk(1)p 2754 4720 36 4 v 2754 4778 a(2)2822 4743 y Fr(+)22 b Fn(K)3003 4758 y Fk(2)3042 4743 y Fn(\025)3099 4758 y Fk(0)3166 4743 y Fn(<)28 b Fr(1)k Fn(:)0 4945 y Fr(By)f(the)f(con)m(traction)h(mapping)g(theorem,)h Fn(F)1707 4909 y Fi(\013;\025)1847 4945 y Fr(has)e(th)m(us)i(a)e (unique)h(\014xed)h(p)s(oin)m(t)e(in)h Fn(B)3296 4960 y Fi(r)3334 4945 y Fr(,)g(whic)m(h)g(w)m(e)0 5066 y(denote)g(b)m(y)h Fn(w)519 5030 y Fi(\013;\025)629 5066 y Fr(.)42 b(It)31 b(remains)h(to)e(sho)m(w)i(that)e Fn(w)1803 5030 y Fi(\013;\025)1944 5066 y Fr(is)h(a)f(smo)s(oth)h(function)g(of)f(\()p Fn(\013)q(;)17 b(\025)p Fr(\))27 b Fm(2)h Fl(R)18 b Fm(\002)g Fr([0)p Fn(;)f(\025)3686 5081 y Fk(0)3725 5066 y Fr(].)0 5186 y(But)35 b(this)g(is)h(a)e(direct)i(consequence)h(of)e(the)g(implicit)h (function)f(theorem,)h(b)s(ecause)g(the)g(map)f Fn(F)3670 5150 y Fi(\013;\025)0 5307 y Fr(dep)s(ends)f(smo)s(othly)f(on)g(\()p Fn(\013)q(;)17 b(\025)p Fr(\))32 b(and)g(the)h(di\013eren)m(tial)619 5536 y Fn(D)700 5551 y Fi(w)756 5536 y Fn(F)833 5495 y Fi(\013;\025)943 5536 y Fr(\()p Fn(w)s Fr(\))44 b(=)64 b(~)-69 b Fn(w)30 b Fm(7!)e Fr(\()p Fm(L)21 b(\000)i Fn(\013)q Fr(\003\))1881 5495 y Ff(\000)p Fk(1)1975 5426 y Fj(\020)2040 5535 y Fr(~)2034 5536 y Fl(v)h Fm(\001)e(r)p Fn(w)i Fr(+)e Fl(v)i Fm(\001)e(r)e Fr(~)-69 b Fn(w)25 b Fm(\000)d Fn(\025)p Fm(M)e Fr(~)-69 b Fn(w)3101 5426 y Fj(\021)1841 5800 y Fr(11)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 0 100 a Fr(satis\014es)28 b Fm(k)p Fn(D)484 115 y Fi(w)541 100 y Fn(F)618 63 y Fi(\013;\025)728 100 y Fr(\()p Fn(w)s Fr(\))p Fm(k)e(\024)j Fn(\024)d Fr(for)g(all)h Fn(w)j Fm(2)e Fn(B)1682 115 y Fi(r)1746 100 y Fr(and)f(all)f(\()p Fn(\013)q(;)17 b(\025)p Fr(\))27 b Fm(2)h Fl(R)10 b Fm(\002)g Fr([0)p Fn(;)17 b(\025)2778 115 y Fk(0)2817 100 y Fr(].)41 b(Th)m(us)29 b Fl(1)10 b Fm(\000)g Fn(D)3388 115 y Fi(w)3444 100 y Fn(F)3521 63 y Fi(\013;\025)3631 100 y Fr(\()p Fn(w)s Fr(\))0 220 y(is)39 b(in)m(v)m(ertible)j(at)c Fn(w)j Fr(=)d Fn(w)965 184 y Fi(\013;\025)1075 220 y Fr(,)i(and)f(the)h(desired)g(conclusion)g (follo)m(ws)g(from)f(the)g(implicit)h(function)0 340 y(theorem.)3329 b Fd(\003)146 535 y Fr(Theorem)48 b(1.1)e(is)h(an)g (immediate)h(consequence)h(of)d(Prop)s(osition)h(4.2:)71 b(w)m(e)48 b(just)e(set)i Fn(!)3543 499 y Fi(\013;\025)3704 535 y Fr(=)0 656 y Fn(\013)q(G)21 b Fr(+)g 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Fn(!)260 1657 y Fk(0)341 1642 y Fr(is)g(not)g(iden)m(tically)i(zero,)h(the)d(Maxim)m (um)i(Principle)f(implies)g(that)f Fn(!)t Fr(\()p Fn(x;)17 b(t)p Fr(\))43 b Fn(>)h Fr(0)d(for)h(all)0 1762 y Fn(x)k Fm(2)h Fl(R)298 1726 y Fk(2)380 1762 y Fr(and)d(all)f Fn(t)j(>)g Fr(0,)g(see)f(e.g.)76 b(\([7)o(],)47 b(Section)d(2.3\).)75 b(Con)m(v)m(ersely)-8 b(,)49 b(if)44 b Fn(!)2962 1777 y Fk(0)3044 1762 y Fr(has)g(non-constan)m(t)0 1882 y(sign,)37 b(the)f(norm)g Fm(k)p Fn(!)t Fr(\()p Fm(\001)p Fn(;)17 b(t)p Fr(\))p Fm(k)1009 1899 y Fi(L)1057 1880 y Fg(1)1130 1882 y Fr(is)37 b(strictly)g(decreasing)g(in)f(time)g(\(for)g Fn(t)d(>)g Fr(0)i(su\016cien)m(tly)k(small\),)e(see)0 2003 y(\([7],)c(Section)h(3.1\).)44 b(Since)34 b Fn(!)1097 1967 y Fi(\013;\025)1235 2003 y Fm(2)29 b Fn(L)1396 1967 y Fk(1)1436 2003 y Fr(\()p Fl(R)1558 1967 y Fk(2)1597 2003 y Fr(\))22 b Fm(\\)h Fn(C)1823 1967 y Fk(0)1862 2003 y Fr(\()p Fl(R)1984 1967 y Fk(2)2023 2003 y Fr(\))33 b(is)h(a)e(stationary)i(solution)f(of)g(\(6\))f(satisfying)0 2043 y Fj(R)47 2158 y Fh(R)108 2139 y Fg(2)163 2123 y Fn(!)228 2087 y Fi(\013;\025)354 2123 y Fr(d)p Fn(x)d Fr(=)e Fn(\013)q Fr(,)32 b(the)h(prop)s(erties)h(ab)s(o)m(v)m(e)f (imply)h(that)e Fn(!)2170 2087 y Fi(\013;\025)2280 2123 y Fr(\()p Fn(x)p Fr(\))c Fn(>)f Fr(0)33 b(for)f(all)g Fn(x)c Fm(2)h Fl(R)3170 2087 y Fk(2)3241 2123 y Fr(if)j Fn(\013)d(>)e Fr(0.)0 2474 y Fo(5)161 b(Stabilit)l(y)52 b(of)i(asymmetric)g(v)l(ortices)0 2718 y Fr(In)43 b(this)g(\014nal)g (section,)j(w)m(e)e(sho)m(w)f(that)g(the)g(asymmetric)h(Burgers)f(v)m (ortex)h Fn(!)3010 2682 y Fi(\013;\025)3162 2718 y Fr(constructed)g(in) 0 2839 y(Section)37 b(4)f(is)h(a)g(stable)g(solution)g(of)f(\(6\))g (with)h(resp)s(ect)g(to)g(p)s(erturbations)g(in)f Fn(X)8 b Fr(,)38 b(pro)m(vided)f Fn(\025)e(>)f Fr(0)0 2959 y(is)42 b(su\016cien)m(tly)i(small.)70 b(Fix)41 b Fn(\013)j Fm(2)f Fl(R)p Fr(,)g Fn(\025)f Fm(2)h Fr([0)p Fn(;)17 b(\025)1860 2974 y Fk(0)1899 2959 y 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3957 y Fg(4)1618 3959 y Fm(\024)28 b Fn(C)7 b Fm(k)h Fr(~)-57 b Fn(!)s Fm(k)1964 3982 y Fi(L)2012 3962 y Fg(4)p Fb(=)p Fg(3)2141 3959 y Fm(\024)28 b Fn(C)7 b Fm(k)h Fr(~)-57 b Fn(!)s Fm(k)2487 3974 y Fi(X)2554 3959 y Fr(,)33 b(see)h(\(20\))e(and)g (\(21\).)146 4105 y(It)h(remains)h(to)f(b)s(ound)g(\()8 b(~)-57 b Fn(!)s(;)17 b Fm(N)1280 4068 y Fi(\013;\025)1397 4105 y Fr(~)-57 b Fn(!)t Fr(\))1492 4120 y Fi(X)1587 4105 y Fr(=)28 b(\()8 b(~)-57 b Fn(!)t(;)17 b Fl(v)1899 4068 y Fi(\013;\025)2031 4105 y Fm(\001)22 b(r)8 b Fr(~)-57 b Fn(!)s Fr(\))2266 4120 y Fi(X)2356 4105 y Fr(+)22 b(\()8 b(~)-57 b Fn(!)s(;)2607 4104 y Fr(~)2600 4105 y Fl(u)22 b Fm(\001)g(r)p Fn(w)2890 4068 y Fi(\013;\025)3000 4105 y Fr(\))3038 4120 y Fi(X)3105 4105 y Fr(.)45 b(The)34 b(\014rst)f(term)0 4225 y(in)48 b(this)g(sum)h(can)f(b)s(e)f(estimated) i(in)f(the)g(same)h(w)m(a)m(y)g(as)f(the)g(nonlinear)g(term)g(ab)s(o)m (v)m(e,)k(namely)0 4345 y Fm(j)p Fr(\()8 b(~)-57 b Fn(!)s(;)17 b Fl(v)235 4309 y Fi(\013;\025)367 4345 y Fm(\001)22 b(r)8 b Fr(~)-57 b Fn(!)s Fr(\))602 4360 y Fi(X)669 4345 y Fm(j)28 b(\024)g Fn(C)7 b Fm(k)p Fn(w)1030 4309 y Fi(\013;\025)1139 4345 y Fm(k)1189 4360 y Fi(X)1256 4345 y Fn(E)f Fr(\()p Fn(f)11 b Fr(\).)43 b(F)-8 b(or)32 b(the)h(second)h(term,)f(w)m(e)h (argue)e(di\013eren)m(tly:)522 4608 y Fm(j)p Fr(\()8 b(~)-57 b Fn(!)s(;)703 4607 y Fr(~)696 4608 y Fl(u)22 b Fm(\001)g(r)p Fn(w)986 4567 y Fi(\013;\025)1096 4608 y Fr(\))1134 4623 y Fi(X)1201 4608 y Fm(j)83 b(\024)1472 4473 y Fj(Z)1528 4698 y Fh(R)1589 4679 y Fg(2)1644 4608 y Fn(G)1721 4567 y Ff(\000)p Fk(1)1815 4608 y Fm(j)8 b Fr(~)-57 b Fn(!)s Fm(jj)1970 4607 y Fr(~)1963 4608 y Fl(u)p Fm(jjr)p Fn(w)2237 4567 y Fi(\013;\025)2346 4608 y Fm(j)17 b Fr(d)p Fn(x)1312 4819 y Fm(\024)83 b(kr)p Fn(w)1678 4778 y Fi(\013;\025)1788 4819 y Fm(k)1838 4834 y Fi(X)1905 4819 y Fm(k)1962 4818 y Fr(~)1955 4819 y Fl(u)p Fm(k)2067 4836 y Fi(L)2115 4817 y Fg(4)2154 4819 y Fm(k)p Fn(f)11 b Fm(k)2313 4836 y Fi(L)2361 4817 y Fg(4)2443 4819 y Fm(\024)45 b Fn(C)7 b Fm(k)p Fn(w)2765 4778 y Fi(\013;\025)2874 4819 y Fm(k)2924 4834 y Fi(Y)2985 4819 y Fn(E)f Fr(\()p Fn(f)11 b Fr(\))32 b Fn(;)0 5033 y Fr(where)44 b(w)m(e)f(ha)m(v)m(e)h(used)f Fm(k)969 5032 y Fr(~)962 5033 y Fl(u)p Fm(k)1074 5050 y Fi(L)1122 5031 y Fg(4)1205 5033 y Fm(\024)i Fn(C)7 b Fm(k)h Fr(~)-57 b Fn(!)s Fm(k)1568 5056 y Fi(L)1616 5036 y Fg(4)p Fb(=)p Fg(3)1761 5033 y Fm(\024)45 b Fn(C)7 b Fm(k)p Fn(f)k Fm(k)2119 5050 y Fi(L)2167 5031 y Fg(2)2250 5033 y Fm(\024)45 b Fn(C)7 b(E)f Fr(\()p Fn(f)11 b Fr(\))2662 4997 y Fk(1)p Fi(=)p Fk(2)2771 5033 y Fr(.)73 b(Th)m(us,)47 b(using)c(Prop)s(osi-)0 5154 y(tion)g(4.2)f(to)h(b)s(ound)g Fn(w)892 5118 y Fi(\013;\025)1001 5154 y Fr(,)j(w)m(e)e(conclude)g(that)e Fm(j)p Fr(\()8 b(~)-57 b Fn(!)s(;)17 b Fm(N)2131 5118 y Fi(\013;\025)2248 5154 y Fr(~)-56 b Fn(!)s Fr(\))2343 5169 y Fi(X)2410 5154 y Fm(j)45 b(\024)h Fn(C)7 b Fm(k)p Fn(w)2806 5118 y Fi(\013;\025)2915 5154 y Fm(k)2965 5169 y Fi(Y)3026 5154 y Fn(E)f Fr(\()p Fn(f)11 b 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