Content-Type: multipart/mixed; boundary="-------------0406151013226" This is a multi-part message in MIME format. ---------------0406151013226 Content-Type: text/plain; name="04-186.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-186.keywords" Navier-Stokes equation, vorticity, uniqueness of solutions, self-similar variables ---------------0406151013226 Content-Type: application/postscript; name="unique_arxiv.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="unique_arxiv.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: unique_arxiv.dvi %%Pages: 32 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips unique_arxiv -o unique_arxiv.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2004.06.15:1724 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S 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b(only)h(on)i Fq(d)p Fr(.)41 b(It)30 b(follo)m(ws)g(that)971 4392 y Fp(k)p Fq(F)1074 4406 y Fl(i;)p Fn(1)1158 4392 y Fr(\()p Fq(\034)10 b Fr(\))p Fp(k)1323 4411 y Fl(L)1371 4392 y Fk(2)1407 4411 y Fn(\()p Fl(m)p Fn(\))1569 4392 y Fp(\024)40 b Fq(C)1745 4406 y Fn(1)1784 4392 y Fr(e)1834 4327 y Ff(\034)p 1835 4339 V 1837 4380 a Fk(2)1914 4392 y Fq(;)106 b Fp(\0001)25 b Fq(<)g(\034)35 b(<)25 b Fr(log)r(\()p Fq(T)13 b Fr(\))30 b Fq(;)785 b Fr(\(5.5\))0 4584 y(where)30 b Fq(C)328 4598 y Fn(1)393 4584 y Fq(>)25 b Fr(0)30 b(dep)s(ends)f(on)h Fq(M)1130 4598 y Fn(pp)1243 4584 y Fr(and)g Fq(d)p Fr(.)141 4697 y(T)-8 b(o)31 b(estimate)g Fq(F)697 4711 y Fl(i;)p Fn(2)781 4697 y Fr(,)f(w)m(e)h(write)f(similarly)281 4985 y Fp(k)p Fq(F)384 4999 y Fl(i;)p Fn(2)468 4985 y Fr(\()p Fq(\034)10 b Fr(\))p Fp(k)633 5005 y Fl(L)681 4986 y Fk(2)717 5005 y Fn(\()p Fl(m)p Fn(\))879 4985 y Fp(\024)40 b Fq(K)1067 4999 y Fn(9)1122 4899 y Fh(X)1132 5096 y Fl(j)t Fi(6)p Fn(=)p Fl(i)1268 4985 y Fp(j)p Fq(\013)1351 4999 y Fl(j)1388 4985 y Fp(j)1428 4861 y Fh(Z)1520 4888 y Fl(\034)1479 5068 y Fi(\0001)1708 4924 y Fr(e)1748 4888 y Fi(\000)1813 4860 y Ff(\034)6 b Fg(\000)p Ff(\034)1931 4839 y Fg(0)p 1813 4873 140 3 v 1868 4914 a Fk(2)p 1634 4964 408 4 v 1634 5070 a Fq(a)p Fr(\()p Fq(\034)31 b Fp(\000)19 b Fq(\034)1928 5044 y Fi(0)1952 5070 y Fr(\))1997 5007 y Fk(1)p 1997 5019 31 3 v 1997 5060 a(2)2052 4881 y Fh(\015)2052 4935 y(\015)2052 4990 y(\015)2106 4985 y Fr(~)-49 b Fq(v)2146 4999 y Fl(j)2183 4985 y Fr(\()p Fq(\030)24 b Fp(\000)c Fr(\()p Fq(z)2450 4999 y Fl(j)2487 4985 y Fp(\000)p Fq(z)2600 4999 y Fl(i)2628 4985 y Fr(\)e)2703 4948 y Fi(\000)2768 4921 y Ff(\034)2803 4900 y Fg(0)p 2769 4933 58 3 v 2782 4974 a Fk(2)2840 4985 y Fq(;)15 b(\034)2930 4948 y Fi(0)2954 4985 y Fr(\))p Fq(b)3028 4948 y Fl(m)3095 4985 y Fq(G)3166 4881 y Fh(\015)3166 4935 y(\015)3166 4990 y(\015)3217 5048 y Fl(L)3265 5030 y Fk(2)3319 4985 y Fr(d)p Fq(\034)3420 4948 y Fi(0)3473 4985 y Fq(:)0 5285 y Fr(Next,)31 b(w)m(e)g(\014x)f 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Fl(j)1911 5530 y Fr(\()p Fq(\034)1996 5493 y Fi(0)2020 5530 y Fr(\))2055 5426 y Fh(\015)2055 5480 y(\015)2055 5535 y(\015)2105 5593 y Fl(L)2153 5575 y Ff(q)2192 5426 y Fh(\015)2192 5480 y(\015)2192 5535 y(\015)2242 5530 y Fq(b)p Fr(\()p Fq(\030)25 b Fp(\000)20 b Fr(\()p Fq(z)2549 5544 y Fl(j)2586 5530 y Fp(\000)p Fq(z)2699 5544 y Fl(i)2727 5530 y Fr(\)e)2802 5493 y Fi(\000)2867 5465 y Ff(\034)2902 5445 y Fg(0)p 2868 5477 58 3 v 2881 5519 a Fk(2)2939 5530 y Fr(\))2984 5458 y Fk(2)p 2984 5470 31 3 v 2984 5511 a Ff(q)3025 5485 y Fi(\000)p Fl(\027)3123 5530 y Fq(b)3162 5493 y Fl(m)3228 5530 y Fq(G)3299 5426 y Fh(\015)3299 5480 y(\015)3299 5535 y(\015)3350 5597 y Fl(L)3432 5531 y Fk(2)p Ff(q)p 3408 5546 109 3 v 3408 5580 a(q)r Fg(\000)p Fk(2)3566 5530 y Fq(:)1844 5800 y Fr(15)p eop %%Page: 16 16 16 15 bop 0 91 a Fr(In)26 b(view)h(of)g(\(2.20\))i(the)f(\014rst)e (factor)i(in)e(the)h(righ)m(t-hand)f(side)g(can)h(b)s(e)g(estimated)g (b)m(y)g Fq(C)7 b Fp(k)19 b Fr(~)-64 b Fq(w)3150 105 y Fl(j)3187 91 y Fr(\()p Fq(\034)3272 58 y Fi(0)3295 91 y Fr(\))p Fp(k)3375 111 y Fl(L)3423 92 y Fk(2)3459 111 y Fn(\()p Fl(m)p Fn(\))3580 91 y Fr(,)28 b(and)0 241 y(a)k(direct)f(calculation)g(sho)m(ws)g(that)i(the)f(second)f(one)h (is)f(b)s(ounded)f(b)m(y)h Fq(C)7 b Fr(e)2583 195 y Fl(\034)2622 172 y Fg(0)2644 195 y Fn(\()2681 168 y Ff(\027)p 2682 180 35 3 v 2684 221 a Fk(2)2726 195 y Fi(\000)2791 168 y Fk(1)p 2791 180 31 3 v 2791 221 a Ff(q)2831 195 y Fn(\))2863 241 y Fr(,)32 b(where)f Fq(C)j(>)27 b Fr(0)32 b(dep)s(ends)0 354 y(on)e Fq(d)p Fr(.)41 b(Th)m(us)485 626 y Fp(k)p Fq(F)588 640 y Fl(i;)p Fn(2)672 626 y Fr(\()p Fq(\034)10 b Fr(\))p Fp(k)837 646 y Fl(L)885 627 y Fk(2)920 646 y Fn(\()p Fl(m)p Fn(\))1125 626 y Fp(\024)83 b Fq(C)1365 540 y Fh(X)1375 737 y Fl(j)t Fi(6)p Fn(=)p Fl(i)1512 626 y Fp(j)p Fq(\013)1595 640 y Fl(j)1632 626 y Fp(j)1672 502 y Fh(Z)1763 529 y Fl(\034)1723 709 y Fi(\0001)1952 565 y Fr(e)1992 529 y Fi(\000)2057 501 y Ff(\034)6 b Fg(\000)p Ff(\034)2175 480 y Fg(0)p 2057 514 140 3 v 2111 555 a Fk(2)p 1877 605 408 4 v 1877 711 a Fq(a)p Fr(\()p Fq(\034)31 b Fp(\000)20 b Fq(\034)2172 685 y Fi(0)2195 711 y Fr(\))2240 648 y Fk(1)p 2240 660 31 3 v 2240 701 a(2)2295 626 y Fp(k)f Fr(~)-64 b Fq(w)2405 640 y Fl(j)2442 626 y Fr(\()p Fq(\034)2527 589 y Fi(0)2551 626 y Fr(\))p Fp(k)2631 646 y Fl(L)2679 627 y Fk(2)2714 646 y Fn(\()p Fl(m)p Fn(\))2836 626 y Fr(e)2876 581 y Fl(\034)2915 557 y Fg(0)2937 581 y Fn(\()2974 554 y Ff(\027)p 2975 566 35 3 v 2977 607 a Fk(2)3019 581 y Fi(\000)3084 554 y Fk(1)p 3084 566 31 3 v 3084 607 a Ff(q)3124 581 y Fn(\))3171 626 y Fr(d)p Fq(\034)3272 589 y Fi(0)1125 897 y Fp(\024)83 b Fq(C)1344 911 y Fn(2)1383 897 y Fq(M)10 b Fr(\(e)1556 860 y Fl(\034)1600 897 y Fr(\)e)1675 852 y Fl(\034)e Fn(\()1751 825 y Ff(\027)p 1752 837 35 3 v 1754 878 a Fk(2)1797 852 y Fi(\000)1862 825 y Fk(1)p 1861 837 31 3 v 1861 878 a Ff(q)1902 852 y Fn(\))1964 897 y Fq(;)1604 b Fr(\(5.6\))0 1101 y(where)30 b Fq(C)328 1115 y Fn(2)393 1101 y Fq(>)25 b Fr(0)30 b(dep)s(ends)f(on)h Fq(M)1130 1115 y Fn(pp)1243 1101 y Fr(and)g Fq(d)p Fr(.)141 1214 y(No)m(w)h(w)m(e)g(consider)e Fq(F)894 1228 y Fl(i;)p Fn(3)978 1214 y Fr(.)41 b(Using)29 b(Prop)s(osition)f(4.6)k(and)e (H\177)-45 b(older's)30 b(inequalit)m(y)-8 b(,)29 b(w)m(e)i(obtain)281 1513 y Fp(k)p Fq(F)384 1527 y Fl(i;)p Fn(3)468 1513 y Fr(\()p Fq(\034)10 b Fr(\))p Fp(k)633 1532 y Fl(L)681 1514 y Fk(2)716 1532 y Fn(\()p Fl(m)p Fn(\))921 1513 y Fp(\024)83 b Fq(K)1152 1527 y Fn(9)1206 1389 y Fh(Z)1297 1415 y Fl(\034)1257 1595 y Fi(\0001)1486 1451 y Fr(e)1526 1415 y Fi(\000)1591 1387 y Ff(\034)6 b Fg(\000)p Ff(\034)1709 1367 y Fg(0)p 1591 1400 140 3 v 1646 1441 a Fk(2)p 1412 1492 408 4 v 1412 1598 a Fq(a)p Fr(\()p Fq(\034)30 b Fp(\000)20 b Fq(\034)1706 1572 y Fi(0)1730 1598 y Fr(\))1775 1535 y Fk(1)p 1775 1547 31 3 v 1775 1588 a(2)1845 1408 y Fh(\015)1845 1463 y(\015)1845 1517 y(\015)1895 1513 y Fr(e)1945 1448 y Ff(\034)1980 1427 y Fg(0)p 1945 1460 58 3 v 1959 1501 a Fk(2)2022 1513 y Fr(~)-50 b Fq(u)2069 1527 y Fn(0)2108 1513 y Fr(\()p Fq(\030)t Fr(e)2237 1448 y Ff(\034)2272 1427 y Fg(0)p 2238 1460 V 2251 1501 a Fk(2)2329 1513 y Fr(+)20 b Fq(z)2462 1527 y Fl(i)2491 1513 y Fq(;)15 b Fr(e)2571 1475 y Fl(\034)2610 1452 y Fg(0)2637 1513 y Fr(\))p Fq(G)2753 1448 y Fk(1)p 2754 1460 31 3 v 2754 1501 a(2)2798 1408 y Fh(\015)2798 1463 y(\015)2798 1517 y(\015)2849 1576 y Fl(L)2897 1557 y Fk(4)2951 1513 y Fp(k)p Fq(b)3035 1475 y Fl(m)3102 1513 y Fq(G)3183 1448 y Fk(1)p 3183 1460 V 3183 1501 a(2)3228 1513 y Fp(k)3273 1532 y Fl(L)3321 1514 y Fk(4)3375 1513 y Fr(d)p Fq(\034)3476 1475 y Fi(0)921 1828 y Fp(\024)83 b Fq(C)1161 1704 y Fh(Z)1252 1730 y Fl(\034)1212 1910 y Fi(\0001)1441 1766 y Fr(e)1481 1730 y Fi(\000)1546 1702 y Ff(\034)6 b Fg(\000)p Ff(\034)1664 1681 y Fg(0)p 1546 1715 140 3 v 1601 1756 a Fk(2)p 1366 1807 408 4 v 1366 1913 a Fq(a)p Fr(\()p Fq(\034)31 b Fp(\000)20 b Fq(\034)1661 1887 y Fi(0)1684 1913 y Fr(\))1729 1850 y Fk(1)p 1730 1862 31 3 v 1730 1903 a(2)1784 1828 y Fq(\025)1837 1842 y Fl(i)1865 1828 y Fr(\(e)1940 1790 y Fl(\034)1979 1767 y Fg(0)2007 1828 y Fr(\))15 b(d)p Fq(\034)2158 1790 y Fi(0)2221 1828 y Fp(\024)41 b Fq(C)2398 1842 y Fn(3)2437 1828 y Fq(\025)2490 1842 y Fl(i)2518 1828 y Fr(\(e)2593 1790 y Fl(\034)2637 1828 y Fr(\))31 b Fq(;)865 b Fr(\(5.7\))0 2097 y(where)30 b Fq(C)328 2111 y Fn(3)393 2097 y Fq(>)25 b Fr(0)30 b(dep)s(ends)f(on)h Fq(M)1130 2111 y Fn(pp)1243 2097 y Fr(and)1133 2350 y Fq(\025)1186 2364 y Fl(i)1215 2350 y Fr(\()p Fq(t)p Fr(\))41 b(=)73 b(sup)1470 2427 y Fn(0)p Fl()e Fr(0)j(dep)s(ends)e(on)h Fq(M)1118 4061 y Fn(pp)1201 4047 y Fr(.)39 b(This)26 b(b)s(ound)f(will)g(b)s(e)h(used) h(later)g(on)g(when)g(estimating)g(the)g(di\013erence)0 4160 y(of)22 b(t)m(w)m(o)h(solutions.)36 b(It)22 b(is)f(not)g (su\016cien)m(t)h(for)f(our)g(presen)m(t)h(purp)s(oses)e(b)s(ecause,)j (unlik)m(e)d(in)h(\(5.6\),)k(the)d(prefactor)0 4273 y(of)35 b Fq(M)196 4287 y Fl(i)224 4273 y Fr(\(e)299 4240 y Fl(\034)343 4273 y Fr(\))g(do)s(es)f(not)h(con)m(v)m(erge)i(to)f(zero)f(as)g Fq(\034)42 b Fp(!)33 b(\0001)p Fr(.)53 b(T)-8 b(o)35 b(estimate)h Fq(F)2603 4287 y Fl(i;)p Fn(4)2721 4273 y Fr(more)f(precisely)-8 b(,)35 b(w)m(e)g(observ)m(e)0 4385 y(that,)c(on)f(the)h(one)g(hand,)889 4626 y Fp(j)p Fq(\030)t Fp(j)25 b(\024)1114 4565 y Fq(d)p 1114 4606 48 4 v 1115 4689 a Fr(2)1171 4626 y(e)1211 4589 y Fi(\000)1276 4562 y Ff(\034)1311 4541 y Fg(0)p 1277 4574 58 3 v 1290 4615 a Fk(2)1464 4626 y Fp(\))116 b(j)p Fq(v)1743 4589 y Fl(G)1803 4626 y Fr(\()p Fq(\030)24 b Fp(\000)c Fr(\()p Fq(z)2070 4640 y Fl(j)2107 4626 y Fp(\000)p Fq(z)2220 4640 y Fl(i)2249 4626 y Fr(\)e)2324 4589 y Fi(\000)2389 4562 y Ff(\034)2424 4541 y Fg(0)p 2389 4574 V 2403 4615 a Fk(2)2461 4626 y Fr(\))p Fp(j)25 b(\024)g Fq(C)7 b Fr(e)2764 4562 y Ff(\034)2799 4541 y Fg(0)p 2764 4574 V 2778 4615 a Fk(2)2866 4626 y Fq(;)0 4856 y Fr(where)30 b Fq(C)i(>)25 b Fr(0)30 b(dep)s(ends)f(on)h Fq(d)p Fr(.)41 b(On)30 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Fq(:)0 5326 y Fr(It)30 b(follo)m(ws)g(that)536 5538 y Fp(k)p Fq(v)628 5500 y Fl(G)688 5538 y Fr(\()p Fq(\030)25 b Fp(\000)20 b Fr(\()p Fq(z)956 5552 y Fl(j)993 5538 y Fp(\000)p Fq(z)1106 5552 y Fl(i)1134 5538 y Fr(\)e)1209 5500 y Fi(\000)1274 5473 y Ff(\034)1309 5452 y Fg(0)p 1275 5485 58 3 v 1288 5526 a Fk(2)1346 5538 y Fr(\))f(~)-64 b Fq(w)1446 5552 y Fl(i)1474 5538 y Fr(\()p Fq(\034)1559 5500 y Fi(0)1583 5538 y Fr(\))p Fp(k)1663 5557 y Fl(L)1711 5539 y Fk(2)1747 5557 y Fn(\()p Fl(m)p Fn(\))1908 5538 y Fp(\024)41 b Fq(C)2092 5437 y Fh(\020)2145 5538 y Fr(e)2195 5473 y Ff(\034)2230 5452 y Fg(0)p 2196 5485 V 2209 5526 a Fk(2)2267 5538 y Fp(k)19 b Fr(~)-64 b Fq(w)2377 5552 y Fl(i)2406 5538 y Fr(\()p Fq(\034)2491 5500 y Fi(0)2514 5538 y Fr(\))p Fp(k)2594 5557 y Fl(L)2642 5539 y Fk(2)2678 5557 y Fn(\()p Fl(m)p Fn(\))2819 5538 y Fr(+)20 b Fq(\020)7 b Fr(\(e)3032 5500 y Fl(\034)3071 5477 y Fg(0)3098 5538 y Fr(\))3133 5437 y Fh(\021)3218 5538 y Fq(;)1844 5800 y Fr(16)p eop %%Page: 17 17 17 16 bop 0 91 a Fr(where)30 b Fq(C)j(>)26 b Fr(0)31 b(dep)s(ends)e(on)h Fq(d)i Fr(and)e Fq(\020)7 b Fr(\()p Fq(t)p Fr(\))26 b(=)f(exp\()p Fp(\000)p Fq(\032=t)p Fr(\))32 b(for)e(some)i Fq(\032)26 b(>)f Fr(0.)43 b(Replacing)30 b(in)m(to)h(\(5.8\),)i(w)m(e)e(th)m(us)0 204 y(\014nd)1060 317 y Fp(k)p Fq(F)1163 331 y Fl(i;)p Fn(4)1247 317 y Fr(\()p Fq(\034)10 b Fr(\))p Fp(k)1412 337 y Fl(L)1460 318 y Fk(2)1496 337 y Fn(\()p Fl(m)p Fn(\))1658 317 y Fp(\024)40 b Fq(C)1834 331 y Fn(4)1873 216 y Fh(\020)1928 317 y Fr(e)1978 252 y Ff(\034)p 1978 264 35 3 v 1980 306 a Fk(2)2027 317 y Fq(M)2115 331 y Fl(i)2143 317 y Fr(\(e)2218 280 y Fl(\034)2262 317 y Fr(\))21 b(+)f Fq(\020)7 b Fr(\(e)2531 280 y Fl(\034)2574 317 y Fr(\))2609 216 y Fh(\021)2694 317 y Fq(;)828 b Fr(\(5.10\))0 502 y(where)30 b Fq(C)328 516 y Fn(4)393 502 y Fq(>)25 b Fr(0)30 b(dep)s(ends)f(on)h Fq(M)1130 516 y Fn(pp)1243 502 y Fr(and)g Fq(d)p Fr(.)141 615 y(T)-8 b(o)31 b(estimate)g Fq(F)697 629 y Fl(i;)p Fn(5)811 615 y Fr(w)m(e)g(ha)m(v)m(e)g(b)m(y)g(Prop)s(osition)d(4.6,)k (for)e(1)c Fq(<)f(p)g(<)g Fr(2,)196 915 y Fp(k)p Fq(F)299 929 y Fl(i;)p Fn(5)383 915 y Fr(\()p Fq(\034)10 b Fr(\))p Fp(k)548 934 y Fl(L)596 915 y Fk(2)632 934 y Fn(\()p Fl(m)p Fn(\))794 915 y Fp(\024)40 b Fq(K)982 929 y Fn(9)1071 801 y Fl(N)1037 828 y Fh(X)1041 1024 y Fl(j)t Fn(=1)1183 915 y Fp(j)p Fq(\013)1266 929 y Fl(j)1303 915 y Fp(j)1343 791 y Fh(Z)1434 817 y Fl(\034)1394 997 y Fi(\0001)1624 853 y Fr(e)1664 817 y Fi(\000)1729 789 y Ff(\034)6 b Fg(\000)p Ff(\034)1847 768 y Fg(0)p 1729 802 140 3 v 1784 843 a Fk(2)p 1549 894 410 4 v 1549 1009 a Fq(a)p Fr(\()p Fq(\034)30 b Fp(\000)20 b Fq(\034)1843 983 y Fi(0)1866 1009 y Fr(\))1912 937 y Fk(1)p 1912 949 33 3 v 1912 990 a Ff(p)1968 915 y Fp(k)s Fr(~)-48 b Fq(v)2057 929 y Fl(j)2094 915 y Fr(\()p Fq(\030)24 b Fp(\000)c Fr(\()p Fq(z)2361 929 y Fl(j)2398 915 y Fp(\000)p Fq(z)2511 929 y Fl(i)2540 915 y Fr(\)e)2615 877 y Fi(\000)2680 850 y Ff(\034)2715 829 y Fg(0)p 2680 862 58 3 v 2694 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Fr(\))2988 5437 y Fh(o)3079 5538 y Fq(;)1844 5800 y Fr(24)p eop %%Page: 25 25 25 24 bop 0 91 a Fr(equipp)s(ed)28 b(with)h(the)i(norm)710 313 y Fp(k)p Fq(w)r Fp(k)867 327 y Fl(X)976 313 y Fr(=)41 b Fp(k)p Fq(w)r Fp(k)1245 333 y Fl(L)1293 314 y Fk(1)1353 313 y Fr(+)20 b(inf)1554 212 y Fh(n)1615 313 y Fp(k)p Fq(f)1705 327 y Fn(1)1744 313 y Fp(k)1789 333 y Fl(L)1837 314 y Fk(1)1897 313 y Fr(+)g Fp(k)p Fq(f)2078 327 y Fn(2)2117 313 y Fp(k)2162 333 y Fl(L)2210 314 y Fk(1)2265 209 y Fh(\014)2265 263 y(\014)2265 318 y(\014)2310 313 y Fq(w)28 b Fr(=)d Fq(@)2547 327 y Fn(1)2586 313 y Fq(f)2631 327 y Fn(1)2691 313 y Fr(+)19 b Fq(@)2829 327 y Fn(2)2869 313 y Fq(f)2914 327 y Fn(2)2953 212 y Fh(o)3044 313 y Fq(:)0 540 y Fr(W)-8 b(e)32 b(also)e(consider)f(the)i(auxiliary)d (equation)i(for)g(the)h(v)m(ector)h(\014eld)d Fq(f)34 b Fr(=)25 b(\()p Fq(f)2621 554 y Fn(1)2660 540 y Fq(;)15 b(f)2745 554 y Fn(2)2785 540 y Fr(\):)1289 726 y Fq(@)5 b(f)p 1289 766 108 4 v 1291 849 a(@)g(\034)1426 787 y Fr(+)20 b 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Fk(2)1893 4000 y Fr(\))i(as)f Fq(\034)41 b Fp(!)31 b(\0001)p Fr(,)j(and)f(using)f (\(2.20\))k(the)e(second)f(one)0 4113 y(can)28 b(b)s(e)f(b)s(ounded)e (b)m(y)j Fq(C)7 b Fp(k)p Fq(w)959 4127 y Fl(j)995 4113 y Fr(\()p Fq(\034)j Fr(\))p Fp(k)1160 4133 y Fl(L)1208 4114 y Fk(2)1244 4133 y Fn(\()p Fl(m)p Fn(\))1391 4113 y Fp(\024)25 b Fq(C)1552 4127 y Fn(2)1591 4113 y Fr(,)k(where)e Fq(C)1970 4127 y Fn(2)2034 4113 y Fq(>)e Fr(0)j(is)f(indep)s(enden)m(t) e(of)j Fq(\034)10 b Fr(.)40 b(On)27 b(the)g(other)h(hand,)0 4253 y(applying)g(Lemma)j(4.2)g(with)e Fq(\037)p Fr(\()p Fq(r)s Fr(\))d(=)f(\(1+)p Fq(r)s Fr(\))1572 4217 y Fi(\000)1637 4190 y Fk(1)p 1637 4202 31 3 v 1637 4243 a(2)1681 4253 y Fr(,)31 b(w)m(e)g(obtain)f(\(with)f Fq(t)c Fr(=)g(e)2590 4220 y Fl(\034)2634 4253 y Fr(\))400 4513 y Fp(k)p Fq(b)484 4475 y Fi(\000)p Fn(1)578 4513 y Fr(e)628 4448 y Ff(\034)p 629 4460 35 3 v 631 4501 a Fk(2)683 4513 y Fr(~)-50 b Fq(u)730 4527 y Fn(0)769 4513 y Fr(\()p Fq(\030)t Fr(e)898 4448 y Ff(\034)p 899 4460 V 901 4501 a Fk(2)968 4513 y Fr(+)20 b Fq(z)1101 4527 y Fl(i)1129 4513 y Fq(;)15 b Fr(e)1209 4475 y Fl(\034)1253 4513 y Fr(\))p Fp(k)1333 4535 y Fl(L)1381 4503 y Ff(q)1381 4559 y(\030)1461 4513 y Fr(=)40 b Fq(t)1615 4440 y Fk(1)p 1615 4452 31 3 v 1615 4493 a(2)1655 4467 y Fi(\000)1720 4440 y Fk(1)p 1720 4452 V 1720 4493 a Ff(q)1765 4408 y Fh(\015)1765 4463 y(\015)1765 4517 y(\015)1821 4513 y Fr(~)-51 b Fq(u)1867 4527 y Fn(0)1907 4513 y Fr(\()p Fq(x;)15 b(t)p Fr(\))2102 4412 y Fh(\020)2157 4513 y Fr(1)20 b(+)2323 4451 y Fp(j)p Fq(x)h Fp(\000)f Fq(z)2554 4465 y Fl(i)2582 4451 y Fp(j)2607 4418 y Fn(2)p 2323 4492 324 4 v 2469 4575 a Fq(t)2657 4412 y Fh(\021)2711 4434 y Fi(\000)2776 4407 y Fk(1)p 2776 4419 31 3 v 2776 4461 a(2)2820 4408 y Fh(\015)2820 4463 y(\015)2820 4517 y(\015)2871 4576 y Fl(L)2919 4543 y Ff(q)2919 4587 y(x)3002 4513 y Fp(\000)-23 b(\000)i(\000)e(!)3055 4571 y Fl(t)p Fi(!)p Fn(0)3279 4513 y Fr(0)31 b Fq(;)0 4762 y Fr(Th)m(us)e Fp(k)p Fq(b)314 4729 y Fi(\000)p 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Fl(i)3628 5038 y Fr(\))26 b Fp(!)0 5150 y Fr(0)d(as)g Fq(\034)35 b Fp(!)25 b(\0001)p Fr(.)38 b(Let)23 b(also)f(\010\()p Fq(\034)10 b Fr(\))1104 5177 y Fl(\034)e Fi(\025)p Fn(0)1260 5150 y Fr(b)s(e)21 b(the)i(semi\015o)m(w)f(in)f Fq(L)2043 5117 y Fn(2)2083 5150 y Fr(\()p Fq(m)p Fr(\))i(de\014ned)e(b)m(y)h(the) h(limiting)d(equation)i(\(6.12\),)0 5263 y(see)31 b(\([10)q(],)g (Theorem)f(3.2\).)43 b(Our)29 b(last)h(result)f(is:)0 5476 y Fo(Lemma)k(6.3)46 b Fe(F)-7 b(or)33 b(any)h Fq(i)25 b Fp(2)g(f)p Fr(1)p Fq(;)15 b(:)g(:)g(:)j(;)d(N)10 b Fp(g)33 b Fe(and)h(any)f Fq(\034)i Fp(\025)25 b Fr(0)p Fe(,)33 b(we)g(have)g Fr(\010\()p Fq(\034)10 b Fr(\))p Fp(A)2721 5490 y Fl(i)2774 5476 y Fr(=)25 b Fp(A)2943 5490 y Fl(i)2971 5476 y Fe(.)1844 5800 y Fr(28)p eop %%Page: 29 29 29 28 bop 0 91 a Fo(Pro)s(of:)48 b Fr(It)41 b(is)f(clearly)g(enough)g (to)i(pro)m(v)m(e)g(the)f(result)e(for)i(0)i Fp(\024)f Fq(\034)53 b Fp(\024)43 b Fr(1.)72 b(If)40 b Fq(w)2839 105 y Fi(1)2957 91 y Fp(2)i(A)3133 105 y Fl(i)3161 91 y Fr(,)i(there)d(exists)f(a)0 204 y(sequence)31 b Fq(\034)417 218 y Fl(n)494 204 y Fr(going)f(to)h Fp(\0001)f Fr(suc)m(h)g(that)h Fp(k)p Fq(w)1551 218 y Fl(i)1580 204 y Fr(\()p Fq(\034)1655 218 y Fl(n)1702 204 y Fr(\))21 b Fp(\000)f Fq(w)1914 218 y Fi(1)1988 204 y Fp(k)2033 224 y Fl(L)2081 205 y Fk(2)2116 224 y Fn(\()p Fl(m)p Fn(\))2263 204 y Fp(!)25 b Fr(0)31 b(as)g Fq(n)24 b Fp(!)i(1)p Fr(.)40 b(F)-8 b(rom)31 b(\(4.14\))i(w)m(e)d(ha)m(v)m(e)291 427 y Fq(w)356 441 y Fl(i)384 427 y Fr(\()p Fq(\034)459 441 y Fl(n)526 427 y Fr(+)20 b Fq(\034)10 b Fr(\))41 b(=)f Fq(S)5 b Fr(\()p Fq(\034)10 b Fr(\))p Fq(w)1100 441 y Fl(i)1129 427 y Fr(\()p Fq(\034)1204 441 y Fl(n)1251 427 y Fr(\))21 b Fp(\000)1398 303 y Fh(Z)1489 330 y Fl(\034)1448 509 y Fn(0)1547 427 y Fr(e)1587 389 y Fi(\000)1652 362 y Fk(1)p 1652 374 31 3 v 1652 416 a(2)1693 389 y Fn(\()p Fl(\034)8 b Fi(\000)p Fl(\034)1853 366 y Fg(0)1875 389 y Fn(\))1907 427 y Fp(r)20 b(\001)g Fq(S)5 b Fr(\()p Fq(\034)31 b Fp(\000)20 b Fq(\034)2356 389 y Fi(0)2379 427 y Fr(\)\()p Fq(v)2493 441 y Fl(i)2522 427 y Fq(w)2587 441 y Fl(i)2636 427 y Fr(+)f Fq(R)2795 441 y Fl(i)2824 427 y Fq(w)2889 441 y Fl(i)2917 427 y Fr(\)\()p Fq(\034)3027 441 y Fl(n)3095 427 y Fr(+)g Fq(\034)3235 389 y Fi(0)3259 427 y Fr(\))c(d)p Fq(\034)3410 389 y Fi(0)3463 427 y Fq(;)0 688 y Fr(for)30 b Fq(\034)35 b Fp(2)25 b Fr([0)p Fq(;)15 b Fr(1].)43 b(On)29 b(the)i(other)f(hand,)g Fq(W)1432 702 y Fl(i)1460 688 y Fr(\()p Fq(\034)10 b Fr(\))1606 637 y Fv(def)1625 688 y Fr(=)44 b(\010\()p Fq(\034)10 b Fr(\))p Fq(w)1991 702 y Fi(1)2096 688 y Fr(satis\014es)397 897 y Fq(W)483 911 y Fl(i)511 897 y Fr(\()p Fq(\034)g Fr(\))41 b(=)f Fq(S)5 b Fr(\()p Fq(\034)10 b Fr(\))p Fq(w)1029 911 y Fi(1)1125 897 y Fp(\000)1216 774 y Fh(Z)1307 800 y Fl(\034)1267 980 y Fn(0)1365 897 y Fr(e)1405 860 y Fi(\000)1470 833 y Fk(1)p 1471 845 V 1471 886 a(2)1511 860 y Fn(\()p Fl(\034)e Fi(\000)p Fl(\034)1671 836 y Fg(0)1694 860 y Fn(\))1725 897 y Fp(r)20 b(\001)h Fq(S)5 b Fr(\()p Fq(\034)30 b Fp(\000)20 b Fq(\034)2174 860 y Fi(0)2197 897 y Fr(\)\()p Fq(V)2320 911 y Fl(i)2349 897 y Fq(W)2435 911 y Fl(i)2463 897 y Fr(\)\()p Fq(\034)2583 860 y Fi(0)2607 897 y Fr(\))15 b(d)p Fq(\034)2758 860 y Fi(0)2812 897 y Fq(;)106 b Fr(0)26 b Fp(\024)f Fq(\034)35 b Fp(\024)25 b Fr(1)31 b Fq(;)0 1108 y Fr(where)e Fq(V)315 1122 y Fl(i)343 1108 y Fr(\()p Fq(\034)10 b Fr(\))30 b(is)e(the)h(v)m(elo)s(cit)m(y)h(\014eld)d(obtained)i(from)f Fq(W)1940 1122 y Fl(i)1968 1108 y Fr(\()p Fq(\034)10 b Fr(\))30 b(via)f(the)g(Biot-Sa)m(v)-5 b(art)31 b(la)m(w.)40 b(No)m(w)30 b(w)m(e)f(compute)0 1221 y(the)d(di\013erence)f(of)h(b)s (oth)e(expressions.)38 b(Using)25 b(Prop)s(osition)f(2.2)i(and)f(pro)s (ceeding)g(as)h(in)e(\([10)q(],)j(Lemma)f(3.1\))0 1334 y(w)m(e)31 b(obtain,)f(for)g(an)m(y)h Fq(p)25 b Fp(2)g Fr(\(1)p Fq(;)15 b Fr(2\),)558 1499 y Fp(k)p Fq(w)668 1513 y Fl(i)696 1499 y Fr(\()p Fq(\034)771 1513 y Fl(n)839 1499 y Fr(+)k Fq(\034)10 b Fr(\))21 b Fp(\000)f Fq(W)1212 1513 y Fl(i)1240 1499 y Fr(\()p Fq(\034)10 b Fr(\))p Fp(k)1405 1519 y Fl(L)1453 1500 y Fk(2)1489 1519 y Fn(\()p Fl(m)p Fn(\))1651 1499 y Fp(\024)40 b Fq(K)1839 1513 y Fn(3)1878 1499 y Fp(k)p Fq(w)1988 1513 y Fl(i)2017 1499 y Fr(\()p Fq(\034)2092 1513 y Fl(n)2139 1499 y Fr(\))21 b Fp(\000)f Fq(w)2351 1513 y Fi(1)2425 1499 y Fp(k)2470 1519 y Fl(L)2518 1500 y Fk(2)2553 1519 y Fn(\()p Fl(m)p Fn(\))739 1695 y Fr(+)p Fq(K)887 1709 y Fn(4)942 1571 y Fh(Z)1033 1597 y Fl(\034)992 1777 y Fn(0)1091 1695 y Fr(e)1131 1657 y Fi(\000)1196 1630 y Fk(1)p 1196 1642 V 1196 1683 a(2)1237 1657 y Fn(\()p Fl(\034)8 b Fi(\000)p Fl(\034)1397 1634 y Fg(0)1419 1657 y Fn(\))1643 1633 y Fr(1)p 1461 1674 410 4 v 1461 1790 a Fq(a)p Fr(\()p Fq(\034)31 b Fp(\000)19 b Fq(\034)1755 1763 y Fi(0)1779 1790 y Fr(\))1825 1717 y Fk(1)p 1824 1729 33 3 v 1824 1770 a Ff(p)1880 1695 y Fp(k)p Fq(R)1994 1709 y Fl(i)2023 1695 y Fr(\()p Fq(\034)2098 1709 y Fl(n)2165 1695 y Fr(+)h Fq(\034)2306 1657 y Fi(0)2329 1695 y Fr(\))p Fq(w)2429 1709 y Fl(i)2458 1695 y Fr(\()p Fq(\034)2533 1709 y Fl(n)2600 1695 y Fr(+)g Fq(\034)2741 1657 y Fi(0)2764 1695 y Fr(\))p Fp(k)2844 1713 y Fl(L)2892 1694 y Ff(p)2929 1713 y Fn(\()p Fl(m)p Fn(\))3066 1695 y Fr(d)p Fq(\034)3167 1657 y Fi(0)739 1972 y Fr(+)p Fq(K)887 1986 y Fn(4)942 1848 y Fh(Z)1033 1874 y Fl(\034)992 2054 y Fn(0)1091 1972 y Fr(e)1131 1934 y Fi(\000)1196 1907 y Fk(1)p 1196 1919 31 3 v 1196 1960 a(2)1237 1934 y Fn(\()p Fl(\034)8 b Fi(\000)p Fl(\034)1397 1911 y Fg(0)1419 1934 y Fn(\))1630 1910 y Fq(C)p 1461 1951 410 4 v 1461 2066 a(a)p Fr(\()p Fq(\034)31 b Fp(\000)19 b Fq(\034)1755 2040 y Fi(0)1779 2066 y Fr(\))1825 1994 y Fk(1)p 1824 2006 33 3 v 1824 2047 a Ff(p)1895 1898 y Fh(\000)1937 1972 y Fp(k)p Fq(w)2047 1986 y Fl(i)2076 1972 y Fr(\()p Fq(\034)2151 1986 y Fl(n)2218 1972 y Fr(+)h Fq(\034)2359 1934 y Fi(0)2382 1972 y Fr(\))p Fp(k)2462 1991 y Fl(L)2510 1972 y Fk(2)2546 1991 y Fn(\()p Fl(m)p Fn(\))2687 1972 y Fr(+)g Fp(k)p Fq(W)2909 1986 y Fl(i)2938 1972 y Fr(\()p Fq(\034)3023 1934 y 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b(Me)-5 b(ch.)32 b(A)n(nal.)e Fo(128)i Fr(\(1994\),)h(329{358.)45 2598 y([2])47 b(H.)40 b(Brezis.)g(Remarks)f(on)h(the)g(preceding)e(pap) s(er)h(b)m(y)h(M.)g(Ben-Artzi:)60 b(\\Global)39 b(solutions)f(of)i(t)m (w)m(o-)187 2711 y(dimensional)30 b(Na)m(vier-Stok)m(es)35 b(and)e(Euler)f(equations".)i Fe(A)n(r)-5 b(ch.)35 b(R)-5 b(ational)38 b(Me)-5 b(ch.)35 b(A)n(nal.)f Fo(128)g Fr(\(1994\),)187 2824 y(359{360.)45 3011 y([3])47 b(M.)31 b(Cannone.)f Fm(Ondelettes,)g(parapro)s(duits)e(et)j(Na)m(vier-Stok)m(es.)h Fr(Diderot)f(Editeur,)e(P)m(aris,)h(1995.)45 3199 y([4])47 b(M.)c(Cannone)g(and)f(F.)h(Planc)m(hon.)g(Self-similar)d(solutions)h (for)i(Na)m(vier-Stok)m(es)h(equations)f(in)e Fo(R)3714 3166 y Fn(3)3754 3199 y Fr(.)187 3312 y Fe(Comm.)33 b(Partial)h (Di\013er.)e(Equations)f Fo(21)g Fr(\(1996\),)i(179{193.)45 3499 y([5])47 b(E.)34 b(A.)g(Carlen)e(and)h(M.)h(Loss.)g(Optimal)e(smo) s(othing)h(and)g(deca)m(y)i(estimates)f(for)g(viscously)e(damp)s(ed)187 3612 y(conserv)-5 b(ation)31 b(la)m(ws,)g(with)f(applications)f(to)j (the)f(2-D)i(Na)m(vier-Stok)m(es)f(equation.)f Fe(Duke)i(Math.)g(J.)e Fo(81)187 3725 y Fr(\(1995\),)i(135{157)g(\(1996\).)45 3913 y([6])47 b(A.)36 b(Carpio.)g(Asymptotic)g(b)s(eha)m(vior)f(for)h (the)g(v)m(orticit)m(y)h(equations)f(in)f(dimensions)e(t)m(w)m(o)k(and) f(three.)187 4026 y Fe(Comm.)d(Partial)h(Di\013er)-5 b(ential)33 b(Equations)e Fo(19)g Fr(\(1994\),)i(827{872.)45 4213 y([7])47 b(A.)24 b(Chorin)d(and)i(J.)h(Marsden.)f Fm(A)h(mathematical)f(in)m(tro)s(duction)f(to)i(\015uid)e(mec)m (hanics.)h Fr(Second)h(edition.)187 4326 y(T)-8 b(exts)31 b(in)e(Applied)f(Mathematics,)k Fo(4)p Fr(.)e(Springer-V)-8 b(erlag,)30 b(New)g(Y)-8 b(ork,)32 b(1990.)45 4514 y([8])47 b(G.-H.)28 b(Cottet.)h(Equations)e(de)g(Na)m(vier-Stok)m(es)i(dans)d (le)h(plan)f(a)m(v)m(ec)j(tourbillon)24 b(initial)h(mesure.)i Fe(C.)i(R.)187 4627 y(A)-5 b(c)g(ad.)33 b(Sci.)g(Paris)g(S)n(\023)-44 b(er.)32 b(I)h(Math.)e Fo(303)g Fr(\(1986\),)i(105{108.)45 4814 y([9])47 b(I.)c(Gallagher)f(and)h(F.)g(Planc)m(hon.)g(On)f(global) g(in\014nite)e(energy)k(solutions)d(to)j(the)f(Na)m(vier{Stok)m(es)187 4927 y(equations)30 b(in)f(t)m(w)m(o)j(dimensions.)c Fe(A)n(r)-5 b(ch.)32 b(R)-5 b(ational)35 b(Me)-5 b(ch.)33 b(A)n(nal.)d Fo(161)h Fr(\(2002\),)i(307{337.)0 5115 y([10])47 b(Th.)39 b(Galla)m(y)h(and)f(C.)h(E.)f(W)-8 b(a)m(yne.)42 b(In)m(v)-5 b(arian)m(t)39 b(manifolds)f(and)h(the)h (long-time)f(asymptotics)h(of)g(the)187 5228 y(Na)m(vier-Stok)m(es)29 b(and)e(v)m(orticit)m(y)h(equations)g(on)f Fo(R)1899 5195 y Fn(2)1939 5228 y Fr(.)h Fe(A)n(r)-5 b(ch.)30 b(R)-5 b(ational)33 b(Me)-5 b(ch.)30 b(A)n(nal.)d Fo(163)i Fr(\(2002\),)i (209{)187 5341 y(258.)1844 5800 y(31)p eop %%Page: 32 32 32 31 bop 0 91 a Fr([11])47 b(Th.)39 b(Galla)m(y)g(and)g(C.)h(E.)f(W)-8 b(a)m(yne.)42 b(Long-time)e(asymptotics)f(of)h(the)g(Na)m(vier-Stok)m (es)h(and)e(v)m(orticit)m(y)187 204 y(equations)34 b(on)f Fo(R)805 171 y Fn(3)845 204 y Fr(.)h Fe(R.)j(So)-5 b(c.)36 b(L)-5 b(ond.)37 b(Philos.)g(T)-7 b(r)i(ans.)37 b(Ser.)f(A)f(Math.)i (Phys.)f(Eng.)f(Sci.)f Fo(360)h Fr(\(2002\),)187 317 y(2155{2188.)0 491 y([12])47 b(Th.)36 b(Galla)m(y)g(and)g(C.)h(E.)f(W) -8 b(a)m(yne.)39 b(Global)d(stabilit)m(y)f(of)h(v)m(ortex)i(solutions)d (of)i(the)g(t)m(w)m(o-dimensional)187 603 y(Na)m(vier-Stok)m(es)52 b(equation.)e Fe(Comm.)i(Math.)f(Phys.)p Fr(,)56 b(to)51 b(app)s(ear.)f(Preprin)m(t)e(v)m(ersion)i(a)m(v)-5 b(ailable)50 b(at)187 716 y Fa(http://www.arXiv.org/mat)o(h.A)o(P/04)o(0244)o(9)p Fr(.)0 890 y([13])d(P)-8 b(.)28 b(Germain.)f(Existence)h(globale)g(de)f (solutions)g(de)g(l')m(\023)-43 b(equation)28 b(de)g(Na)m(vier-Stok)m (es)h(2D)g(a)m(v)m(ec)h(donn)m(\023)-43 b(ees)187 1003 y(initiales)28 b(dans)h Fq(@)5 b Fv(BMO)p Fr(.)31 b(In)e(preparation.)0 1176 y([14])47 b(Y.)27 b(Giga)g(and)f(T.)g(Miy)m(ak)-5 b(a)m(w)m(a.)29 b(Solutions)c(in)g Fq(L)1812 1143 y Fl(r)1876 1176 y Fr(of)i(the)g(Na)m(vier-Stok)m(es)h(initial)c(v)-5 b(alue)26 b(problem.)f Fe(A)n(r)-5 b(ch.)187 1289 y(R)g(ational)35 b(Me)-5 b(ch.)32 b(A)n(nal.)e Fo(89)h Fr(\(1985\),)i(267{281.)0 1463 y([15])47 b(Y.)38 b(Giga)h(and)f(T.)g(Kam)m(b)s(e.)g(Large)g(time) g(b)s(eha)m(vior)g(of)g(the)g(v)m(orticit)m(y)h(of)f(t)m(w)m (o-dimensional)f(viscous)187 1575 y(\015o)m(w)30 b(and)g(its)g (application)e(to)k(v)m(ortex)f(formation.)f Fe(Comm.)k(Math.)f(Phys.)e Fo(117)g Fr(\(1988\),)i(549{568.)0 1749 y([16])47 b(Y.)32 b(Giga,)h(T.)e(Miy)m(ak)-5 b(a)m(w)m(a,)35 b(and)c(H.)h(Osada.)g(Tw)m (o-dimensional)d(Na)m(vier-Stok)m(es)k(\015o)m(w)f(with)f(measures)187 1862 y(as)f(initial)e(v)m(orticit)m(y)-8 b(.)32 b Fe(A)n(r)-5 b(ch.)32 b(R)-5 b(ational)35 b(Me)-5 b(ch.)33 b(A)n(nal.)d Fo(104)h Fr(\(1988\),)i(223{250.)0 2035 y([17])47 b(Y.)31 b(Giga)g(and)f(T.)h(Miy)m(ak)-5 b(a)m(w)m(a.)32 b(Na)m(vier-Stok)m(es)h (\015o)m(w)d(in)g Fo(R)2226 2002 y Fn(3)2296 2035 y Fr(with)g(measures) g(as)h(initial)d(v)m(orticit)m(y)j(and)187 2148 y(Morrey)g(spaces.)g Fe(Comm.)i(Partial)h(Di\013er)-5 b(ential)33 b(Equations)e Fo(14)g Fr(\(1989\),)i(577{618.)0 2322 y([18])47 b(M.-H.)28 b(Giga)f(and)g(Y.)g(Giga.)g Fm(Nonlinear)f(partial)f(di\013eren)m(tial) h(equations)g({)i(asymptotic)f(b)s(eha)m(viour)e(of)187 2434 y(solutions)k(and)g(self-similar)f(solutions)p Fr(.)h(Bo)s(ok)i (in)e(preparation.)0 2608 y([19])47 b(D.)24 b(Henry)-8 b(.)24 b Fe(Ge)-5 b(ometric)28 b(the)-5 b(ory)28 b(of)f(semiline)-5 b(ar)28 b(p)-5 b(ar)g(ab)g(olic)30 b(e)-5 b(quations)p Fr(.)25 b(Springer-V)-8 b(erlag,)24 b(Berlin,)f(1981.)0 2781 y([20])47 b(T.)39 b(Kato.)h(Strong)f Fq(L)942 2748 y Fl(p)982 2781 y Fr(-solutions)f(of)h(the)h(Na)m(vier-Stok)m(es)h (equation)e(in)f Fo(R)2835 2748 y Fl(m)2902 2781 y Fr(,)j(with)d (applications)g(to)187 2894 y(w)m(eak)31 b(solutions.)e Fe(Math.)k(Z.)d Fo(187)h Fr(\(1984\),)i(471{480.)0 3068 y([21])47 b(T.)33 b(Kato.)h(The)e(Na)m(vier-Stok)m(es)i(equation)f(for) g(an)f(incompressible)e(\015uid)h(in)g Fo(R)2981 3035 y Fn(2)3054 3068 y Fr(with)g(a)i(measure)g(as)187 3181 y(the)d(initial)e(v)m(orticit)m(y)-8 b(.)32 b Fe(Di\013er)-5 b(ential)33 b(Inte)-5 b(gr)g(al)34 b(Equations)d Fo(7)f Fr(\(1994\),)j(949{966.)0 3354 y([22])47 b(H.)23 b(Ko)s(c)m(h)h(and)e (D.)i(T)-8 b(ataru.)24 b(W)-8 b(ell-p)s(osedness)22 b(for)h(the)g(Na)m (vier{Stok)m(es)i(equations.)f Fe(A)-5 b(dvanc)g(es)26 b(in)g(Math-)187 3467 y(ematics)31 b Fo(157)g Fr(\(2001\),)i(22{35.)0 3640 y([23])47 b(O.)23 b(Ladyzhensk)-5 b(a)m(y)m(a,)27 b Fm(The)22 b(mathematical)i(theory)g(of)g(viscous)e(incompressible)f (\015o)m(w.)j Fr(Second)f(English)187 3753 y(edition.)28 b(Mathematics)i(and)e(its)h(Applications,)e(V)-8 b(ol.)30 b(2.)g(Gordon)f(and)f(Breac)m(h,)j(New)e(Y)-8 b(ork-London-)187 3866 y(P)m(aris,)30 b(1969.)0 4040 y([24])47 b(J.)28 b(Lera)m(y)-8 b(.)30 b(Etude)f(de)f(div)m(erses)e(\023)-43 b(equations)29 b(in)m(t)m(\023)-43 b(egrales)29 b(non)f(lin)m(\023)-43 b(eaires)28 b(et)h(de)g(quelques)e(probl)m(\022)-43 b(emes)29 b(que)187 4153 y(p)s(ose)h(l'h)m(ydro)s(dynamique.)d Fe(J.)33 b(Math.)g(Pur)-5 b(es.)32 b(Appl.)f Fo(12)g Fr(\(1933\),)i(1{82.)0 4326 y([25])47 b(Y.)41 b(Mey)m(er.)g Fm(W)-8 b(a)m(v)m(elets,)46 b(P)m(arapro)s(ducts)39 b(and)h(Na)m (vier{Stok)m(es)p Fr(.)i(Curren)m(t)e(Dev)m(elopmen)m(ts)h(in)e(Mathe-) 187 4439 y(matics,)31 b(In)m(ternational)e(Press,)h(Cam)m(bridge,)g (Massac)m(h)m(ussets,)i(1996.)0 4612 y([26])47 b(H.)c(Osada.)h (Di\013usion)d(pro)s(cesses)i(with)f(generators)i(of)g(generalized)f (div)m(ergence)g(form.)g Fe(J.)h(Math.)187 4725 y(Kyoto)34 b(Univ.)29 b Fo(27)i Fr(\(1987\),)i(597{619.)0 4899 y([27])47 b(E.)30 b(Stein.)g Fm(Harmonic)g(Analysis)p Fr(.)f(Princeton)g(Univ)m (ersit)m(y)h(Press,)g(1993.)0 5072 y([28])47 b(R.)36 b(T)-8 b(emam.)37 b Fm(Na)m(vier-Stok)m(es)g(equations.)f(Theory)g(and) f(n)m(umerical)g(analysis)p Fr(.)f(Reprin)m(t)h(of)i(the)f(1984)187 5185 y(edition.)29 b(AMS)h(Chelsea)g(Publishing,)d(Pro)m(vidence,)j (RI,)h(2001.)0 5358 y([29])47 b(F.)30 b(W)-8 b(eissler.)30 b(The)f(Na)m(vier-Stok)m(es)i(initial)c(v)-5 b(alue)30 b(problem)e(in)g Fq(L)2457 5326 y Fl(p)2497 5358 y Fr(.)i Fe(A)n(r)-5 b(ch.)32 b(R)-5 b(ational)34 b(Me)-5 b(ch.)32 b(A)n(nal.)e Fo(74)187 5471 y Fr(\(1980\),)j(219{230.)1844 5800 y(32)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0406151013226--