This is a multi-part message in MIME format. ---------------9909301607703 Content-Type: text/plain; name="99-367.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-367.keywords" quantum hydrodynamics, dispersive dissipative equations, ---------------9909301607703 Content-Type: application/postscript; name="final.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="final.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: final.dvi %%Pages: 21 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSCommandLine: dvips -o final.ps final %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 1999.03.01:1101 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 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Fj(f)2369 5325 y Ff(2)2413 5366 y Fn(\))22 b Fh(\000)g Fj(k)k Fn(+)c Fj(K)29 b Fn(+)3033 5299 y Fj(J)p 2976 5343 180 4 v 2976 5435 a(\034)3020 5450 y Ff(0)3064 5435 y Fj(")3110 5406 y Ff(2)1144 5645 y Fh(\000)1367 5578 y Fj(\027)6 b(")p 1255 5622 326 4 v 1255 5714 a Fn(\(2)p Fj(")p Fn(\))1429 5685 y Fg(\015)t Ff(+1)1593 5505 y Fi(\025)1662 5645 y Fj(dx;)p eop %%Page: 11 11 11 10 bop 787 -139 a Fl(POSITIVE)38 b(SOLUTIONS)g(TO)e(SINGULAR)j(EQUA) -7 b(TIONS)704 b(11)0 100 y Fn(using)34 b(the)f(monotonicit)m(y)f(of)h Fj(h)h Fn(and)f(Lemma)f(2.3.)43 b(Therefore)0 339 y Fj(\016)48 298 y Ff(2)109 203 y Fi(Z)165 429 y Ff(\012)243 258 y Fi(\000)289 339 y Fn(\()p Fj(w)14 b Fh(\000)e Fj(f)f Fn(\))601 298 y Fc(\000)601 363 y Fg(x)668 258 y Fi(\001)714 279 y Ff(2)775 339 y Fj(dx)28 b Fh(\024)1017 203 y Fi(Z)1072 429 y Ff(\012)1134 339 y Fn(\()p Fh(\000)p Fn(\()p Fj(w)14 b Fh(\000)e Fj(f)f Fn(\))1562 298 y Fc(\000)1629 339 y Fn(\))p Fj(w)1742 198 y Fi(\024)1822 271 y Fj(J)1887 235 y Ff(2)p 1807 316 141 4 v 1807 407 a Fn(2)p Fj(")1903 378 y Ff(4)1971 339 y Fn(+)h Fj(T)i(h)p Fn(\(4)p Fj(")2324 298 y Ff(2)2370 339 y Fn(\))e Fh(\000)g Fj(k)j Fn(+)d Fj(K)18 b Fn(+)2928 271 y Fj(J)p 2871 316 180 4 v 2871 407 a(\034)2915 422 y Ff(0)2959 407 y Fj(")3005 378 y Ff(2)3074 339 y Fh(\000)3297 271 y Fj(\027)p 3175 316 300 4 v 3175 407 a Fn(2)3225 378 y Fg(\015)t Ff(+1)3377 407 y Fj(")3423 378 y Fg(\015)3486 198 y Fi(\025)3555 339 y Fj(dx)28 b(:)0 578 y Fn(W)-8 b(e)25 b(claim)d(that)i(for)h (su\016cien)m(tly)g(small)e Fj(")28 b(>)g Fn(0,)e(the)e(expression)i (in)e(the)h(brac)m(k)m(ets)g(is)f(non)h(p)s(ositiv)m(e,)0 757 y(whic)m(h)30 b(implies)f Fj(w)s Fn(\()p Fj(x)p Fn(\))e Fh(\025)h Fj(f)11 b Fn(\()p Fj(x)p Fn(\))27 b Fh(\025)h Fj(")g(>)g Fn(0)h(in)h(\012,)g(i.e.,)f(w)m(e)h(get)f(the)h(assertion)g (of)g(the)g(theorem)f(after)0 936 y(taking)j Fj(m)p Fn(\()p Fj(\027)6 b Fn(\))28 b(=)h Fj(")p Fn(.)44 b(No)m(w)33 b(c)m(ho)s(ose)h Fj(")28 b Fh(2)g Fn(\(0)p Fj(;)17 b Fn(1\))32 b(suc)m(h)i(that)f(\(see)h(\(H1\)\))364 1203 y Fj(")28 b Fh(\024)543 1063 y Fi(\022)704 1136 y Fn(1)p 628 1180 202 4 v 628 1271 a(2)678 1243 y Fg(\015)t Ff(+1)1480 1136 y Fj(\027)p 870 1180 1276 4 v 870 1271 a(J)935 1243 y Ff(2)980 1271 y Fj(=)p Fn(2)22 b(+)g Fj(J)t(=\034)1354 1286 y Ff(0)1421 1271 y Fn(+)g(max)o(\()p Fj(K)28 b Fh(\000)23 b Fj(k)s(;)17 b Fn(0\))2158 1063 y Fi(\023)2231 1083 y Ff(1)p Fg(=)p Ff(\()p Fg(\015)t Fc(\000)p Ff(4\))2610 1203 y Fn(and)67 b Fj(h)p Fn(\(4)p Fj(")3029 1162 y Ff(2)3074 1203 y Fn(\))28 b Fh(\024)g Fn(0)33 b Fj(:)0 1442 y Fn(Notice)g(that)g Fj(\015)f(>)d Fn(4.)43 b(Then,)34 b(since)g Fj(")28 b Fh(\024)g Fn(1,)752 1614 y Fj(J)817 1577 y Ff(2)p 736 1658 141 4 v 736 1749 a Fn(2)p Fj(")832 1721 y Ff(4)911 1681 y Fn(+)23 b Fj(T)14 b(h)p Fn(\(4)p Fj(")1275 1640 y Ff(2)1320 1681 y Fn(\))22 b(+)g Fj(K)29 b Fh(\000)23 b Fj(k)i Fn(+)1940 1614 y Fj(J)p 1882 1658 180 4 v 1882 1749 a(\034)1926 1764 y Ff(0)1971 1749 y Fj(")2017 1721 y Ff(2)2096 1681 y Fh(\000)2329 1614 y Fj(\027)p 2208 1658 300 4 v 2208 1749 a Fn(2)2258 1721 y Fg(\015)t Ff(+1)2409 1749 y Fj(")2455 1721 y Fg(\015)951 1953 y Fh(\024)1089 1885 y Fn(1)p 1068 1930 92 4 v 1068 2021 a Fj(")1114 1992 y Ff(4)1188 1812 y Fi(\022)1273 1885 y Fj(J)1338 1849 y Ff(2)p 1273 1930 110 4 v 1303 2021 a Fn(2)1417 1953 y(+)1540 1885 y Fj(J)p 1528 1930 89 4 v 1528 2021 a(\034)1572 2036 y Ff(0)1651 1953 y Fn(+)d(max)o(\(0)p Fj(;)17 b(K)27 b Fh(\000)c Fj(k)s Fn(\))f Fh(\000)2682 1885 y Fj(\027)p 2509 1930 402 4 v 2509 2021 a Fn(2)2559 1992 y Fg(\015)t Ff(+1)2711 2021 y Fj(")2757 1992 y Fg(\015)t Fc(\000)p Ff(4)2923 1812 y Fi(\023)951 2162 y Fh(\024)28 b Fn(0)33 b Fj(:)943 2494 y Fm(3.)55 b(Existence)35 b(of)j(solutions)f (f)n(or)g Fj(\027)52 b Fn(=)45 b(0)0 2733 y(In)36 b(the)f(case)h(of)g (v)-6 b(anishing)36 b(viscosit)m(y)e(w)m(e)i(can)g(only)f(exp)s(ect)g (to)g(get)h(existence)g(of)f(solutions)h(for)0 2912 y(su\016cien)m(tly) g(small)d Fj(J)41 b(>)31 b Fn(0,)k(corresp)s(onding)i(to)d(a)h (subsonic)i(condition)e(for)g(the)h(h)m(ydro)s(dynamic)0 3092 y(equations)d(\()p Fj(\016)f Fn(=)c(0\).)120 3271 y(In)k(this)f(section)h(w)m(e)g(need)h(the)f(follo)m(wing)f (assumption.)43 b(Assume)32 b(that)f(there)i(exists)e Fj(m)3492 3286 y Ff(0)3565 3271 y Fj(>)d Fn(0)0 3450 y(suc)m(h)35 b(that)761 3562 y(1)p 761 3607 50 4 v 761 3698 a(2)823 3630 y Fj(T)14 b(p)945 3589 y Fc(0)973 3630 y Fn(\()p Fj(m)1099 3589 y Ff(2)1099 3654 y(0)1144 3630 y Fn(\))22 b(+)h Fj(T)14 b(h)p Fn(\()p Fj(m)1560 3589 y Ff(2)1560 3654 y(0)1605 3630 y Fn(\))22 b(+)1797 3562 y(1)p 1777 3607 89 4 v 1777 3698 a Fj(\034)1821 3713 y Ff(0)1878 3508 y Fi(q)p 1977 3508 360 4 v 1977 3630 a Fj(T)14 b(p)2099 3601 y Fc(0)2127 3630 y Fn(\()p Fj(m)2253 3595 y Ff(2)2253 3656 y(0)2298 3630 y Fn(\))22 b(+)h Fj(K)29 b Fh(\000)22 b Fj(k)31 b Fh(\024)d Fn(0)33 b Fj(:)544 b Fn(\(3.1\))0 3852 y(Since)28 b Fj(K)17 b Fh(\000)10 b Fj(k)31 b Fn(=)d Fh(\000)p Fj(E)782 3867 y Ff(0)837 3852 y Fn(+)10 b Fj(\025)982 3816 y Fc(\000)p Ff(2)1090 3852 y Fn(\()p Fj(M)1237 3816 y Ff(2)1291 3852 y Fh(\000)g(k)p Fj(C)d Fh(k)1556 3870 y Fg(L)1611 3850 y Fb(1)1650 3870 y Ff(\(\012\))1774 3852 y Fn(\),)27 b(this)h(assumption)f(is)g (satis\014ed)h(if,)g(for)f(instance,)174 4055 y(\(i\))89 b(lim)329 4116 y Fg(s)p Fc(!)p Ff(0+)563 4055 y Fj(h)p Fn(\()p Fj(s)p Fn(\))28 b(=)g Fh(\0001)p Fn(,)33 b(or)146 4234 y(\(ii\))49 b Fj(E)403 4249 y Ff(0)475 4234 y Fj(>)28 b Fn(0)33 b(is)g(su\016cien)m(tly)h(large.)0 4437 y(Indeed,)48 b(in)43 b(case)i(\(i\),)g(the)f(condition)g(\(3.1\))f(b)s(ecomes)h (true)g(for)g(su\016cien)m(tly)g(small)e Fj(m)3429 4452 y Ff(0)3520 4437 y Fj(>)k Fn(0,)0 4616 y(observing)36 b(that)g Fj(p)718 4580 y Fc(0)782 4616 y Fn(is)g(non-decreasing)i(suc)m (h)g(that)e Fj(p)2061 4580 y Fc(0)2089 4616 y Fn(\()p Fj(m)2215 4580 y Ff(2)2215 4641 y(0)2260 4616 y Fn(\))f(cannot)i(b)s(e) f(arbitrarily)f(large)h(near)0 4795 y(zero.)50 b(In)35 b(case)h(\(ii\))d(w)m(e)j(can)f(c)m(ho)s(ose,)h(for)f(instance,)h Fj(m)2099 4810 y Ff(0)2175 4795 y Fj(>)31 b Fn(0)k(suc)m(h)h(that)f Fj(h)p Fn(\()p Fj(m)2995 4759 y Ff(2)2995 4820 y(0)3040 4795 y Fn(\))c Fh(\024)g Fn(0,)k(and)g(then)0 4975 y(tak)m(e)e Fj(E)290 4990 y Ff(0)362 4975 y Fj(>)28 b Fn(0)33 b(large)g(enough)h (suc)m(h)h(that)624 5214 y Fj(E)698 5229 y Ff(0)770 5214 y Fh(\025)887 5146 y Fn(1)p 887 5191 50 4 v 887 5282 a(2)948 5214 y Fj(T)14 b(p)1070 5172 y Fc(0)1099 5214 y Fn(\()p Fj(m)1225 5172 y Ff(2)1225 5238 y(0)1269 5214 y Fn(\))22 b(+)1461 5146 y(1)p 1442 5191 89 4 v 1442 5282 a Fj(\034)1486 5297 y Ff(0)1542 5092 y Fi(q)p 1642 5092 360 4 v 122 x Fj(T)14 b(p)1764 5185 y Fc(0)1792 5214 y Fn(\()p Fj(m)1918 5179 y Ff(2)1918 5240 y(0)1963 5214 y Fn(\))22 b(+)g Fj(\025)2181 5172 y Fc(\000)p Ff(2)2288 5214 y Fn(\()p Fj(M)2435 5172 y Ff(2)2501 5214 y Fh(\000)h(k)p Fj(C)7 b Fh(k)2779 5232 y Fg(L)2834 5212 y Fb(1)2873 5232 y Ff(\(\012\))2997 5214 y Fn(\))33 b Fj(:)0 5453 y Fn(No)m(w,)g(de\014ne)1360 5632 y Fj(m)28 b Fn(=)g(min)p Fh(f)p Fj(w)1868 5647 y Ff(0)1912 5632 y Fj(;)17 b(w)2028 5647 y Ff(1)2073 5632 y Fj(;)g(m)2205 5647 y Ff(0)2249 5632 y Fh(g)33 b Fj(:)p eop %%Page: 12 12 12 11 bop 0 -139 a Fl(12)949 b(IRENE)37 b(GAMBA)h(AND)f(ANSGAR)i(J)2394 -160 y(\177)2383 -139 y(UNGEL)0 100 y Fo(Theorem)h(3.1.)58 b Fd(L)-5 b(et)38 b(the)f(assumptions)g Fn(\(H1\){\(H3\))f Fd(hold)i(and)g(let)g Fj(\027)g Fn(=)32 b(0)p Fd(.)51 b(F)-8 b(urthermor)j(e,)39 b(let)0 279 y Fj(J)e(>)29 b Fn(0)35 b Fd(b)-5 b(e)35 b(such)h(that)1315 458 y Fj(J)i Fh(\024)28 b Fj(J)1568 473 y Ff(0)1629 402 y(def)1643 458 y Fn(=)i Fj(m)1837 417 y Ff(2)1882 368 y Fi(p)p 1982 368 360 4 v 90 x Fj(T)14 b(p)2104 430 y Fc(0)2132 458 y Fn(\()p Fj(m)2258 430 y Ff(2)2303 458 y Fn(\))35 b Fj(:)1110 b Fn(\(3.2\))0 684 y Fd(Then)36 b(ther)-5 b(e)35 b(exists)g(a)g(classic)-5 b(al)35 b(solution)h Fn(\()p Fj(w)s(;)17 b(V)k Fn(\))28 b Fh(2)g Fn(\()p Fj(C)2146 648 y Ff(2)2191 684 y Fn(\()p 2230 604 72 4 v(\012)o(\)\))2379 648 y Ff(2)2459 684 y Fd(to)36 b Fn(\(1.10\){\(1.12\))c Fd(satisfying)1252 930 y Fn(0)27 b Fj(<)i(m)f Fh(\024)g Fj(w)s Fn(\()p Fj(x)p Fn(\))f Fh(\024)h Fj(M)82 b Fd(in)71 b Fn(\012)35 b Fj(:)0 1224 y Fd(R)-5 b(emark.)50 b Fn(The)d(condition)f (\(3.2\))f(can)h(b)s(e)h(in)m(terpreted)g(as)f(a)g(\\subsonic)i (condition")e(since)h(it)0 1403 y(implies)32 b(that)h(the)h(v)m(elo)s (cit)m(y)e Fj(J)t(=n)h Fn(satis\014es)959 1582 y Fj(J)p 959 1626 65 4 v 962 1718 a(n)1063 1649 y Fn(=)1207 1582 y Fj(J)p 1180 1626 119 4 v 1180 1718 a(w)1254 1689 y Ff(2)1339 1649 y Fh(\024)1490 1582 y Fj(J)p 1456 1626 133 4 v 1456 1718 a(m)1543 1689 y Ff(2)1628 1649 y Fh(\024)1733 1559 y Fi(p)p 1832 1559 360 4 v 1832 1649 a Fj(T)14 b(p)1954 1621 y Fc(0)1983 1649 y Fn(\()p Fj(m)2109 1621 y Ff(2)2154 1649 y Fn(\))27 b Fh(\024)2325 1559 y Fi(p)p 2425 1559 288 4 v 90 x Fj(T)14 b(p)2547 1621 y Fc(0)2575 1649 y Fn(\()p Fj(n)p Fn(\))32 b Fj(:)0 1895 y Fd(Pr)-5 b(o)g(of.)49 b Fn(By)44 b(Prop)s(osition)f(2.2,)j(there)f(exists)f(a)g(solution)g (\()p Fj(w)s(;)17 b(V)22 b Fn(\))44 b(to)g(the)g(truncated)h(problem)0 2075 y(\(2.3\){\(2.4\),)e(\(1.12\))f(with)h Fj(")i Fn(=)g Fj(m)g(>)g Fn(0.)74 b(It)43 b(remains)g(to)f(sho)m(w)i(that)f Fj(w)48 b Fh(\025)d Fj(m)f Fn(in)f(\012.)74 b(Using)0 2254 y(\()p Fj(w)25 b Fh(\000)e Fj(m)p Fn(\))361 2218 y Fc(\000)461 2254 y Fn(as)34 b(test)f(function)h(giv)m(es)191 2500 y Fj(\016)239 2459 y Ff(2)301 2364 y Fi(Z)356 2591 y Ff(\012)434 2419 y Fi(\000)480 2500 y Fn(\()p Fj(w)25 b Fh(\000)e Fj(m)p Fn(\))841 2459 y Fc(\000)841 2525 y Fg(x)908 2419 y Fi(\001)954 2440 y Ff(2)1015 2500 y Fj(dx)418 2779 y Fn(=)523 2644 y Fi(Z)579 2870 y Ff(\012)641 2779 y Fn(\()p Fh(\000)p Fn(\()p Fj(w)i Fh(\000)d Fj(m)p Fn(\))1117 2738 y Fc(\000)1185 2779 y Fn(\))p Fj(w)1298 2639 y Fi(\024)1486 2712 y Fj(J)1551 2676 y Ff(2)p 1362 2756 358 4 v 1362 2847 a Fn(2)p Fj(t)1448 2862 y Fg(m)1524 2847 y Fn(\()p Fj(w)s Fn(\))1676 2819 y Ff(4)1754 2779 y Fn(+)g Fj(T)14 b(h)p Fn(\()p Fj(w)2095 2738 y Ff(2)2140 2779 y Fn(\))22 b Fh(\000)h Fj(V)44 b Fn(+)23 b Fj(K)29 b Fn(+)22 b Fj(J)2798 2644 y Fi(Z)2897 2668 y Fg(x)2853 2870 y Ff(0)3108 2712 y Fj(ds)p 2976 2756 363 4 v 2976 2847 a(\034)11 b(t)3067 2862 y Fg(m)3143 2847 y Fn(\()p Fj(w)s Fn(\))3295 2819 y Ff(2)3351 2639 y Fi(\025)3420 2779 y Fj(dx)418 3062 y Fh(\024)523 2927 y Fi(Z)579 3153 y Ff(\012)641 3062 y Fn(\()p Fh(\000)p Fn(\()p Fj(w)25 b Fh(\000)d Fj(m)p Fn(\))1117 3021 y Fc(\000)1185 3062 y Fn(\))p Fj(w)1298 2922 y Fi(\024)1398 2995 y Fj(J)1463 2959 y Ff(2)1453 3020 y(0)p 1362 3039 182 4 v 1362 3131 a Fn(2)p Fj(m)1499 3102 y Ff(4)1578 3062 y Fn(+)h Fj(T)14 b(h)p Fn(\()p Fj(m)1933 3021 y Ff(2)1978 3062 y Fn(\))22 b Fh(\000)h Fj(k)i Fn(+)d Fj(K)29 b Fn(+)2601 2995 y Fj(J)2656 3010 y Ff(0)p 2541 3039 221 4 v 2541 3131 a Fj(\034)2585 3146 y Ff(0)2629 3131 y Fj(m)2716 3102 y Ff(2)2773 2922 y Fi(\025)2842 3062 y Fj(dx)418 3341 y Fn(=)523 3206 y Fi(Z)579 3432 y Ff(\012)641 3341 y Fn(\()p Fh(\000)p Fn(\()p Fj(w)c Fh(\000)d Fj(m)p Fn(\))1117 3300 y Fc(\000)1185 3341 y Fn(\))p Fj(w)1298 3201 y Fi(\024)1362 3274 y Fn(1)p 1362 3318 50 4 v 1362 3410 a(2)1424 3341 y Fj(T)14 b(p)1546 3300 y Fc(0)1574 3341 y Fn(\()p Fj(m)1700 3300 y Ff(2)1745 3341 y Fn(\))22 b(+)g Fj(T)14 b(h)p Fn(\()p Fj(m)2160 3300 y Ff(2)2206 3341 y Fn(\))22 b Fh(\000)g Fj(k)j Fn(+)e Fj(K)29 b Fn(+)2788 3274 y(1)p 2768 3318 89 4 v 2768 3410 a Fj(\034)2812 3425 y Ff(0)2869 3251 y Fi(p)p 2968 3251 360 4 v 2968 3341 a Fj(T)14 b(p)3090 3312 y Fc(0)3118 3341 y Fn(\()p Fj(m)3244 3312 y Ff(2)3289 3341 y Fn(\))3328 3201 y Fi(\025)3397 3341 y Fj(dx)418 3620 y Fh(\024)523 3485 y Fi(Z)579 3711 y Ff(\012)641 3620 y Fn(\()p Fh(\000)p Fn(\()p Fj(w)25 b Fh(\000)d Fj(m)p Fn(\))1117 3579 y Fc(\000)1185 3620 y Fn(\))p Fj(w)1298 3480 y Fi(\024)1362 3553 y Fn(1)p 1362 3597 50 4 v 1362 3689 a(2)1424 3620 y Fj(T)14 b(p)1546 3579 y Fc(0)1574 3620 y Fn(\()p Fj(m)1700 3579 y Ff(2)1700 3645 y(0)1745 3620 y Fn(\))22 b(+)g Fj(T)14 b(h)p Fn(\()p Fj(m)2160 3579 y Ff(2)2160 3645 y(0)2206 3620 y Fn(\))22 b Fh(\000)g Fj(k)j Fn(+)e Fj(K)29 b Fn(+)2788 3553 y(1)p 2768 3597 89 4 v 2768 3689 a Fj(\034)2812 3704 y Ff(0)2869 3499 y Fi(q)p 2968 3499 360 4 v 2968 3620 a Fj(T)14 b(p)3090 3591 y Fc(0)3118 3620 y Fn(\()p Fj(m)3244 3586 y Ff(2)3244 3647 y(0)3289 3620 y Fn(\))3328 3480 y Fi(\025)3397 3620 y Fj(dx)418 3829 y Fh(\024)28 b Fn(0)p Fj(;)0 4075 y Fn(in)33 b(view)g(of)g(the)h(de\014nition)g(\(3.1\))e(of)h Fj(m)1512 4090 y Ff(0)1557 4075 y Fn(.)44 b(This)34 b(implies)e(that)h Fj(w)e Fh(\025)d Fj(m)33 b Fn(in)h(\012.)120 4399 y(Next,)e(w)m(e)h (pro)m(v)m(e)h(that)f(ev)m(ery)g(w)m(eak)g(solution)g(is)g(necessarily) h(strictly)e(p)s(ositiv)m(e:)0 4626 y Fo(Prop)s(osition)51 b(3.2.)58 b Fd(L)-5 b(et)46 b Fn(\()p Fj(w)s(;)17 b(V)k Fn(\))46 b Fh(2)h Fn(\()p Fj(H)1615 4590 y Ff(1)1659 4626 y Fn(\(\012\)\))1848 4590 y Ff(2)1938 4626 y Fd(b)-5 b(e)46 b(a)g(we)-5 b(ak)45 b(solution)h(to)g Fn(\(1.10\){\(1.12\))d Fd(with)0 4806 y Fj(\027)i Fh(\025)39 b Fn(0)i Fd(and)g Fj(w)574 4769 y Fc(\000)p Ff(3)719 4806 y Fh(2)e Fj(L)892 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Ff(1)1414 5740 y(0)1519 5649 y Fj(w)1590 5664 y Fg(x)1641 5649 y Fn(\()p Fj(\025x)g Fn(+)g(\(1)g Fh(\000)h Fj(\025)p Fn(\))p Fj(x)2281 5664 y Ff(0)2325 5649 y Fn(\))17 b Fj(dx)32 b(;)216 b(x)28 b Fh(2)g Fn(\012)33 b Fj(;)p eop %%Page: 13 13 13 12 bop 787 -139 a Fl(POSITIVE)38 b(SOLUTIONS)g(TO)e(SINGULAR)j(EQUA) -7 b(TIONS)704 b(13)0 100 y Fn(and)629 162 y Fi(Z)728 187 y Ff(1)684 388 y(0)901 230 y Fj(dx)p 802 275 309 4 v 802 366 a Fh(j)p Fj(w)s Fn(\()p Fj(x)p Fn(\))p Fh(j)1067 337 y Ff(3)1150 298 y Fh(\025)1255 162 y Fi(Z)1354 187 y Ff(1)1310 388 y(0)1399 157 y Fi(\022)1473 162 y(Z)1572 187 y Ff(1)1528 388 y(0)1633 298 y Fh(j)p Fj(w)1732 313 y Fg(x)1783 298 y Fn(\()p Fj(s)p Fn(\))p Fh(j)17 b Fj(ds)2052 157 y Fi(\023)2123 178 y Fc(\000)p Ff(3)2230 298 y Fh(j)p Fj(x)22 b Fh(\000)g Fj(x)2493 313 y Ff(0)2538 298 y Fh(j)2566 256 y Fc(\000)p Ff(3)2689 298 y Fj(dx)28 b Fn(=)g Fh(1)33 b Fj(;)0 572 y Fn(con)m(tradicting)i(the)g(in)m(tegrabilit)m(y)f(of)h (1)p Fj(=w)1634 535 y Ff(3)1678 572 y Fn(.)49 b(Hence)36 b Fj(w)d(>)e Fn(0)k(in)g(\012,)g(and)g(since)h Fj(w)i Fn(is)d(con)m(tin)m(uous)0 751 y(in)p 116 671 72 4 v 33 w(\012,)e(there)h(exists)f Fj(m)28 b(>)g Fn(0)33 b(suc)m(h)i(that)e Fj(w)s Fn(\()p Fj(x)p Fn(\))27 b Fh(\025)h Fj(m)34 b Fn(for)f(all)g Fj(x)27 b Fh(2)h Fn(\012.)833 1081 y Fm(4.)55 b(Non-existence)35 b(of)i(solutions)g(f)n(or)g Fj(\027)50 b Fn(=)44 b(0)0 1326 y(W)-8 b(e)42 b(sho)m(w)h(that,)g(under)g(some)e (condition)h(on)g Fj(h)p Fn(,)i(a)d(w)m(eak)h(solution)g(to)f (\(1.10\){\(1.12\))e(cannot)0 1505 y(exist)i(if)h Fj(J)52 b(>)43 b Fn(0)e(is)h(large)g(enough.)71 b(F)-8 b(or)41 b(this,)j(w)m(e)e(ha)m(v)m(e)g(to)g(pro)m(v)m(e)g(that)g Fj(w)i Fn(is)e(b)s(ounded)i(from)0 1685 y(ab)s(o)m(v)m(e)33 b(indep)s(enden)m(tly)i(of)f Fj(J)10 b Fn(.)0 1942 y Fo(Lemma)33 b(4.1.)58 b Fd(L)-5 b(et)32 b Fn(\()p Fj(w)s(;)17 b(V)k Fn(\))31 b Fd(b)-5 b(e)31 b(any)h(we)-5 b(ak)32 b(solution)f(to)h Fn(\(1.10\){\(1.12\))d Fd(with)i Fj(\027)j Fn(=)28 b(0)p Fd(,)k Fj(K)j Fh(2)28 b Fe(R)42 b Fd(and)0 2122 y Fj(J)37 b(>)29 b Fn(0)p Fd(.)45 b(F)-8 b(urthermor)j(e,)36 b(let)845 2439 y Fj(h)p Fn(\()p Fj(s)p Fn(\))27 b Fh(\025)i Fj(c)1203 2454 y Ff(0)1247 2439 y Fn(\()p Fj(s)1333 2398 y Fg(\013)p Fc(\000)p Ff(1)1514 2439 y Fh(\000)22 b Fn(1\))71 b Fd(for)g Fj(s)27 b Fh(\025)h Fn(0)36 b Fj(;)87 b Fd(with)71 b Fj(\013)28 b(>)g Fn(2)p Fj(:)640 b Fn(\(4.1\))0 2757 y Fd(Then)36 b(ther)-5 b(e)35 b(exists)g Fj(M)878 2772 y Ff(0)950 2757 y Fj(>)28 b Fn(0)p Fd(,)35 b(indep)-5 b(endent)34 b(of)i Fj(J)10 b Fd(,)35 b(such)g(that)h Fj(w)s Fn(\()p Fj(x)p Fn(\))27 b Fh(\024)h Fj(M)2840 2772 y Ff(0)2920 2757 y Fd(for)35 b(al)5 b(l)37 b Fj(x)27 b Fh(2)h Fn(\012)p Fd(.)0 3014 y(R)-5 b(emark.)50 b Fn(The)34 b(b)s(ound)g Fj(M)1024 3029 y Ff(0)1102 3014 y Fn(do)s(es)f(not)g(dep)s (end)i(on)f Fj(K)40 b Fn(if)33 b Fj(K)h(>)29 b Fn(0.)0 3272 y Fd(Pr)-5 b(o)g(of.)49 b Fn(T)-8 b(ak)m(e)33 b(\()p Fj(w)25 b Fh(\000)e Fj(\026)p Fn(\))888 3236 y Ff(+)987 3272 y Fn(with)33 b Fj(\026)27 b Fh(\025)h Fn(max)o(\()p Fj(w)1702 3287 y Ff(0)1747 3272 y Fj(;)17 b(w)1863 3287 y Ff(1)1907 3272 y Fn(\))33 b(as)g(test)g(function)i(in)e(\(1.10\))f (to)g(get)275 3594 y Fj(\016)323 3552 y Ff(2)384 3458 y Fi(Z)439 3684 y Ff(\012)501 3594 y Fn(\()p Fj(w)25 b Fh(\000)e Fj(\026)p Fn(\))835 3552 y Ff(+2)835 3618 y Fg(x)957 3594 y Fj(dx)k Fn(=)1198 3458 y Fi(Z)1254 3684 y Ff(\012)1316 3594 y Fn(\()p Fj(w)e Fh(\000)d Fj(\026)p Fn(\))1649 3552 y Ff(+)1715 3594 y Fj(w)1789 3453 y Fi(\024)1842 3594 y Fh(\000)1961 3526 y Fj(J)2026 3490 y Ff(2)p 1931 3571 169 4 v 1931 3662 a Fn(2)p Fj(w)2055 3633 y Ff(4)2134 3594 y Fh(\000)g Fj(J)2315 3458 y Fi(Z)2414 3483 y Fg(x)2370 3684 y Ff(0)2531 3526 y Fj(ds)p 2493 3571 174 4 v 2493 3662 a(\034)11 b(w)2622 3633 y Ff(2)2701 3594 y Fh(\000)22 b Fj(h)p Fn(\()p Fj(w)2970 3552 y Ff(2)3015 3594 y Fn(\))g Fh(\000)h Fj(K)3268 3453 y Fi(\025)3337 3594 y Fj(dx)1287 3864 y Fn(+)1387 3728 y Fi(Z)1442 3955 y Ff(\012)1504 3864 y Fn(\()p Fj(w)i Fh(\000)d Fj(\026)p Fn(\))1837 3823 y Ff(+)1903 3864 y Fj(w)s(V)39 b(dx)33 b(:)1271 b Fn(\(4.2\))0 4181 y(The)34 b(main)e(di\016cult)m(y)h(is)g(to)g (estimate)f(the)i(last)e(in)m(tegral.)44 b(F)-8 b(rom)32 b(\(2.6\))g(it)g(follo)m(ws)h(that)926 4499 y Fj(V)23 b Fn(\()p Fj(x)p Fn(\))k Fh(\024)h Fj(V)1332 4514 y Ff(0)1399 4499 y Fn(+)23 b(max)n(\()p Fh(\000)p Fj(E)1874 4514 y Ff(0)1918 4499 y Fj(;)17 b Fn(0\))k(+)i Fj(\025)2231 4458 y Fc(\000)p Ff(2)2355 4363 y Fi(Z)2410 4589 y Ff(\012)2489 4499 y Fj(w)2563 4458 y Ff(2)2624 4499 y Fj(dx)33 b(:)0 4816 y Fn(Since)287 5134 y Fj(w)361 5093 y Ff(2)433 5134 y Fn(=)28 b(\()p Fj(w)d Fn(+)e Fj(\026)p Fn(\)\()p Fj(w)h Fh(\000)f Fj(\026)p Fn(\))f(+)g Fj(\026)1386 5093 y Ff(2)1459 5134 y Fh(\024)28 b Fn(\()p Fj(w)d Fn(+)e Fj(\026)p Fn(\)\()p Fj(w)h Fh(\000)f Fj(\026)p Fn(\))2231 5093 y Ff(+)2319 5134 y Fn(+)f Fj(\026)2478 5093 y Ff(2)2551 5134 y Fh(\024)28 b Fn(2)p Fj(w)s Fn(\()p Fj(w)d Fh(\000)d Fj(\026)p Fn(\))3113 5093 y Ff(+)3201 5134 y Fn(+)h Fj(\026)3361 5093 y Ff(2)3406 5134 y Fj(;)0 5451 y Fn(w)m(e)34 b(get)559 5649 y Fj(V)22 b Fn(\()p Fj(x)p Fn(\))27 b Fh(\024)h Fj(V)964 5664 y Ff(0)1031 5649 y Fn(+)23 b(max)n(\()p Fh(\000)p Fj(E)1506 5664 y Ff(0)1551 5649 y Fj(;)17 b Fn(0\))k(+)h(2)p Fj(\025)1913 5608 y Fc(\000)p Ff(2)2037 5513 y Fi(Z)2092 5740 y Ff(\012)2171 5649 y Fj(w)s Fn(\()p Fj(w)j Fh(\000)d Fj(\026)p Fn(\))2578 5608 y Ff(+)2661 5649 y Fj(dx)g Fn(+)g Fj(\026)2951 5608 y Ff(2)2996 5649 y Fj(\025)3054 5608 y Fc(\000)p Ff(2)p eop %%Page: 14 14 14 13 bop 0 -139 a Fl(14)949 b(IRENE)37 b(GAMBA)h(AND)f(ANSGAR)i(J)2394 -160 y(\177)2383 -139 y(UNGEL)0 115 y Fn(and,)33 b(setting)g Fj(V)606 130 y Ff(1)668 58 y(def)681 115 y Fn(=)e Fj(V)847 130 y Ff(0)914 115 y Fn(+)22 b(max)o(\()p Fh(\000)p Fj(E)1389 130 y Ff(0)1433 115 y Fj(;)17 b Fn(0\),)390 244 y Fi(Z)446 471 y Ff(\012)507 380 y Fn(\()p Fj(w)25 b Fh(\000)e Fj(\026)p Fn(\))841 339 y Ff(+)907 380 y Fj(w)s(V)39 b(dx)27 b Fh(\024)1319 244 y Fi(Z)1375 471 y Ff(\012)1437 380 y Fn(\()p Fj(w)d Fh(\000)f Fj(\026)p Fn(\))1770 339 y Ff(+)1836 380 y Fj(w)s Fn(\()p Fj(V)2007 395 y Ff(1)2074 380 y Fn(+)f Fj(\026)2233 339 y Ff(2)2278 380 y Fj(\025)2336 339 y Fc(\000)p Ff(2)2443 380 y Fn(\))17 b Fj(dx)1408 675 y Fn(+)23 b(2)p Fj(\025)1616 634 y Fc(\000)p Ff(2)1722 535 y Fi(\022)1796 540 y(Z)1851 766 y Ff(\012)1913 675 y Fn(\()p Fj(w)i Fh(\000)e Fj(\026)p Fn(\))2247 634 y Ff(+)2312 675 y Fj(w)d(dx)2512 535 y Fi(\023)2585 556 y Ff(2)1214 946 y Fh(\024)28 b Fj(V)1377 961 y Ff(1)1439 810 y Fi(Z)1494 1036 y Ff(\012)1556 946 y Fn(\()p Fj(w)d Fh(\000)e Fj(\026)p Fn(\))1890 904 y Ff(+)1955 946 y Fj(w)d(dx)i Fn(+)h Fj(\025)2335 904 y Fc(\000)p Ff(2)2458 810 y Fi(Z)2514 1036 y Ff(\012)2575 946 y Fn(\()p Fj(w)i Fh(\000)e Fj(\026)p Fn(\))2909 904 y Ff(+)2975 946 y Fj(w)3049 904 y Ff(3)3110 946 y Fj(dx)1408 1212 y Fn(+)g(2)p Fj(\025)1616 1171 y Fc(\000)p Ff(2)1739 1076 y Fi(Z)1794 1303 y Ff(\012)1856 1212 y Fn(\()p Fj(w)i Fh(\000)e Fj(\026)p Fn(\))2190 1171 y Ff(+2)2295 1212 y Fj(w)2369 1171 y Ff(2)2431 1212 y 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Fj(w)s Fn([)p Fj(V)1831 2024 y Ff(1)1898 2009 y Fn(+)g(3)p Fj(\025)2105 1968 y Fc(\000)p Ff(2)2212 2009 y Fj(w)2286 1968 y Ff(2)2353 2009 y Fh(\000)h Fj(K)29 b Fh(\000)22 b Fj(c)2709 2024 y Ff(0)2754 2009 y Fn(\()p Fj(w)2867 1968 y Ff(2)p Fg(\013)p Fc(\000)p Ff(2)3087 2009 y Fh(\000)h Fn(1\)])17 b Fj(dx)32 b(:)0 2274 y Fn(Since)i(2)p Fj(\013)22 b Fh(\000)h Fn(2)k Fj(>)h Fn(2,)33 b(there)h(exists)f Fj(M)1416 2289 y Ff(0)1488 2274 y Fj(>)28 b Fn(0)33 b(suc)m(h)i(that)973 2539 y Fj(c)1016 2554 y Ff(0)1061 2539 y Fj(M)1169 2497 y Ff(2)p Fg(\013)p Fc(\000)p Ff(2)1158 2566 y(0)1389 2539 y Fh(\000)23 b Fn(3)p Fj(\025)1597 2498 y Fc(\000)p Ff(2)1704 2539 y Fj(M)1812 2498 y Ff(2)1801 2564 y(0)1878 2539 y Fh(\000)f Fj(V)2035 2554 y Ff(1)2102 2539 y Fn(+)h Fj(K)29 b Fh(\000)23 b Fj(c)2459 2554 y Ff(0)2531 2539 y Fh(\025)28 b Fn(0)33 b Fj(;)768 b Fn(\(4.3\))0 2805 y(whic)m(h)34 b(implies,)e(after)h (taking)g Fj(\026)27 b Fn(=)h Fj(M)1483 2820 y Ff(0)1528 2805 y Fn(,)33 b(that)g Fj(w)d Fh(\024)e 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Fh(\000)22 b Fj(w)2949 1705 y Ff(2)2946 1771 y(2)2994 1746 y Fn(\))p Fj(dx)373 b Fn(\(5.3\))1563 2013 y Fh(\000)22 b Fj("T)1797 1877 y Fi(Z)1853 2103 y Ff(\012)1915 2013 y Fn(\()p Fj(h)p Fn(\()p Fj(w)2124 1972 y Ff(2)2121 2037 y(1)2168 2013 y Fn(\))g Fh(\000)h Fj(h)p Fn(\()p Fj(w)2499 1972 y Ff(2)2496 2037 y(2)2544 2013 y Fn(\)\)\()p Fj(w)2735 1972 y Ff(2)2732 2037 y(1)2801 2013 y Fh(\000)f Fj(w)2974 1972 y Ff(2)2971 2037 y(2)3019 2013 y Fn(\))p Fj(dx)1563 2279 y Fn(+)1662 2143 y Fi(Z)1718 2370 y Ff(\012)1780 2279 y Fn(\()p Fj(V)1877 2294 y Ff(1)1943 2279 y Fh(\000)h Fj(V)2101 2294 y Ff(2)2146 2279 y Fn(\)\()p Fj(w)2298 2238 y Ff(2)2295 2304 y(1)2364 2279 y Fh(\000)g Fj(w)2538 2238 y Ff(2)2535 2304 y(2)2582 2279 y Fn(\))p Fj(dx:)0 2498 y Fn(No)m(w)36 b(m)m(ultiply)e(the)i(di\013erence)i(of)e(Eq.)f (\(1.11\))g(for)h Fj(V)2058 2513 y Ff(1)2139 2498 y Fn(and)g Fj(V)2393 2513 y Ff(2)2474 2498 y Fn(b)m(y)g Fj(V)2673 2513 y Ff(1)2742 2498 y Fh(\000)25 b Fj(V)2902 2513 y Ff(2)2947 2498 y 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Fh(\025)g Fj(\025)1824 5715 y Ff(1)1869 5700 y Fj(f)1918 5715 y Ff(0)1984 5700 y Fn(+)23 b Fj(g)t Fn(\(0+\))e(+)i(max)n(\(0)p Fj(;)17 b(K)7 b(a)2926 5715 y Ff(0)2969 5700 y Fn(\))27 b(=)h Fj(L)133 b Fn(in)33 b(\012)p Fj(:)p eop %%Page: 20 20 20 19 bop 0 -139 a Fl(20)949 b(IRENE)37 b(GAMBA)h(AND)f(ANSGAR)i(J)2394 -160 y(\177)2383 -139 y(UNGEL)0 100 y Fn(No)m(w)33 b(de\014ne)967 279 y Fj(p)p Fn(\()p Fj(x)p Fn(\))28 b(=)1314 212 y Fj(L)p 1297 256 102 4 v 1297 347 a Fn(2)p Fj(d)1410 279 y Fh(j)p Fj(x)22 b Fh(\000)h Fj(x)1674 294 y Ff(0)1718 279 y Fh(j)1746 238 y Ff(2)1813 279 y Fh(\000)f Fn(1)133 b(for)34 b Fj(x)27 b Fh(2)h Fj(B)2502 294 y Fg(r)2546 279 y Fn(\()p Fj(x)2642 294 y Ff(0)2686 279 y Fn(\))p Fj(:)0 504 y Fn(Then)578 683 y(\001)p Fj(p)g Fn(=)g Fj(L)100 b Fn(in)33 b Fj(B)1204 698 y Fg(r)1247 683 y Fn(\()p Fj(x)1343 698 y Ff(0)1388 683 y Fn(\))p Fj(;)215 b(p)28 b Fn(=)g Fj(r)1901 642 y Ff(2)1945 683 y Fj(L=)p Fn(2)p Fj(d)22 b Fh(\000)h Fn(1)k Fh(\025)h Fn(0)100 b(on)33 b Fj(@)6 b(B)2892 698 y Fg(r)2935 683 y Fn(\()p Fj(x)3031 698 y Ff(0)3075 683 y Fn(\))p Fj(:)0 908 y Fn(This)33 b(implies)255 1152 y(\001\()p Fj(u)22 b Fh(\000)g Fj(p)p Fn(\))28 b Fh(\025)g Fn(0)100 b(in)33 b Fj(B)1119 1167 y Fg(r)1163 1152 y Fn(\()p Fj(x)1259 1167 y Ff(0)1303 1152 y Fn(\))p Fj(;)215 b(u)22 b Fh(\000)h Fj(p)28 b Fh(\024)g Fj(M)33 b Fh(\000)22 b Fj(r)2224 1111 y Ff(2)2269 1152 y Fj(L=)p Fn(2)p Fj(d)f Fn(+)i(1)k Fh(\024)h Fn(0)100 b(on)33 b Fj(@)6 b(B)3215 1167 y Fg(r)3258 1152 y Fn(\()p Fj(x)3354 1167 y Ff(0)3399 1152 y Fn(\))p Fj(:)0 1397 y Fn(By)44 b(the)h(maxim)m(um)c(principle,) 48 b(w)m(e)e(conclude)g(that)e Fj(u)30 b Fh(\000)g Fj(p)47 b Fh(\024)h Fn(0)c(in)h Fj(B)2800 1412 y Fg(r)2844 1397 y Fn(\()p Fj(x)2940 1412 y Ff(0)2984 1397 y Fn(\).)79 b(In)45 b(particular,)0 1576 y Fj(u)p Fn(\()p Fj(x)153 1591 y Ff(0)197 1576 y Fn(\))28 b Fh(\024)g Fj(p)p Fn(\()p Fj(x)515 1591 y Ff(0)559 1576 y Fn(\))g(=)g Fh(\000)p Fn(1)g Fj(<)g Fn(0,)33 b(con)m(tradiction.)1570 1851 y Fm(References)-160 2113 y Fn([ALMS98])50 b(A.)33 b(Arnold,)h(J.)g (Lop)m(\023)-47 b(ez,)35 b(P)-8 b(.)33 b(Mark)m(o)m(wic)m(h,)h(J.)g (Soler.)46 b(An)34 b(analysis)g(of)g(quan)m(tum)g(F)-8 b(okk)m(er-)329 2293 y(Planc)m(k)33 b(mo)s(dels.)43 b(Preprin)m(t,)33 b(TU)g(Berlin,)g(German)m(y)-8 b(,)32 b(1998.)-14 2472 y([BP98])48 b(A.)g(Bertozzi,)53 b(M.)c(Pugh.)91 b(Long-w)m(a)m(v)m(e)50 b(instabilities)e(and)i(saturation)f(in)g(thin)g(\014lm)329 2651 y(equations.)44 b Fd(Comm.)36 b(Pur)-5 b(e)35 b(Appl.)h(Math.)44 b Fn(51)33 b(\(1998\),)f(625-661.)-64 2831 y([BLS94])49 b(P)-8 b(.)42 b(Bleher,)j(J.)e(Leb)s(o)m(witz,)j(E.)c(Sp)s(eer.)75 b(Existence)43 b(and)g(p)s(ositivit)m(y)f(of)h(solutions)g(of)g(a)329 3010 y(fourth-order)49 b(nonlinear)g(PDE)f(describing)h(in)m(terface)g (\015uctuations.)90 b Fd(Comm.)50 b(Pur)-5 b(e)329 3189 y(Appl.)35 b(Math.)44 b Fn(47)33 b(\(1994\),)f(923-942.)-24 3369 y([BO86])49 b(H.)35 b(Br)m(\023)-47 b(ezis,)35 b(L.)h(Osw)m(ald.) 51 b(Remarks)35 b(on)g(sublinear)h(elliptic)f(equations.)50 b Fd(Nonlin.)38 b(A)m(nal.)329 3548 y Fn(10)33 b(\(1986\),)e(55-64.)-43 3727 y([DM93])49 b(P)-8 b(.)38 b(Degond)h(and)g(P)-8 b(.)38 b(Mark)m(o)m(wic)m(h.)62 b(A)38 b(steady)h(state)f(p)s(oten)m (tial)h(\015o)m(w)g(mo)s(del)e(for)j(semi-)329 3907 y(conductors.)45 b Fd(A)m(nn.)36 b(Mat.)f(Pur)-5 b(a)36 b(Appl.)44 b Fn(165)33 b(\(1993\),)e(87-98.)-30 4086 y([F)-8 b(ey72])49 b(R.)40 b(F)-8 b(eynman.)64 b Fd(Statistic)-5 b(al)41 b(Me)-5 b(chanics,)42 b(A)g(Set)g(of)g(L)-5 b(e)g(ctur)g(es.)64 b Fn(F)-8 b(ron)m(tiers)41 b(in)f(Ph)m(ysics,)329 4265 y(W.A.)32 b(Benjamin,)h(1972.)-87 4445 y([Gam92])48 b(I.)35 b(Gam)m(ba.)50 b(Stationary)34 b(transonic)i(solutions)f(of)h(a)f (one-dimensional)h(h)m(ydro)s(dynamic)329 4624 y(mo)s(del)c(for)h (semiconductors.)45 b Fd(Comm.)36 b(P.)f(D.)g(E.)e Fn(17)g(\(1992\),)e (553-577.)-87 4803 y([Gam97])48 b(I.)38 b(Gam)m(ba.)60 b(Sharp)40 b(uniform)e(b)s(ounds)i(for)f(steady)g(p)s(oten)m(tial)f 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b(Gasser,)k(P)-8 b(.)43 b(Mark)m(o)m(wic)m(h.)73 b(Quan)m(tum)44 b(h)m(ydro)s(dynamics,)g(Wigner)g(transforms)e(and)329 638 y(the)33 b(classical)g(limit.)42 b Fd(Asympt.)36 b(A)m(nal.)45 b Fn(14)33 b(\(1997\),)e(97-116.)-119 817 y([GMR96])50 b(I.)43 b(Gasser,)j(P)-8 b(.)42 b(Mark)m(o)m(wic)m(h,)k (C.)d(Ringhofer.)76 b(Closure)43 b(conditions)h(for)f(classical)g(and) 329 996 y(quan)m(tum)35 b(momen)m(t)f(hierarc)m(hies)j(in)f(the)g (small)e(temp)s(erature)h(limit.)49 b Fd(T)-8 b(r)j(ansp.)38 b(The)-5 b(ory)329 1176 y(Stat.)35 b(Phys.)44 b Fn(25)32 b(\(1996\),)g(409-423.)-26 1355 y([GT83])49 b(D.)c(Gilbarg,)k(N.)c(T)-8 b(rudinger.)83 b Fd(El)5 b(liptic)47 b(Partial)g(Di\013er)-5 b(ential)46 b(Equations)h(of)g(Se)-5 b(c)g(ond)329 1534 y(Or)g(der)p Fn(.)44 b(Springer,)34 b(Berlin)e(1983.)-113 1714 y([GyJ)s(\177)-53 b(u98])49 b(M.)39 b(Gyi,)g(A.)f(J)s(\177)-53 b(ungel.)62 b(A)39 b(quan)m(tum)f(regularization)g(of)h(the)h (one-dimensional)e(h)m(ydro-)329 1893 y(dynamic)32 b(mo)s(del)g(for)i (semiconductors.)44 b(Submitted)33 b(for)h(publication,)f(1998.)-38 2072 y([J)s(\177)-53 b(un97])50 b(A.)30 b(J)s(\177)-53 b(ungel.)44 b(A)31 b(note)g(on)h(curren)m(t-v)m(oltage)f(c)m (haracteristics)h(from)e(the)i(quan)m(tum)e(h)m(ydro-)329 2252 y(dynamic)i(equations)h(for)h(semiconductors.)44 b Fd(Appl.)36 b(Math.)f(L)-5 b(etters)33 b Fn(10)g(\(1997\),)f(29-34.) -38 2431 y([J)s(\177)-53 b(un98])50 b(A.)36 b(J)s(\177)-53 b(ungel.)55 b(A)37 b(steady-state)g(quan)m(tum)f(Euler-Poisson)h (system)f(for)h(p)s(oten)m(tial)g(\015o)m(ws.)329 2610 y Fd(Comm.)46 b(Math.)g(Phys.)d Fn(194)33 b(\(1998\),)f(463-479.)-96 2790 y([KPV93])49 b(C.)42 b(Kenig,)k(G.)d(P)m(once,)j(L.)d(V)-8 b(ega.)72 b(W)-8 b(ell-p)s(osedness)45 b(and)f(scattering)f(results)g (for)h(the)329 2969 y(generalized)36 b(Kortew)m(eg-de)g(V)-8 b(ries)35 b(equation)f(via)h(the)g(con)m(traction)g(principle.)50 b Fd(Comm.)329 3148 y(Pur)-5 b(e)35 b(Appl.)h(Math.)44 b Fn(46)33 b(\(1993\),)e(527-620.)-30 3328 y([LM93])50 b(M.)37 b(Lo\013redo,)h(L.)e(Morato.)55 b(On)38 b(the)f(creation)g(of)g (quan)m(tum)f(v)m(ortex)g(lines)h(in)g(rotating)329 3507 y(HeI)s(I.)32 b Fd(Il)37 b(nouvo)f(cimento)d Fn(108B)f(\(1993\),)f (205-215.)-56 3686 y([Mar91])49 b(P)-8 b(.)36 b(Mark)m(o)m(wic)m(h.)58 b(On)38 b(steady)f(state)h(Euler-Poisson)g(mo)s(dels)e(for)i (semiconductors.)57 b Fd(Z.)329 3866 y(A)m(ngew.)35 b(Math.)h(Physik)c Fn(42)h(\(1991\),)e(385-407.)-43 4045 y([Lau97])49 b(C.)30 b(Laurey)-8 b(.)44 b(The)31 b(Cauc)m(h)m(y)h(problem)f(for)g(a)g(third) g(order)h(nonlinear)f(Sc)m(hr\177)-50 b(odinger)33 b(equa-)329 4224 y(tion.)43 b Fd(Nonlin.)36 b(A)m(nal.)44 b Fn(29)33 b(\(1997\),)f(121-158.)-100 4403 y([PGG98])49 b(R.)42 b(Dal)f(P)m(asso,)j(H.)d(Garc)m(k)m(e,)j(G.)e(Gr)s(\177)-53 b(un.)71 b(On)43 b(a)e(fourth-order)i(degenerate)g(parab)s(olic)329 4583 y(equation:)51 b(global)36 b(en)m(trop)m(y)i(estimates,)e (existence,)j(and)e(qualitativ)m(e)f(b)s(eha)m(vior)g(of)i(so-)329 4762 y(lutions.)44 b Fd(SIAM)36 b(J.)g(Math.)f(A)m(nal.)45 b Fn(29)33 b(\(1998\),)e(321-342.)-20 4941 y([Sta66])49 b(G.)35 b(Stampacc)m(hia.)49 b Fd(Equations)37 b(el)5 b(liptiques)37 b(du)h(se)-5 b(c)g(ond)36 b(or)-5 b(dr)g(e)37 b(\022)-51 b(a)37 b(c)-5 b(o)g(e\016cients)36 b(disc)-5 b(onti-)329 5121 y(nus)p Fn(,)33 b(Les)h(Presses)g(de)g(l'Univ)m(ersit) m(\023)-47 b(e)33 b(Mon)m(tr)m(\023)-47 b(eal,)33 b(Canada,)g(1966.)12 5300 y([ZJ96])49 b(B.)43 b(Zhang,)k(J.)e(Jerome.)76 b(On)46 b(a)e(steady-state)g(quan)m(tum)g(h)m(ydro)s(dynamic)f(mo)s(del)h(for) 329 5479 y(semiconductors.)g Fd(Nonlin.)35 b(A)m(nal.)45 b Fn(26)33 b(\(1996\),)f(845-856.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9909301607703--