流体力学数学原理 第二讲.pdf

n Hilbert m Navier Stokes 6N n 5 2 Stokes f 1 m K f mP u mD C 0 D mD0 2 m u mH u D n divu 0 Sobolev mX H1 n X0 H1 0 n m H div u L2 n divu L2 1 1 D S S p u v H div u v u v divu divv kuk 2 u u KH div Hilbert m m H H L2 n V H H1 n H3L2 H1 e O3H V 1 2 Rn C2ak m8 0 H1 H 1 2 f oKer 0 H1 0 0 3m l H 1 2 H1 0l I3H 1 2 kuk1 2 ckuk1 kl k1 ck k1 2 u H 1 H 1 2 aq n H div H 1 2 n k nu u n u C n k y n H div H 1 2 k 5 f R n H 1 2 Ker n H0 div u H div u n 0 1 3 0 n Stokes u gradw divu w u H div w H 1 H H3L2 n 4 V H3H1 0 n 4 n5 2 1 fi D0 i 1 2 n 2 f f1 f2 fn 2 p D0 F f gradp 0 v H 5 grad D0 H 0 1 4 n5 2 2 Rn k m8 kLipschitz Y o 1 XJ2 p D0 k Dip L2 op L2 kpkL2 R c kgradpk0 2 XJ2 p D0 k Dip H 1 Kp L2 kpkL2 R c kgradpkH 1 XJ Kp L2 loc kpkL2 R inf c R kp ck0 kp meas 1 Z pdxk0 1 5 n5 2 3 u H1 u L2 o u n H 1 2 gradu gradv u v v H 1 n5 2 4 Rn Lipschitzk m8 o H u H0 div divu 0 u L2 n divu 0 u n 0 H u L2 n u gradp p H 1 L2 n H H y 12 13 Sobolev m k e 11 divu 0 u H0 div d Xu H div u n 0 11 12 u H0 div divu 0 Hw 1 6 eu H duH H3L2 n 4 3S um H um u 3L2 n duum H kdivum 03L2 n l k k umk kumk0 2 um 3H div uu u u H div du n H div H 1 2 5 Y f d num 0 nu u n 0 l k u u H0 div divu 0 n5 2 1 Rn C2aq m8 K L2 n H H1 H2 H1 L2 n u gradp p H 1 p 0 H2 u L2 n u gradp p H 1 0 1 7 y w H1 H2 H H1 H2 u L2 n q H1 0 q divu H 1 dLax Milgram n d q H1 0 u2 gradq H2 w divu2 q divu e u1 u u2 Kdiv u1 0 l Z n u1ds 0 KNeumann K p H1 p 0 p n n u1 k p e u1 gradp Kdivu1 p 0 u1 H1 u0 u u1 u2 K divu0 0 nu0 nu nu1 nu2 n u1 u1 0 l u0 H 1 8 n5 2 2 k m8 Lipschtz Y o V u H 1 0 n divu 0 y V u H1 0 n divu 0 yV V Banach mV f mH3V du V 0 3H 73V e 5 H3V L V 5 Y H1 0 n L l1 l2 ln li H1 0 0 H 1 e 0 v H Kd n5 2 1 3p D0 li p xi H 1 L gradp 1 9 d n5 2 2 p L2 l L w p divw 0 w V 1 10 L2 K f U n5 2 4 P L2 n H Q I P n5 2 5 P H1 0 n H 1 n 5k f y u H1 0 n 9 K p H 1 0 p divu dudivu L2 K 3 p kpk2 ckdivuk0 ckuk1 u2 gradp Kdivu2 p divu u d u u2 H1 n div u u2 0 n u u2 H 1 2 9 K q 0 q n n u u2 H 1 2 1 11 dudiv u u2 0 0 K Kd u1 gradq H1 n K u0 u u1 u2 H1 n divu0 0 nu0 u u1 u2 n u u2 n q n 0 l n5 2 4 u0 Pu H ku0k1 ku u1 u2k1 kuk1 ku1k1 ku2k1 kuk1 kpk2 kqk2 d K K5 ku0k1 ckuk1 ck n u u2 k1 2 c kuk1 ku2k1 ckuk1 n5 2 6 1w oP Hs n Hs n 5k f ps 0 1 12 2 Stokes f 5 2 1 Stokes fA A D A H H A P D A H 2 n V n5 2 7 Stokes f g 5 Au v a0 u v y y H H H 5 Y d D A 5 Y P x x y Ax KA f A H D A A y x u D A 3f H H a0 v u v D A 1 13 f A u dStokes 3 5 u D A Au f A u f Au K A A n5 2 3 Stokes fA fA 1 H f 5 duA g A 1 H g f H g f A 1k l 3 i 0 i 1 i i 0 9H X wi A 1wi iwi eP i 1 i K Awi iwi wi D A i 1 2 0 0 A A f XJ A u X i 1 j ujwj u X j 1 ujwj D A D A u H u X j 1 ujwj X j 1 2 j uj 2 0 PV V m fA V 2 V D A H V V 0 V 1 c1kuk kA 2uk0 c2kuk u V n5 2 4 H D A 0 T 0 D A 3H y dD A w D A D A p H dudiv 0 KA H u Ap H du X wj H X Ap Fourier m Ap X j 1 Ap wj wj X j 1 p j wj wj L2 n 1 17 CVg 8 C XJ kd fN 9 x x 0 k 0 P x x meas R dx L L meas 1 n LL f kfk2 m X m meas 2 m n Z f 2dx y meas m n 1 2kfkm Lm n 2kfkm C kfkm aquL n 2 m n y L2Au u C AaquL 2 1 18 meas 2 n j C j j 1 C j meas 1 2wj C j n5 2 5 P0 s1 n 2 0 kfk Rn ckfk1 t s1 Rnkfk t s2 Rn f H s2 Rn y I f C 0 Rn y f x 2 n Z Rn eix f d 2 n Z R d 2 n Z R d I1 I2 1 19 w I1 Z R f d Z R 1 2 s1 2 1 2 s1 2 f d Z R 1 2 s1d 1 2kfks1 p kfks Rn Z Rn 1 2 s f 2d 1 2 f H s Rn 1 20 nPn K Z R 1 2 s1d n Z R 0 1 r2 s1rn 1dr n Z R 0 1 1 r2 s1rn 2s1 1dr n n 2s1 Rn 2s1 l I1 r n n 2s1 R n 2s1 2 kfks1 n y I2 r n 2s2 nR n 2s2 2 kfks2 R kfks2 kfks1 1 s2 s1 K R n 2 s1kfks1 R n 2 s2kfks2 dd 1 21 n5 2 6 Agmon Clak m8 l n 2 s 0 kfk ckfk1 t s kfkt l f H l t n 2 s l s n5 2 7 Rn C2ak m8 o 3 C c0 0 Stokes fA A v j c0j 2 n 1 n 2 3 1 22 n5 2 8 R