Dispersive Sysems 1) Schrödinger eqaion ) Cbic Schrödinger 3) KdV 4) Discreerised hyperbolic eqaion 5) Discree sysems KdV + + ε =, = ( ) ( )
d d + = d d =, =. ( ) = ( ) DISCONTINUITY, prescribed cri Collision of characerisic lines ( ) > ( ) 1 No coninos solion for > cri
Solions of eq.-s of compressible flow develop disconiniies (shocks). Sokes, Airy, Riemann: Conservaion laws + f + g + h = y z ( ) ( ) f = f, g = g, ec.
Inegrae over G: + = d dv F nds d G G F = f, g, h fl ( ), Weak solions, disribions + f = d d b a d + f f = a b a. y (), dy b d = s
y b y b d d + d = [ ] d + d + s l r = a y a y d y a b [ ] fd fd+ s = l r y f + f f + f + s[ ], l a b r l r So by he conservaion law, s = [ f ] [ ]
Rankine-Hgonio condiion Eample: + = f ( ) ( 1 ) 1 + = = [ ] [ ] e r e+ r 1 s = f = = e r Enropy condiion: e > r
1 + = ( ) + = log a + = RH condiion s = log e + r e e r log r
Mliply eqaion + = ( ) 3 3 R-H condiion s + = + + = = 3 3 3 3 3 e r e e r r e r e + r Which conservaion law??
Goodman-La; in + = discreise space:, k kδ ( ) ( ) Replace by qoien: symmeric difference k + k = Δ d =. d k+ 1 k 1 Dispersive Conservaion form
f k f f k+ 1 1 k+ + = Δ = k+ k k+ 1 1 3 (,) = + cos Nmerical eperimens reveal oscillaions and weak convergence for >, cri
1. Eplain oscillaory behavior. Prove weak convergence 3. Deermine weak limi 4. Wha eqaion does he weak limi saisfy?
f k k k + + k+ 1 k = k Δ f f k+ 1 1 k + = Δ 1 = 1 k+ 1 Divide by : k log + k+ 1 k 1 = k Δ k
g g k 1 1 log + k + = k Δ g k + + = k k 1 1 +
( Δ) If wold end srongly o, boh and ( ) + f = ( ) g( ) log + = wold be saisfied. B hey are incompaible for disconinos solions.
The solion of he difference eqaion desperaely ries o saisfy wo incompaible conservaion laws. Hence he oscillaions.
Limis of Semigrops. Denoe by U(, ε ) he solion operaor for he KdV eqaion + + (1) ε =, ha is, he operaor U, :,;, ; ( ε ) ( ε) ( ε) ha relaes iniial vales of solions of (1) o heir vales a. Clearly, he operaors
U(, ε ) form a semigrop (even a grop) in he parameer : (, ) (, ) = ( +, ) U s U U s ε ε ε () La & Levermore have shown ha he limi ε of U(, ε ) eis in he weak b for > cri no in he srong opology: U (, ) ( ) ε U (3). Noe ha since eqaion (1), wrien in conservaion form
1 ( ) + + ε =, is nonlinear, a weak limi of is solions ha is no a srong limi is no a solion of he limiing eqaion 1 ( ) + =. (4) So U( ) in () is no he solion of (4) wih iniial vale. Qesion. Do he limi operaors U( ) defined in (3) form a semigrop:
( ) ( ) ( ). U U s U + s (5) L & L have derived an eplici formla for U ( ), which shows ha U ( ) is no coninos in he weak opology. Tha makes i very dbios ha he limi as ε ends zero of () is (5). The qesion can be easily decided by a single calclaion.
There is an analoge of his qesion in he heory of rblence. Here operaors o be considered are he solions operaors U(, R ) for he Navier-Sokes eqaion. I srmise ha as he Reynolds nmber R, he operaors U(, R ) end weakly b no srongly o a limi U( ).
Since he limiing Eler eqaions are nonlinear conservaion laws, U ( ) is no he solion operaor for he incompressible Eler eqaion b a descripion of rblen flow. In analogy wih he zero dispersion limi of he KdV eqaion I wold srmise ha he operaors U ( ), he limis of U(, R ), do no form a semigrop, as conjecred by Heinz Kreiss.
Compacness,, + = can be wrien as conservaion law ( 1 ) + =. Iniial vale (,) ( ) zero oside a finie inerval. =, bonded and Viscosiy mehod + = ε ( ), ε >, =, ; ε,
( ) ( ) ( ) ( ),; ε =. lim, ; ε =,, ε bondedly, in 1 L norm. How does depend on iniial? and se Define ( ) ( ) U = y dy, = sp U 1. For any o solions, υ wih iniial vales, υ,
(, ) (, ) υ 1 υ 1 Proof. Inegrae viscos eqaion: and + 1 =, U U εu + 1 =. V V εv Sbrac hese eqaions, and denoe U V as D: D + 1 U + V D= ε D ( ) By maimm principle,
as claimed. ( n) D(, ) sp D, Le be any seqence of iniial dae ha end weakly, ha is in he sense of he nom, o ; we saw ha hen 1 for every ends weakly o (, ) ( n) eqaion and ( n ) (, ) >. saisfy he basic differenial
( n) ( ) ( n) ( n) + = 1 1 ( ) + = ends o in he sense of disribion. ) Therefore ( n ends o. disribion. So ( n ) ends o, in he sense of he sense of disribion. Since boh n and end o zero as, he + cons in
n consan is zero. B hen ms end o srongly, ha is in he L 1 norm. Smmary If he iniial vales of ( n) end o he iniial vales of weakly, n (, ) ends o (, ) srongly, for every >. The mapping from iniial vales (,) o heir vales (, ), >, is a compac mapping.