On a Kinetic Equation for Coalescing Particles
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. On a Kinetic Equation for Coalescing Particles Miguel Escobedo — Philippe Laurençot — Stéphane Mischler. N° 4748 Février 2003. N 0249-6399. ISRN INRIA/RR--4748--FR+ENG. THÈME 4. apport de recherche.
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(118) .8<!!6 ghirghendkacVWY7y©in?m eryqgq`fv dbY7p s YJv yhY7enpqY7p}#`¥ghTEgq`bWY φ(y) = 1 .,enp4YXwamHY7v gqY s 0 Z Z Z Z Z Z Z Z .Ao§Îr% 0 1 a f f dy dydxdt = f (x, y) dydx. f (t, x, y) dydx + Ω. Y. α. . t. Ω. Y. 2. 0. 0. Ω. Y. Y. 0. in. Ω. Y. ÷¡û<ü¨÷äØ.
(119)
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(123) !" #. ~}kVirghTVYXy]v iQk puYJ´2cVYXk vXY«`fpgqTeg ¢ `j}pqcVmVm f (x, y) ⊂ M := {(x, y) ∈ Ω × Y , m > δ} ¢ gqTVY7k pqcVmVm f (t) ⊂ M ¡iQy:erkat t ≥ 0 >§ =³k s YXY s£¢ `jgC`fp?pqc «v `bYXk2g?ghikVirgh`bvXY4gqT erg φ(y) = 1 (m) phegq`fpu¤ Y7p .Ao§În 0 §#S4T eg}WYJerk p4ghT eg©kVi«m enyugh`bvXdjYin¬pq`bªXYpqWendjdbYXyGgqT enk v7erkEHY v yhY7ergqY s `j¬gqT YXyhYeryhYkVinkVY `bkV`jgq`ferdbdjtQ§ ~ Y7en nYXyQYXypu`bink irgqTV`fpÓeQv°g«`fpgqTVY½¡indbdji#`bkV ?lδ ¡iQy«erkat kViQko²Å`jk vXyqYJenpq`jk ¡cVkv°gq`bink .,pucvTleQp φ(m) = m ¢ α < 00 ¢ YTePnY φ:R →R Z Z .Ao§ÎZ 0 1 Y (f (t, .)) + Y (f (τ )) dxdτ ≤ Y (f ) , 2 #`jgqT in. δ. &. δ. +. [0,δ]. α. +. t. φ. φ. 0. Ω. Z Z. Yφ (f ) :=. Y. •. f (x, y) φ(m) dydx,. ZΩ ZY. Yφ (f ) :=. in. φ. a φ(m) f f 0 dydx .. Y. inyGerkat«k inko² s Y7vXyqYJenpq`jk erk s kViQkVkVY7Qegh`jQY}¡c k v°gh`jiQk. φ : R + → R+. ¢ Y[kVirgh`bvXY[gqT erg. |p + p0 | ≤ |p| + |p0 | ≤ (m + m0 ) max(|v|, |v 0 |),. enk s gqTac p |v | ≤ max(|v|, |v |) §S4TVYEWinkVingqink `bvX`¥g{t in φ gqTVY7k `bWmVdj`bY7p¦gqT erg φ ¡cVdj¤ dfp ,. V§ Q 0°§ ,CiQk pqY7´2cVYXk2ghdjt ¢ Z Z 00. 0. f (t, x, y) φ(|v|) dydx. t 7−→. f` pekViQko²Å`jk vXyqYJenpq`jk ¡cVk v°gh`jiQk in?gq`bWYn§~}pe¤ yhpugvXink pqY7´2cVY7k v Yir¬ghTV`bp©ÓeQv°g ¢ Y yhY7endj`bªXY gqT erg ¢ `j ¢ ghTVYXkRpuc mVm f (t) ⊂ V ¡iny[erkat t ≥ 0 .ÓenmVmVdbtlgqTVY pqcVmVm m yqY72`binfcpC⊂yhY7pqVcVdjg#:=#`¥gh{(x, TghTVy)YvTV∈iQΩ`bvXY ×φY,=|v|1 ≤ R}0 § ,Cink pq` s YXyCkViekViQkVkVY7Qegh`jQYGerk s kVinkV²A`bk v yhY7eQpu`bkV[¡cVkv°gq`bink φ : R → R erk s ekVink kVYX2egq`bnY • s enk v inkaQY w¦¡cVk v gq`bink φ : R → R §?S4TVYXk Ω. in. Y. R. R. [R,+∞). 2. 1. 3. t 7−→. Z Z. +. f (t, x, y) m φ1 (m) φ2 (v) dydx. `fpGekViQko²Å`jk vXyqYJenpq`jk ¡c k v°gh`jiQkEir<gq`bW]YQ§ =³k s YXY s£¢ mVcVgugq`bkV φ(y) = m φ (m) φ (p/m) ¢ y ∈ Y ¢ ghTVY vXinkanYXwo`¥g{t¦in φ erk s ghTVYWinkVingqink `bvX`¥g{t«in φ YXkpucVyhYgqT erg Ω. Y. 1. 2. 2. 1. m0 m v + v0 φ(y + y ) = (m + m ) φ1 (m + m ) φ2 m + m0 m + m0 ≤ φ1 (m + m0 ) (m φ2 (v) + m0 φ2 (v 0 )) 0. 0. 0. . ≤ m φ1 (m) φ2 (v) + m0 φ1 (m0 ) φ2 (v 0 ) ,. enk s gqTVY[¡cVkv°gq`bink φ pqergq`fp{¤ YJp ,. V§ Q 0°§ üü ÇAVÚ hÚ Ý . .
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(128)
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