SLIDE 1
❏♦✉r♥é❡s ❞❡ Pr♦❜❛❜✐❧✐tés ✷✵✵✼ ▲❛ ▲♦♥❞❡✱ ✶✵✲✶✹ s❡♣t❡♠❜r❡ ✷✵✵✼
Pr✐♥❝✐♣❡ ❞❡ ▼❛①✐♠✉♠ ❡t ❚❤é♦rè♠❡ ❞❡ ❈♦♠♣❛r✐s♦♥ ♣♦✉r ❧❡s s♦❧✉t✐♦♥s ❞✬❊❉P❙ q✉❛s✐✲❧✐♥é❛✐r❡s ❙P❉❊✬s ❆✳ ▼❛t♦✉ss✐ ✭❯♥✐✈❡rs✐té ❞✉ ▼❛✐♥❡✱ ▲❡ ▼❛♥s✮
✫ ▲✳ ❉❡♥✐s ✭❯♥✐✈❡rs✐té ❞✬❊✈r②✮ ✫ ▲✳ ❙t♦✐❝❛ ✭❯♥✐✈❡rs✐té ❞❡ ❇✉❝❤❛r❡st✱ ❘♦✉♠❛♥✐❡✮ ✵✲✵
SLIDE 2 Pr♦❜❧❡♠ ✿
❲❡ st✉❞② t❤❡ ❢♦❧❧♦✇✐♥❣ st♦❝❤❛st✐❝ ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ✭✐♥ s❤♦rt ❙P❉❊✮ ❢♦r ❛ r❡❛❧ ✲✈❛❧✉❡❞ r❛♥❞♦♠ ✜❡❧❞ ut (x) := u (t, x) , dut (x) = Lut (x) dt + ft (x, ut (x) , ∇ut (x)) dt +
d
∂igi,t (x, ut (x) , ∇ut (x)) dt +
d1
hj,t (x, ut (x) , ∇ut (x)) dBj
t
✭✶✮ ✇✐t❤ ❛ ❣✐✈❡♥ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ u0 = ξ, ✇❤❡r❡ L ✐s ❛ s②♠♠❡tr✐❝ s❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦r ❞❡✜♥❡❞ ✐♥ s♦♠❡ ❜♦✉♥❞❡❞ ♦♣❡♥ ❞♦♠❛✐♥ O ⊂ Rd ❛♥❞ f, gi, i = 1, ..., d, hj, j = 1, ..., d1 ❛r❡ ♥♦♥❧✐♥❡❛r r❛♥❞♦♠ ❢✉♥❝t✐♦♥s✳ ❲❡ st✉❞② ✿ ✲ t❤❡ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❢♦r t❤❡ ❙P❉❊ (E) ✲ ❝♦♠♣❛r✐s♦♥ t❤❡♦r❡♠✳ ✲ ❡①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ st♦❝❤❛st✐❝ ❇✉r❣❡r ❡q✉❛t✐♦♥✳
✵✲✶
SLIDE 3
❚❤❡ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❢♦r q✉❛s✐❧✐♥❡❛r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ✭t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ❝❛s❡ ✿ h = 0✮ ✇❛s ♣r♦✈❡❞ ❜② ❆r♦♥s♦♥ ✲❙❡rr✐♥ ✭✶✾✻✼✮ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠ ✿ ❚❤❡♦r❡♠ ✿ ▲❡t u ❜❡ ❛ ✇❡❛❦ s♦❧✉t✐♦♥ ♦❢ ❛ q✉❛s✐❧✐♥❡❛r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❢♦r♠ ∂tu = divA (t, x, u, ∇u) + B (t, x, u, ∇u) ✐♥ t❤❡ ❜♦✉♥❞❡❞ ❝②❧✐♥❞❡r ]0, T[×O ⊂ Rd+1. ■❢ u ≤ M ♦♥ t❤❡ ♣❛r❛❜♦❧✐❝ ❜♦✉♥❞❛r② {[0, T[×∂O} ∪ {{0} × O}✱ t❤❡♥ ♦♥❡ ❤❛s u ≤ M + Ck (A, B), ✇❤❡r❡ C ❞❡♣❡♥❞s ♦♥❧② ♦♥ T, t❤❡ ✈♦❧✉♠❡ ♦❢ O ❛♥❞ t❤❡ str✉❝t✉r❡ ♦❢ t❤❡ ❡q✉❛t✐♦♥✱ ✇❤✐❧❡ k (A, B) ✐s ❞✐r❡❝t❧② ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ s♦♠❡ q✉❛♥t✐t✐❡s r❡❧❛t❡❞ t♦ t❤❡ ❝♦❡✣❝✐❡♥ts A ❛♥❞ B. ❚❤❡ ♠❡t❤♦❞ ♦❢ ♣r♦♦❢ ✇❛s ❜❛s❡❞ ♦♥ ▼♦s❡r✬s ✐t❡r❛t✐♦♥ s❝❤❡♠❡ ❛❞❛♣t❡❞ t♦ t❤❡ ♥♦♥❧✐♥❡❛r ❝❛s❡✳ ❚❤✐s ♠❡t❤♦❞ ✇❛s ❢✉rt❤❡r ❛❞❛♣t❡❞ t♦ t❤❡ st♦❝❤❛st✐❝ ❢r❛♠❡✇♦r❦ ✐♥ ❉❡♥✐s✱ ▼✳ ❛♥❞ ❙t♦✐❝❛ ✭✷✵✵✺✮✱ ♦❜t❛✐♥✐♥❣ s♦♠❡ Lp ❛ ♣r✐♦r✐ ❡st✐♠❛t❡s ❢♦r t❤❡ ✉♥✐❢♦r♠ ♥♦r♠ ♦❢ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ st♦❝❤❛st✐❝ q✉❛s✐❧✐♥❡❛r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥✳
✵✲✷
SLIDE 4 ❲❡ ♣r♦✈❡ t❤❡ st♦❝❤❛st✐❝ ✈❡rs✐♦♥ ♦❢ t❤❡ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ♦❢ ❆r♦♥s♦♥ ✲❙❡rr✐♥ ✿ ❚❤❡♦r❡♠ ✿ ▲❡t p ≥ 2 ❛♥❞ u ❜❡ ❛ s♦❧✉t✐♦♥ ♦❢ ✭✶✮ ✐♥ t❤❡ ✇❡❛❦ s❡♥s❡✳ ❆ss✉♠❡ t❤❛t u ≤ M ♦♥ t❤❡ ♣❛r❛❜♦❧✐❝ ❜♦✉♥❞❛r② {[0, T[×∂O} ∪ {{0} × O}✱ t❤❡♥ ❢♦r ❛❧❧ t ∈ [0, T] ✿ E
p
∞,∞;t ≤ k (p, t) E
∞ +
∗p
θ,t +
∗p/2
θ;t
+
∗p/2
θ;t
f 0,M(t, x) = f(t, x, M, 0), g0,M(t, x) = g(t, x, M, 0), h0,M(t, x) = h(t, x, M, 0) ❛♥❞ k ✐s ❛ ❢✉♥❝t✐♦♥ ✇❤✐❝❤ ♦♥❧② ❞❡♣❡♥❞s ♦♥ t❤❡ str✉❝t✉r❡ ❝♦♥st❛♥ts ♦❢ t❤❡ ❙P❉❊✱ ·∞,∞;t ✐s t❤❡ ✉♥✐❢♦r♠ ♥♦r♠ ♦♥ [0, t]×O ❛♥❞ ·∗
θ;t ✐s ❛ ❝❡rt❛✐♥ ♥♦r♠ ✇❤✐❝❤ ✐s ♣r❡❝✐s❡❧② ❞❡✜♥❡❞
❜❡❧♦✇✳
✵✲✸
SLIDE 5 ❍②♣♦t❤❡s✐s ❛♥❞ ❞❡✜♥✐t✐♦♥s ✿ ⋄ O ⊂ Rd ♦♣❡♥ ❜♦✉♥❞❡❞ s❡t✳ ⋄ (Bt)t d1✲❞✐♠❡♥s✐♦♥❛❧ ❇▼ ❞❡✜♥❡❞ ♦♥ (Ω, F, (Ft)t, P)✱ ⋄ A := −L := − ∂i(ai,j∂j) ✿ s②♠♠❡tr✐❝ s❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦r✱ ⋄ a := (aij)i,j ✐s ❛ ♠❡❛s✉r❛❜❧❡ ❛♥❞ s②♠♠❡tr✐❝ ♠❛tr✐① ❛♥❞ s❛t✐s✜❡s ✉♥✐❢♦r♠ ❡❧❧✐♣t✐❝✐t② ✿ λ|ζ|2 ≤
d
ai,j(x)ζi ζj ≤ Λ|ζ|2, ∀x ∈ O, ζ ∈ Rd ✇❤❡r❡ λ ❛♥❞ Λ ❛r❡ ♣♦s✐t✐✈❡ ❝♦♥st❛♥ts✳ ⋄ ξ ∈ L2(Ω × O)✳ ⋄ T > 0✳
✵✲✹
SLIDE 6 ❲❡ ❛r❡ ❣✐✈❡♥ ♣r❡❞✐❝t❛❜❧❡ ❢✉♥❝t✐♦♥s ✿ f : R+ × Ω × O × R × Rd → R , h : R+ × Ω × O × R × Rd → Rd1 g = (¯ g1, ..., ¯ gd) : R+ × Ω × O × R × Rd → Rd. s✉❝❤ t❤❛t ✿ ✶✳ |f(t, ω, x, y, z) − f(t, ω, x, y′, z′)| ≤ C(|y − y′| + |z − z′|) ✷✳ d1
j=1 |hj(t, ω, x, y, z) − hj(t, ω, x, y′, z′)|2 1
2 ≤ C |y − y′| + β |z − z′|✱
✸✳ d
i=1 |gi(t, ω, x, y, z) − gi(t, ω, x, y′, z′)|2 1
2 ≤ C|y − y′| + α |z − z′|✱
✇❤❡r❡ C, α, β ❛r❡ ♥♦♥ ♥❡❣❛t✐✈❡ ❝♦♥st❛♥ts✳ ❈♦♥tr❛❝t✐♦♥ ❤②♣♦t❤❡s✐s ✿ α + 1 2β2 < λ.
✵✲✺
SLIDE 7 ❲❡❛❦ s♦❧✉t✐♦♥s ♦❢ ❙P❉❊✬s ✿
0(O)✲✈❛❧✉❡❞ ♣r❡❞✐❝t❛❜❧❡ ♣r♦❝❡ss❡s u s✳t✳
uE,T :=
0≤t≤T
ut2 + T E E (ut, ut) dt 1/2 < ∞ . ✇❤❡r❡ E ✐s t❤❡ ❡♥❡r❣② ✭❉✐r✐❝❤❧❡t ❢♦r♠ ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦r A✮ ✿ E(u, v) :=
d
ai,j∂iu ∂jv dx, ∀u ∈ H1
loc(O), ∀v ∈ H1 0(O).
loc(O)✲✈❛❧✉❡❞ ♣r❡❞✐❝t❛❜❧❡ ♣r♦❝❡ss❡s s✉❝❤ t❤❛t ❢♦r ❛♥② ❝♦♠♣❛❝t s✉❜s❡t K ✐♥ O ✿
uE,K,T :=
0≤t≤T
ut(x)2 dx + E T
|∇ut(x)|2 dxdt 1/2 < ∞ .
✵✲✻
SLIDE 8
❉❡✜♥✐t✐♦♥ ✿ u ∈ Hloc ✐s ❛ ✇❡❛❦ s♦❧✉t✐♦♥ ♦❢ (E)✱ ✇✐t❤ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ u0 = ξ✱ ✐❢ ❢♦r ❡❛❝❤ t❡st ❢✉♥❝t✐♦♥ ϕ ∈ D := C∞
c ([0, T)) ⊗ C2 c (O).
T [(us, ∂sϕ) − E (us, ϕs) +(f (s, us, ∇us) , ϕs) − (gi (s, us, ∇us) , ∂iϕs)]ds + T (hj (us, ∇us) , ϕs) dBj
s + (ξ, ϕ0) = 0.
✇❤❡r❡ ( , ) ✐s t❤❡ ✐♥♥❡r ♣r♦❞✉❝t ✐♥ L2(O)✳ ❲❡ ❞❡♥♦t❡ ❜② Uloc(ξ, f, g, h) t❤❡ s❡t ♦❢ s✉❝❤ s♦❧✉t✐♦♥✳ ■❢ u ∈ H0 ✐s ❛ ✇❡❛❦ s♦❧✉t✐♦♥✱ ✇❡ s❛② t❤❛t ✐t s♦❧✈❡s (E) ✇✐t❤ ③❡r♦ ❉✐r✐❝❤❧❡t ❝♦♥❞✐t✐♦♥ ♦♥ ∂O ❛♥❞ ✇❡ ❞❡♥♦t❡ u = U0(ξ, f, g, h) ✳
✵✲✼
SLIDE 9 ❋✉♥❝t✐♦♥❛❧ s♣❛❝❡s ✿
- ❲❡ s❤❛❧❧ ✉s❡ t❤❡ ♥♦t❛t✐♦♥
(u, v) =
u(x)v(x) dx, ✇❤❡r❡ u✱ v ❛r❡ ♠❡❛s✉r❛❜❧❡ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ✐♥ O ❛♥❞ uv ∈ L1(O)✳
0 (O) t❤❡ ❍✐❧❜❡rt s♣❛❝❡ ✿ t❤❡ ✜rst ♦r❞❡r ❙♦❜♦❧❡✈ s♣❛❝❡ ♦❢ ❢✉♥❝t✐♦♥s ✈❛♥✐s❤✐♥❣ ❛t t❤❡ ❜♦✉♥❞❛r②✱
■ts ♥❛t✉r❛❧ s❝❛❧❛r ♣r♦❞✉❝t ❛♥❞ ♥♦r♠ ❛r❡ (u, v)H1
0(O) = (u, v) +
d
(∂iu (x)) (∂iv (x)) dx, uH1
0(O) =
2 + ∇u2 2
1
2 .
loc(O) t❤❡ s♣❛❝❡ ♦❢ ❢✉♥❝t✐♦♥s ✇❤✐❝❤ ❛r❡ ❧♦❝❛❧❧② sq✉❛r❡ ✐♥t❡❣r❛❜❧❡ ✐♥ O ❛♥❞ ✇❤✐❝❤ ❛❞♠✐t ✜rst
♦r❞❡r ❞❡r✐✈❛t✐✈❡s t❤❛t ❛r❡ ❛❧s♦ ❧♦❝❛❧❧② sq✉❛r❡ ✐♥t❡❣r❛❜❧❡✳
- ❋♦r ❡❛❝❤ t > 0 ❛♥❞ ❢♦r ❛❧❧ r❡❛❧ ♥✉♠❜❡rs p, q ≥ 1✱ ✇❡ ❞❡♥♦t❡ ❜② Lp,q([0, t] × O) t❤❡ s♣❛❝❡ ♦❢
✭❝❧❛ss❡s ♦❢✮ ♠❡❛s✉r❛❜❧❡ ❢✉♥❝t✐♦♥s u : [0, t] × O − → R s✉❝❤ t❤❛t up,q; t := t
|u(t, x)|p dx q/p dt 1/q ✐s ✜♥✐t❡✳ ❚❤❡ ❧✐♠✐t✐♥❣ ❝❛s❡s ✇✐t❤ p ♦r q t❛❦✐♥❣ t❤❡ ✈❛❧✉❡ ∞ ❛r❡ ❛❧s♦ ❝♦♥s✐❞❡r❡❞ ✇✐t❤ t❤❡ ✉s❡ ♦❢ t❤❡
✵✲✽
SLIDE 10 ❡ss❡♥t✐❛❧ s✉♣ ♥♦r♠✳ ❲❡ ✐❞❡♥t✐❢② t❤✐s s♣❛❝❡✱ ✐♥ ❛♥ ♦❜✈✐♦✉s ✇❛②✱ ✇✐t❤ t❤❡ s♣❛❝❡ Lq ([0, t] ; Lp (O)) , ❝♦♥s✐st✐♥❣ ♦❢ ❛❧❧ ♠❡❛s✉r❛❜❧❡ ❢✉♥❝t✐♦♥s u : [0, t] → Lp (O) s✉❝❤ t❤❛t t usq
p ds < ∞. ❚❤✐s
✐❞❡♥t✐✜❝❛t✐♦♥ ✐♠♣❧✐❡s t❤❛t t usq
p ds
1
q
= up,q; t.
- ❚❤❡ s♣❛❝❡ ♦❢ ♠❡❛s✉r❛❜❧❡ ❢✉♥❝t✐♦♥s u : R+ → L2 (O) s✉❝❤ t❤❛t u2,2;t < ∞, ❢♦r ❡❛❝❤ t ≥ 0, ✐s
❞❡♥♦t❡❞ ❜② L2
loc (R+; L2 (O)) .
loc (R+; H1 0 (O)) ❝♦♥s✐sts ♦❢ ❛❧❧ ♠❡❛s✉r❛❜❧❡ ❢✉♥❝t✐♦♥s u : R+ → H1 0 (O)
s✉❝❤ t❤❛t u2,2;t + ∇u2,2;t < ∞, ❢♦r ❛♥② t ≥ 0.
- ◆❡①t ✇❡ ❛r❡ ❣♦✐♥❣ t♦ ✐♥tr♦❞✉❝❡ s♦♠❡ ♦t❤❡r s♣❛❝❡s ♦❢ ❢✉♥❝t✐♦♥s ♦❢ ✐♥t❡r❡st ✇❤✐❝❤ ❤❛✈❡ ❛❧r❡❛❞②
❜❡❡♥ ✉s❡❞ ✐♥ ❆r♦♥s♦♥ ❛♥❞ ❙❡rr✐♥ ✿ ▲❡t (p1, q1) , (p2, q2) ∈ [1, ∞]2 ❜❡ ✜①❡❞ ❛♥❞ s❡t I = I (p1, q1, p2, q2) :=
- (p, q) ∈ [1, ∞]2 / ∃ ρ ∈ [0, 1] s.t. 1
p = ρ 1 p1 + (1 − ρ) 1 p2 , 1 q = ρ 1 q1 + (1 − ρ) 1 q2
✵✲✾
SLIDE 11 ❚❤✐s ♠❡❛♥s t❤❛t t❤❡ s❡t ♦❢ ✐♥✈❡rs❡ ♣❛✐rs
p, 1 q
- , (p, q) ❜❡❧♦♥❣✐♥❣ t♦ I, ✐s ❛ s❡❣♠❡♥t ❝♦♥t❛✐♥❡❞ ✐♥
t❤❡ sq✉❛r❡ [0, 1]2 , ✇✐t❤ t❤❡ ❡①tr❡♠✐t✐❡s
p1, 1 q1
p2, 1 q2
- .
- ❚❤❡r❡ ❛r❡ t✇♦ s♣❛❝❡s ♦❢ ✐♥t❡r❡st ❛ss♦❝✐❛t❡❞ t♦ I. ❖♥❡ ✐s t❤❡ ✐♥t❡rs❡❝t✐♦♥ s♣❛❝❡
LI;t =
Lp,q ([0, t] × O) . ❍ö❧❞❡r✬s ✐♥❡q✉❛❧✐t② ❧❡❛❞ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥❝❧✉s✐♦♥ ✿ Lp1,q1 ([0, t] × O) ∩ Lp2,q2 ([0, t] × O) ⊂ Lp,q ([0, t] × O) , ❢♦r ❡❛❝❤ (p, q) ∈ I, ❛♥❞ t❤❡ ✐♥❡q✉❛❧✐t② up,q;t ≤ up1,q1;t ∨ up2,q2;t ❢♦r ❛♥② u ∈ Lp1,q1 ([0, t] × O) ∩ Lp2,q2 ([0, t] × O) .
- ▼♦r❡♦✈❡r✱ ❜② ❍ö❧❞❡r✬s ✐♥❡q✉❛❧✐t②✱ ✐t ❢♦❧❧♦✇s t❤❛t ♦♥❡ ❤❛s
t
u (s, x) v (s, x) dxds ≤ uI;t vI′;t , ✭✷✮ ❢♦r ❛♥② u ∈ LI;t ❛♥❞ v ∈ LI′;t. ❚❤✐s ✐♥❡q✉❛❧✐t② s❤♦✇s t❤❛t t❤❡ s❝❛❧❛r ♣r♦❞✉❝t ♦❢ L2 ([0, t] × O) ❡①t❡♥❞s t♦ ❛ ❞✉❛❧✐t② r❡❧❛t✐♦♥ ❢♦r t❤❡ s♣❛❝❡s LI;t ❛♥❞ LI′;t.
✵✲✶✵
SLIDE 12
- ◆♦✇ ❧❡t ✉s r❡❝❛❧❧ t❤❛t t❤❡ ❙♦❜♦❧❡✈ ✐♥❡q✉❛❧✐t② st❛t❡s t❤❛t
u2∗ ≤ cS ∇u2, ❢♦r ❡❛❝❤ u ∈ H1
0 (O) , ✇❤❡r❡ cS > 0 ✐s ❛ ❝♦♥st❛♥t t❤❛t ❞❡♣❡♥❞s ♦♥ t❤❡ ❞✐♠❡♥s✐♦♥ ❛♥❞ 2∗ = 2d d−2
✐❢ d > 2, ✇❤✐❧❡ 2∗ ♠❛② ❜❡ ❛♥② ♥✉♠❜❡r ✐♥ ]2, ∞[ ✐❢ d = 2 ❛♥❞ 2∗ = ∞ ✐❢ d = 1.
u2∗,2;t ≤ cS ∇u2,2;t , ❢♦r ❡❛❝❤ t ≥ 0 ❛♥❞ ❡❛❝❤ u ∈ L2
loc (R+; H1 0 (O)) .
✵✲✶✶
SLIDE 13
loc (R+; L2 (O) ) L2 loc (R+; H1 0 (O)) , ♦♥❡ ❤❛s
u2,∞;t ∨ u2∗,2;t ≤ c1
2,∞;t + ∇u2 2,2;t
1
2 ,
✇✐t❤ c1 = cS ∨ 1.
- ❖♥❡ ♣❛rt✐❝✉❧❛r ❝❛s❡ ♦❢ ✐♥t❡r❡st ❢♦r ✉s ✐♥ r❡❧❛t✐♦♥ ✇✐t❤ t❤✐s ✐♥❡q✉❛❧✐t② ✐s ✇❤❡♥ p1 = 2, q1 = ∞
❛♥❞ p2 = 2∗, q2 = 2. ■❢ I = I (2, ∞, 2∗, 2) , t❤❡♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❡t ♦❢ ❛ss♦❝✐❛t❡❞ ❝♦♥❥✉❣❛t❡ ♥✉♠❜❡rs ✐s I′ = I′ (2, ∞, 2∗, 2) = I
2∗ 2∗−1, 2
- , ✇❤❡r❡ ❢♦r d = 1 ✇❡ ♠❛❦❡ t❤❡ ❝♦♥✈❡♥t✐♦♥ t❤❛t
2∗ 2∗−1 = 1.
- ■♥ t❤✐s ♣❛rt✐❝✉❧❛r ❝❛s❡ ✇❡ s❤❛❧❧ ✉s❡ t❤❡ ♥♦t❛t✐♦♥ L#;t := LI;t ❛♥❞ L∗
#;t := LI′;t ❛♥❞ t❤❡ r❡s♣❡❝t✐✈❡
♥♦r♠s ✇✐❧❧ ❜❡ ❞❡♥♦t❡❞ ❜② u#;t := uI;t = u2,∞;t ∨ u2∗,2;t , u∗
#;t := uI′;t .
❚❤✉s ✇❡ ♠❛② ✇r✐t❡ u#;t ≤ c1
2,∞;t + ∇u2 2,2;t
1
2 ,
✭✸✮ ❢♦r ❛♥② u ∈ L∞
loc (R+; L2 (O) ) L2 loc (R+; H1 0 (O)) ❛♥❞ t ≥ 0 ❛♥❞ t❤❡ ❞✉❛❧✐t② ✐♥❡q✉❛❧✐t② ❜❡❝♦♠❡s
t
u (s, x) v (s, x) dxds ≤ u#;t v∗
#;t ,
❢♦r ❛♥② u ∈ L#;t ❛♥❞ v ∈ L∗
#;t.
✵✲✶✷
SLIDE 14 ❍②♣♦t❤❡s❡s ♦♥ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ✿
❋♦r ❛ ❝❡rt❛✐♥ p ≥ 2 ✿
- ξ ∈ Lp(Ω; L∞(O))✳
- ❚❤❡r❡ ❡①✐sts θ ∈ (0, 1) s✉❝❤ t❤❛t f 0∗
θ;T✱
θ;T
1/2 ✱
θ;T
1/2 ❛r❡ ✐♥ Lp (Ω, P) ✳ ⋄ f 0 := f(., ., ., 0, 0) ∈ L2([0, T] × Ω × O) ✭r❡s♣✳ g0 ❛♥❞ h0✮✳
- ❆ss✉♠♣t✐♦♥ ✭❍❉✮ ❧♦❝❛❧ ✐♥t❡❣r❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥s ♦♥ f 0✱ g0 ❛♥❞ h0 ✿
E t
(|f 0
t (x)| + |g0 t (x)|2 + |h0 t|2 )dxdt < ∞
❢♦r ❛♥② ❝♦♠♣❛❝t s❡t K ⊂ O✱ ❛♥❞ ❢♦r ❛♥② t ≥ 0✳
- ❆ss✉♠♣t✐♦♥ ✭❍■✮ ❧♦❝❛❧ ✐♥t❡❣r❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥ ♦♥ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✿
E
|ξ(x)|2dx < ∞ ❢♦r ❛♥② ❝♦♠♣❛❝t s❡t K ⊂ O✳
✵✲✶✸
SLIDE 15
E
∗
#;t
2 +
2
2,2;t +
2
2,2;t
❢♦r ❡❛❝❤ t ≥ 0. ❙♦♠❡t✐♠❡s ✇❡ s❤❛❧❧ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ str♦♥❣❡r ❢♦r♠s ♦❢ t❤❡s❡ ❝♦♥❞✐t✐♦♥s ✿
E
2
2,2;t +
2
2,2;t +
2
2,2;t
❢♦r ❡❛❝❤ t ≥ 0.
- ❆ss✉♠♣t✐♦♥ ✭❍■✷✮ ✐♥t❡❣r❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥ ♦♥ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✿
Eξ2 < ∞.
✵✲✶✹
SLIDE 16 ■✳ ❙♦❧✉t✐♦♥ ♦❢ (E) ✇✐t❤ ❩❡r♦ ❉✐r✐❝❤❧❡t ❝♦♥❞✐t✐♦♥
- ▲✳ ❉❡♥✐s✱ ❆✳ ▼✳ ❛♥❞ ▲✳ ❙t♦✐❝❛ ✿ Lp ❡st✐♠❛t❡s ❢♦r t❤❡ ✉♥✐❢♦r♠ ♥♦r♠ ♦❢ s♦❧✉t✐♦♥s ♦❢ q✉❛s✐❧✐♥❡❛r
❙P❉❊✬s✳ Pr♦❜✳ ❚❤❡♦r✳ ❘❡❧❛t❡❞ ❋✐❡❧❞s ✭✷✵✵✺✮✳ ❚❤❡♦r❡♠ ✶ ❊q✉❛t✐♦♥ (E) ❛❞♠✐ts ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥✱ u ✐♥ H0✳ ❚❤✐s s♦❧✉t✐♦♥ ❤❛s L2(O)✲❝♦♥t✐♥✉♦✉s tr❛❥❡❝t♦r✐❡s ❛♥❞ ✐t s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❡st✐♠❛t❡ ∀t ∈ [0, T] ✿ E up
∞,∞;t ≤ k (p, t) E
∞ +
∗p
θ,t +
∗p/2
θ;t
+
∗p/2
θ;t
✇❤❡r❡ k ✐s ❛ ❢✉♥❝t✐♦♥ ✇❤✐❝❤ ♦♥❧② ❞❡♣❡♥❞s ♦♥ C, α ❛♥❞ β✳
✵✲✶✺
SLIDE 17 ▼❛✐♥ t♦♦❧s ❢♦r t❤❡ ♣r♦♦❢ ✿
▲❡♠♠❛ ✷ ✿ ▲❡t ϕ : R − → R ❜❡ C2 ✇✐t❤ ❜♦✉♥❞❡❞ ❞❡r✐✈❛t✐✈❡s✳ ❚❤❡♥ ❛✳s✳✱ ❢♦r ❛❧❧ t ≥ 0
ϕ(ut(x)) dx + t E(ϕ′(us), us) ds =
ϕ(ξ) dx + t (ϕ′(us), fs) ds −
d
t
ϕ′′(us(x))∂ius(x) gi(s, x) dx ds +
d1
t (ϕ′(us), hj(s)) dBj
s
+1 2
d1
t
ϕ′′(us(x))h2
j(s, x) dx ds .
✵✲✶✻
SLIDE 18
▲❡♠♠❛ ✸ ❋♦r ❛❧❧ l ≥ 2✱ P✲❛❧♠♦st s✉r❡❧②✱ ❢♦r ❛❧❧ t ≥ 0
|ut(x)|l dx + t E (l (us)l−1 sgn(us), us) ds =
|ξ(x)|l dx + l t
sgn(us)|us(x)|l−1f(s, x, us, ∇us) dxds − l(l − 1)
d
t
|us(x)|l−2∂ius(x) gi(s, x, us, ∇us) dx ds + l
d1
t
sgn(us)|ut(x)|l−1hj(s, x, us, ∇us) dxdBj
s
+ l(l − 1) 2
d1
t
|ut(x)|l−2h2
j(s, x, us, ∇us) dx ds .
✇❤❡r❡ E (l (us)l−1 sgn(us), us) = l(l − 1)
d
|us(x)|l−2aij(x) ∂ius(x) ∂jus(x) dx.
✵✲✶✼
SLIDE 19
- ❙♦❜♦❧❡✈✬s ✐♥❡q✉❛❧✐t②✳
- ❊st✐♠❛t❡s ♦❢ t❤❡ st♦❝❤❛st✐❝ ♣❛rt t❤❛♥❦s t♦ t❤❡ t❤❡♦r② ♦❢ ❞♦♠✐♥❛t✐♦♥ ♦❢ ♣r♦❝❡ss❡s✳
- ❆r♦♥s♦♥✲❙❡rr✐♥✬s ■t❡r❛t✐♦♥ ✭♦r ▼♦s❡r✬s s❝❤❡♠❡✮
✵✲✶✽
SLIDE 20
❆ ❝♦♠♣❛r✐s♦♥ ❚❤❡♦r❡♠ ✿
❲❡ ❦❡❡♣ s❛♠❡ ❤②♣♦t❤❡s❡s✳ ❲❡ ❛r❡ ❣✐✈❡♥ ❛♥♦t❤❡r ❢✉♥❝t✐♦♥ ξ′ ∈ Lp(Ω, L∞(O)) ❛♥❞ ❛♥♦t❤❡r ❝♦❡✣❝✐❡♥t ¯ f ✇❤✐❝❤ s❛t✐s❢② t❤❡ s❛♠❡ ❛ss✉♠♣t✐♦♥s ❛s f✳ ❲❡ st✐❧❧ ❝♦♥s✐❞❡r u = U0(ξ, f, g, h) ❛♥❞ ✇❡ s❡t v = U0(ξ′, ¯ f, g, h)✳ ❚❤❡♦r❡♠ ✹ ❆ss✉♠❡ ξ ≥ ξ′ dP ⊗ dx✲❛✳❡✳ ❛♥❞ f(t, w, x, ut(x), ∇ut(x)) ≥ ¯ f(t, w, x, ut(x), ∇ut(x)), dt ⊗ dP ⊗ dx❛✳❡ ❚❤❡♥✱ ❢♦r ❛❧❧ t ∈ [0, T]✱ ut ≥ vt; dP ⊗ dx − a.e. ■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢ ✿ ❆♣♣❧② ■t♦✬s ❢♦r♠✉❧❛ t♦ |(ut − vt)+|2 ❛♥❞ ●r♦♥✇❛❧❧✬s ▲❡♠♠❛✳
✵✲✶✾
SLIDE 21 ■■✳ ▲♦❝❛❧ ❙♦❧✉t✐♦♥ ♦❢ (E) ✇✐t❤ ♥♦♥ ❩❡r♦ ❇♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥
- ▲✳ ❉❡♥✐s✱ ❆✳ ▼✳ ❛♥❞ ▲✳ ❙t♦✐❝❛ ✿ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❛♥❞ ❈♦♠♣❛r✐s♦♥ ❚❤❡♦r❡♠ ❢♦r ◗✉❛s✐❧✐♥❡❛r
❙P❉❊✬s✳ Pr❡♣r✐♥t ✭✷✵✵✼✮✱ s✉❜♠✐tt❡❞✳
- ❲❡ ❝♦♥s✐❞❡r t❤❡ ♣❛r❛❜♦❧✐❝ ❜♦✉♥❞❛r② ✿ Γ = {∂O × [0, T]} {O × (t = 0)}✳
- M✱ ❛♥ R✲✈❛❧✉❡❞ ♣r♦❝❡ss ❣✐✈❡♥ ❜②
∀t ≥ 0, Mt = m + t bs ds + t σs dBs, ✇❤❡r❡ b ✱ σ ❛r❡ ♣r❡❞✐❝t❛❜❧❡ ♣r♦❝❡ss❡s s✉❝❤ t❤❛t ✿ E T |bs|
1 1−θ ds
p < +∞ ❛♥❞ E T |σs|
2 1−θ ds
p < +∞. ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡s❡ ❝♦♥❞✐t✐♦♥s ✐♠♣❧② t❤❛t E(M∗p
θ,T) < +∞✳
- ▲❡t ✉s ♥♦✇ ♣r❡❝✐s❡ t❤❡ s❡♥s❡ ✐♥ ✇❤✐❝❤ ❛ s♦❧✉t✐♦♥ ✐s ❞♦♠✐♥❛t❡❞ ♦♥ t❤❡ ❧❛t❡r❛❧ ❜♦✉♥❞❛r② ✿ ❆ss✉♠❡
t❤❛t v ❜❡❧♦♥❣s t♦ H1
loc(O′) ✇❤❡r❡ O′ ✐s ❛ ❧❛r❣❡r ♦♣❡♥ s❡t s✉❝❤ t❤❛t O ⊂ O′✳ ❚❤❡♥ t❤❡ ❝♦♥❞✐t✐♦♥
v+
|O ∈ H1 0(O) ❡①♣r❡ss❡s t❤❡ ❜♦✉♥❞❛r② r❡❧❛t✐♦♥ v ≤ 0 ♦♥ ∂O✳
❙✐♠✐❧❛r❧②✱
✵✲✷✵
SLIDE 22 ❉❡✜♥✐t✐♦♥ ✺ ✐❢ ❛ ♣r♦❝❡ss u ❜❡❧♦♥❣s t♦ Hloc(O′), t❤❡♥ t❤❡ ❝♦♥❞✐t✐♦♥ u+
|O ∈ H0 ❡♥s✉r❡s t❤❡ ✐♥✲
❡q✉❛❧✐t② u ≤ 0 ♦♥ t❤❡ ❧❛t❡r❛❧ ❜♦✉♥❞❛r② {[0, ∞[×∂O}✳ ❚❤❡♦r❡♠ ✻ ▲❡t u ∈ Uloc(ξ, f, g, h)✳ ❆ss✉♠❡ t❤❛t u ≤ M ♦♥ Γ✱ t❤❡♥ ❢♦r ❛❧❧ t ∈ [0, T] ✿ E
p
∞,∞;t ≤ k (t) E
∞ +
∗p
θ,t +
∗p/2
θ;t
+
∗p/2
θ;t
- ✇❤❡r❡ f M(t, x) = f(t, x, Mt, 0) ✱ gM(t, x) = g(t, x, Mt, 0)✱ hM(t, x) = h(t, x, Mt, 0)✳
❘❡♠❛r❦ ✶ ■❢ M ✐s ❛ ❝♦♥st❛♥t✱ t❤❡♥ ✉s✐♥❣ t❤❡ ▲✐♣s❝❤✐t③ ❝♦♥❞✐t✐♦♥ ♦♥ t❤❡ ❝♦❡✣❝✐❡♥ts✱ ✇❡ ❣❡t ❛♥ ❡st✐♠❛t❡ s✐♠✐❧❛r t♦ t❤❡ ❝❧❛ss✐❝❛❧ ♦♥❡ ❢♦r q✉❛s✐❧✐♥❡❛r P❉❊ ♦❜t❛✐♥❡❞ ❜❛② ❆rr♦♥s♦♥ ❛♥❞ ❙❡rr✐♥ ✳
✵✲✷✶
SLIDE 23 ❙t❡♣ ✶ ✿ ❊st✐♠❛t❡s ❢♦r s♦❧✉t✐♦♥s ✇✐t❤ ♥✉❧❧ ❉✐r✐❝❤❧❡t ❝♦♥❞✐t✐♦♥s ✉♥❞❡r ✇❡❛❦❡r L1✲✐♥t❡❣r❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥ ♦♥ f 0 ❚❤❡♦r❡♠ ✼ ❚❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥ ♦❢ ✭✶✮ ✐♥ H0. ▼♦r❡♦✈❡r✱ t❤✐s s♦❧✉t✐♦♥ ❤❛s ❛ ✈❡rs✐♦♥ ✇✐t❤ L2(O)✲❝♦♥t✐♥✉♦✉s tr❛❥❡❝t♦r✐❡s ❛♥❞ ✐t s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❡st✐♠❛t❡s E
2,∞;t + ∇u2 2,2;t
2 +
∗
#;t
2 +
2
2,2;t +
2
2,2;t
❢♦r ❡❛❝❤ t ≥ 0, ✇❤❡r❡ k (t) ✐s ❛ ❝♦♥st❛♥t t❤❛t ♦♥❧② ❞❡♣❡♥❞s ♦♥ t❤❡ str✉❝t✉r❡ ❝♦♥st❛♥ts ❛♥❞ t. Pr♦♦❢ ✿ ❲❡ st❛rt ❜② ✇r✐t✐♥❣ ■t♦✬s ❢♦r♠✉❧❛ ❢♦r t❤❡ s♦❧✉t✐♦♥ ✐♥ t❤❡ ❢♦r♠ ut2
2 + 2
t E (us, us) ds = ξ2
2 + 2
t (us, fs (us, ∇us)) ds − 2 t
d
(∂ius, gi,s (us, ∇us)) ds + t hs (us, ∇us)2
2 ds
+ 2
d1
t (us, hj,s (us, ∇us)) dBj
s,
✭✹✮
✵✲✷✷
SLIDE 24 ❡q✉❛❧✐t② ✇❤✐❝❤ ❤♦❧❞s ❛✳s✳ ❚❤❡ ▲✐♣s❝❤✐t③ ❝♦♥❞✐t✐♦♥ ❛♥❞ t❤❡ ✐♥❡q✉❛❧✐t② ✭✷✮ ❧❡❛❞ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡st✐♠❛t❡ t (us, fs (us, ∇us)) ds ≤ ε ∇u2
2,2;t + cε u2 2,2;t + δ u2 #;t + cδ
∗
#;t
2 , ✇❤❡r❡ ε, δ > 0 ❛r❡ t✇♦ s♠❛❧❧ ♣❛r❛♠❡t❡rs t♦ ❜❡ ❝❤♦s❡♥ ❧❛t❡r ❛♥❞ cε, cδ ❛r❡ ❝♦♥st❛♥ts ❞❡♣❡♥❞✐♥❣ ♦❢ t❤❡♠✳ ❙✐♠✐❧❛r ❡st✐♠❛t❡s ❤♦❧❞ ❢♦r t❤❡ ♥❡①t t✇♦ t❡r♠s − t
d
(∂ius, gi,s (us, ∇us)) ds ≤ (α + ε) ∇u2
2,2;t + cε u2 2,2;t + cε
2
2,2;t ,
t hs (us, ∇us)2
2 ds ≤
2,2;t + cε u2 2,2;t + cε
2
2,2;t .
❈♦r♦❧❧❛r② ✽ ▲❡t ✉s ❛ss✉♠❡ t❤❡ ❤②♣♦t❤❡s❡s ♦❢ t❤❡ ♣r❡❝❡❞✐♥❣ ❚❤❡♦r❡♠ ✇✐t❤ t❤❡ s❛♠❡ ♥♦t❛t✐♦♥s✳ ▲❡t ϕ : R → R ❜❡ ❛ ❢✉♥❝t✐♦♥ ♦❢ ❝❧❛ss C2 ❛♥❞ ❛ss✉♠❡ t❤❛t ϕ′′ ✐s ❜♦✉♥❞❡❞ ❛♥❞ ϕ′ (0) = 0. ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥ ❤♦❧❞s ❛✳s✳ ❢♦r ❛❧❧ t ≥ 0 ✿
ϕ (ut (x)) dx + t E (ϕ′ (us) , us) ds =
ϕ (ξ (x)) dx + t (ϕ′ (us) , fs(us, ∇us) ds
✵✲✷✸
SLIDE 25 − t
d
(∂i (ϕ′ (us)) , gi,s(us, ∇us) ds + 1 2 t
- ϕ′′ (us) , |hs(us, ∇us)|2
ds +
d1
t (ϕ′ (us) , hj,s(us, ∇us)) dBj
s.
✵✲✷✹
SLIDE 26 ❙t❡♣ ✷ ✿ ❊st✐♠❛t❡s ♦❢ t❤❡ ♣♦s✐t✐✈❡ ♣❛rt ♦❢ t❤❡ s♦❧✉t✐♦♥ ❲❡ ♥❡①t ♣r♦✈❡ ❛♥ ❡st✐♠❛t❡ ❢♦r t❤❡ ♣♦s✐t✐✈❡ ♣❛rt u+ ♦❢ t❤❡ s♦❧✉t✐♦♥ u = U (ξ, f, g, h) . ❋♦r t❤✐s ✇❡ ♥❡❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦t❛t✐♦♥ ✿ f u,0 = 1{u>0}f 0, gu,0 = 1{u>0}g0, hu,0 = 1{u>0}h0, f u = f − f 0 + f u,0, gu = g − g0 + gu,0, hu = h − h0 + hu,0 f u,0+ = 1{u>0}
❚❤❡♦r❡♠ ✾ ❚❤❡ ♣♦s✐t✐✈❡ ♣❛rt ♦❢ t❤❡ s♦❧✉t✐♦♥ s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❡st✐♠❛t❡ E
2
2,∞;t +
2
2,2;t
2
2 +
∗
#;t
2 +
2
2,2;t +
2
2,2;t
Pr♦♦❢ ✿ ❚❤❡ ✐❞❡❛ ✐s t♦ ❛♣♣❧② ■t♦✬s ❢♦r♠✉❧❛ t♦ t❤❡ ❢✉♥❝t✐♦♥ ψ ❞❡✜♥❡❞ ❜② ψ (y) = (y+)2 , ❢♦r ❛♥② y ∈ R.
- ❙✐♥❝❡ t❤✐s ❢✉♥❝t✐♦♥ ✐s ♥♦t ♦❢ t❤❡ ❝❧❛ss C2 ✇❡ s❤❛❧❧ ♠❛❦❡ ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ❛s ❢♦❧❧♦✇s✳ ▲❡t ϕ ❜❡
❛ C∞ ❢✉♥❝t✐♦♥ s✉❝❤ t❤❛t ϕ (y) = 0 ❢♦r ❛♥② y ∈] − ∞, 1] ❛♥❞ ϕ (y) = 1 ❢♦r ❛♥② y ∈ [2, ∞[. ❲❡ s❡t ψn (y) = y2ϕ (ny) , ❢♦r ❡❛❝❤ y ∈ R ❛♥❞ ❛❧❧ n ∈ N∗. ■t ✐s ❡❛s② t♦ ✈❡r✐❢② t❤❛t (ψn)n∈N∗ ❝♦♥✈❡r❣❡s ✉♥✐❢♦r♠❧② t♦ t❤❡ ❢✉♥❝t✐♦♥ ψ ❛♥❞ t❤❛t lim
n→∞ ψ′ n (y) = 2y+, lim n∞ ψ′′ n (y) = 2 · 1{y>0},
✵✲✷✺
SLIDE 27 ❢♦r ❛♥② y ∈ R. ▼♦r❡♦✈❡r ✇❡ ❤❛✈❡ t❤❡ ❡st✐♠❛t❡s 0 ≤ ψn (y) ≤ ψ (y) , 0 ≤ ψ′ (y) ≤ Cy, |ψ′′
n (y)| ≤ C,
❢♦r ❛♥② y ≥ 0 ❛♥❞ ❛❧❧ n ∈ N∗, ✇❤❡r❡ C ✐s ❛ ❝♦♥st❛♥t✳ ❙t❡♣ ✸ ✿ ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ■t♦✬s ❢♦r♠✉❧❛ ❚❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡♦r❡♠ r❡♣r❡s❡♥ts ❛ ❦❡② t❡❝❤♥✐❝❛❧ r❡s✉❧t ✇❤✐❝❤ ❧❡❛❞s t♦ ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❡st✐♠❛t❡s ♦❢ t❤❡ ♣♦s✐t✐✈❡ ♣❛rt✳ ▲❡t u ∈ Uloc (ξ, f, g, h) ❜❡ ❛ s♦❧✉t✐♦♥ ❛♥❞ u+ ✐ts ♣♦s✐t✐✈❡ ♣❛rt ❛♥❞ s❡t f u,0 = 1{u>0}f 0, gu,0 = 1{u>0}g0, hu,0 = 1{u>0}h0, f u,0+ = 1{u>0}
✵✲✷✻
SLIDE 28 ❚❤❡♦r❡♠ ✶✵ ❆ss✉♠❡ t❤❛t u+ ❜❡❧♦♥❣s t♦ H ❛♥❞ ❛ss✉♠❡ t❤❛t t❤❡ ❞❛t❛ s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥t❡❣r❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥s E
2
2 < ∞, E
∗
#;t
2 < ∞, E
2
2,2;t < ∞, E
2
2,2;t < ∞,
❢♦r ❡❛❝❤ t ≥ 0. ▲❡t ϕ : R → R ❜❡ ❛ ❢✉♥❝t✐♦♥ ♦❢ ❝❧❛ss C2, ✇❤✐❝❤ ❛❞♠✐ts ❛ ❜♦✉♥❞❡❞ s❡❝♦♥❞ ♦r❞❡r ❞❡r✐✈❛t✐✈❡ ❛♥❞ s✉❝❤ t❤❛t ϕ′ (0) = 0. ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥ ❤♦❧❞s✱ ❛✳s✳✱ ❢♦r ❡❛❝❤ t ≥ 0,
ϕ
t (x)
t E
u+
s
s
ϕ
t
u+
s
s , ∇u+ s
− t
d
u+
s
s , gi,s
s , ∇u+ s
2 t
u+
s
s , ∇u+ s
ds +
d1
t
u+
s
s , ∇u+ s
s.
✵✲✷✼
SLIDE 29 ❙t❡♣ ✹ ✿ ❊st✐♠❛t❡s ❈♦r♦❧❧❛r② ✶✶ ❯♥❞❡r t❤❡ ❤②♣♦t❤❡s❡s ♦❢ t❤❡ ❛❜♦✈❡ t❤❡♦r❡♠ ✇✐t❤ s❛♠❡ ♥♦t❛t✐♦♥s✱ ♦♥❡ ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ❡st✐♠❛t❡s E
2
2,∞;t +
2
2,2;t
2
2 +
∗
#;t
2 +
2
2,2;t +
2
2,2;t
❆s ❛ ❈♦♥s❡q✉❡♥❝❡ ✿ ▼♦r❡ ❣❡♥❡r❛❧ ❝♦♠♣❛r✐s♦♥ ❚❤❡♦r❡♠
❚❤❡♦r❡♠ ✶✷ ❆ss✉♠❡ t❤❛t f 1, f
2 ❛r❡ t✇♦ ❢✉♥❝t✐♦♥s s✐♠✐❧❛r t♦ f ✇❤✐❝❤ s❛t✐s❢② t❤❡ ▲✐♣s❝❤✐t③
❝♦♥❞✐t✐♦♥ ❛♥❞ s✉❝❤ t❤❛t ❜♦t❤ tr✐♣❧❡s (f 1, g, h) ❛♥❞ (f 2, g, h) s❛t✐s❢② ♦✉r ❛ss✉♠♣t✐♦♥s✳ ❆ss✉♠❡ t❤❛t ξ1, ξ2 ❛r❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s s✐♠✐❧❛r t♦ ξ✳ ▲❡t ui ∈ Uloc (ξi, f i, g, h) , i = 1, 2 ❛♥❞ s✉♣♣♦s❡ t❤❛t t❤❡ ♣r♦❝❡ss (u1 − u2)+ ❜❡❧♦♥❣s t♦ H0 ❛♥❞ t❤❛t ♦♥❡ ❤❛s E
u2, ∇u2 − f 2 u2, ∇u2 ∗
#;t
2 < ∞, ❢♦r ❡❛❝❤ t ≥ 0. ■❢ ξ1 ≤ ξ2 ❛✳s✳ ❛♥❞ f 1 (u2, ∇u2) ≤ f 2 (u2, ∇u2) ❛✳s✳✱ t❤❡♥ ♦♥❡ ❤❛s u1 ≤ u2 ❛✳s✳
✵✲✷✽
SLIDE 30 ■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢ ✿ ❚❤❡ ❞✐✛❡r❡♥❝❡ v = u1 − u2 ❜❡❧♦♥❣s t♦ Uloc
ξ = ξ1 − ξ2, f (t, ω, x, y, z) = f 1 t, ω, x, y + u2
t (x) , z + ∇u2 t (x)
t, ω, x, u2
t (x) , ∇u2 t (x)
g (t, ω, x, y, z) = g
t (x) , z + ∇u2 t (x)
t (x) , ∇u2 t (x)
h (t, ω, x, y, z) = h
t (x) , z + ∇u2 t (x)
t (x) , ∇u2 t (x)
❚❤❡ r❡s✉❧t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ♣r❡❝❡❞✐♥❣ ❝♦r♦❧❧❛r②✱ s✐♥❝❡ ξ ≤ 0 ❛♥❞ f
0 ≤ 0 ❛♥❞ g0 = h 0 = 0. ✷
✵✲✷✾
SLIDE 31 ❘❡❢❡r❡♥❝❡s ✿
■✮ ❈❛s❡ ✇✐t❤♦✉t ❞✐✈❡r❣❡♥❝❡ t❡r♠ ✭ ✐✳❡✳ g = 0✮
- P❛r❞♦✉① ❛♥❞ P❡♥❣ ✿ ❈❧❛ss✐❝❛❧ s♦❧✉t✐♦♥s ♦❢ ❙P❉❊✬s ❛♥❞ t❤❡ ❧✐♥❦ ✇✐t❤ ❇❛❝❦✇❛r❞ ❉♦✉❜❧② ❙❉❊✬s
✭s♠♦♦t❤ ❝♦❡✣❝✐❡♥ts✮✳ Pr♦❜✳ ❚❤❡♦r✳ ❘✳ ❋✐❡❧❞s ✭✶✾✾✹✮
- ❱✳ ❇❛❧❧② ❛♥❞ ❆✳ ▼✳ ✿ ❙♦❜♦❧❡✈ s♦❧✉t✐♦♥s ♦❢ s❡♠✐❧✐♥❡❛r ❙P❉❊✳ ❏✳ ❚❤❡♦r✳ Pr♦❜✳ ✭✵✶✮✳
- ❆✳ ▼✳ ✫ ❙❝❤❡✉t③♦✇ ✿ ❙♦❜♦❧❡✈ s♦❧✉t✐♦♥s ✇✐t❤ ❑✉♥✐t❛✲♥♦✐s❡✳ ❏✳ ❚❤❡♦r✳ Pr♦❜✳ ✭✵✷✮✳
- ❘♦③❦♦s③ ✿ ❇❙❉❊✬s ❛♥❞ P❉❊✬s ✐♥ ❞✐✈❡r❣❡♥❝❡ ❢♦r♠✳ Pr♦❜✳ ❚❤❡♦r✳ ❘✳ ❋✐❡❧❞s ✭✵✸✮✳
- ▲✳ ❉❡♥✐s ✿ ❙♦❧✉t✐♦♥s ♦❢ ❙P❉❊✬s ❝♦♥s✐❞❡r❡❞ ❛s ❉✐r✐❝❤❧❡t ♣r♦❝❡ss❡s✳ ❇❡r♥♦✉❧❧✐ ✭✵✹✮✳
■■✮ ❈❛s❡ ✇✐t❤ ❞✐✈❡r❣❡♥❝❡ t❡r♠ ✿
- ●②¨
- ♥❣② ❛♥❞ ❘♦✈✐r❛ ✿ Lp✲♥♦r♠ ❡st✐♠❛t❡s ❢♦r t❤❡ s♦❧✉t✐♦♥s ♦❢ ❙P❉❊✬s ✭♥♦✲❣r❛❞✐❡♥t ❞❡♣❡♥❞❡♥❝❡
✐♥ t❤❡ ❝♦❡✣❝✐❡♥ts✮✳ ❙P❆ ✭✵✵✮✳
- ▲✳ ❉❡♥✐s ❛♥❞ ▲✳ ❙t♦✐❝❛ ✿ ❆ ❣❡♥❡r❛❧ ❛♥❛❧②t✐❝❛❧ r❡s✉❧t ❢♦r ♥♦♥✲❧✐♥❡❛r
❙P❉❊✬s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✳ ❊❏P ✭✵✹✮✳
- ▲✳ ❉❡♥✐s✱ ❆✳ ▼✳ ❛♥❞ ▲✳ ❙t♦✐❝❛ ✿ Lp ❡st✐♠❛t❡s ❢♦r t❤❡ ✉♥✐❢♦r♠ ♥♦r♠ ♦❢ s♦❧✉t✐♦♥s ♦❢ q✉❛s✐❧✐♥❡❛r
❙P❉❊✬s✳ Pr♦❜✳ ❚❤❡♦r✳ ❘✳ ❋✐❡❧❞s ✭✷✵✵✺✮✳
✵✲✸✵
