❏♦✉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✱ ❘♦✉♠❛♥✐❡✮ ✵✲✵
Pr♦❜❧❡♠ ✿ ❲❡ st✉❞② t❤❡ ❢♦❧❧♦✇✐♥❣ st♦❝❤❛st✐❝ ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ✭✐♥ s❤♦rt ❙P❉❊✮ ❢♦r ❛ r❡❛❧ ✲✈❛❧✉❡❞ r❛♥❞♦♠ ✜❡❧❞ u t ( x ) := u ( t, x ) , d � du t ( x ) = Lu t ( x ) dt + f t ( x, u t ( x ) , ∇ u t ( x )) dt + ∂ i g i,t ( x, u t ( x ) , ∇ u t ( x )) dt i =1 d 1 � h j,t ( x, u t ( x ) , ∇ u t ( x )) dB j + ✭✶✮ t j =1 ✇✐t❤ ❛ ❣✐✈❡♥ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ u 0 = ξ, ✇❤❡r❡ L ✐s ❛ s②♠♠❡tr✐❝ s❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦r ❞❡✜♥❡❞ ✐♥ s♦♠❡ ❜♦✉♥❞❡❞ ♦♣❡♥ ❞♦♠❛✐♥ O ⊂ R d ❛♥❞ f, g i , i = 1 , ..., d, h j , j = 1 , ..., d 1 ❛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✐♦♥✳ ✵✲✶
❚❤❡ ♠❛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♠ ∂ t u = div A ( t, x, u, ∇ u ) + B ( t, x, u, ∇ u ) ✐♥ t❤❡ ❜♦✉♥❞❡❞ ❝②❧✐♥❞❡r ]0 , T [ ×O ⊂ R d +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♦♠❡ L p ❛ ♣r✐♦r✐ ❡st✐♠❛t❡s ❢♦r t❤❡ ✉♥✐❢♦r♠ ♥♦r♠ ♦❢ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ st♦❝❤❛st✐❝ q✉❛s✐❧✐♥❡❛r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥✳ ✵✲✷
❲❡ ♣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 ] ✿ � p � � ∗ p � ∗ p/ 2 � ∗ p/ 2 � � ( u − M ) + � � � ( ξ − M ) + � p � � ( f 0 ,M ) + � � | g 0 ,M | 2 � � � � | h 0 ,M | 2 � E ∞ , ∞ ; t ≤ k ( p, t ) E ∞ + θ,t + + θ ; t θ ; t ✇❤❡r❡ ✿ f 0 ,M ( t, x ) = f ( t, x, M, 0) , g 0 ,M ( t, x ) = g ( t, x, M, 0) , h 0 ,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 ❞❡✜♥❡❞ ❜❡❧♦✇✳ ✵✲✸
❍②♣♦t❤❡s✐s ❛♥❞ ❞❡✜♥✐t✐♦♥s ✿ ⋄ O ⊂ R d ♦♣❡♥ ❜♦✉♥❞❡❞ s❡t✳ ⋄ ( B t ) t d 1 ✲❞✐♠❡♥s✐♦♥❛❧ ❇▼ ❞❡✜♥❡❞ ♦♥ (Ω , F , ( F t ) t , P ) ✱ ⋄ A := − L := − � ∂ i ( a i,j ∂ j ) ✿ s②♠♠❡tr✐❝ s❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦r✱ ⋄ a := ( a ij ) i,j ✐s ❛ ♠❡❛s✉r❛❜❧❡ ❛♥❞ s②♠♠❡tr✐❝ ♠❛tr✐① ❛♥❞ s❛t✐s✜❡s ✉♥✐❢♦r♠ ❡❧❧✐♣t✐❝✐t② ✿ d λ | ζ | 2 ≤ a i,j ( x ) ζ i ζ j ≤ Λ | ζ | 2 , � ∀ x ∈ O , ζ ∈ R d i,j =1 ✇❤❡r❡ λ ❛♥❞ Λ ❛r❡ ♣♦s✐t✐✈❡ ❝♦♥st❛♥ts✳ ⋄ ξ ∈ L 2 (Ω × O ) ✳ ⋄ T > 0 ✳ ✵✲✹
❲❡ ❛r❡ ❣✐✈❡♥ ♣r❡❞✐❝t❛❜❧❡ ❢✉♥❝t✐♦♥s ✿ R + × Ω × O × R × R d → R , f : R + × Ω × O × R × R d → R d 1 h : g d ) : R + × Ω × O × R × R d → R d . g = (¯ g 1 , ..., ¯ s✉❝❤ t❤❛t ✿ ✶✳ | f ( t, ω, x, y, z ) − f ( t, ω, x, y ′ , z ′ ) | ≤ C ( | y − y ′ | + | z − z ′ | ) j =1 | h j ( t, ω, x, y, z ) − h j ( t, ω, x, y ′ , z ′ ) | 2 � 1 2 ≤ C | y − y ′ | + β | z − z ′ | ✱ � � d 1 ✷✳ i =1 | g i ( t, ω, x, y, z ) − g i ( t, ω, x, y ′ , z ′ ) | 2 � 1 2 ≤ C | y − y ′ | + α | z − z ′ | ✱ � � d ✸✳ ✇❤❡r❡ C, α, β ❛r❡ ♥♦♥ ♥❡❣❛t✐✈❡ ❝♦♥st❛♥ts✳ ❈♦♥tr❛❝t✐♦♥ ❤②♣♦t❤❡s✐s ✿ α + 1 2 β 2 < λ. ✵✲✺
❲❡❛❦ s♦❧✉t✐♦♥s ♦❢ ❙P❉❊✬s ✿ • H 0 ✿ s❡t ♦❢ H 1 0 ( O ) ✲✈❛❧✉❡❞ ♣r❡❞✐❝t❛❜❧❡ ♣r♦❝❡ss❡s u s✳t✳ � T � 1 / 2 � � u t � 2 + � u � E,T := E sup E E ( u t , u t ) dt < ∞ . 0 ≤ t ≤ T 0 ✇❤❡r❡ E ✐s t❤❡ ❡♥❡r❣② ✭❉✐r✐❝❤❧❡t ❢♦r♠ ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦r A ✮ ✿ d � � a i,j ∂ i u ∂ j v dx, ∀ u ∈ H 1 loc ( O ) , ∀ v ∈ H 1 E ( u, v ) := 0 ( O ) . O i,j =1 • H loc ✿ s❡t ♦❢ H 1 loc ( O ) ✲✈❛❧✉❡❞ ♣r❡❞✐❝t❛❜❧❡ ♣r♦❝❡ss❡s s✉❝❤ t❤❛t ❢♦r ❛♥② ❝♦♠♣❛❝t s✉❜s❡t K ✐♥ O ✿ � T � 1 / 2 � � � u t ( x ) 2 dx + E |∇ u t ( x ) | 2 dxdt � u � E,K,T := E sup < ∞ . 0 ≤ t ≤ T K 0 K ✵✲✻
❉❡✜♥✐t✐♦♥ ✿ u ∈ H loc ✐s ❛ ✇❡❛❦ s♦❧✉t✐♦♥ ♦❢ ( E ) ✱ ✇✐t❤ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ u 0 = ξ ✱ ✐❢ ❢♦r ❡❛❝❤ t❡st ❢✉♥❝t✐♦♥ ϕ ∈ D := C ∞ c ([0 , T )) ⊗ C 2 c ( O ) . � T [( u s , ∂ s ϕ ) − E ( u s , ϕ s ) +( f ( s, u s , ∇ u s ) , ϕ s ) − ( g i ( s, u s , ∇ u s ) , ∂ i ϕ s )] ds 0 � T ( h j ( u s , ∇ u s ) , ϕ s ) dB j + s + ( ξ, ϕ 0 ) = 0 . 0 ✇❤❡r❡ ( , ) ✐s t❤❡ ✐♥♥❡r ♣r♦❞✉❝t ✐♥ L 2 ( O ) ✳ ❲❡ ❞❡♥♦t❡ ❜② U loc ( ξ, f, g, h ) t❤❡ s❡t ♦❢ s✉❝❤ s♦❧✉t✐♦♥✳ ■❢ u ∈ H 0 ✐s ❛ ✇❡❛❦ s♦❧✉t✐♦♥✱ ✇❡ s❛② t❤❛t ✐t s♦❧✈❡s ( E ) ✇✐t❤ ③❡r♦ ❉✐r✐❝❤❧❡t ❝♦♥❞✐t✐♦♥ ♦♥ ∂ O ❛♥❞ ✇❡ ❞❡♥♦t❡ u = U 0 ( ξ, f, g, h ) ✳ ✵✲✼
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