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▼❡t❤♦❞ ♦❢ ❝❤❛r❛❝t❡r✐st✐❝s✿ t❤❡♦r❡t✐❝❛❧ ❛♥❞ ♥✉♠❡r✐❝❛❧ ❛♣♣❧✐❝❛t✐♦♥s ❨♦❤❛♥ P❡♥❡❧ ✶ ✶ ❚❡❛♠ ❆◆●❊ ✭❈❊❘❊▼❆ ✕ ■♥r✐❛ ✕ ❯P▼❈ ✕ ❈◆❘❙✮ ■♥ ❝♦❧❧❛❜♦r❛t✐♦♥ ✇✐t❤ ❙✳ ❉❡❧❧❛❝❤❡r✐❡ ✭❈❊❆ ❙❛❝❧❛② ✫ P♦❧②t❡❝❤♥✐q✉❡ ▼♦♥tré❛❧✮✱ ❏✳ ❏✉♥❣ ✭P❛✉✮ ●✳ ❋❛❝❝❛♥♦♥✐ ✭❚♦✉❧♦♥✮ ✫ ❇✳ ●r❡❝ ✭P❛r✐s ✺✮ ❈❊▼❘❆❈❙ ✷✵✶✺ ❈■❘▼ ✕ ✷✽ ❥✉✐❧❧❡t ✷✵✶✺

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ❖✉t❧✐♥❡ Pr❡s❡♥t❛t✐♦♥ ♦❢ Pr♦❥❡❝t ❉✐P▲♦▼❛ ✶ ■♥tr♦❞✉❝t✐♦♥ t♦ t❤❡ ♠❡t❤♦❞ ♦❢ ❝❤❛r❛❝t❡r✐st✐❝s ✷ ❘❡s✉❧t✐♥❣ ❢❛♠✐❧② ♦❢ ♥✉♠❡r✐❝❛❧ s❝❤❡♠❡s ✸ ❊①t❡♥s✐♦♥s ♦❢ t❤❡ ▼❖❈ s❝❤❡♠❡s ✹ ❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✷ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s Pr❡s❡♥t❛t✐♦♥ ♦❢ Pr♦❥❡❝t ❉✐P▲♦▼❛ ❉✐P▲♦▼❛ ♦r✐❣✐♥s ❖♥❝❡ ✉♣♦♥ ❛ t✐♠❡ ✐♥ ✷✵✶✶✳✳✳ t❤❡ ✏❜❛s♠❛❝✑ ♣r♦❥❡❝t ✇❛s ♣r♦♣♦s❡❞ t♦ t❤❡ ✷✵✶✶ ❈❊▼❘❆❈❙ s❡ss✐♦♥✳ ■t ✇❛s t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ ❛♥ ❛♠❛③✐♥❣ r❡s❡❛r❝❤ ❛❝t✐✈✐t②✳ ■t ✇❛s ❛✐♠❡❞ ❛t st✉❞②✐♥❣ t❤❡ ❧♦✇ ▼❛❝❤ ♥✉♠❜❡r r❡❣✐♠❡✳ ❙♦ ✇❛s ❜♦r♥ t❤❡ ❈❉▼❆❚❍ ❣r♦✉♣ ✳ ❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✸ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s Pr❡s❡♥t❛t✐♦♥ ♦❢ Pr♦❥❡❝t ❉✐P▲♦▼❛ ▼♦❞❡❧s ❛♥❞ ♥✉♠❡r✐❝❛❧ ❛s♣❡❝ts ▲▼◆❈ ♠♦❞❡❧ ✭✈❡rs✐♦♥ ✷✵✶✶✮  ∇· ✉ = β ( ❤ , ♣ ✵ ) Φ , ✭✶❛✮    ♣ ✵   � � ρ ( ❤ , ♣ ✵ ) × ∂ t ❤ + ✉ · ∇ ❤ = Φ , ✭✶❜✮      � � ρ ( ❤ , ♣ ✵ ) × ∂ t ✉ + ( ✉ · ∇ ) ✉ − ∇· σ ( ✉ ) + ∇ ¯ ♣ = ρ ( ❤ , ♣ ✵ ) ❣ . ✭✶❝✮ ❊q✉❛t✐♦♥ ♦❢ st❛t❡ ✿ st✐✛❡♥❡❞ ❣❛s ❧❛✇ ❢♦r ❛ ♠♦♥♦♣❤❛s✐❝ ✢✉✐❞ γ ♣ ✵ + π ρ ( ❤ , ♣ ✵ ) = γ − ✶ ❤ − q ❉✐♠❡♥s✐♦♥ ✿ ✶ ◆✉♠❡r✐❝❛❧ s❝❤❡♠❡ ✿ ▼❖❈ ✭▼❛t❧❛❜✮ ❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✹ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s Pr❡s❡♥t❛t✐♦♥ ♦❢ Pr♦❥❡❝t ❉✐P▲♦▼❛ ▼♦❞❡❧s ❛♥❞ ♥✉♠❡r✐❝❛❧ ❛s♣❡❝ts ▲▼◆❈ ♠♦❞❡❧ ✭✈❡rs✐♦♥ ✷✵✶✷✮  ∇· ✉ = β ( ❤ , ♣ ✵ ) Φ , ✭✶❛✮    ♣ ✵   � � ρ ( ❤ , ♣ ✵ ) × ∂ t ❤ + ✉ · ∇ ❤ = Φ , ✭✶❜✮      � � ρ ( ❤ , ♣ ✵ ) × ∂ t ✉ + ( ✉ · ∇ ) ✉ − ∇· σ ( ✉ ) + ∇ ¯ ♣ = ρ ( ❤ , ♣ ✵ ) ❣ . ✭✶❝✮ ❊q✉❛t✐♦♥ ♦❢ st❛t❡ ✿ st✐✛❡♥❡❞ ❣❛s ❧❛✇ ✇✐t❤ ♣❤❛s❡ ❝❤❛♥❣❡ γ ( ❤ ) ♣ ✵ + π ( ❤ ) ρ ( ❤ , ♣ ✵ ) = γ ( ❤ ) − ✶ ❤ − q ( ❤ ) ❉✐♠❡♥s✐♦♥ ✿ ✶ ◆✉♠❡r✐❝❛❧ s❝❤❡♠❡ ✿ ■◆❚▼❖❈ ✭❋♦rtr❛♥✮ ❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✹ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s Pr❡s❡♥t❛t✐♦♥ ♦❢ Pr♦❥❡❝t ❉✐P▲♦▼❛ ▼♦❞❡❧s ❛♥❞ ♥✉♠❡r✐❝❛❧ ❛s♣❡❝ts ▲▼◆❈ ♠♦❞❡❧ ✭✈❡rs✐♦♥ ✷✵✶✸✮  ∇· ✉ = β ( ❤ , ♣ ✵ ) Φ , ✭✶❛✮    ♣ ✵   � � ρ ( ❤ , ♣ ✵ ) × ∂ t ❤ + ✉ · ∇ ❤ = Φ , ✭✶❜✮      � � ρ ( ❤ , ♣ ✵ ) × ∂ t ✉ + ( ✉ · ∇ ) ✉ − ∇· σ ( ✉ ) + ∇ ¯ ♣ = ρ ( ❤ , ♣ ✵ ) ❣ . ✭✶❝✮ ❊q✉❛t✐♦♥ ♦❢ st❛t❡ ✿ t❛❜✉❧❛t❡❞ ❧❛✇ ρ ∈ R ✼ [ ❤ ] ❉✐♠❡♥s✐♦♥ ✿ ✶ ◆✉♠❡r✐❝❛❧ s❝❤❡♠❡ ✿ ▼❖❈ ✭❋♦rtr❛♥✮ ❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✹ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s Pr❡s❡♥t❛t✐♦♥ ♦❢ Pr♦❥❡❝t ❉✐P▲♦▼❛ ▼♦❞❡❧s ❛♥❞ ♥✉♠❡r✐❝❛❧ ❛s♣❡❝ts ▲▼◆❈ ♠♦❞❡❧ ✭✈❡rs✐♦♥ ✷✵✶✹✮  ∇· ✉ = β ( ❤ , ♣ ✵ ) Φ , ✭✶❛✮    ♣ ✵   � � ρ ( ❤ , ♣ ✵ ) × ∂ t ❤ + ✉ · ∇ ❤ = Φ , ✭✶❜✮      � � ρ ( ❤ , ♣ ✵ ) × ∂ t ✉ + ( ✉ · ∇ ) ✉ − ∇· σ ( ✉ ) + ∇ ¯ ♣ = ρ ( ❤ , ♣ ✵ ) ❣ . ✭✶❝✮ ❊q✉❛t✐♦♥ ♦❢ st❛t❡ ✿ st✐✛❡♥❡❞ ❣❛s✴t❛❜✉❧❛t❡❞ ❧❛✇ γ ( ❤ ) ♣ ✵ + π ( ❤ ) ρ ( ❤ , ♣ ✵ ) = / ρ ∈ R ✼ [ ❤ ] γ ( ❤ ) − ✶ ❤ − q ( ❤ ) ❉✐♠❡♥s✐♦♥ ✿ ✷ ◆✉♠❡r✐❝❛❧ s❝❤❡♠❡ ✿ ❋r❡❡❋❡♠✰✰ ✭❝♦♥✈❡❝t✮ ❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✹ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s Pr❡s❡♥t❛t✐♦♥ ♦❢ Pr♦❥❡❝t ❉✐P▲♦▼❛ ▼♦❞❡❧s ❛♥❞ ♥✉♠❡r✐❝❛❧ ❛s♣❡❝ts ▲▼◆❈ ♠♦❞❡❧ ✭✈❡rs✐♦♥ ✷✵✶✺✮  ∇· ✉ = β ( ❤ , ♣ ✵ ) � � �� Φ + ∇· λ ( ❤ , ♣ ✵ ) ∇ ❚ ( ❤ , ♣ ✵ ) , ✭✶❛✮    ♣ ✵   � � � � ρ ( ❤ , ♣ ✵ ) × ∂ t ❤ + ✉ · ∇ ❤ = Φ + ∇· λ ( ❤ , ♣ ✵ ) ∇ ❚ ( ❤ , ♣ ✵ ) , ✭✶❜✮      � � ρ ( ❤ , ♣ ✵ ) × ∂ t ✉ + ( ✉ · ∇ ) ✉ − ∇· σ ( ✉ ) + ∇ ¯ ♣ = ρ ( ❤ , ♣ ✵ ) ❣ . ✭✶❝✮ ❊q✉❛t✐♦♥ ♦❢ st❛t❡ ✿ st✐✛❡♥❡❞ ❣❛s✴t❛❜✉❧❛t❡❞ ❧❛✇ γ ( ❤ ) ♣ ✵ + π ( ❤ ) ρ ( ❤ , ♣ ✵ ) = / ρ ∈ R ✼ [ ❤ ] γ ( ❤ ) − ✶ ❤ − q ( ❤ ) ❉✐♠❡♥s✐♦♥ ✿ ✶✴✷✴✸ ◆✉♠❡r✐❝❛❧ s❝❤❡♠❡ ✿ ▼❖❈ ✭❋♦rtr❛♥✮✴❋r❡❡❋❡♠✰✰ ✭❝♦♥✈❡❝t✮ ❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✹ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s Pr❡s❡♥t❛t✐♦♥ ♦❢ Pr♦❥❡❝t ❉✐P▲♦▼❛ ▼♦❞❡❧s ❛♥❞ ♥✉♠❡r✐❝❛❧ ❛s♣❡❝ts ▲▼◆❈ ♠♦❞❡❧ ✭✈❡rs✐♦♥ ✷✵✶❄✮  � � P ′ ✵ ( t ) � + β ❤ , P ✵ ( t )   ∇· ✉ = − Φ , ✭✶❛✮  � � ❝ ✷ �  ρ ❤ , P ✵ ( t ) ❤ , P ✵ ( t ) P ✵ ( t )   � � � � = Φ + P ′ ρ ❤ , P ✵ ( t ) × ∂ t ❤ + ✉ · ∇ ❤ ✵ ( t ) , ✭✶❜✮      � � � � � �  ρ ❤ , P ✵ ( t ) × ∂ t ✉ + ( ✉ · ∇ ) ✉ − ∇· σ ( ✉ ) + ∇ ¯ ♣ = ρ ❤ , P ✵ ( t ) ❣ . ✭✶❝✮ ❊q✉❛t✐♦♥ ♦❢ st❛t❡ ✿ st✐✛❡♥❡❞ ❣❛s✴t❛❜✉❧❛t❡❞ ❧❛✇ � � � � γ ❤ , P ✵ ( t ) P ✵ ( t ) + π ❤ , P ✵ ( t ) � � ρ ❤ , P ✵ ( t ) = / ρ ∈ R ✼ [ ❤ , P ✵ ( t )] � � � � γ ❤ , P ✵ ( t ) − ✶ ❤ − q ❤ , P ✵ ( t ) ❉✐♠❡♥s✐♦♥ ✿ ✶✴✷✴✸ ◆✉♠❡r✐❝❛❧ s❝❤❡♠❡ ✿ ▼❖❈ ✭❋♦rtr❛♥✮✴❋r❡❡❋❡♠✰✰ ✭❝♦♥✈❡❝t✮ ❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✹ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s Pr❡s❡♥t❛t✐♦♥ ♦❢ Pr♦❥❡❝t ❉✐P▲♦▼❛ ❋r♦♠ P❛r✐s ✇✐t❤ ❧♦✈❡ ❲❤❡♥❄ ✺✳✕✻✳ ◆♦✈❡♠❜❡r ✷✵✶✺ ❲❤❡r❡❄ ❯♥✐✈✳ P❛r✐s ❉❡s❝❛rt❡s ❆❜♦✉t❄ ▲♦✇ ▼❛❝❤ ❛♥❞ ❧♦✇ ❋r♦✉❞❡ ✢♦✇s ❆❞❞r❡ss❡❞ t♦ ✇❤♦♠❄ ❆♥②❜♦❞② ✐♥t❡r❡st❡❞ ✐♥ t❤❡♦r❡t✐❝❛❧ ❛♥❞ ♥✉♠❡r✐❝❛❧ ❛s♣❡❝ts ❆♥❞❄ P♦st❡r s❡ss✐♦♥ ✭❛♣♣❧✐❝❛t✐♦♥s ❜❡❢♦r❡ ✷✺✳ ❙❡♣t❡♠❜❡r✮ ❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✺ ✴ ✶✾ / / :