❆❞❛♣t✐✈❡ ♠❡❛s✉r❡✲✈❛❧✉❡❞ ❝♦✉♣❧✐♥❣ ♦❢ ♥♦♥✲❧✐♥❡❛r ❤②♣❡r❜♦❧✐❝ P❉❊s ❋❧♦r❡♥t ❘❡♥❛❝ ✶ ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❈❧❛✉❞❡ ▼❛r♠✐❣♥♦♥ ✶ ❛♥❞ ❋ré❞ér✐❝ ❈♦q✉❡❧ ✷ ✶ ❖◆❊❘❆ ❚❤❡ ❋r❡♥❝❤ ❆❡r♦s♣❛❝❡ ▲❛❜ ✭❋r❛♥❝❡✮ ✷ ❈◆❘❙ ✲ ❈▼❆P✱ ❊❝♦❧❡ P♦❧②t❡❝❤♥✐q✉❡ ✭❋r❛♥❝❡✮ ❍❨P✷✵✶✷ ❏✉♥❡ ✷✺✲✷✾✱ ✷✵✶✷ P❛❞♦✈❛ THE FRENCH AEROSPACE LAB ❋✳ ❘❡♥❛❝ ❡t ❛❧✳ ✭❖◆❊❘❆ ✲ ❈▼❆P✮ ❆❞❛♣t✐✈❡ ❝♦✉♣❧✐♥❣ ♦❢ ❤②♣❡r❜♦❧✐❝ P❉❊s ❏✉♥❡ ✷✺✲✷✾✱ ✷✵✶✷✱ P❛❞♦✈❛ ✶ ✴ ✷✻
◆✉♠❡r✐❝❛❧ s✐♠✉❧❛t✐♦♥ ♦❢ s②st❡♠s ✇✐t❤ ♠✉❧t✐s❝❛❧❡ ♣❤❡♥♦♠❡♥❛ ❈♦♥t❡①t✿ ❝♦♠♣❧❡① s②st❡♠s ❞❡❝♦♠♣♦s❡❞ ✐♥t♦ s✉❜✲s②st❡♠s s✐♠✉❧❛t❡❞ ❜② s♣❡❝✐✜❝ s♦❢t✇❛r❡s ✇✐t❤ ❞✐✛❡r❡♥t ♠♦❞❡❧✐♥❣s ♥❡❡❞ t♦ ❝♦✉♣❧❡ ♣r♦❜❧❡♠s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❞✐st✐♥❝t ♠♦❞❡❧✐♥❣ ❧❡✈❡❧s ❊①❛♠♣❧❡s✿ ♠✉❧t✐♣❤❛s❡ ✢♦✇s✱ t✉r❜✉❧❡♥t ✢♦✇s✱ ♣♦r♦✉s✴♥♦♥✲♣♦r♦✉s ♠❡❞✐❛✱ ❦✐♥❡t✐❝✴❤②❞r♦❞②♥❛♠✐❝ r❡♣r❡s❡♥t❛t✐♦♥s✱ ❡t❝✳ ❆✐♠✿ s✐♠✉❧❛t✐♥❣ t❤❡ ✇❤♦❧❡ s②st❡♠ ✇✐t❤ ❛♥ ❡✣❝✐❡♥t ❛♥❞ r❡❧✐❛❜❧❡ ❝♦✉♣❧✐♥❣ ♠❡t❤♦❞ ❬●♦❞❧❡✇s❦✐ ❡t ❛❧✳ ✬✵✹ ✬✵✺✱ ❆♠❜r♦s♦ ❡t ❛❧✳ ✬✵✽❪ ❝♦✉♣❧✐♥❣ ❤②♣❡r❜♦❧✐❝ s②st❡♠s ♦❢ ❝♦♥s❡r✈❛t✐♦♥ ❧❛✇s ✇✐t❤ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s ❝❧♦s✉r❡ ❧❛✇s ✭✐✳❡✳ ❞❡♣❡♥❞✐♥❣ ♦♥ s♣❛❝❡ ✈❛r✐❛❜❧❡s✮ ❆♣♣❧✐❝❛t✐♦♥✿ ♠♦❞❡❧❧✐♥❣ ✇✐t❤ ❛❞❛♣t✐✈❡ s✐♠♣❧✐✜❡❞ ❝❧♦s✉r❡ ❧❛✇s ❛♣♣❧✐❝❛t✐♦♥ t♦ ✐♥❝♦♠♣❧❡t❡ ❊❖❙ ❬▼❡♥✐❦♦✛ ✫ P❧♦❤r ✬✽✾❪ THE FRENCH AEROSPACE LAB ❋✳ ❘❡♥❛❝ ❡t ❛❧✳ ✭❖◆❊❘❆ ✲ ❈▼❆P✮ ❆❞❛♣t✐✈❡ ❝♦✉♣❧✐♥❣ ♦❢ ❤②♣❡r❜♦❧✐❝ P❉❊s ❏✉♥❡ ✷✺✲✷✾✱ ✷✵✶✷✱ P❛❞♦✈❛ ✷ ✴ ✷✻
❖✉t❧✐♥❡ ▼♦❞❡❧ ♣r♦❜❧❡♠ ✶ ❊✉❧❡r ❡q✉❛t✐♦♥s ◆♦♥✲❤♦♠♦❣❡♥❡♦✉s ❡q✉❛t✐♦♥ ♦❢ st❛t❡ ◆✉♠❡r✐❝❛❧ ♠❡t❤♦❞ ✷ ❈♦✉♣❧✐♥❣ ♣r♦❝❡❞✉r❡ ❘❡❧❛①❛t✐♦♥ ♠❡t❤♦❞ ❉✐s❝r❡t✐③❛t✐♦♥ ♠❡t❤♦❞ ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ✸ ❙✐♠♣❧✐✜❡❞ ❊❖❙ ❙♣❛❝❡ t✐♠❡ ❛❞❛♣t✐✈❡ ❊❖❙ ❈♦♥❝❧✉❞✐♥❣ r❡♠❛r❦s ✹ THE FRENCH AEROSPACE LAB ❋✳ ❘❡♥❛❝ ❡t ❛❧✳ ✭❖◆❊❘❆ ✲ ❈▼❆P✮ ❆❞❛♣t✐✈❡ ❝♦✉♣❧✐♥❣ ♦❢ ❤②♣❡r❜♦❧✐❝ P❉❊s ❏✉♥❡ ✷✺✲✷✾✱ ✷✵✶✷✱ P❛❞♦✈❛ ✸ ✴ ✷✻
❖✉t❧✐♥❡ ▼♦❞❡❧ ♣r♦❜❧❡♠ ✶ ❊✉❧❡r ❡q✉❛t✐♦♥s ◆♦♥✲❤♦♠♦❣❡♥❡♦✉s ❡q✉❛t✐♦♥ ♦❢ st❛t❡ ◆✉♠❡r✐❝❛❧ ♠❡t❤♦❞ ✷ ❈♦✉♣❧✐♥❣ ♣r♦❝❡❞✉r❡ ❘❡❧❛①❛t✐♦♥ ♠❡t❤♦❞ ❉✐s❝r❡t✐③❛t✐♦♥ ♠❡t❤♦❞ ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ✸ ❙✐♠♣❧✐✜❡❞ ❊❖❙ ❙♣❛❝❡ t✐♠❡ ❛❞❛♣t✐✈❡ ❊❖❙ ❈♦♥❝❧✉❞✐♥❣ r❡♠❛r❦s ✹ THE FRENCH AEROSPACE LAB ❋✳ ❘❡♥❛❝ ❡t ❛❧✳ ✭❖◆❊❘❆ ✲ ❈▼❆P✮ ❆❞❛♣t✐✈❡ ❝♦✉♣❧✐♥❣ ♦❢ ❤②♣❡r❜♦❧✐❝ P❉❊s ❏✉♥❡ ✷✺✲✷✾✱ ✷✵✶✷✱ P❛❞♦✈❛ ✹ ✴ ✷✻
❊✉❧❡r ❡q✉❛t✐♦♥s ✇✐t❤ ❤❡t❡r♦❣❡♥❡✐t② ✐♥ s♣❛❝❡ ∂ t ✉ + ∂ ① ❢ ( ✉ , ① ) = ✵ , ∀ ① ∈ D ( t ) , t > ✵ ✉ ( ① , ✵ ) = ✉ ✵ ( ① ) , ∀ ① ∈ R ❚✐♠❡✲❞❡♣❡♥❞❛♥t s♣❛❝❡ ❞♦♠❛✐♥✿ ♥ ( t ) � � � D ( t ) = ❛ ✐ − ✶ ( t ) , ❛ ✐ ( t ) ✇✐t❤ ❛ ✵ ( t ) = −∞ , ❛ ♥ ( t ) ( t ) = + ∞ , t > ✵ ✐ = ✶ ❈♦♥s❡r✈❛t✐✈❡ ✈❛r✐❛❜❧❡s✿ ✉ = ( ρ, ρ ✉ , ρ ❊ ) ⊤ ❍❡t❡r♦❣❡♥❡♦✉s ♣❤②s✐❝❛❧ ✢✉①❡s ✐♥ ① t❤r♦✉❣❤ ♣r❡ss✉r❡ ❞❡✜♥✐t✐♦♥✿ ρ ✉ ρ ✉ ✷ + ♣ ( ✉ , ① ) ❢ ( ✉ , ① ) = � � ρ ❊ + ♣ ( ✉ , ① ) ✉ ❙❡t ♦❢ ❛❞♠✐ss✐❜❧❡ st❛t❡s✿ ✉ ∈ R ✸ : ρ > ✵ , ✉ ∈ R , ❡ = ❊ − ✉ ✷ � � Ω = ✷ > ✵ THE FRENCH AEROSPACE LAB ❋✳ ❘❡♥❛❝ ❡t ❛❧✳ ✭❖◆❊❘❆ ✲ ❈▼❆P✮ ❆❞❛♣t✐✈❡ ❝♦✉♣❧✐♥❣ ♦❢ ❤②♣❡r❜♦❧✐❝ P❉❊s ❏✉♥❡ ✷✺✲✷✾✱ ✷✵✶✷✱ P❛❞♦✈❛ ✺ ✴ ✷✻
◆♦♥✲❤♦♠♦❣❡♥❡♦✉s ❡q✉❛t✐♦♥ ♦❢ st❛t❡ ✭❊❖❙✮ � ♣ ❡① ( ρ, ❡ ) ✐❢ ✉ ( ① , t ) ∈ Ω \ Ω s✐♠♣ ♣ ( ✉ , ① ) = ♣ s✐♠♣ ( ρ, ρ ❡ ) ✐❢ ✉ ( ① , t ) ∈ Ω s✐♠♣ ρ e Ω simp p ex ( ρ ,e) p simp,1 ( ρ , ρ e) p simp,2 ( ρ , ρ e) p ex ( ρ ,e) p=p simp,2 ( ρ , ρ e) a i-1 (t) a i (t) a i+1 (t) x p=p simp,1 ( ρ , ρ e) Ω p=p ex ( ρ ,e) ρ ◮ ❡①❛❝t ✭❜✉t ❝♦♠♣❧❡①✮ ❊❖❙ ♣ ❡① ( ρ, ❡ ) ❊❖❙✿ ◮ ✐♥❝♦♠♣❧❡t❡ ✭❜✉t s✐♠♣❧❡✮ ❊❖❙ ♣ s✐♠♣ ( ρ, ρ ❡ ) THE FRENCH AEROSPACE LAB ❋✳ ❘❡♥❛❝ ❡t ❛❧✳ ✭❖◆❊❘❆ ✲ ❈▼❆P✮ ❆❞❛♣t✐✈❡ ❝♦✉♣❧✐♥❣ ♦❢ ❤②♣❡r❜♦❧✐❝ P❉❊s ❏✉♥❡ ✷✺✲✷✾✱ ✷✵✶✷✱ P❛❞♦✈❛ ✻ ✴ ✷✻
❖✉t❧✐♥❡ ▼♦❞❡❧ ♣r♦❜❧❡♠ ✶ ❊✉❧❡r ❡q✉❛t✐♦♥s ◆♦♥✲❤♦♠♦❣❡♥❡♦✉s ❡q✉❛t✐♦♥ ♦❢ st❛t❡ ◆✉♠❡r✐❝❛❧ ♠❡t❤♦❞ ✷ ❈♦✉♣❧✐♥❣ ♣r♦❝❡❞✉r❡ ❘❡❧❛①❛t✐♦♥ ♠❡t❤♦❞ ❉✐s❝r❡t✐③❛t✐♦♥ ♠❡t❤♦❞ ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ✸ ❙✐♠♣❧✐✜❡❞ ❊❖❙ ❙♣❛❝❡ t✐♠❡ ❛❞❛♣t✐✈❡ ❊❖❙ ❈♦♥❝❧✉❞✐♥❣ r❡♠❛r❦s ✹ THE FRENCH AEROSPACE LAB ❋✳ ❘❡♥❛❝ ❡t ❛❧✳ ✭❖◆❊❘❆ ✲ ❈▼❆P✮ ❆❞❛♣t✐✈❡ ❝♦✉♣❧✐♥❣ ♦❢ ❤②♣❡r❜♦❧✐❝ P❉❊s ❏✉♥❡ ✷✺✲✷✾✱ ✷✵✶✷✱ P❛❞♦✈❛ ✼ ✴ ✷✻
❈♦✉♣❧✐♥❣ ♣r♦❝❡❞✉r❡ ❈♦♥s✐❞❡r t❤❡ ♠♦❞❡❧ ♣r♦❜❧❡♠ � ♣ − ( ✉ ) ✐❢ ① < ✵ ♣ ( ✉ , ① ) = ♣ + ( ✉ ) ✐❢ ① > ✵ ∂ t u + ∂ x f ( u , x ) = 0 ∂ t u + ∂ x f ( u , x ) = 0 p ( u ) = p - ( u ) p ( u ) = p + ( u ) 0 x < 0 x > 0 x ❈♦✉♣❧✐♥❣ ♣r❡s❝r✐❜❡❞ ❜② ❛ ✈❡❝t♦r✲✈❛❧✉❡❞ ❉✐r❛❝ ♠❡❛s✉r❡ ❝♦♥❝❡♥tr❛t❡❞ ❛t ① = ✵✿ ✉ ( ✵ + , t ) , ✉ ( ✵ − , t ) � � ∂ t ✉ + ∂ ① ❢ ( ✉ , ① ) = δ ① = ✵ , ∀ ① ∈ R , t > ✵ M ✉ ( ① , ✵ ) = ✉ ✵ ( ① ) , ∀ ① ∈ R THE FRENCH AEROSPACE LAB ❋✳ ❘❡♥❛❝ ❡t ❛❧✳ ✭❖◆❊❘❆ ✲ ❈▼❆P✮ ❆❞❛♣t✐✈❡ ❝♦✉♣❧✐♥❣ ♦❢ ❤②♣❡r❜♦❧✐❝ P❉❊s ❏✉♥❡ ✷✺✲✷✾✱ ✷✵✶✷✱ P❛❞♦✈❛ ✽ ✴ ✷✻
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