Mathematical studies for a system

᪩✄⏣኱Ꮫ኱Ꮫ㝔 ඛ㐍⌮ᕤᏛ◊✲⛉

༤

༤㻌 ኈ㻌 ㄽ㻌 ᩥ㻌 ᴫ㻌 せ

ㄽ ᩥ 㢟 ┠

Mathematical studies for a system

describing the double-diffusive convection

஧㔜ᣑᩓᑐὶࢆグ㏙ࡍࡿ᪉⛬ᘧ⣔ࡢ

ᩘᏛⓗゎᯒ

⏦ ㄳ ⪅

Shun UCHIDA

≀⌮Ꮫཬᛂ⏝≀⌮Ꮫᑓᨷ ᩘ⌮≀⌮Ꮫ◊✲

2015 ᖺ 10 ᭶

༹ ࣭ Λ ؚ Ή ྲྀ ମ ಺ ʹ ͓ ͍ ͯ ɼ্ ෦ ͕ ߴ Թ ߴ ೱ ౓ ͔ ͭ Լ ෦ ͕ ௿ Թ ௿ ೱ ౓ Ͱ ͋ Δ

৔ ߹ ʹ ɼ௨ ৗ ͷ ֦ ࢄ Ϟ σ ϧ Ͱ ͸ આ ໌ ͞ Ε ͳ ͍ ɼ༹ ࣭ ͕ ࢦ ͷ ܗ ঢ় Ͱ ௜ ߱ ͢ ΔSalt fingeringͱ ݺ ͹ Ε Δ ݱ ৅ ͕100೥ Ҏ ্ ΋ લ ͔ Β ؍ ࡯ ͞ Ε ͯ ͍ ͨ ɽ͜ ͷ ݱ ৅ ͷ ϝ Χ χ ζ Ϝ ͸1960೥M. Sternʹ Α Γ ॳ Ί ͯ ղ ໌ ͞ Ε ɼු ྗ ٴ ͼ Թ ౓ ༹ ࣭ ೱ ౓ ؒ ͷ ૬ ޓ

࡞ ༻ ͕ ͦ ͷ ཁ Ҽ Ͱ ͋ Δ ͜ ͱ ͕ ൑ ໌ ͠ ͨ ɽҰ ൠ ʹ ྲྀ ମ ಺ ʹ ҟ ͳ Δ ֦ ࢄ ଎ ౓ Λ ࣋ ͭ ̎

੒ ෼ ͕ ࠞ ࡏ ͠ ɼ͔ ͭ ͦ ͷ ෼ ෍ ͕ ඇ Ұ ༷ Ͱ ͋ Δ ঢ় گ Ͱ ੜ ͡ Δ ݱ ৅ ͸ ೋ ॏ ֦ ࢄ ର ྲྀ (Double-diffusive convection)ͱ ݺ ͹ Ε ɼSternͷ ઌ ۦ త ͳ ݚ ڀ Ҏ ߱ ༷ ʑ ͳ ෼ ໺ Ͱ ݚ ڀ

͞ Ε ͯ ͍ Δ ɽͱ Γ Θ ͚ ଟ ޸ ࣭ ഔ ࣭ த ͷ ೋ ॏ ֦ ࢄ ର ྲྀ ͸ ɼ౔ ৕ Ԛ છ Ϟ σ ϧ ɼ৮ ഔ ಺ ͷ Խ ֶ ൓ Ԡ Ϟ σ ϧ ౳ ͷ Ԡ ༻ ੑ ʹ ෋ Έ ɼಛ ʹ ॏ ཁ ͳ ݚ ڀ ର ৅ ͱ ͳ ͬ ͯ ͍ Δ ɽҰ ํ ɼ

͜ ͷ ݱ ৅ Λ ه ड़ ͢ Δ ํ ఔ ࣜ ܥ ͷ ਺ ֶ త ݚ ڀ ʹ ؔ ͠ ͯ ͸ ɼՄ ղ ੑ ౳ ͷ ॏ ཁ ͳ ໰ ୊ ͕ ະ ղ ܾ ͷ · · ࢒ ͞ Ε ͯ ͍ ͨ ɽຊ ࿦ จ Ͱ ͸ ɼଟ ޸ ࣭ ഔ ࣭ த ͷ ೋ ॏ ֦ ࢄ ର ྲྀ ݱ ৅ Λ ه ड़ ͢ Δ ํ ఔ ࣜ ܥ ʹ ର ͢ Δ ภ ඍ ෼ ํ ఔ ࣜ ࿦ త ͳ ࢹ ఺ ͔ Β ͷ ݚ ڀ ʹ ͭ ͍ ͯ ड़ ΂ Δ ɽ

ຊ ࿦ จ ͷ ߏ ੒ ٴ ͼ ֓ ཁ ͸ Ҏ Լ ͷ ௨ Γ Ͱ ͋ Δ ɽ

ୈ ̍ ষ Ͱ ͸ ଟ ޸ ࣭ ഔ ࣭ த ͷ ೋ ॏ ֦ ࢄ ର ྲྀ ݱ ৅ ʹ ؔ ͢ Δ ෺ ཧ త എ ܠ ɼ਺ ֶ త ख ๏ ʹ ج ͮ ͘ ઌ ߦ ݚ ڀ ʹ ͭ ͍ ͯ ड़ ΂ ɼຊ ࿦ จ ͷ ୈ ̎ ষ Ҏ ߱ ͷ ಺ ༰ Λ ֓ ؍ ͢ Δ ɽ

ୈ ̎ ষ Ͱ ͸ ࣍ ষ Ҏ ߱ ༻ ͍ Δ ه ๏ ͷ ఆ ٛ ٴ ͼ ਺ ֶ త ج ຊ ࣄ ߲ ͕ ड़ ΂ Β Ε ͯ ͍ Δ ɽ

ୈ ̏ ষ Ҏ ߱ Ͱ ͸ ࣍ ͷ ํ ఔ ࣜ ܥ ͕ ߟ ࡯ ͞ Ε Δ ɽ

(DCBF)

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

∂_tu=νΔu−au− ∇p+gT +hC+f₁ (x, t)∈Ω×[0, S],

∂_tT +u·∇T = ΔT+f₂ (x, t)∈Ω×[0, S],

∂_tC+u·∇C = ΔC+ρΔT+f₃ (x, t)∈Ω×[0, S],

∇·u= 0 (x, t)∈Ω×[0, S].

͜ ͜ Ͱ ɼΩ͸N ࣍ ݩEuclidۭ ؒR^N ಺ ͷ ྖ Ҭ ͱ ͠ ɼS ͸ ࣌ ؒ ؒ ִ Λ ද ͢ ਖ਼ ఆ ਺ Ͱ ͋ Γ ɼྲྀ ଎u= (u¹, u²,· · ·, u^N)^tɼྲྀ ମ Թ ౓Tɼ༹ ࣭ ೱ ౓Cٴ ͼ ѹ ྗp͸ ະ ஌ ؔ ਺ Ͱ ͋ Δ ɽ

· ͨ ط ஌ ͷ σ ʔ λ ͱ ͠ ͯ ɼν,a,ρ͸ ਖ਼ ఆ ਺ ɼg= (g¹, g²,· · ·, g^N)^t,h= (h¹, h²,· · ·, h^N)^t

͸ ఆ ϕ Ϋ τ ϧ ɼf₁= (f₁¹, f₁²,· · ·, f₁^N)^t,f₂,f₃͸ ֎ ྗ Λ ද ͢ ؔ ਺ Ͱ ͋ Δ ɽ

ୈ Ұ ࣜ ͸ ଟ ޸ ࣭ ഔ ࣭ த ͷ ྲྀ ଎ ͷ ڍ ಈ Λ ه ड़ ͢ ΔBrinkman-Forchheimerํ ఔ ࣜ Λ ɼ ద ੾ ͳ ෺ ཧ త ৚ ݅ ͷ Լ Ͱ ઢ ܕ Խ ͠ ͨ ΋ ͷ Ͱ ͋ Δ ɽgTɼhC ͸ Oberbeck-Boussinesq

ۙ ࣅ ʹ Α Γ ݱ Ε Δ ු ྗ ߲ Ͱ ͋ Γ ɼߋ ʹ ྲྀ ମ ͷ ඇ ѹ ॖ ੑ Λ ه ड़ ͢ Δ ҝ ɼྲྀ ଎ ʹ ͸

∇ ·u = 0ͷ ৚ ݅ ͕ ՝ ͞ Ε ͯ ͍ Δ ɽୈ ೋ ɼୈ ࡾ ࣜ ʹ ͸ ඇ ઢ ܕ ߲ ͱ ͠ ͯ Ҡ ྲྀ ߲u· ∇Tɼ u· ∇C ͕ ؚ · Ε ͯ ͓ Γ ɼํ ఔ ࣜ ܥ ͷ ਺ ֶ త औ ѻ ͍ Λ ࠔ ೉ ʹ ͢ Δ ࠷ ΋ େ ͖ ͳ ཁ Ҽ ͱ ͳ ͬ ͯ ͍ Δ ɽ· ͨ ɼୈ ࡾ ࣜ ʹ ͸ Թ ౓ ͱ ༹ ࣭ ೱ ౓ ͱ ͷ ૬ ޓ ࡞ ༻ ͷ Ұ ͭ Ͱ ͋ ΔSoretޮ Ռ Λ ද ͢ ߲ρΔT ͕ ؚ · Ε ͯ ͍ Δ ɽೋ ॏ ֦ ࢄ ର ྲྀ ʹ ͓ ͍ ͯ ͸ Թ ౓ ͱ ༹ ࣭ ೱ ౓ ͷ ෼ ෍

͸ ඇ Ұ ༷ ͱ ͳ Δ ҝ ɼԹ ౓ ޯ ഑ Λ ཁ Ҽ ͱ ͢ ΔSoretޮ Ռ Λ ແ ࢹ ͢ Δ ͜ ͱ ͸ Ͱ ͖ ͳ ͍ ɽ (DCBF)ͷ ୈ Ұ ࣜ ͕Navier-Stokesํ ఔ ࣜ ʹ ஔ ͖ ׵ ͑ Β Ε ͨ ํ ఔ ࣜ ܥ ʹ ؔ ͠ ͯ ͸ ɼ

̎ ࣍ ݩ ௕ ํ ܗ ྖ Ҭ ্ Ͱ ͷ Մ ղ ੑ ͱ େ Ҭ Ξ τ ϥ Ϋ λ ʔ ͷ ଘ ࡏ ʹ ͭ ͍ ͯ ड़ ΂ ͨ M.

Piniewski(2008)౳ ɼط ʹ ͍ ͘ ͭ ͔ ͷ ઌ ߦ ݁ Ռ ͕ ಘ Β Ε ͯ ͍ Δ ɽ͠ ͔ ͠ ͜ Ε Β ͷ ݚ ڀ Ͱ ͸Navier-Stokesํ ఔ ࣜ ͷ औ ѻ ͍ ʹ ٞ ࿦ ͕ ू த ͠ ͯ ͓ Γ ɼҠ ྲྀ ߲ ɼSoretޮ Ռ ٴ ͼ

1

ු ྗ ߲ ʹ ى Ҽ ͢ Δ ࠔ ೉ ͷ ղ ܾ ʹ ର ͠ ͯ ͸ ৮ Ε Β Ε ͯ ͍ ͳ ͍ ɽຊ ݚ ڀ Ͱ ͸ ͜ Ε Λ ౿

· ͑ ɼ޻ ֶ ܥ Ͱ ྑ ͘ ༻ ͍ Β Ε Δ ɼྲྀ ଎ Λ ه ड़ ͢ Δ ํ ఔ ࣜ Λ ୯ ७ Խ ͠ ͨ Ϟ σ ϧ ʹ ର

͠ ͯ ɼҠ ྲྀ ߲ ɼSoretޮ Ռ ٴ ͼ ු ྗ ߲ ͕ ํ ఔ ࣜ ܥ ͷ Մ ղ ੑ ΍ ղ ͷ ઴ ۙ ڍ ಈ ʹ Ͳ ͷ Α

͏ ͳ Ө ڹ Λ ٴ ΅ ͢ ͔ Λ ਺ ֶ త ʹ ղ ໌ ͢ Δ ͜ ͱ Λ ໨ ඪ ͱ ͠ ͯ ͍ Δ ɽ

(DCBF)ͷ Մ ղ ੑ ʹ ؔ ͢ Δ ݁ Ռ ͸ ɼ̏ ࣍ ݩ ༗ ք ྖ Ҭ ্ Ͱ ੪ ࣍Dirichletڥ ք ৚ ݅ Λ

՝ ͠ ͨ ॳ ظ ஋ ڥ ք ஋ ໰ ୊ ʹ ର ͢ Δ ࣌ ؒ େ Ҭ ղ ͷ Ұ ҙ ଘ ࡏ Λ ࣔ ͠ ͨK. Terasawa–M.

Otani(2010)ˆ ͕ ࠷ ॳ Ͱ ͋ Δ ɽ͜ ͜ Ͱ ͸ ɼM. ˆOtani(1982)ʹ Α Δ ྼ ඍ ෼ ࡞ ༻ ૉ ʹ ର ͢ Δ

ඇ ୯ ௐ ઁ ಈ ཧ ࿦ Λ ద ༻ ͢ Δ ͜ ͱ ʹ Α Γ ɼղ ͷ ଘ ࡏ ͕ ࣔ ͞ Ε ͯ ͍ Δ ɽ͜ ͷ ࣄ ࣮ ͸ ɼ̏

࣍ ݩ ྖ Ҭ ʹ ͓ ͚ Δ େ Ҭ Մ ղ ੑ ͕ ະ ղ ܾ Ͱ ͋ ΔNavier-Stokesํ ఔ ࣜ ͷ ඇ ઢ ܕ ߲ ʹ ྨ

ࣅ ͢ Δ Ҡ ྲྀ ߲ Λ(DCBF)΋ อ ༗ ͠ ͯ ͍ Δ ͱ ͍ ͏ ؍ ఺ ͔ Β ͢ Ε ͹ ඇ ৗ ʹ ڵ ຯ ਂ ͍ ΋ ͷ Ͱ ͋ Γ ɼํ ఔ ࣜ ܥ(DCBF)ͷ ݚ ڀ ʹ ର ͢ Δ ߋ ͳ Δ ൃ ల ͷ Մ ೳ ੑ Λ ࣔ ࠦ ͠ ͯ ͍ Δ ɽ

Ҏ ্ Λ ౿ · ͑ ɼୈ ̏ ষ Ҏ ߱ Ͱ ͸ ࣍ ͷ ໰ ୊ Λ औ Γ ѻ ͏ ɽ

ୈ ̏ ষ Ͱ ͸ · ͣ ɼK. Terasawa–M. ˆOtani(2010)ͷ ख ๏ ʹ Α Γ ɼ̏ ࣍ ݩ ༗ ք ྖ Ҭ ্ Ͱ ͷ ॳ ظ ஋ ڥ ք ஋ ໰ ୊ ͕ ੪ ࣍Neumannڥ ք ৚ ݅ Լ Ͱ ΋ Ұ ҙ త େ Ҭ Մ ղ ੑ Λ ༗ ͢ Δ ͱ

͍ ͏ ݁ Ռ ͕ ࣔ ͞ Ε Δ ɽߋ ʹ ͜ ͷ ख ๏ Λ ൃ ల ͞ ͤ ɼ̏ ࣍ ݩ ༗ ք ྖ Ҭ ্ Ͱ ͷ ࣌ ؒ प ظ

໰ ୊ ͕ ੪ ࣍Dirichlet৚ ݅ ٴ ͼ ੪ ࣍Neumann৚ ݅ ͷ ૒ ํ ʹ ର ͠ Մ ղ ੑ Λ ༗ ͢ Δ ͱ ͍

͏ ݁ Ռ ͕ ࣔ ͞ Ε Δ ɽΑ Γ ۩ ମ త ʹ ͸ ɼ؇ ࿨ ߲ ͕ ෇ Ճ ͞ Ε ͨ(DCBF)ͷ ۙ ࣅ ํ ఔ ࣜ ʹ ɼM. ˆOtani(1984)ʹ Α Δ ɼྼ ඍ ෼ ࡞ ༻ ૉ Λ ओ ཁ ߲ ͱ ͢ Δ ඇ ୯ ௐ ઁ ಈ ߲ Λ ؚ Ή ൃ ల

ํ ఔ ࣜ ͷ ࣌ ؒ प ظ ໰ ୊ ͷ Մ ղ ੑ ͷ ݁ Ռ Λ ద ༻ ͢ Δ ͜ ͱ Ͱ ۙ ࣅ ղ ͷ ଘ ࡏ Λ อ ূ ͠ ɼ

ۙ ࣅ ղ ͷ ऩ ଋ ੑ Λ ٞ ࿦ ͢ Δ ͜ ͱ ʹ Α Γ ࣌ ؒ प ظ ղ ͷ ଘ ࡏ ͕ ࣔ ͞ Ε Δ ɽ͜ ͜ Ͱ ɼM.

Otani(1982)(1984)ˆ ͷ ݁ Ռ Λ ద ༻ ͢ Δ ࡍ ʹ ͸Rellich–Kondrachovͷ ఆ ཧ ͷ د ༩ ͕ େ ͖

͘ ɼۭ ؒ ྖ Ҭ ͷ ༗ ք ੑ ͸ ຊ ࣭ త ͳ Ծ ఆ Ͱ ͋ Δ ͜ ͱ ʹ ཹ ҙ ͢ Δ ɽ

ୈ ̐ ষ Ͱ ͸ ඇ ༗ ք ۭ ؒ ྖ Ҭ ্ Ͱ ͷ(DCBF)ͷ ॳ ظ ஋ ڥ ք ஋ ໰ ୊ ʹ ର ͢ Δ Մ ղ ੑ

͕ ߟ ࡯ ͞ Ε Δ ɽલ ड़ ͷ ௨ Γ ɼୈ ̏ ষ ʹ ͓ ͚ Δ ख ๏ Λ ඇ ༗ ք ྖ Ҭ ্ ͷ ໰ ୊ ʹ ௚ ઀ ద

༻ ͢ Δ ͜ ͱ ͸ ࠔ ೉ Ͱ ͋ Δ ɽͦ ͜ Ͱ ୈ ̐ ষ Ͱ ͸ ɼBanachͷ ෆ ಈ ఺ ఆ ཧ ͷ ద ༻ ʹ Α Γ ղ ͷ ߏ ੒ Λ ߦ ͍ ɼۭ ؒ ࣍ ݩN ͕ ̐ Ҏ Լ ͔ ͭ ྖ ҬΩ͕Sobolevͷ ຒ ଂ ఆ ཧ ٴ ͼLaplace

࡞ ༻ ૉ ͱStokes࡞ ༻ ૉ ͷ ପ ԁ ܕ ਖ਼ ଇ ੑ Λ ੒ ཱ ͞ ͤ Δ ৚ ݅ʢ ྫ ͑ ͹uniform C²-regular classʣͷ Լ Ͱ ɼDirichletڥ ք ৚ ݅ ٴ ͼNeumannڥ ք ৚ ݅ ͷ ૒ ํ ʹ ର ͠ ͯ(DCBF)ʹ Ұ ҙ େ Ҭ ղ ͕ ଘ ࡏ ͢ Δ ͱ ͍ ͏ ࣄ ࣮ ͕ ࣔ ͞ Ε Δ ɽNavier-Stokesํ ఔ ࣜ ͱ ൺ ֱ ͠ ͨ ৔ ߹

͜ ͷ ݁ Ռ ͸ ɼҠ ྲྀ ߲ ͷ Մ ੵ ෼ ੑ ͕ อ ূ ͞ Ε Δ ۭ ؒ ࣍ ݩ ̐ · Ͱ ͷ Մ ղ ੑ ͕ ॳ ظ ஋ ɼ

֎ ྗ ߲ ͷ খ ͞ ͞ Λ Ծ ఆ ͤ ͣ ʹ ಘ Β Ε Δ ͱ ͍ ͏ ڵ ຯ ਂ ͍ ஌ ݟ Λ ༩ ͑ ͯ ͍ Δ ɽ

ୈ ̑ ষ Ͱ ͸ શ ۭ ؒR^{N ্ ͷ}(DCBF)ͷ ࣌ ؒ प ظ ໰ ୊ ͕ ߟ ࡯ ͞ Ε ͯ ͍ Δ ɽલ ষ ͷ ݁ Ռ Λ ؑ Έ Ε ͹ ɼ࣌ ؒ प ظ ໰ ୊ ʹ ؔ ͠ ͯ ΋Large dataʢ ֎ ྗ ߲ ʹ খ ͞ ͞ Λ ՝ ͞ ͳ ͍ ʣ ʹ ର ͢ Δ Մ ղ ੑ ͕ ظ ଴ ͞ Ε Δ ɽ͠ ͔ ͠ ɼඇ ༗ ք ྖ Ҭ ্ ͷ ࣌ ؒ प ظ ໰ ୊ ͷLarge data ʹ ର ͢ Δ Մ ղ ੑ ʹ ؔ ͢ Δ ݁ Ռ ͸ ඇ ৗ ʹ গ ͳ ͍ ɽඇ ༗ ք ྖ Ҭ ্ Ͱ ͷ ࣌ ؒ प ظ ໰ ୊ ͷ Մ ղ ੑ ͸ ɼྫ ͑ ͹Navier-Stokesํ ఔ ࣜ ʹ ର ͠ ͯ ͸ ɼP. Maremonti(1991)ɼH. Kozono–M.

Nakao(1996)ʹ ͓ ͍ ͯ ط ʹ ಘ Β Ε ͯ ͍ Δ ɽ͜ Ε Β ͷ ࿦ จ Ͱ ͸ ɼղ ͷ ߏ ੒ ʹ ஞ ࣍ ۙ ࣅ

Λ ༻ ͍ ͯ ͓ Γ ɼͦ ͷ ऩ ଋ Λ อ ূ ͢ Δ ҝ ʹ ֎ ྗ ߲ ͷ খ ͞ ͞ ͸ ຊ ࣭ త ͳ ৚ ݅ ͱ ͳ ͬ ͯ

͍ Δ ɽҰ ํ ɼ࣌ ؒ प ظ ໰ ୊ ͷLarge dataʹ ର ͢ Δ Մ ղ ੑ ͸ ɼ์ ෺ ܕ ந ৅ ൃ ల ํ ఔ ࣜ ͷ ࿮ ૊ Έ Ͱ ଟ ͘ ͷ ݁ Ռ ͕ ಘ Β Ε ͯ ͍ Δ ɽ͠ ͔ ͠ ͜ Ε Β ͷ ࿦ จ Ͱ ͸ ɼۭ ؒ ྖ Ҭ ͕ ༗ ք Ͱ ͋ Δ ͜ ͱ Λ ૝ ఆ ͠ ͨ ৚ ݅ʢ ྫ ͑ ͹ ྼ ඍ ෼ ࡞ ༻ ૉ ʹ ର ͢ Δ ڧ ѹ ੑ ٴ ͼ ఆ ٛ Ҭ ʹ

ؔ ͢ Δ ί ϯ ύ Ϋ τ ੑ ʣͷ Լ Ͱ ٞ ࿦ ͕ ͳ ͞ Ε ͯ ͓ Γ ɼඇ ༗ ք ྖ Ҭ ͷ ৔ ߹ ʹ ͜ Ε Λ ద

༻ ͢ Δ ͜ ͱ ͸ ࠔ ೉ Ͱ ͋ Δ ɽ͜ ͷ Α ͏ ʹ ɼඇ ୯ ௐ ઁ ಈ ߲u· ∇T,u· ∇CΛ ؚ Ή ์ ෺ ܕ

ํ ఔ ࣜ Ͱ ͋ Δ(DCBF)ʹ ର ͢ ΔR^N ্ ͷ ࣌ ؒ प ظ ໰ ୊ ͷLarge dataʹ ର ͢ Δ Մ ղ ੑ ʹ ͭ ͍ ͯ ͸ ɼઌ ߦ ݚ ڀ ͷ ख ๏ ͕ ௚ ઀ ద ༻ ग़ དྷ ͣ ɼ৽ ͠ ͍ Ξ ϓ ϩ ʔ ν ͕ ඞ ཁ ͱ ͳ Δ ɽ

ୈ ̑ ষ Ͱ ͸ ɼҎ Լ ͷ ख ๏ ʹ Α Γ ࣌ ؒ प ظ ղ ͷ ߏ ੒ Λ ߦ ͏ ɽ· ͣ શ ۭ ؒ Λ ൒ ܘn ͷ ։ ٿ Ͱ ஔ ͖ ׵ ͑ ͨ ۙ ࣅ ໰ ୊ Λ ߟ ࡯ ͢ Δ ɽ༗ ք ྖ Ҭ ্ ͷ ໰ ୊ ͸ ઌ ߦ ݚ ڀ ʹ ͓ ͍ ͯ ط ʹ ߟ ࡯ ͞ Ε ͯ ͓ Γ ɼ೚ ҙ ͷ େ ͖ ͞ ͷ ֎ ྗ ߲ ʹ ର ͢ Δ ղ ͷ ߏ ੒ ͕ Մ ೳ Ͱ ͋ Δ ɽ࣍

ʹ ֤nʹ ର ͠ ͯ ಘ Β Ε ͨ ղ ٴ ͼ ํ ఔ ࣜ ܥ ͷn→ ∞ʹ ͓ ͚ Δ ۃ ݶ ͕ ɼAscoliͷ ఆ ཧ Λ ద ༻ ͠ ͨ ہ ॴ ڧ ऩ ଋ ੑ ͱ ର ֯ ઢ ࿦ ๏ Λ ༻ ͍ Δ ͜ ͱ ʹ Α Γ શ ۭ ؒ ʹ ͓ ͚ Δ ղ ٴ ͼ ํ ఔ ࣜ ܥ ͱ ͳ Δ ࣄ Λ ࣔ ͢ ɽ͜ ͷ ࡍ ʹ ۙ ࣅ ղ ͷ Ұ ༷ ༗ ք ੑ Λ Ξ ϓ Ϧ Φ Ϧ ධ Ձ ʹ Α Γ ٻ Ί Δ ͕ ɼ͜ ͜ ʹ ֎ ྗ ߲ ͷ খ ͞ ͞ ͸ ඞ ཁ ͱ ͞ Ε ͳ ͍ ɽҎ ্ ͷ ํ ๏ ʹ Α Γ ɼۭ ؒ ࣍ ݩ

͕N = 3,4ͷ ৔ ߹ ɼLarge dataʹ ର ͢ Δ(DCBF)ͷ ࣌ ؒ प ظ ղ ͷ ଘ ࡏ ͕ ࣔ ͞ Ε Δ ɽ

ୈ ̒ ষ Ͱ ͸ ɼ༗ ք ྖ Ҭ ্ ͷ(DCBF)ͷ ղ ͷ ࣌ ؒ େ Ҭ త ڍ ಈ ͕ ɼେ Ҭ Ξ τ ϥ Ϋ λ ʔ ٴ ͼ ࢦ ਺ Ξ τ ϥ Ϋ λ ʔ ͷ ଘ ࡏ ͱ ͍ ͏ ؍ ఺ ͔ Β ߟ ࡯ ͞ Ε ͯ ͍ Δ ɽຊ ࿦ จ Ͱ ͸ ಛ ʹ ֎

ྗ ߲ ͕ ࣌ ؒ ʹ ґ ଘ ͠ ͳ ͍ ৔ ߹ʢ ࣗ ྭ ܥ ʣʹ ͭ ͍ ͯ ߟ ࡯ ͞ Ε ͯ ͍ Δ ɽΞ τ ϥ Ϋ λ ʔ ͷ ߏ ੒ ͸ ந ৅ ࿦ ͷ ݁ Ռ ʹ ج ͮ ͍ ͯ ߦ Θ Ε Δ ɽଈ ͪ ୈ ̐ ষ ͷ Մ ղ ੑ ͷ ݁ Ռ ʹ ج ͮ ͖ ఆ ٛ ͞ Ε Δ ൒ ܈{S(t)}_t≥0ʢ ॳ ظ ஋ ʹt࣌ ؒ ޙ ͷ ղ Λ ର Ԡ ͞ ͤ Δ ࣸ ૾ ଒ ʣʹ ର ͠ ɼେ

Ҭ Ξ τ ϥ Ϋ λ ʔʢ શ ͯ ͷ ղ ي ಓ Λ Ҿ ͖ ෇ ͚ Δ ί ϯ ύ Ϋ τ ू ߹ Ͱ ࠷ খ ͷ ΋ ͷ ʣͷ ଘ ࡏ ͸ ɼ֤ ࣸ ૾S(t)ͷ ࿈ ଓ ੑ ͱ ί ϯ ύ Ϋ τ ٵ ऩ ू ߹ʢ ೚ ҙ ͷ ༗ ք ू ߹ ͔ Β ग़ ൃ ͠ ͨ ղ ي ಓ Λ ༗ ݶ ࣌ ؒ Ͱ Ҿ ͖ ࠐ Ή ू ߹ ʣͷ ଘ ࡏ ʹ Α Γ อ ূ ͞ Ε Δ ɽ· ͨ ࢦ ਺ Ξ τ ϥ Ϋ λ ʔʢ ϑ ϥ Ϋ λ ϧ ࣍ ݩ ͕ ༗ ݶ Ͱ ɼҾ ͖ ࠐ Έ ͷ ଎ ౓ ͕ ࢦ ਺ ؔ ਺ త Ͱ ͋ Δ ू ߹ ʣͷ ଘ ࡏ ͸ ɼSmoothing propertyͱ ݺ ͹ Ε Δ ੑ ࣭ ͕ ຬ ͨ ͞ Ε Δ ͜ ͱ ʹ Α Γ อ ূ ͞ Ε Δ ɽ

M. Piniewski(2008)Ͱ ͸ ̎ ࣍ ݩ ͷ ৔ ߹ ͕ ߟ ࡯ ͞ Ε ͯ ͍ Δ ͕ ɼ̏ ࣍ ݩ Ҏ ্ ͷ ৔ ߹ ʹ

͸ ɼί ϯ ύ Ϋ τ ٵ ऩ ू ߹ ͷ ଘ ࡏ ٴ ͼSmoothing propertyΛ ݕ ূ ͢ Δ ҝ ʹ ɼલ ষ · Ͱ ʹ ཁ ٻ ͞ Ε ͳ ͔ ͬ ͨ Α Γ ߴ ͍ ਖ਼ ଇ ੑ ʹ ؔ ͢ Δ ධ Ձ Λ ߦ ͏ ඞ ཁ ͕ ͋ Γ ɼͦ ͷ ख ๏ ͸

े ෼ Ͱ ͸ ͳ ͍ ɽୈ ̒ ষ Ͱ ͸ ͜ Ε Λ ղ ܾ ͢ Δ ҝ ʹ ɼඇ ઢ ܕ ൃ ల ํ ఔ ࣜ ࿦ ʹ ͓ ͍ ͯ ྑ

͘ ஌ Β Ε ͯ ͍ ΔH. Br´ezis(1973)ͷ ख ๏ Λ վ ྑ ɼ֦ ு ͠ ͨ ݁ Ռ ͕ ༩ ͑ Β Ε ͯ ͍ Δ ɽ͜

Ε Λ ओ ͨ Δ ಓ ۩ ͱ ͠ ͯ ɼ੪ ࣍Dirichletڥ ք ৚ ݅ Λ ՝ ͠ ͨ(DCBF)ʹ ର ͢ Δ େ Ҭ Ξ τ ϥ Ϋ λ ʔ ٴ ͼ ࢦ ਺ Ξ τ ϥ Ϋ λ ʔ ͷ ଘ ࡏ ͕ ɼۭ ؒ ࣍ ݩ ͕ ̐ Ҏ Լ ͷ ৔ ߹ ʹ ࣔ ͞ Ε Δ ɽ

· ͨ ୈ ̒ ষ Ͱ ͸ ɼ੪ ࣍Neumannڥ ք ৚ ݅ Λ ՝ ͠ ͨ ৔ ߹ ʹ ͭ ͍ ͯ ΋ ߟ ࡯ ͞ Ε ͯ ͍ Δ ɽ͜ ͷ ࣌T, C͸Mass conservation propertyΛ ຬ ͨ ͢ ҝ ɼ௨ ৗ ͷ ҙ ຯ Ͱ ͷ Ξ τ ϥ Ϋ λ ʔ ͸ ଘ ࡏ ͠ ಘ ͳ ͍ ɽ͜ ͜ Ͱ ͸ ॳ ظ ஋ ͱ ֎ ྗ ߲ ʹ ੍ ݶ Λ ՝ ͢ ͜ ͱ Ͱ ͜ ͷ ໰ ୊ Λ ղ ফ ͠ ɼҰ ൠ Խ ͞ Ε ͨ Ξ τ ϥ Ϋ λ ʔ ͷ ଘ ࡏ ͕ ࣔ ͞ Ε Δ ɽ

3

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㸬஧㔜ᣑᩓᑐὶ᪉⛬ᘧࡢ୍⯡㡿ᇦୖ࡟࠾ࡅࡿ᫬㛫኱ᇦゎࡢᏑᅾ࡟ࡘ࠸࡚㸪

➨ ᅇᛂ⏝ゎᯒ◊✲఍ࢩ࣏ࣥࢪ࣒࢘㸦㛤ദᆅ㸸⟽᰿㸧㸪 ᖺ ᭶㸪ෆ⏣ಇ㸬

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㸬஧㔜ᣑᩓᑐὶ᪉⛬ᘧࡢᣦᩘ࢔ࢺࣛࢡࢱ࣮࡟ࡘ࠸࡚㸪➨ ᅇᛂ⏝ゎᯒ◊✲఍ࢩ࣏ࣥࢪ

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㸬඲✵㛫ୖࡢ஧㔜ᣑᩓᑐὶ᪉⛬ᘧ࡟ᑐࡍࡿ᫬㛫࿘ᮇゎࡢᏑᅾ࡟ࡘ࠸࡚㸪

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