[ ] (TW) c 2 t q = c 2 ( (1+( q) 2 ) 1/2 q ) q : I R (t, x) q(t, x) R (TW) c q quasilinear partial differential (TW)

Ұ࣍ݩԣ೾໛ܕͱͯ͠ͷ೾ಈํఔࣜ

ฏ੒ 31 ೥ 3 ݄ খᖒɹప http://www.ozawa.phys.waseda.ac.jp/index2.html

ۭؒҰ࣍ݩʢۭ࣌ೋ࣍ݩʣॎܕ໛ܕͷߟ࡯ʢʰҰ࣍ݩॎ೾໛ܕͱͯ͠ͷ೾ಈํఔࣜʱҎ Լ [ॎ] ͱͯ͠Ҿ༻͢ΔʣʹҾ͖ଓ͖ԣ೾໛ܕͱͯ͠ͷ೾ಈํఔࣜͷॳظ஋໰୊ͷղ๏ʹब

͍ͯߟ͑Δɻ

̍ɽԣ೾໛ܕɹ

Ұ࣍ݩతͳݭ stringʢ҃͸Ꭲ৚ springʣΛ఻೻͢Δԣ೾໛ܕ transversal wave model ͱ

ͯ࣍͠ͷํఔࣜΛߟ͑Δɿ

(TW)_c ∂_t²q = c²∂

(1 + (∂q)²)^−1/2∂q

͜͜ʹ q ͸ೋ࣍ݩۭ࣌ R × R ͷ෦෼ྖҬ I × R (I ⊂ R ͸ॳظ࣌ࠁ 0 ΛؚΉ࣌ؒ۠ؒ) ʹ ԙ͚ΔεΧϥʔ৔

q : I × R  (t, x) −→ q(t, x) ∈ R

Ͱ͋Γ (TW)_c͸ q ʹब͍ͯͷೋ֊४ઢܕภඍ෼ํఔࣜ quasilinear partial differential equation of second order ͷܗΛऔ͍ͬͯΔɻ࣌ؒ࣠ͷई౓ม׵t −→ c^±1t ʹґΓ c = 1 ͷ

৔߹

(TW) ∂_t²q = ∂

(1 + (∂q)²)^−1/2∂q

Λߟ࡯ͷର৅ͱͯ͠΋ҰൠੑΛࣦΘͳ͍ͷͰɺҎԼͰ͸ઐΒ (TW) ͷղ๏ʹब͍ͯ࿦͡Δɻ

̎ɽҰ֊૒ۂܥ΁ͷؼண

(TW) ͷҰ֊ԽΛਤΔҝɺ৽ͨͳະ஌വ਺ u ٴͼ v Λ

u = ∂tq, v = ∂q (2.1)

ͱஔ͍ͯ (TW) ͔Β৽ͨʹ u, v ͷຬͨ͢΂͖ํఔࣜ (ͷܥ) ΛٻΊΑ͏ɻ(TW) ͸ (2.1) ʹ ґΓ

∂_tu = ∂

(1 + v²)^−1/2v

(2.2) ͱද͞ΕΔɻ(2.2) ͷӈลͷඍ෼Λ࣮ߦ͢Δͱ

∂_tu = (1 + v²)^−1/2∂v − (1 + v²)^−3/2v²∂v

= (1 + v²)^−3/2

(1 + v²)− v²

∂v

= (1 + v²)^−3/2∂v (2.3)

ͳΔҰ֊ͷํఔ͕ࣜಘΒΕΔɻͦ͜ͰϕΫτϧ஋ະ஌വ਺

U =

u v



: I × R  (t, x) −→

u(t, x) v(t, x)



∈ R²

Λಋೖ͢Δͱɺͦͷຬͨ͢΂͖ํఔࣜܥ͸

(HS) ∂tU + A(U)∂U = 0

ͷܗͰ༩͑ΒΕΔɻ͜͜ʹ

A(U) =



0 −(1 + v²)^−3/2

−1 0



(2.4)

Ͱ͋Γ (HS) ͸ (2.3) ʹՃ͑ͯࣗ໌ͱظ଴͞ΕΔؔ܎ࣜ

∂tv = ∂u (2.5)

ΛऔΓೖΕͨํఔࣜܥʹ֎ͳΒͳ͍ࣄ͕෼͔Δɻ(2.4) Λର֯Խͯ͠ɺೋͭͷ୯ಠҰ֊ภ ඍ෼ํఔ͔ࣜΒ੒Δ৽͍͠ܥʹॻ͖׵͑Α͏ɻߦྻ A(U) ͷݻ༗஋͸ ±(1 + v²)^−3/4Ͱ͋

Δɻ࣮ࡍ

det (λI − A(U)) = det



λ (1 + v²)^−3/2

1 λ



= λ²− (1 + v²)^−3/2 ͱͳΔ͔ΒͰ͋Δɻͦ͜Ͱ λ₋< 0 < λ+ͱͳΔ༷ʹ

λ±:=±(1 + v²)^−3/4 (2.5)±

ͱஔ͜͏ɻରԠ͢Δݻ༗ϕΫτϧ e_±(U) ͸

e±(U) :=

∓(1 + v²)^−3/4 1



(2.6)±

Ͱ༩͑ΒΕΔɻ࣮ࡍ

A(U)e±(U) =



0 −(1 + v²)^−3/2

−1 0

 ∓(1 + v²)^−3/4 1



=

−(1 + v²)^−3/2

±(1 + v²)^−3/4



=±(1 + v²)^−3/4

∓(1 + v²)^−3/4 1



= λ_±e_±(U)

ͱͳΔ͔ΒͰ͋Δɻೋͭͷݻ༗ۭؒRe±(U) ͷ੒͢௚࿨෼ղ R² = 

±

Re±(U) ʹର͢Δ

U ͷ෼ղΛߟ͑ɺͦͷ܎਺ʹ૬౰͢Δ੒෼Λۭ࣌ೋม਺വ਺ͱͯ͠ v±ͱදͦ͏:

U =

u v



=

±

v±e±(U) = v+e+(U) + v−e−(U) =

(1 + v²)^−3/4(v−− v+) v++ v−



͜ΕΑΓ

(1 + v²)^3/4u = v₋− v₊ v = v++ v−

(2.7)

͕ै͍ v±͸ u ٴͼ v ʹґͬͯ

v±= 1

2(v ∓ (1 + v²)^3/4u) (2.8)±

ͱද͞ΕΔɻ೾ಈവ਺ U ͷ (ۭؒ) ಋവ਺ ∂U ͷ Re_±(U) ্΁ͷ෼ղ΋ಉ༷ʹߟ͑ɺͦͷ

܎਺ʹ૬౰͢Δവ਺Λ w_±ͱදͦ͏ɻ

∂U =

∂u

∂v



=

±

w_±e_±(U) =

(1 + v²)^−3/4(w−− w+) w++ w−



͜ΕΑΓ

(1 + v²)^3/4∂u = w−− w+

∂v = w++ w−

(2.9)

͕ै͍ w_±͸ ∂u ٴͼ ∂v ʹґͬͯ

w± = 1

2(∂v ∓ (1 + v²)^3/4∂u) (2.10)±

ͱද͞Εɺߋʹ v_±ʹґͬͯ

w_± = 1 2

∂(v₊+ v₋)∓ (1 + v²)^3/4∂

(1 + v²)^−3/4(v₋− v₊)

= 1 2

∂(v++ v−)± ∂(v+− v−)± [(1 + v²)^3/4∂((1 + v²)^−3/4)](v+− v−)

= ∂v±∓ 3

4(1 + v²)⁻¹v∂v(v+− v−)

= ∂v±∓ 3

4(1 + v²)⁻¹(v++ v−)(∂v++ ∂v−)(v+− v−)

= 1

4(1 + v²)⁻¹[4(1 + v²)∂v±∓ 3(v₊² − v₋²)(∂v++ ∂v−)]

= 1

4(1 + v²)⁻¹[4(1 + v²₊+ 2v+v−+ v²₋)∂v±∓ 3(v²₊− v²₋)∂v+∓ 3(v₊² − v₋²)∂v−]

= 1

4(1 + v²)⁻¹[(4 + v²_±+ 7v_∓² + 8v+v−)∂v±∓ 3(v²₊− v²₋)∂v∓] (2.11)±

ͱද͞ΕΔɻ

u ٴͼ v ͷํఔࣜ (HS) ͔Β (2.7)-(2.10)_±ʹґΓ v_±, w_±ͷํఔࣜΛಋग़͠Α͏ɻA(U) ͷݻ༗஋ λ±͕ (2.5)_±Ͱ༩͑ΒΕΔͱӠ͏ࣄ৘ʹؑΈ v±, w±ʹ ∂t+ λ±∂ Λ࡞༻ͤ͞

(∂t± (1 + v²)^−3/4∂)v±

= 1

2(∂t± (1 + v²)^−3/4∂)(v ∓ (1 + v²)^3/4u)

= 1

2(∂tv ± (1 + v²)^−3/4∂v) ∓ 1

2(1 + v²)^3/4(∂tu ± (1 + v²)^−3/4∂u)

∓ 1 2

(∂t± (1 + v²)^−3/4∂)(1 + v²)^3/4 u

= 1

2(∂u ± (1 + v²)^−3/4∂v) ∓1

2(1 + v²)^3/4((1 + v²)^−3/2∂v ± (1 + v²)^−3/4∂u)

∓ 3 4

(1 + v²)^−1/4v∂tv ± (1 + v²)⁻¹v∂v u

= 1

2(∂u ± (1 + v²)^−3/4∂v) ∓1

2(1 + v²)^−3/4∂v − 1 2∂u

∓ 3

4(1 + v²)^−1/4vu∂u − 3

4(1 + v²)⁻¹vu∂v

=− 3

4(1 + v²)⁻¹

±(1 + v²)^3/4∂u + ∂v uv

=− 3

2(1 + v²)⁻¹w∓

(1 + v²)^−3/4(v−− v+)

(v++ v−)

= 3

2(1 + v²)^−7/4(v₊² − v₋²)w±

= 3 2

1 + (v++ v−)²)_−7/4

(v²₊− v₋²)w±, (2.12)±

(∂_t± (1 + v²)^−3/4∂)w_±

= 1

2(∂_t± (1 + v²)^−3/4∂)(∂v ∓ (1 + v²)^3/4∂u)

= 1

2(∂_t∂v ± (1 + v²)^−3/4∂²v) ∓ 1

2(1 + v²)^3/4(∂_t∂u ± (1 + v²)^−3/4∂²u)

∓ 1

2[(∂_t± (1 + v²)^−3/4∂)(1 + v²)^3/4]∂u

= 1

2(∂²u ± (1 + v²)^−3/4∂²v) ∓ 1

2(1 + v²)^3/4∂

(1 + v²)^−3/2∂v

−1 2∂²u

∓ 3 4

(1 + v²)^−1/4v∂_tv ± (1 + v²)⁻¹v∂v

∂u

=∓ 1 2

(1 + v²)^3/4∂

(1 + v²)^−3/2

∂u

∓ 3

4(1 + v²)^−1/4v(∂u)²− 3

4(1 + v²)⁻¹v∂v∂u

=± 3

2(1 + v²)^−7/4v(∂v)²∓ 3

4(1 + v²)^−1/4v(∂u)²−3

4(1 + v²)⁻¹v∂v∂u

=± 3

2(1 + v²)^−7/4v(w++ w−)²∓ 3

4(1 + v²)^−1/4v

(1 + v²)^−3/4(w+− w−)₂ +3

4(1 + v²)⁻¹v(w++ w−)(1 + v²)^−3/4(w+− w−)

=± 3

4(1 + v²)^−7/4v

2(w++ w−)²− (w+− w−)²± (w₊² − w₋²)

(2.13)±

ΛಘΔɻ

Ҏ্ΛవΊΔͱ࣍ͷ༷ʹͳΔ:

ɹԣ೾໛ܕ ∂_t²q = ∂((1 + (∂q)²)^−1/2∂q) (TW) Ұ֊Խ ? u = ∂tq, v = ∂q

Ұ֊૒ۂܥ ∂_tu = (1 + v²)^−3/2∂v

∂tv = ∂u (HS)

ର֯Խ

ඍ෼ଛࣦফ໓Խ

?

v± = 1

2(v ∓ (1 + v²)^3/4u) ⇔

(1 + v²)^3/4 u = v−− v+

v = v++ v−

w±= 1

2(∂v ∓ (1 + v²)^3/4∂u) ⇔

(1 + v²)^3/4 ∂u = w₋− w₊

∂v = w++ w−

(DHS) ඍ෼ඇଛࣦܕ

Ұ֊૒ۂܥ

(∂t± (1 + v²)^−3/4∂)v± = 3

2(1 + v²)^−7/4(v₊² − v₋²)w±

(∂_t± (1 + v²)^−3/4∂)w_±

=±3

4(1 + v²)^−7/4v

2(w++ w−)²− (w₊− w₋)²± (w₊² − w₋²)

̏ɽඍ෼ඇଛࣦܕҰ֊૒ۂܥͷॳظ஋໰୊ʹؔ͢Δجૅఆཧ

ඍ෼ඇଛࣦܕҰ֊૒ۂܥ (DHS) Λಛੑۂઢ๏Λ༻͍ͯߋʹॻ͖׵͑Α͏ɻ࣌ؒ۠ؒ

I = [−T, T ] ্ͷ C¹∩ W_∞¹ ஋࿈ଓവ਺ v_± ∈ C (I; (C¹∩ W_∞¹)(R)) ͕Ұ૊༩͑ΒΕͨ΋ͷ ͱͯ͠ɺҰ֊ඇઢܕඍ෼ํఔࣜͷॳظ஋໰୊

⎧⎪

⎪⎨

⎪⎪

⎩

∂_tξ_±(t, x) = ∓ (1 + v(t, ξ_±(t, x))²)^−3/4ɹ

=∓ (1 + (v+(t, ξ±(t, x)) + v−(t, ξ±(t, x)))²)^−3/4, ξ±(0, x) = x

(3.1)±

Λ༩͑Δɻ͜Ε͸ [ॎ] ୈ 2 અ (2.2) ͷ α Λ

α = ∓(1 + v²)^−3/4=∓(1 + (v₊+ v₋)²)^−3/4

ͱͨ͠΋ͷʹ֎ͳΒͳ͍ɻT > 0 Λॆ෼খ͘͞औΕ͹ɺ[ॎ] ఆཧ̍ٴͼ̎ΑΓ (3.1)±͸ I = [−T, T ] ্ʹҰҙతͳղ ξ± ∈ X = C¹(I × R) ∩ C(I; ˙W_∞¹) Λ࣋ͪ

T_t^±(x) = ξ_±(t, x), (t, x) ∈ I × R

Ͱఆ·ΔҰܘ਺ม׵଒ (T_t^±; t ∈ I) ͸ [ॎ] ఆཧ̎ͷੑ࣭Λ I0 = I = [−T, T ] ্Ͱຬͨͯ͠

͍Δࣄ͕෼͔Δɻߋʹ [ॎ] ఆཧ̏ͷܥ̍Λ (DHS) ʹద༻͢Ε͹ (v_±, w_±) ͷຬͨ͢΂͖ੵ

෼ํఔࣜ

v±(t) = v±(t, ·)

= v_±⁰ ◦ (T_t^±)⁻¹+

 _t

0 F_±⁽¹⁾(v±, w±)(s, T_s^±◦ (T_t^±)⁻¹(·))ds, (3.2)±

w±(t) = w±(t, ·)

= w_±⁰ ◦ (T_t^±)⁻¹+

 _t

0 F_±⁽²⁾(v±, w±)(s, T_s^±◦ (T_t^±)⁻¹(·))ds (3.3)±

͕ಋ͔ΕΔɻ͜͜ʹ F_±⁽¹⁾(v±, w±) ٴͼ F_±⁽²⁾(v±, w±) ͸ F_±⁽¹⁾(v±, w±) = 3

2(1 + (v++ v−)²)^−7/4(v₋² − v₋²)w±, (3.4)±

F_±⁽²⁾(v±, w±) =±3

4(1 + (v++ v−)²)^−7/4(v++ v−)

2(w++ w−)²− (w+− w−)²± (w₊² − w₋²) (3.5)±

ͱ͠ (v_±⁰, w_±⁰) ͸ (v_±, w_±) ͷॳظ஋ͱ͢Δ:

v_±⁰(x), w⁰_±(x)

= (v±(0, x), w±(0, x)) , x ∈ R

I ্ͷ C¹∩ W_∞¹ ஋࿈ଓവ਺Λ 4 ͭͷ੒෼ͱ͢ΔۭؒΛ X¹ = X¹(I) ͱ͢Δ:

X¹(I) = C

I; (C¹∩ W_∞¹)(R; R⁴)

={(v±, w±); v±, w± ∈ C

I; (C¹∩ W_∞¹)(R; R) } (v±, w±)∈ X¹(I) ʹର͠

(v±, w±) =



±

v±; L^∞(I; W_∞¹)



∨



±

w±; L^∞(I; W_∞¹)



ͱஔ͖ X¹(I) ͷดٿ X_R¹ = X_R¹(I) Λ

X_R¹(I) = {(v_±, w_±)∈ X¹(I); (v_±, w_±) ≤ R}

ͱఆٛ͢ΔɻI ্ͷ C¹∩ W_∞¹ ∩ H¹஋࿈ଓവ਺Λ 4 ͭͷ੒෼ͱ͢ΔۭؒΛ Y¹ = Y¹(I) ͱ

͢Δ:

Y¹(I) = X¹(I) ∩ C

I; H¹(R; R⁴)

= C

I; (C¹∩ W_∞¹ ∩ H¹)(R; R⁴) } (v_±, w_±)∈ Y¹(I) ʹର͠

|||(v±, w±)||| =



±

v±; L^∞(I; H¹∩ W_∞¹)



∨



±

w±; L^∞(I; H¹∩ W_∞¹)



ͱஔ͖ Y¹(I) ͷดٿ Y_R¹ = Y_R¹(I) Λ

Y_R¹(I) = {(v±, w±)∈ Y¹(I); |||(v±, w±)||| ≤ R}

ͱఆٛ͢ΔɻҰ֊ͣͭ׈Β͔͞Λ্ۭ͛ͨؒΛ෉ʑX² = X²(I) ٴͼ Y² = Y²(I) ͱ͠ɺ

ͦͷดٿΛ෉ʑX_R² = X_R²(I) ٴͼ Y_R² = Y_R²(I) ͱ͢Δ:

X²(I) = C

I; (C²∩ W_∞²)(R; R⁴) ,

X_R²(I) = {(v_±, w_±)∈ X²(I); (v_±, w_±) ∨ (∂v_±, ∂w_±) ≤ R},

Y²(I) = C

I; (C²∩ W_∞² ∩ H²)(R; R⁴) ,

Y_R²(I) = {(v±, w±)∈ Y²(I); |||(v±, w±)||| ∨ |||(∂v±, ∂w±)||| ≤ R}

X_R¹, X_R², Y_R¹, Y_R¹͸

d ((v±, w±), (˜v±, ˜w±)) =



±

v±− ˜v±; L^∞(I; L^∞)



∨



±

w±− ˜w±; L^∞(I; L^∞)



ʹґͬͯఆ·Δڑ཭ d Ͱ׬උڑ཭ۭؒͱͳΔɻ

ੵ෼ํఔࣜܥ (3.2)_±-(3.3)_±ͷ࣌ؒہॴղͷଘࡏͱҰҙੑɺॳظ஋ʹؔ͢Δղͷ࿈ଓґ ଘੑɺղͷਖ਼ଇੑ (׈Β͔͞)ɺ࣌ؒۃେղͷଘࡏͱҰҙੑʹब͍ͯɺఆཧͷܗͰవΊͯஔ

͜͏ɻ

ఆཧ̍ (࣌ؒہॴ W_∞¹ ղͷଘࡏͱҰҙੑ)

೚ҙͷ ρ > 0 ʹର͠ T = T (ρ) > 0 ͕ଘࡏ͠



±

v⁰_±; W_∞¹



∨



±

w⁰_±; W_∞¹



≤ ρ

ͳΔ೚ҙͷ (v_±⁰, w_±⁰)∈ (C¹∩ W_∞¹)(R; R⁴) ʹର͠ (3.2)_±ٴͼ (3.3)_±͔Β੒Δੵ෼ํఔࣜܥ

͸ I = [−T, T ] ্

(v_±, w_±)∈ X¹(I)

ͳΔղΛ།Ұͭ࣋ͭɻߋʹ (v±, w±)∈ C¹(I; C ∩ L^∞) Ͱ͋Γ (v±, w±) ͸Ұ֊૒ۂܥ (DHS) Λ I × R ্ຬͨ͢ɻ

ఆཧ̎ (ॳظ஋ʹର͢Δղͷ L^∞࿈ଓґଘੑ)

I ⊂ R Λॳظ࣌ࠁ 0 ΛؚΉ༗քด۠ؒͱ͠ (v±, w±)∈ X¹(I) Λ

(v_±(0), w_±(0)) = (v⁰_±, w⁰_±)∈ C¹∩ W_∞¹ ͳΔ (3.2)_±-(3.3)_±ͷղͰ͋Δͱ͠

(v_±n⁰ , w_±n⁰ ); n ≥ 1

⊂ C¹ ∩ W_∞¹ Λ W_∞¹ ʹԙ͍ͯ༗քͰ L^∞ʹԙ͍ͯ (v_±⁰, w_±⁰) ʹऩଋ͢

Δྻͱ͢Δɻ͜ͷͱ͖೚ҙͷ n ʹର͠ (v±n(0), w±n(0)) = (v⁰_±n, w⁰_±n) ͳΔରԠ͢Δղ (v_±n, w_±n)∈ X¹(I) ͷྻ͕ଘࡏ͠ C(I; L^∞) ʹԙ͍ͯ (v_±, w_±) ʹऩଋ͢Δɻ

ఆཧ̏ (ղͷਖ਼ଇੑ (׈Β͔͞))

I ⊂ R Λॳظ࣌ࠁ 0 ΛؚΉ༗քด۠ؒͱ͠ (v_±, w_±)∈ X¹(I) Λ

(v±(0), w±(0)) = (v⁰_±, w⁰_±)∈ C¹∩ W_∞¹ ͳΔ (3.2)_±-(3.3)_±ͷղͱ͢Δɻ͜ͷͱ͖

(1) (v_±⁰, w_±⁰)∈ C²∩ W_∞² ͳΒ͹ (v±, w±)∈ X²(I) ͱͳΔɻ (2) (v_±⁰, w_±⁰)∈ H¹ͳΒ͹ (v_±, w±)∈ Y¹(I) ͱͳΔɻ

(3) (v_±⁰, w_±⁰)∈ C²∩ W_∞² ∩ H²ͳΒ͹ (v±, w±)∈ Y²(I) ͱͳΔɻ

ఆཧ̐ (ॳظ஋ʹର͢Δղͷ W_∞¹, L^∞∩ L², W_∞¹ ∩ H¹࿈ଓґଘੑ) I ⊂ R Λॳظ࣌ࠁ 0 ΛؚΉ༗քด۠ؒͱ͢Δɻ

(1) (v±, w±) ∈ X²(I) Λ (v±(0), w±(0)) = (v⁰_±, w⁰_±) ∈ C² ∩ W_∞² ͳΔ (3.2)_±-(3.3)_± ͷ ղͰ͋Δͱ͠

(v_±n⁰ , w⁰_±n); n ≥ 1

⊂ C² ∩ W_∞² Λ W_∞² ʹԙ͍ͯ༗քͰ W_∞¹ ʹԙ

͍ͯ (v_±⁰, w_±⁰) ʹऩଋ͢Δྻͱ͢Δɻ͜ͷͱ͖೚ҙͷ n ʹର͠ (v±n(0), w±n(0)) = (v_±n⁰ , w⁰_±n) ͳΔରԠ͢Δղ (v±n, w±n) ∈ X²(I) ͷྻ͕ଘࡏ͠ C(I; W_∞¹) ʹԙ͍ͯ

(v_±, w_±) ʹऩଋ͢Δɻ

(2) (v±, w±)∈ Y¹(I) Λ (v±(0), w±(0)) = (v_±⁰, w_±⁰)∈ C¹∩W_∞¹ ∩H¹ͳΔ (3.2)_±-(3.3)_±ͷ ղͰ͋Δͱ͠

(v_±n⁰ , w_±n⁰ ); n ≥ 1

⊂ C¹∩W_∞¹∩H¹Λ W_∞¹∩H¹ʹԙ͍ͯ༗քͰ L^∞∩L² ʹԙ͍ͯ (v_±⁰, w⁰_±) ʹऩଋ͢Δྻͱ͢Δɻ͜ͷͱ͖೚ҙͷ n ʹର͠ (v±n(0), w±n(0)) = (v_±n⁰ , w⁰_±n) ͳΔରԠ͢Δղ (v±n, w±n) ∈ Y¹(I) ͷྻ͕ଘࡏ͠ C(I; L^∞∩ L²) ʹԙ͍

ͯ (v_±, w_±) ʹऩଋ͢Δɻ

(3) (v±, w±) ∈ Y²(I) Λ (v±(0), w±(0)) = (v_±⁰, w_±⁰) ∈ C²∩ W_∞² ∩ H²ͳΔ (3.2)_±-(3.3)_± ͷղͰ͋Δͱ͠

(v⁰_±n, w_±n⁰ ); n ≥ 1

⊂ C² ∩ W_∞² ∩ H² Λ W_∞² ∩ H² ʹԙ͍ͯ༗

քͰ W_∞¹ ∩ H¹ ʹԙ͍ͯ (v_±⁰, w_±⁰) ʹऩଋ͢Δྻͱ͢Δɻ͜ͷͱ͖೚ҙͷ n ʹର͠

(v±n(0), w±n(0)) = (v_±n⁰ , w⁰_±n) ͳΔରԠ͢Δղ (v±n, w±n) ∈ Y²(I) ͷྻ͕ଘࡏ͠

C(I; W_∞¹ ∩ H¹) ʹԙ͍ͯ (v±, w±) ʹऩଋ͢Δɻ

ఆཧ̑ (࣌ؒۃେ W_∞¹ ղͷଘࡏͱҰҙੑ)

೚ҙͷ (v_±⁰, w_±⁰) ∈ (C¹ ∩ W_∞¹)(R; R⁴) ʹର͠ɺ(3.2)_±-(3.3)_±͸ॳظ࣌ࠁ 0 ΛؚΉ։۠ؒ

(T₋^∗, T₊^∗) ্

(v±, w±)∈ C

(T₋^∗, T₊^∗); (C¹∩ W_∞¹)(R; R⁴) ͳΔۃେղΛ།Ұͭ࣋ͭɻ·ͨ (v_±, w_±)∈ C¹

(T₋^∗, T₊^∗); C ∩ L^∞

Ͱ͋Γ (v_±, w_±) ͸Ұ֊

૒ۂܥ (DHS) Λ (T₋^∗, T₊^∗)× R ্ຬͨ͢ɻߋʹ

• T₊^∗ < +∞ ͳΒ͹ɹ lim

t↑T₊^∗



±

v_±(t); W_∞¹



∨



±

w_±(t); W_∞¹



= +∞

• T₋^∗ > −∞ ͳΒ͹ɹ lim

t↓T₋^∗



±

v±(t); W_∞¹



∨



±

w±(t); W_∞¹



= +∞

࣍અҎ߱ɺఆཧ̍-̑ͷূ໌Λ༩͑Δɻ

̐ɽඍ෼ඇଛࣦܕҰ֊૒ۂܥͷجૅఆཧͷূ໌(ͦͷ 1ɿิॿఆཧ)

؆୯ͷҝ I = [0, T ] ͱ͠ t ≥ 0 ͷ৔߹ͷΈߟ͑Δɻ࣍ͷิ୊͸͠͹͠͹༻͍Δɻ

ิ୊̍ɹ༗քด۠ؒ I = [0, T ] ⊂ R ্ͷ (C¹∩ W_∞¹)(R; R) ஋࿈ଓവ਺

v±, ˜v±∈

I; (C¹∩ W_∞¹)(R; R) ʹର͠ [ॎ] ఆཧ̍ɼ̎Ͱఆ·ΔҰܘ਺ม׵଒Λ෉ʑ

(T_t^±; t ∈ I), ( T_t^±; t ∈ I) ͱ͢Δɻ͜ͷͱ͖࣍ͷධՁ͕੒ཱͭɻ

(1) ೚ҙͷ t ∈ I, x, y ∈ R ʹର͠

∂T_t^±_∞≤ exp

3 2

±

 _t

0 ∂v_±(t^)_∞dt^



, (4.1)_±

∂ T_t^±_∞≤ exp

3 2

±

 _t

0 ∂˜v_±(t^)_∞dt^



, (4.2)_±

|T_t^±(x) − T_t^±(y)| ≤ exp

3 2

±

 _t

0 ∂v_±(t^)_∞dt^



|x − y|

+3 2

±

 _t

0

exp

3 2

±

 _t

t^ ∂v_±(t^)_∞dt^



v_±(t^)− ˜v_±(t^)_∞dt^ (4.3)±

(2) M = M(T ), M = M(T ), N = N(T ), N = N(T ) Λ M = M(T ) = 3

2

±

 _T

0 ∂v±(t^)∞dt, M =  M(T ) = 3

2

±

 _T

0 ∂˜v_±(t^)_∞dt, N = N(T ) = Me^M = M(T ) exp(M(T )), N =  N(T ) = Me^M= M(T ) exp( M(T ))

ͱஔ͘ɻT₀ > 0 ͕ଘࡏ͠ N(T₀)∨ N(T₀) < 1

2ͱͳΔɻ͜ͷͱ͖೚ҙͷ T ≤ T₀ ٴͼ

೚ҙͷ t ∈ [0, T ] ʹର͠ T_t^±ٴͼ T_t^±ͷٯ (T_t^±)⁻¹ٴͼ ( T_t^±)⁻¹͕ଘࡏ࣍͠ͷධՁ͕೚

ҙͷ t, s ∈ [0, T ] ʹରͯ͠੒ཱͭɻ

∂(T_t^±)⁻¹− 1_∞ ≤ N(T )

1− N(T ), (4.4)_±

∂( T_t^±)⁻¹− 1∞ ≤ N(T )

1− N(T ), (4.5)±

(T_t^±)⁻¹− (T_s^±)⁻¹_∞ ≤ |t − s|

1− N(T ), (4.6)_±

(T_t^±)⁻¹− ( T_t^±)⁻¹∞ ≤ 3 2

1 + N(T ) 1− N(T )

±

 _t

0 v±(t^)− ˜v±(t^)∞dt^, (4.7)±

T_s^±◦ (T_t^±)⁻¹− T_s^±◦ ( T_t^±)⁻¹∞

≤ 3 exp(M(T )) 1 1− N(T )

±

 _t∨s

0 v_±(t^)− ˜v_±(t^)_∞dt^, (4.8)_± sup

θ∈[0,1]

∂

T_s^±◦ ( T_t^±)⁻¹+ θ(T_s^±◦ (T_t^±)⁻¹− T_s^±◦ ( T_t^±)⁻¹)

₋₁

∞

≤ 1

1− 2(N(T ) ∨ N(T )) (4.9)±

ิ୊̎ɹ༗քด۠ؒ I = [0, T ] ্ͷ (C²∩ W_∞²)(R; R) ஋࿈ଓവ਺

v±, ˜v± ∈ C

I; (C²∩ W_∞²)(R; R) ʹର͠ิ୊̍ͱಉ༷ͷઃఆͷԼͰ࣍ͷධՁ͕੒ཱͭɻ

(1) ೚ҙͷ t ∈ I ʹର͠

∂²T_t^±∞

≤ 8 exp

3 2

±

 _t

0 ∂v_±(t^)_∞dt^



±

 _t

0

∂²v_±(t^)_∞+∂v_±(t^)²_∞ dt^,

(4.10)±

∂²T_t^±∞

≤ 8 exp

3 2

±

 _t

0 ∂˜v_±(t^)_∞dt^



±

 _t

0

∂²v˜_±(t^)_∞+∂˜v_±(t^)²_∞ dt^,

(4.11)±

|∂T_t^±(x) − ∂ T_t^±(y)|

≤ 12 exp

3 2

±

 _t

0

(∂v_±(t^)_∞+∂˜v_±(t^)_∞)dt^



·

±

 _t

0

∂²v±(t^)∞+∂v±(t^)²_∞

dt^·

±

 _t

0 v±(t^)− ˜v±(t^)∞dt^ (4.12)±

(2) ೚ҙͷ T ≤ T0ٴͼ೚ҙͷ t ∈ [0, T ] ʹର͠

∂²(T_t^±)⁻¹_∞ ≤ 4 e^{M(T )} (1− N(T ))³

±

 _T

0

∂²v_±(t^)_∞+∂v_±(t^)²_∞

dt^, (4.13)_±

∂(T_t^±)⁻¹− ∂( T_t^±)⁻¹_∞

≤ 24 e^M+M (1− N)³

±

 _t

0

(∂²v_±(t^)_∞+∂v_±(t^)²_∞)dt^·

±

 _t

0 v_±(t^)− ˜v_±(t^)_∞dt^ + 18 e^M+M

(1− N)²

±

∂˜v_±L^∞_(L^∞₎+∂v_±L^∞_(L^∞₎

·

±

 _t

0

(∂²v±(t^)∞+∂v±(t^)²_∞)dt^·

±

 _t

0 v±(t^)− ˜v±(t^)∞dt^ + 3

2

e^M (1− N)²

±

 _t

0 ∂v_±(t^)− ∂˜v_±(t^)_∞dt^ (4.14)_±

(ิ୊̍ͷূ໌) ɹ (3.1)_±ʹݟ༷ͨʹ

α = α_± =∓(1 + v²)^−3/4 =∓(1 + (v₊+ v₋)²)^−3/4 ͱ͓͚͹೚ҙͷ (t, x) ∈ I × R ʹର͠

T_t^±(x) = x +

 _t

0 α±(t^, T_t^±(x))dt^ (4.15)±

͕੒ཱͭͷͰ྆ลΛඍ෼͢Δͱ౳ࣜ

∂T_t^±(x) = 1 +

 _t

0 ∂α±(t^, T_t^±(x))∂T_t^±(x)dt^ (4.16)±

͕ಘΒΕΔɻ͜ΕΑΓෆ౳ࣜ

|∂T_t^±(x) − 1| ≤

 _t

0 ∂α±(t^)∞|∂T_t^±(x) − 1|dt^+

 _t

0 ∂α±(t^)∞dt^ (4.17)±

͕ै͏ɻ(4.17)_±ʹάϩϯ΢Υʔϧͷิ୊Λద༻ͯ͠ಘΒΕΔෆ౳ࣜ

|∂T_t^±(x) − 1| ≤ exp

 _t

0 ∂α±(t^)∞dt^



− 1 ΑΓ

∂T_t^±− 1_∞≤ exp

 _t

0 ∂α_±(t^)_∞



− 1 (4.18)±

ΛಘΔɻ͜͜Ͱ

∂α_± =±3

2(1 + v²)^−7/4v∂v, ∂α_±(t)_∞≤ 3

2∂v(t)_∞≤ 3

2(∂v₊(t)_∞+∂v₋(t)_∞) = 3 2

±

∂v_±(t)_∞