Dispersive estimates for rotating fluids and stably stratified fluids

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1 特別講演 17 : msjmeeting-17sep-5i4 Dispersive estimates for rotating fluids and stably stratified fluids ( ) 1. Navier-Stokes (1.1) Boussinesq (1.) t v + (v )v = v q t >, x R, v = t >, x R, t v + (v )v = v q + ηe t >, x R, t η + (v )η = η t >, x R, v = t >, x R. (1.1) (1.) v = (v 1 (t, x), v (t, x), v (t, x)) T, q = q(t, x), η = η(t, x) e = (,, 1) T (1.1), (1.) (1.1) ( ) (v Ω, q Ω ) v Ω (x) = Ωe x, q Ω (x) = Ω (x 1 + x ). Ω R \ } q Ω O(t) cos t sin t O(t) = sin t cos t 1 (1.1) (v, q) u(t, x) = O(Ωt) T v(t, O(Ωt)x) v Ω (x), p(t, x) = q(t, O(Ωt)x) q Ω (x) (v Ω, q Ω ) (u, p) (u, p) Coriolis Navier-Stokes t u + (u )u = u Ωe u p t >, x R, u = t >, x R. (1.) ( :15H546) takada@math.kyushu-u.ac.jp

2 (1.) (1.) q s = η s (v s, η s, q s ) v s, η s (x ) = a + bx, q s (x ) = c + ax + b x (a, b, c R). b = η s > N = b (1.) (η, q) (η s, q s ) θ(t, x) = η(t, x) η s (x ), p(t, x) = q(t, x) q s (x ) (v, θ, p) t v + (v )v = v p + θe t >, x R, t θ + (v )θ = θ N v t >, x R, v = t >, x R. (1.4) (1.), (1.4) Coriolis Ωe u (θe, N v ) T Helmholtz ( ) (δ jk + R j R k ) P = (δ jk + R j R k ) 1 j,k P = 1 j,k 1 R j = xj ( ) 1/ Riesz Coriolis J r, J s 1 J r = 1, J s = 1 1 (1.4) u = (v, θ/n) T = (, ) T Helmholtz (1.), (1.4) Ω 1 : t u u + ΩPJ r Pu + P(u u) =, u =, (1.5) u(, x) = u (x), t u u + NPJ s Pu + P(u u) =, u =, u(, x) = u (x) = (v (x), θ (x)/n). (1.6) PJ r P PJ s P [ ] σ PJ r P = ±ip r (ξ), }, p r (ξ) = ξ ξ, [ ] σ PJ s P = ±ip s (ξ),, }, p s (ξ) = ξ h ξ, ξ h = (ξ 1, ξ )

3 + ΩPJ r P + NPJ s P e t( ΩPJ rp) u = 1 et e iωtp r(d) (I + R)u + 1 et e iωtp r(d) (I R)u e t( NPJsP) u = e t e intps(d) P + u + e t e intps(d) P u + e t P u. I R R R = R R 1, Pj u (ξ) = û (ξ), a j (ξ) C 4 a j (ξ) (j = ±, ), R R 1 a + (ξ) = 1 ξh ξ iξ 1 ξ iξ ξ i ξ h ξ h ξ, a (ξ) = a + (ξ), a (ξ) = 1 ξ h ξ ξ 1 (1.7) a ± (ξ), a (ξ)} PJ s P ±ip s (ξ), } e ±iωtpr(d) e ±intps(d) Fourier e ±iωtpr(d) f(x) = e ix ξ±iωtpr(ξ) f(ξ) dξ, pr (ξ) = ξ R ξ, (1.8) e ±intps(d) f(x) =. R e ix ξ±intp s(ξ) f(ξ) dξ, ps (ξ) = ξ h ξ. (1.9) Coriolis Navier-Stokes (1.5) Babin- Mahalov-Nicolaenko [1] T (1.5) u H s (T ) (s > 1/) Ω = Ω (u ) Ω Ω (1.5) u C([, ); H s (T )) L (, ; H s+1 (T )) Resonant Kishimoto-Yoneda [15] R Chemin- Desjardins-Gallagher-Grenier [5] u Ḣ 1 (R ) Coriolis u Ḣs (R ) (1/ < s < /4) Ω u H s [11] Ω R Giga-Inui-Mahalov-Saal [8], Hieber-Shibata [9], Konieczny-Yoneda [18], Ito-Kato [14] [1] Navier-Stokes Sawada [1], Giga-Inui-Mahalov-Matsui [7] u Ḣs (R ) (1/ < s < 5/4) [1] [1] T Ω

4 Boussinesq (1.6) Charve [,4] (1.6) Coriolis Ωe v [,4] (1.7) u = u = P + u +P u +P u P ± u Ḣ 1 (R ), P u H 1 +ε (R ) (ε > ) u N = N (u ) Ω = Ω (u ) N N Ω Ω (1.6) u L (, ; Ḣ 1 (R )) L (, ; Ḣ (R )) Koba-Mahalov-Yoneda [16] N Ω u Ḣ 1 (R ) Ibrahim-Yoneda [1] Babin-Mahalov-Nicolaenko [] (1.8), (1.9) p r (ξ) p s (ξ) Littlewood-Paley U r ± (t)f(x) = e ix ξ±itpr(ξ) ψ(ξ) f(ξ) dξ, p r (ξ) = ξ R ξ, (.1) U s ± (t)f(x) = R e ix ξ±itps(ξ) ψ(ξ) f(ξ) dξ, p s (ξ) = ξ h ξ, (t, x) R1+. (.) ψ S (R ) supp ψ ξ } ψ(ξ) = 1 ( 1 ξ ) Dutrifoy [6] [1] U ± r (t) U ± r (t)f L C log(e + t ) 1 + t } 1 fl 1.1. C = C(ψ) U r ± (t)f L C(1 + t ) 1 f L 1 t R f L 1 (R ) 1 U s ± (t).. C = C(ψ) U ± s (t)f L C(1 + t ) 1 fl 1 t R f L 1 (R ) 1/ Coriolis Navier- Stokes (1.5).1 [11] u Ḣs (R ), 1/ < s < /4

5 .. (s, p, q) 1 < s < 9 1, 1 + s 9 1 p < 7 1 s 6, max, 1 p + s 1 } < 1 q < 5 8 p + s 4, 4 p 1 q < p. δ = δ(s, p, q) u Ḣs δ Ω 1 (s 1 ) (.) Ω R \ } u Ḣs (R ) ( u = ) (1.5) u C([, ); Ḣs (R )) L q (, ; Ẇ s,p (R )).4.. (.) u Ḣs (R ) Ω = ( ) δ 1 s 1/ u Ḣs Ω Ω (1.5) u.1 [11] s < /4 s < 9/1 Boussinesq (1.6).5. 1/ < s 5/8 δ 1 = δ 1 (s) δ P + u Ḣs + P u Ḣs δ 1 N 1 (s 1 ), P u Ḣ 1 δ (.4) N > u Ḣ 1 (R ) Ḣs (R ) ( u = ) (1.6) u C([, ); Ḣ 1 (R )) L 4 (, ; Ẇ 1, (R )).6..5 (.4) (v, θ ) ( h ) 1 ( ) 1 xj x v,j Ḣs δ 1N 1 (s 1 ) (j = 1, ), ( h ) 1 ( ) 1 v, δ Ḣs 1N 1 (s 1 ), θ Ḣs δ 1N 1 (s 1 )+1, ( h ) 1 ( x1 v, x v,1 ) Ḣ 1 δ. δ 1 = δ 1(s) δ 1 N > (1.6) θ Ḣs -..1,. p r (ξ) = ξ / ξ ξ } p s (ξ) = ξ h / ξ ξ h = }.1 p r (ξ) C ( ξ }) Littman

6 .1 ([], [, Corollary., page 4]). ψ C (R d ) p C (supp ψ; R) supp ψ rank p(ξ) k C = C(d, ψ, p) e ix ξ+itp(ξ) ψ(ξ) dξ C(1 + k t ) R d (t, x) R 1+d.1.1 ξ } rank p r (ξ) p r (ξ) = 1 ξ (ξ1 ξ ) ξ 1 ξ ξ ξ 1 (ξ ξ ) ξ 5 ξ 1 ξ ξ ξ (ξ ξ ) ξ (ξ ξ ) ξ 1 (ξ ξ ) ξ (ξ ξ ) ξ ξ h det p r (ξ) = (ξ 1 + ξ)ξ p ξ 9 r (ξ) ξ h = } ξ = } p r (, ξ ) = 1 ξ 5 ξ ξ, p r (ξ h, ) = 1 ξ h ξ 1 ξ ξ 1 ξ rank p r (ξ) p s (ξ) = ξ h / ξ ξ h = }.1 det p s (ξ) = ξ4 ξ 9 ξ h ξ h = } ξ = } Littlewood-Paley ψ h S (R ), ψ S (R) supp ψ h 1 ξ h }, supp ψ 1 ξ } ψ ( k ξ ) = 1 (ξ h, ξ ) j Z ψ h ( j ξ h ) = k Z U s ± (t) U s ± (t)f(x) = e ix ξ±itps(ξ) ψ h ( j ξ h )ψ ( k ξ )ψ(ξ) f(ξ) dξ j,k Z R ( = + + ) e ix ξ±itps(ξ) ψ h ( j ξ h )ψ ( k ξ )ψ(ξ) f(ξ) dξ j, k j, R k 4 k, j 4 =: I 1 (t, x) + I (t, x) + I (t, x). I 1 (det p s (ξ) ) I I I ξ = } k k N e ix ξ±itps(ξ) ψ h (ξ h )ψ ( k ξ ) dξ C(1 + 1 t ) R k k

7 ξ k ξ e ix ξ±itps(ξ) ψ h (ξ h )ψ ( k ξ ) dξ = k R R e i(x h, k x ) ξ±itp s (ξ h, k ξ ) ψ h (ξ h )ψ (ξ ) dξ p s (ξ h, k ξ ) Taylor k 1 p s (ξ h, k ξ ) } ( ) 1 } = 1 k 1 + k ξ = 1 ξ ξ h ξ h + E k(ξ), E k C N (supp ψ h ψ ) C N k (N N }) p (ξ) = 1 ξ h supp ψ h ψ ξ det p (ξ) = ξ4 ξ h 1 k N e ix ξ±itps(ξ) ψ h (ξ h )ψ ( k ξ ) dξ k k R = k e k k R i(x h, k x ) ξ±itp s (ξ h, k ξ ) ψ h (ξ h )ψ (ξ ) dξ = k e ±it e k k R i(x h, k x ) ξ i k t k 1 p s (ξ h, k ξ )} ψh (ξ h )ψ (ξ ) dξ = k e k k R i(x h, k x ) ξ i k tp (ξ)+e k (ξ)} ψ h (ξ h )ψ (ξ ) dξ C } k min 1, (1 + k t ) k k C } min k, k (1 + t ) k k C(1 + t ) 1 I (t, x) j N e ix ξ±itps(ξ) ψ h ( j ξ h )ψ (ξ ) dξ C(1 + t ) R j j I 1 (t, x) det p s (ξ) (1 + t ). Youngwoo Koh Kongju National University Sanghyuk Lee Seoul National University [17, 19]

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