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Global solvability of the rotating Navier-Stokes-Boussinesq equation with stratification effect with decaying initial data Internatinal Workshop on Mathematical Waseda University, March 8-16, 2010

Hajime Koba

Graduate School of Mathematical Sciences, the University of Tokyo This is a joint work with Tsuyoshi Yoneda (University of Minnesota, IMA)

Hajime Koba (Univ.Tokyo) Mathematical 1 / 17 Features of geophysical fluids: Rotation and Stratification.

Rotation: the earth’s rotation ().

Stratification: Density and Temperature.

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Geophysical fluid dynamics

Fluids: gases and liquids as water and air.

Geophysical fluids: large-scale fluids as the earth’s atmosphere, oceans and climate.

Hajime Koba (Univ.Tokyo) Mathematical Geophysics 2 / 17 ......

Geophysical fluid dynamics

Fluids: gases and liquids as water and air.

Geophysical fluids: large-scale fluids as the earth’s atmosphere, oceans and climate.

Features of geophysical fluids: Rotation and Stratification.

Rotation: the earth’s rotation (Coriolis force).

Stratification: Density and Temperature.

Hajime Koba (Univ.Tokyo) Mathematical Geophysics 2 / 17 . . . . .

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Purpose and Goal

Purpose To study the influence of rotational and stratification effects. . . Goal .. To construct a global-in-time unique smooth solution of. a equation governing geophysical fluids (The equation is a rotating . Navier-Stokes Boussinesq equation with stratification effects).

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Hajime Koba (Univ.Tokyo) Mathematical Geophysics 3 / 17

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Coriolis force

Rotating Navier-Stokes equations:   ∂t u − ν∆u + (u, ∇)u + ∇p = −Ωe3 × u, (RNS)  ∇ · u = 0,

2 2 2 t where ∆ = ∂1 + ∂2 + ∂3，∇ = (∂1, ∂2, ∂3)，×: outer product, u = t (u1, u2, u3)：the velocity， p：the pressure， ν > 0：the viscosity coefficient ， t e3 = (0, 0, 1): the axis of earth, Ω: the rotation rate(rotation parameter), ∇ · u = 0: incompressibility condition.

Hajime Koba (Univ.Tokyo) Mathematical Geophysics 4 / 17 ......

Stratification effects

Navier-Stokes Boussinesq-type equations:   ∂t u − ν∆u + (u, ∇)u + ∇p = ge3θ,  2 3 ∂t θ − κ∆θ + (u, ∇)θ = −N u ,

where θ: the temperature (distribution)， κ > 0: the heat diffusion rate， g > 0: gravity， N: Brunt-Vais¨ al¨ a¨ frequency(stratification parameter).

Hajime Koba (Univ.Tokyo) Mathematical Geophysics 5 / 17 ......

Our system

A rotating Navier-Stokes-Boussinesq equation with stratification effects:  ∂ u − ν∆u + (u, ∇)u + Ωd × u + ∇p = ge θ,  t 3  2 3 3 ∂t θ − κ∆θ + (u, ∇)θ = −N u , (t > 0, x ∈ R ),  (S) ∇ · u = 0, x ∈ R3,  u| = t (u1, u2, u3), x ∈ R3,  t=0 0 0 0  3 θ|t=0 = θ0, x ∈ R .

References: B.Cushman-Roisin, “Introduction to geophysical fluid dynamics”, Prentice Hall, 1994. J.Pedolosky, “Geophysical Fluid Dynamics”, volume 2nd edition. Springer verlag,1987.

Hajime Koba (Univ.Tokyo) Mathematical Geophysics 6 / 17 ......

Preliminary results

Global solvability of (RNS);

Giga-Inui-Mahalov-Saal2008, Hieber-Shibata2009: The case when initial data is sufficiently small.

Chemin-Desjardins-Gallagher-Grenier2006: The case when initial data is not restricted, but Ω >> 1.

Hajime Koba (Univ.Tokyo) Mathematical Geophysics 7 / 17 ......

Formulation √ t 1 2 3 4 t 1 2 3 g v = (v , v , v , v ) := (u , u , u , N θ)  dv  + Pe(A + B)v = −Pe(v, ∇)v, (S)  dt  e v|t=0 = Pv0, √ t 1 2 3 4 t 1 2 3 g v0 = (v0 , v0 , v0 , v0 ) := (u0, u0, u0, N θ0),

    −ν∆ 0 0 0  0 −Ω 0 0       0 −ν∆ 0 0  Ω 0 0 0  A =   , B =  √  ,  0 0 −ν∆ 0  0 0 0 −N g    √  0 0 0 −κ∆ 0 0 N g 0

Pe = P ⊕ 1 with P: a Helmholtz projection, i.e. Pe :[L p]4 3 v 7→ Pve ∈ [L p]4 with ∇ · (Pve ) = 0.

Hajime Koba (Univ.Tokyo) Mathematical Geophysics 8 / 17 . .

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Main result

. Theorem 1 ˙ 2 Let ν, κ, Ω, N, g > 0 such that ν = κ. Assume that u0 ∈ Hσ and 1 ˙ 2 1 2 3 θ0 ∈ H with ∂2u0 − ∂1u0 = 0 in R . Then there exists a positive number M0 = M0(ν, ku0k 1 , kθ0k 1 ) such that there exists a H˙ 2 H˙ 2 global-in-time unique solution (u, θ) to the system (S) with initial data (u0, θ0),  1 1  2 (u, θ) ∈ C([0, ∞); H˙ × H˙ 2 ),  σ  2 1 (∇u, ∇θ) ∈ L (R+ : H˙ 2 ), √ √ for N g > M0 with Ω < N g.

s ˙ s 3 0 2 H (R ) ={u ∈ S : kukH˙ s ≡ k(−∆) ukL 2 < ∞},. ˙ s 3 ˙ s 1 2 3 Hσ(R ) ={u ∈ H : ξ1uˆ + ξ2uˆ + ξ3uˆ = 0}.

Hajime Koba (Univ.Tokyo) Mathematical Geophysics 9 / 17 .

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Outline the proof

Linearized system:  dw  + Pe(A + B)w = 0, (LS)  dt  e w|t=0 = w0 = PPr,R v0. Perturbation system: Set δ := v − w,  dδ  + Pe(A + B)δ = −Pe(w, ∇)w − Pe(δ, ∇)w  dt (PS)  − Pe(w, ∇)δ − Pe(δ, ∇)δ,   e δ|t=0 = δ0 = P(I − Pr,R )v0, I is an 4 × 4 identity matrix.

Hajime Koba (Univ.Tokyo) Mathematical Geophysics 10 / 17 . .

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Fourier cut-off operators Pr,R

. Definition s Let w ∈ H˙ (s ∈ R). The operators Pr,R are called Fourier cut-off operators if [ (1) supp Pr,R w ⊂ Dr,R , s (2)Pr,R w → w in H˙ as r → 0 and R → ∞, 3 with Dr,R := {ξ ∈ R ; |ξ| ≤ R, |ξ3| ≥ r, |ξh| ≥ r}. . Here ξ = (ξ1, ξ2, ξ3) and ξh = (ξ1, ξ2).

Remark

By the definition of Pr,R , we see that 0 (3)kPr,R wkH˙ s0 ≤ Cr,R,s,s0 kwkH˙ s for any s, s ∈ R, . where Cr,R,s,s0 does not depend on w. .. .

Hajime Koba (Univ.Tokyo) Mathematical Geophysics 11 / 17 . .

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A priori estimate

Multiplying (PS) by the solution δ to (PS), we obtain

Z t 1 2 2 1 2 kδ(t)k 1 + ν k∇δ(s)k 1 ds = kδ0k 1 ˙ ˙ ˙ 2 H 2 0 H 2 2 H 2 Z t − ((w, ∇)w + (δ, ∇)w + (w, ∇)δ + (δ, ∇)δ, δ) 1 (s)ds. H˙ 2 0 To use absorbing argument, we need estimates of the solution w of (LS). (1)Energy inequality of (LS). (2)Strichartz-type estimates of (LS).

Hajime Koba (Univ.Tokyo) Mathematical Geophysics 12 / 17 . .

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Energy estimate on [T0, ∞)

Proposition (A) . Let ε, ν, r, R, Ω, N, g be any positive numbers such that r < R. 1 1 ˙ 2 ˙ 2 Assume that v0 ∈ Hσ × H and w be the solution of (LS) with initial data Pr,R v0. Then there exist positive constants r0 = r0(ε, ν, kv0k 1 ), H˙ 2 R0 = R0(ε, ν, kv0k 1 , r0) and T0 = T0(ε, ν, kv0k 1 , r0, R0) such that for H˙ 2 H˙ 2 any fixed r < r0 and R > R0

Z ∞ 1 2 ||∇w(τ)|| 1 dτ < ε. 2 H˙ 2 T0 .

Hajime Koba (Univ.Tokyo) Mathematical Geophysics 13 / 17 .

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Strichartz-type estimate on [0, T0] . Proposition (B) For any positive numbers (ν, T, r, R, Ω, N, g) such that r < R and √ 1 1 ˙ 2 ˙ 2 Ω < N g. Let v0 ∈ Hσ × H and w be the solution of (LS) with initial data Pr,R v0. Then there exist positive constants C1 = C1(ν, r, R) and C2 = C2(ν, r, R, g) such that then for all p∈[1, ∞],

1 1 p 1− p ||w||L p ([0,T];L∞(R3)) ≤ M1 kw0k 1 , H˙ 2 with

M1 :=M1(ν, r, R, T, N, g, w0) . 1 1 2 4 C2 :=C1T 4 (k∂2w − ∂1w kL 2 + kw k 1 ) + ||w0|| 1 . 0 0 0 H˙ 2 1 H˙ 2 N 4

Hajime Koba (Univ.Tokyo) Mathematical Geophysics 14 / 17 .

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What is Strichartz estimate?

Strichartz estimate: a basic estimate on dispersive equations.

The symbol of (LS):

−ν|ξ|2t −ν|ξ|2t −ν|ξ|2t wb = [M1e cos αt + M2e sin αt + M3e ]wb0,

q 2 2 2 2 2 2 N gξ1 + N gξ2 + Ω ξ3 α = q , Mj : 4 × 4 coefficient matrix. 2 2 2 ξ1 + ξ2 + ξ3

Hajime Koba (Univ.Tokyo) Mathematical Geophysics 15 / 17 The case when initial data satisfies the condition: 1 2 ∂2w0 − ∂1w0 = 0 (1 ≤ p ≤ ∞) 1 + T β kwkL∞([0,T];L p (R3)) ≤ C( )kw0k 1 , Nα H˙ 2 C = C(r, R), α = α(p), β = β(p) ≥ 0.

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Our Strichartz-type estimate

The case when initial data with no condition: (1 ≤ p ≤ ∞)

1 β kwkL∞([0,T];L p (R3)) ≤ C( + T )kw0k 1 , Nα H˙ 2 C = C(r, R), α = α(p), β = β(p) ≥ 0.

Hajime Koba (Univ.Tokyo) Mathematical Geophysics 16 / 17 ......

Our Strichartz-type estimate

The case when initial data with no condition: (1 ≤ p ≤ ∞)

1 β kwkL∞([0,T];L p (R3)) ≤ C( + T )kw0k 1 , Nα H˙ 2 C = C(r, R), α = α(p), β = β(p) ≥ 0. The case when initial data satisfies the condition: 1 2 ∂2w0 − ∂1w0 = 0 (1 ≤ p ≤ ∞) 1 + T β kwkL∞([0,T];L p (R3)) ≤ C( )kw0k 1 , Nα H˙ 2 C = C(r, R), α = α(p), β = β(p) ≥ 0.

Hajime Koba (Univ.Tokyo) Mathematical Geophysics 16 / 17 ......

Conclusion For all T 0 > 0, we have

Z T 0 E(T 0) := kδ(T 0)k2 + k∇δ(s)k2 ds ˙ 1 ˙ 1 H 2 0 H 2 Z ∞ 0 2 2 ∞ 2 ≤ C(T )(k(I−Pr,R )v0k 1 +kwkL ([0,T0];L ) + k∇w(s)k 1 ds), H˙ 2 H˙ 2 T0

0 0 with C(T ) = C(1 + E(T )), C = C(kv0k 1 ) > 0. H˙ 2 Let ε be any positive number fixed.

By the definition of Pr,R , we choose r0 << 1 and R0 >> 1. By Proposition(A), we can take T0 >> 1. Finally, Proposition(B), we take N >> 1. Then we get the desired result:

1 1 kδk ∞ + k∇δk 2 < ε. L ([0,∞);H˙ 2 ) L (R+:H˙ 2 )

Hajime Koba (Univ.Tokyo) Mathematical Geophysics 17 / 17