The quasi-geostrophic equation C regularity for the critical case - - PowerPoint PPT Presentation

The quasi-geostrophic equation

Cα regularity for the critical case

Luis Caffarelli and Alexis Vasseur

caffarel@math.utexas.edu, vasseur@math.utexas.edu

Department of Mathematics University of Texas at Austin

The quasi-geostrophic equation – p. 1/17

Outline of the talk

• The equation
• Discussion on the model
• The result
• The method
• Brief sketch of the proof

The quasi-geostrophic equation – p. 2/17

The equation

We consider the temperature function θ : R2 → R solution to: ∂ ∂tθ + u · ∇θ = −Λθ, (1) u = R⊥θ.

Λθ = |ξ|ˆ θ : models the Ekman pumping,

• R⊥θ = (R2θ, −R1θ): perp of the Riesz transform

where:

• Riθ = ξi

|ξ| ˆ θ. Note that: div u = 0 (incompressible flow).

The quasi-geostrophic equation – p. 3/17

Energy inequality

We consider solutions with finite initial energy (no smallness assumption):

• θ2

0 dx < +∞,

which verify the energy inequality: ∂ ∂t

• θ2 dx + 2
• |Λ1/2θ|2 dx ≤ 0.

(2)

• Solution "a la Leray".
• Weak solutions have been constructed in

Resnick 1995

The quasi-geostrophic equation – p. 4/17

Discussion on the model

• Toy model for the study of possible blow-up in

3D fluid dynamics.

Constantin, Wu 1999; Cordoba, Cordoba 2004...

• If the dissipation term is replaced by −Λαθ,

α > 1 (sub-critical case): existence of global smooth solutions.

Constantin, Wu 1999.

• For the critical case, α = 1 (case studied here):

global solutions were known to exist ONLY for small initial data in L∞.

Constantin, Cordoba, Wu 2001.

The quasi-geostrophic equation – p. 5/17

The result

Theorem 1 (critical case: α = 1) Let θ0 ∈ L2(R2). Then for every t0 > 0, θ lies in Cβ((t0, ∞) × R2) for a β > 0.

• Condition on initial data is finite energy.
• No condition of smallness.
• The results holds true in RN for every N > 1

provided that each component of u is a linear combination of Riesz transform verifying div u = 0.

The quasi-geostrophic equation – p. 6/17

The method

• De Giorgi (57): Cα regularity for elliptic

equation with rough coefficients.

• reminescent of results of Nash in the parabolic

case. Other application of the method in fluid dynamics:

• V.: A new proof of partial regularity for

incompressible Navier-Stokes equations (To appear in NoDEA),

• Mellet, V.: Lp bounds for quantities advected by

a compressible fluid (preprint)

motivated by the study of the velocity field near vacuum for compressible Navier-Stokes equations.

The quasi-geostrophic equation – p. 7/17

First step: L∞ bound

Lemma 1 Any solution to (1)(2) with θ0 ∈ L2(R2) verifies for every ε > 0: θ ∈ L∞((ε, +∞) × R2). It relies on:

• De Giorgi techniques,
• "Maximum principle" due to Cordoba and

Cordoba (2004).

The quasi-geostrophic equation – p. 8/17

Maximum principle

• (Cordoba and Cordoba 2004)

For any convex function φ we have (despite the non-locality of the diffusion term !): −φ′(θ)Λθ ≤ −Λ(φ(θ)).

• Then we can derive an equation on the De Giorgi

truncations: θk = (θ − Ck)+. We have: ∂ ∂t

• θ2

k dx + 2

• |Λ1/2θk|2 dx ≤ 0.

The quasi-geostrophic equation – p. 9/17

2nd step: local energy inequality

• Cα regularity is a LOCAL property.
• We need a local version (in x and t) of the energy

equality on the truncation θk.

• The "maximum principle" of Cordoba and

Cordoba is not sufficient.

• We need to extend θ with a new variable z > 0 to

keep memory of the non locality of the diffusion term.

The quasi-geostrophic equation – p. 10/17

Extension for z > 0

• θ(t, x, z) defined by:

∆x,zθ = 0 x ∈ R2, z > 0, θ(t, x, 0) = θ(t, x).

• We have: Λθ(t, x) = ∂zθ(t, x, 0).
• We get a nice local estimate:

sup

−1≤t≤0

• B1

θ2

k dx

• +

−1

• B1

1 |∇θk|2 dz dx dt ≤ Cp

−2

• B2

θp

k dx dt + −2

• B2

2 θ

2 k dx dt

• .

The quasi-geostrophic equation – p. 11/17

Remarks

• The "maximum principle" of Cordoba and

Cordoba follows naturally in the extension framework.

• The local energy inequality allows to use De

Giorgi techniques.

• However, the degeneracy of the estimates makes

the proof more tedious (some of the terms hold

• nly on the trace θ of θ at z = 0).

The quasi-geostrophic equation – p. 12/17

Oscillation lemma

Q1 = (−1, 0) × (−1, 1)2, Q1/2 = (−1/2, 0) × (−1/2, 1/2)2. uL∞(−1,0;BMO((−1,1)2)) ≤ C,

• Q1 u dx dt
• ≤ C.

Lemma 2 There exists 0 < λ < 1 s.t. for any θ solution to (1)(2) for (t, x) ∈ Q1, with

• sup

Q1

θ − inf

Q1 θ

• ≤ 1.

Then:

• sup

Q1/2

θ − inf

Q1/2 θ

• ≤ λ.

The quasi-geostrophic equation – p. 13/17

Scaling

• The final point is to use the invariant scaling of

the equation: θε(t, x) = θ(εt, εx).

• Applying the oscillation lemma recursively on θεn

with εn = 2−n gives: sup

(t,x),(s,y)∈Qn

|u(t, x) − u(s, y)| ≤ λn, Qn dyadic cubes centered on a fixed t0, x0.

• Finally θ ∈ Cα for α = ln2(1/λ).

The quasi-geostrophic equation – p. 14/17

scaling of the velocity u

from the Riesz transform: u = R⊥θ Thus L∞ bound on θ gives that for any ε > 0: u ∈ L∞((ε, ∞), BMO(R2)).

• BMO estimate is invariant by the scaling.
• But it does not control the mean values !
• Additional difficulties to control the mean drift

from scale to scale !

The quasi-geostrophic equation – p. 15/17

conclusion

• CLAIM: De Giorgi method is a powerful tool for

the study of possible blow-up in PDEs.

• The L∞ part (the easiest one !) works also for

SYSTEM.

The quasi-geostrophic equation – p. 16/17

Proof of the L∞ bound

• Fix t0 > 0 and set Tk = t0(1 − 2−k).
• Consider Ck = M(1 − 2−k), M > 0 chosen later.
• Uk =

∞

Tk

• |Λ1/2θk|2 dx dt +
• supt≥Tk
• θ2

k dx.

• We can compare the energy level Uk from the

previous one Uk−1 in a non linear way: Uk ≤ C M (1 + 1/t0)2CkU γ

k ,

for a γ > 1.

• So for M big enough (depending on t0):

lim

k→∞ Uk = 0,

so θ ≤ M.

The quasi-geostrophic equation – p. 17/17