Introduction
Results
ProofsNavier-Stokes Flow in Spatially Periodic Domains
Jonas Sauer
IRTG Darmstadt Tokyo
Department of Mathematics
Technische Universität Darmstadt

November 5, 2013

Jonas Sauer

Spatially Periodic Navier-Stokes FlowIntroduction
Results
ProofsThe Group Setup
Stokes resolvent problem
λu − ∆u + ∇p = f(R)div u = gin H,in H,in Lqω (H)-settingwith λ ∈ Σθ , θ ∈ (0, π).

Jonas Sauer

Spatially Periodic Navier-Stokes FlowIntroduction
Results
ProofsThe Group Setup
Stokes resolvent problem
λu − ∆u + ∇p = f(R)div u = gin H,in H,in Lqω (H)-settingwith λ ∈ Σθ , θ ∈ (0, π).
H := Rn−1 × R/Z locally compact abelian group

Jonas Sauer

Spatially Periodic Navier-Stokes FlowIntroduction
Results
ProofsThe Group Setup
Stokes resolvent problemλu − ∆u + ∇p = f(R)div u = gin H,in H,in Lqω (H)-settingwith λ ∈ Σθ , θ ∈ (0, π).
H := Rn−1 × R/Z locally compact abelian groupω ≥ 0 weight function in Muckenhoupt class Aq (H)Jonas Sauer

Spatially Periodic Navier-Stokes FlowIntroduction
ResultsProofs

The Group Setup
Stokes resolvent problem
λu − ∆u + ∇p = f(R)div u = gin H,in H,in Lqω (H)-settingwith λ ∈ Σθ , θ ∈ (0, π).
H := Rn−1 × R/Z locally compact abelian groupω ≥ 0 weight function in Muckenhoupt class Aq (H)Use (simultaneous) Fourier transform on group H,
not sequentially performing Fourier series expansion on R/Z
and Fourier transform on Rn−1

Jonas Sauer

Spatially Periodic Navier-Stokes FlowIntroduction
Results
ProofsThe Group Setup
Stokes resolvent problem
λu − ∆u + ∇p = f(R)div u = gin H,in H,in Lqω (H)-settingwith λ ∈ Σθ , θ ∈ (0, π).
H := Rn−1 × R/Z locally compact abelian groupω ≥ 0 weight function in Muckenhoupt class Aq (H)Use (simultaneous) Fourier transform on group H,
not sequentially performing Fourier series expansion on R/Zand Fourier transform on Rn−1

Jonas Sauer

Spatially Periodic Navier-Stokes FlowIntroduction
Results
ProofsMuckenhoupt weights

Definition
1 < q < ∞. ω ≥ 0 is in Muckenhoupt class Aq (Rn ), ifAq (ω) := supU⊂Rn1|U|Zω dx
U1|U| q00Zω− qqqdx

U

where the supremum runs over all balls U in Rn .
Jonas Sauer

Spatially Periodic Navier-Stokes Flow

< ∞,Introduction
ResultsProofs

Muckenhoupt weights
Assume G LCA group furnished with ball-like sets around zero:

Definition
1 < q < ∞. ω ≥ 0 is in Muckenhoupt class Aq (Rn ), if
Aq (ω) := supU⊂Rn

1|U|

Z
ω dx
U1|U|

 q00Zω− qqqdx

U

where the supremum runs over all balls U in Rn .
Jonas Sauer

Spatially Periodic Navier-Stokes Flow< ∞,

Introduction
Results
ProofsMuckenhoupt weights
Assume G LCA group furnished with ball-like sets around zero:Uk , k ∈ Z nested, relatively compact,

Definition
1 < q < ∞. ω ≥ 0 is in Muckenhoupt class Aq (Rn ), if
Aq (ω) := supU⊂Rn

1
|U|

Z
ω dx
U1|U|

 q00Zω− qqqdx

U

where the supremum runs over all balls U in Rn .
Jonas Sauer

Spatially Periodic Navier-Stokes Flow< ∞,

Introduction
Results
ProofsMuckenhoupt weights
Assume G LCA group furnished with ball-like sets around zero:Uk , k ∈ Z nested, relatively compact,
form a local base of 0 ∈ G ,Definition
1 < q < ∞. ω ≥ 0 is in Muckenhoupt class Aq (Rn ), if
Aq (ω) := supU⊂Rn

1
|U|

Z
ω dx
U1
|U|

 q00Zω− qqqdx

U

where the supremum runs over all balls U in Rn .
Jonas Sauer

Spatially Periodic Navier-Stokes Flow< ∞,

Introduction
Results
ProofsMuckenhoupt weights
Assume G LCA group furnished with ball-like sets around zero:Uk , k ∈ Z nested, relatively compact,
form a local base of 0 ∈ G ,satisfy the doubling propertyµ(Uk+1 ) ≤ Aµ(Uk ),A ≥ 1.

Definition
1 < q < ∞. ω ≥ 0 is in Muckenhoupt class Aq (Rn ), if
Aq (ω) := sup

U⊂Rn

1|U|

Z
ω dx
U

1|U|

 q00Zω− qqqdx

U

where the supremum runs over all balls U in Rn .
Jonas Sauer

Spatially Periodic Navier-Stokes Flow< ∞,

Introduction
Results
ProofsMuckenhoupt weights
Assume G LCA group furnished with ball-like sets around zero:Uk , k ∈ Z nested, relatively compact,
form a local base of 0 ∈ G ,satisfy the doubling propertyµ(Uk+1 ) ≤ Aµ(Uk ),A ≥ 1.

Definition
1 < q < ∞. ω ≥ 0 is in Muckenhoupt class Aq ( G ), if
Aq (ω) := supU⊂G1µ(U)

Z
ω dµ
U1µ(U)

 q00Zω− qqqdµU

where the supremum runs over all ball-like sets U in G .
Jonas Sauer

Spatially Periodic Navier-Stokes Flow< ∞,

Introduction
ResultsProofs

Properties of Muckenhoupt weights
Proposition
1Open-ended property : If ω ∈ Aq (Rn ), then there is p < qsuch that ω ∈ Ap (Rn ) (note that this is trivial for p > q)

Jonas Sauer

Spatially Periodic Navier-Stokes FlowIntroduction
Results
ProofsProperties of Muckenhoupt weights
Proposition
12Open-ended property : If ω ∈ Aq (Rn ), then there is p < qsuch that ω ∈ Ap (Rn ) (note that this is trivial for p > q)
Boundedness of maximal operator : ω ∈ Aq (Rn ) if and only if
the maximal operator
Z
1
MRn f (x) := sup|f | dx
Rn ⊃U3x |U|Uis bounded in Lqω (Rn )Jonas Sauer

Spatially Periodic Navier-Stokes FlowIntroduction
Results
ProofsProperties of Muckenhoupt weights
Proposition
12Open-ended property : If ω ∈ Aq (Rn ), then there is p < qsuch that ω ∈ Ap (Rn ) (note that this is trivial for p > q)
Boundedness of maximal operator : ω ∈ Aq (Rn ) if and only if
the maximal operator
Z
1
MRn f (x) := sup|f | dx
Rn ⊃U3x |U|
Uis bounded in Lqω (Rn )3

Extrapolation property : Tλ family of linear operators,uniformly bounded in Lqω (Rn ) for one 1 < q < ∞ and allweights ω ∈ Aq (Rn ) ⇒ R-bounded in Lpν (Rn ) for all
exponents 1 < p < ∞ and all weights ν ∈ Ap (Rn )Jonas Sauer

Spatially Periodic Navier-Stokes FlowIntroduction
Results
ProofsProperties of Muckenhoupt weights
Theorem 1 (S. 2013)
1

Open-ended property : If ω ∈ Aq ( G ), then there is p < qsuch that ω ∈ Ap ( G ) (note that this is trivial for p > q)2

Boundedness of maximal operator : ω ∈ Aq ( G ) if and only ifthe maximal operator
Z1MG f (x) := sup|f | dµG ⊃U3x µ(U) U
is bounded in Lqω ( G )3Extrapolation property : Tλ family of linear operators,
uniformly bounded in Lqω ( G ) for one 1 < q < ∞ and allweights ω ∈ Aq ( G ) ⇒ R-bounded in Lpν ( G ) for all
exponents 1 < p < ∞ and all weights ν ∈ Ap ( G )Jonas Sauer

Spatially Periodic Navier-Stokes FlowIntroduction
Results
ProofsResolvent Problem
Theorem 2, S. 2013n ≥ 2, 0 < θ < π
1 < q < ∞, ω ∈ Aq (H)Jonas Sauer

Spatially Periodic Navier-Stokes FlowIntroduction
Results
ProofsResolvent Problem
Theorem 2, S. 2013n ≥ 2, 0 < θ < π
1 < q < ∞, ω ∈ Aq (H)1,qf ∈ Lqω (H), g ∈ Wω,div(H) and λ ∈ ΣθJonas Sauer

Spatially Periodic Navier-Stokes FlowIntroduction
Results
ProofsResolvent Problem
Theorem 2, S. 2013n ≥ 2, 0 < θ < π
1 < q < ∞, ω ∈ Aq (H)1,qf ∈ Lqω (H), g ∈ Wω,div(H) and λ ∈ Σθ

⇒ unique solution (u, p) ∈ Wω2,q (H) × Ŵω1,q (H) to (R) satisfying
kλu, ∇2 u,∇pkLqω (H)≤ c kf kLqω (H) + (1 + |λ|)kg kW 1,q (H)ω,divJonas Sauer

Spatially Periodic Navier-Stokes FlowIntroduction
Results
ProofsResolvent Problem
Theorem 2, S. 2013n ≥ 2, 0 < θ < π
1 < q < ∞, ω ∈ Aq (H)1,qf ∈ Lqω (H), 0 ∈ Wω,div(H) and λ ∈ Σθ

⇒ unique solution (u, p) ∈ Wω2,q (H) × Ŵω1,q (H) to (R) satisfying
kλu, ∇2 u,∇pkLqω (H)≤ c kf kLqω (H) + (1 + |λ|)kg kW 1,q (H)ω,divJonas Sauer

Spatially Periodic Navier-Stokes FlowIntroduction
Results
ProofsResolvent Problem
Theorem 2, S. 2013n ≥ 2, 0 < θ < π
1 < q < ∞, ω ∈ Aq (H)1,qf ∈ Lqω (H), 0 ∈ Wω,div(H) and λ ∈ Σθ

⇒ unique solution (u, p) ∈ Wω2,q (H) × Ŵω1,q (H) to (R) satisfying
kλu, ∇2 u,∇pkLqω (H)≤ c kf kLqω (H) + (1 + |λ|)kg kW 1,q (H)ω,divTheorem 3, S. 2013
The Stokes operator Aq,ω : Wω2,q (H) ∩ Lqω,σ (H) → Lqω,σ (H) has
maximal Lp -regularity.
Jonas Sauer

Spatially Periodic Navier-Stokes FlowIntroduction
Results
Proofs

Proof of Theorem 2 (ω = 1)
Ingredients
Solution formula (choose η = (ξ 0 , 2πk) as phase variable)1η ⊗ η ˆ iη−1
u=F
id −f + 2 ĝ
λ + |η|2|η|2|η|Jonas Sauer

Spatially Periodic Navier-Stokes FlowIntroduction
Results
Proofs

Proof of Theorem 2 (ω = 1)
Ingredients
Solution formula (choose η = (ξ 0 , 2πk) as phase variable)1η ⊗ η ˆ iη
−1u=F
id −
f + 2 ĝλ + |η|2|η|2|η|g = 0: Transference principle and Mikhlin’s multiplier
theorem.

Jonas Sauer

Spatially Periodic Navier-Stokes FlowIntroduction
Results
Proofs

Proof of Theorem 2 (ω = 1)
Ingredients
Solution formula (choose η = (ξ 0 , 2πk) as phase variable)1η ⊗ η ˆ iη
−1u=F
id −
f + 2 ĝλ + |η|2|η|2|η|g = 0: Transference principle and Mikhlin’s multiplier
theorem.
f = 0: Split function spaces with projection
1Pg (x) :=LZ L

g (x 0 , xn ) dxn = F −1 χ{k=0} ĝ (x)0Jonas Sauer

Spatially Periodic Navier-Stokes FlowIntroduction
Results
Proofs

Proof of Theorem 2 (ω = 1)
Ingredients
Solution formula (choose η = (ξ 0 , 2πk) as phase variable)1η ⊗ η ˆ iη
−1u=F
id −f + 2 ĝ
λ + |η|2|η|2|η|g = 0: Transference principle and Mikhlin’s multiplier
theorem.
f = 0: Split function spaces with projection
1Pg (x) :=LZ L

g (x 0 , xn ) dxn = F −1 χ{k=0} ĝ (x)0

Lq (H) = Lq (Rn−1 ) ⊕ (id − P)Lq (H) and similar for Sobolev
spacesJonas Sauer

Spatially Periodic Navier-Stokes FlowIntroduction
Results
ProofsThank you very much for your attention!

Questions?

Jonas Sauer

Spatially Periodic Navier-Stokes Flow