Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova,...

Nonlocal evolution equations, SQG

Peter Constantin

IMA, June 2016

Collaborators

I Tarek Elgindi (Princeton)I Mihaela Ignatova, (Princeton)I Vlad Vicol (Princeton)

Nonlocality in fluids:

I incompressibility: pressureI irrotationality: Dirichlet-to Neumann map in water wavesI anisotropy: dissipative fractional Laplacians: sqg, mhdI confined matter, unconfined fields: electroconvection

Nonlocality in fluids:I incompressibility: pressure

I irrotationality: Dirichlet-to Neumann map in water wavesI anisotropy: dissipative fractional Laplacians: sqg, mhdI confined matter, unconfined fields: electroconvection

Nonlocality in fluids:I incompressibility: pressureI irrotationality: Dirichlet-to Neumann map in water waves

I anisotropy: dissipative fractional Laplacians: sqg, mhdI confined matter, unconfined fields: electroconvection

Nonlocality in fluids:I incompressibility: pressureI irrotationality: Dirichlet-to Neumann map in water wavesI anisotropy: dissipative fractional Laplacians: sqg, mhd

I confined matter, unconfined fields: electroconvection

Nonlocality in fluids:I incompressibility: pressureI irrotationality: Dirichlet-to Neumann map in water wavesI anisotropy: dissipative fractional Laplacians: sqg, mhdI confined matter, unconfined fields: electroconvection

SQG without boundaries

∂tθ + u · ∇θ = 0

Active scalar:u = R⊥θ

in Rd or Td .R = ∇(−∆)−

12

R⊥ = MR with M invertible, antisymmetric. In d = 2, rotation by 90degrees. Makes u divergence-free. In Fourier

R⊥θ(k) = ik⊥

|k |θ(k)

SQG– geophysical origin: Charney. Held, SwansonC-Majda-Tabak: analogies to 3D Euler.


∂tθ + u · ∇θ = 0

Active scalar:

u = R⊥θ

in Rd or Td .R = ∇(−∆)−

12


R⊥θ(k) = ik⊥

|k |θ(k)



∂tθ + u · ∇θ = 0


in Rd or Td .R = ∇(−∆)−

12


R⊥θ(k) = ik⊥

|k |θ(k)



∂tθ + u · ∇θ = 0


in Rd or Td .

R = ∇(−∆)−12


R⊥θ(k) = ik⊥

|k |θ(k)



∂tθ + u · ∇θ = 0


in Rd or Td .R = ∇(−∆)−

12

R⊥ = MR with M invertible, antisymmetric.

In d = 2, rotation by 90degrees. Makes u divergence-free. In Fourier

R⊥θ(k) = ik⊥

|k |θ(k)



∂tθ + u · ∇θ = 0


in Rd or Td .R = ∇(−∆)−

12

R⊥ = MR with M invertible, antisymmetric. In d = 2, rotation by 90degrees.

Makes u divergence-free. In Fourier

R⊥θ(k) = ik⊥

|k |θ(k)



∂tθ + u · ∇θ = 0


in Rd or Td .R = ∇(−∆)−

12

R⊥ = MR with M invertible, antisymmetric. In d = 2, rotation by 90degrees. Makes u divergence-free.

In Fourier

R⊥θ(k) = ik⊥

|k |θ(k)



∂tθ + u · ∇θ = 0


in Rd or Td .R = ∇(−∆)−

12


R⊥θ(k) = ik⊥

|k |θ(k)



∂tθ + u · ∇θ = 0


in Rd or Td .R = ∇(−∆)−

12


R⊥θ(k) = ik⊥

|k |θ(k)

SQG– geophysical origin: Charney.

Held, SwansonC-Majda-Tabak: analogies to 3D Euler.


∂tθ + u · ∇θ = 0


in Rd or Td .R = ∇(−∆)−

12


R⊥θ(k) = ik⊥

|k |θ(k)

SQG– geophysical origin: Charney. Held, Swanson

C-Majda-Tabak: analogies to 3D Euler.


∂tθ + u · ∇θ = 0


in Rd or Td .R = ∇(−∆)−

12


R⊥θ(k) = ik⊥

|k |θ(k)


Motivation: 3D Euler analogy

∇⊥θ like 3D vorticity.

Levels of theta = lines frozen in the flow:

[(∂t + u · ∇), (∇⊥θ · ∇

)] = 0

Kinetic energy conserved:

ddt

|u(x , t)|2dx = 0

∂t (∇⊥θ) + u · ∇(∇⊥θ) = (∇u)(∇⊥θ)

Stretching term like 3D Euler, 6= 0. Blow-up problem, open. Directionof level lines locally nice⇒ depletion of nonlinearity.


∇⊥θ like 3D vorticity. Levels of theta = lines frozen in the flow:

[(∂t + u · ∇), (∇⊥θ · ∇

)] = 0


ddt

|u(x , t)|2dx = 0

∂t (∇⊥θ) + u · ∇(∇⊥θ) = (∇u)(∇⊥θ)




[(∂t + u · ∇), (∇⊥θ · ∇

)] = 0


ddt

|u(x , t)|2dx = 0

∂t (∇⊥θ) + u · ∇(∇⊥θ) = (∇u)(∇⊥θ)

Stretching term like 3D Euler, 6= 0. Blow-up problem, open.

Directionof level lines locally nice⇒ depletion of nonlinearity.



[(∂t + u · ∇), (∇⊥θ · ∇

)] = 0


ddt

|u(x , t)|2dx = 0

∂t (∇⊥θ) + u · ∇(∇⊥θ) = (∇u)(∇⊥θ)

Stretching term like 3D Euler, 6= 0. Blow-up problem, open. Directionof level lines locally nice

⇒ depletion of nonlinearity.



[(∂t + u · ∇), (∇⊥θ · ∇

)] = 0


ddt

|u(x , t)|2dx = 0

∂t (∇⊥θ) + u · ∇(∇⊥θ) = (∇u)(∇⊥θ)


Critical (dissipative) SQG, Rd

∂tθ + u · ∇θ + Λθ = 0,u = R⊥θ

Λ = (−∆)12 , R = ∇Λ−1

In Fourier:Λθ(k) = |k |θ(k), Rθ(k) =

ik|k |θ(k).

I transport + nonlocal diffusion ⇒ L∞

I L∞ not good for CZ operatorsI quasilinear, critical: no room


∂tθ + u · ∇θ + Λθ = 0,u = R⊥θ

Λ = (−∆)12 , R = ∇Λ−1


ik|k |θ(k).


I L∞ not good for CZ operators

I quasilinear, critical: no room


∂tθ + u · ∇θ + Λθ = 0,u = R⊥θ

Λ = (−∆)12 , R = ∇Λ−1


ik|k |θ(k).


I L∞ not good for CZ operatorsI quasilinear, critical: no room

Regularity and Uniqueness

Regularity and uniqueness: with critical dissipation: Cordoba-Wu-C =small data in L∞.

Large data: many methods (by now):1. Kiselev-Nazarov-Volberg: Maximum priciple for a modulus ofcontinuity. adequate h(r) so that

|θ0(x)− θ0(y)| < h(|x − y |)⇒ |θ(x , t)− θ(y , t)| < h(|x − y |)

2. Caffarelli-Vasseur: de Giorgi strategy: from L2 to L∞, from L∞ toCα, from Cα to C∞.3. Kiselev-Nazarov: duality method, co-evolving molecules.4. C-Vicol: nonlinear maximum principle, stability of the “only smallshocks” condition5. C-Tarfulea-Vicol: nonlinear maximum principle, small Holderexponent.


Regularity and uniqueness: with critical dissipation: Cordoba-Wu-C =small data in L∞. Large data: many methods (by now):

1. Kiselev-Nazarov-Volberg: Maximum priciple for a modulus ofcontinuity. adequate h(r) so that




Regularity and uniqueness: with critical dissipation: Cordoba-Wu-C =small data in L∞. Large data: many methods (by now):1. Kiselev-Nazarov-Volberg: Maximum priciple for a modulus ofcontinuity.

adequate h(r) so that




Regularity and uniqueness: with critical dissipation: Cordoba-Wu-C =small data in L∞. Large data: many methods (by now):1. Kiselev-Nazarov-Volberg: Maximum priciple for a modulus ofcontinuity. adequate h(r) so that






2. Caffarelli-Vasseur: de Giorgi strategy:

from L2 to L∞, from L∞ toCα, from Cα to C∞.3. Kiselev-Nazarov: duality method, co-evolving molecules.4. C-Vicol: nonlinear maximum principle, stability of the “only smallshocks” condition5. C-Tarfulea-Vicol: nonlinear maximum principle, small Holderexponent.




2. Caffarelli-Vasseur: de Giorgi strategy: from L2 to L∞, from L∞ toCα, from Cα to C∞.

3. Kiselev-Nazarov: duality method, co-evolving molecules.4. C-Vicol: nonlinear maximum principle, stability of the “only smallshocks” condition5. C-Tarfulea-Vicol: nonlinear maximum principle, small Holderexponent.




2. Caffarelli-Vasseur: de Giorgi strategy: from L2 to L∞, from L∞ toCα, from Cα to C∞.3. Kiselev-Nazarov: duality method, co-evolving molecules.

4. C-Vicol: nonlinear maximum principle, stability of the “only smallshocks” condition5. C-Tarfulea-Vicol: nonlinear maximum principle, small Holderexponent.




2. Caffarelli-Vasseur: de Giorgi strategy: from L2 to L∞, from L∞ toCα, from Cα to C∞.3. Kiselev-Nazarov: duality method, co-evolving molecules.4. C-Vicol: nonlinear maximum principle, stability of the “only smallshocks” condition

5. C-Tarfulea-Vicol: nonlinear maximum principle, small Holderexponent.

Forced Critical SQG

SQG+f

∂tθ + (R⊥θ) · ∇θ + Λθ = f

in T2, with θ0 ∈ H1,

H1(T2) = θ |

T2θdx = 0,

T2|∇θ|2dx <∞

f ∈ H1 ∩ L∞. Let S(t)θ0 denote the solution.

Theorem(C, Tarfulea, Vicol, ’13). ∃!X ⊂ H1,

limt→∞

distH1 (S(t)θ0,X ) = 0,

uniform for θ0 in bounded sets, X invariant S(t)X = X, compact, asnice (C∞) as forces permit, and dF (X ) <∞.

Forced Critical SQG

SQG+f

∂tθ + (R⊥θ) · ∇θ + Λθ = f

in T2,

with θ0 ∈ H1,

H1(T2) = θ |

T2θdx = 0,

T2|∇θ|2dx <∞



limt→∞

distH1 (S(t)θ0,X ) = 0,


Forced Critical SQG

SQG+f

∂tθ + (R⊥θ) · ∇θ + Λθ = f


H1(T2) = θ |

T2θdx = 0,

T2|∇θ|2dx <∞



limt→∞

distH1 (S(t)θ0,X ) = 0,


Forced Critical SQG

SQG+f

∂tθ + (R⊥θ) · ∇θ + Λθ = f


H1(T2) = θ |

T2θdx = 0,

T2|∇θ|2dx <∞

f ∈ H1 ∩ L∞.

Let S(t)θ0 denote the solution.


limt→∞

distH1 (S(t)θ0,X ) = 0,


Forced Critical SQG

SQG+f

∂tθ + (R⊥θ) · ∇θ + Λθ = f


H1(T2) = θ |

T2θdx = 0,

T2|∇θ|2dx <∞



limt→∞

distH1 (S(t)θ0,X ) = 0,


Forced Critical SQG

SQG+f

∂tθ + (R⊥θ) · ∇θ + Λθ = f


H1(T2) = θ |

T2θdx = 0,

T2|∇θ|2dx <∞



limt→∞

distH1 (S(t)θ0,X ) = 0,

uniform for θ0 in bounded sets,

X invariant S(t)X = X, compact, asnice (C∞) as forces permit, and dF (X ) <∞.

Forced Critical SQG

SQG+f

∂tθ + (R⊥θ) · ∇θ + Λθ = f


H1(T2) = θ |

T2θdx = 0,

T2|∇θ|2dx <∞



limt→∞

distH1 (S(t)θ0,X ) = 0,

uniform for θ0 in bounded sets, X invariant S(t)X = X,

compact, asnice (C∞) as forces permit, and dF (X ) <∞.

Forced Critical SQG

SQG+f

∂tθ + (R⊥θ) · ∇θ + Λθ = f


H1(T2) = θ |

T2θdx = 0,

T2|∇θ|2dx <∞



limt→∞

distH1 (S(t)θ0,X ) = 0,

uniform for θ0 in bounded sets, X invariant S(t)X = X, compact, asnice (C∞) as forces permit, and

dF (X ) <∞.

Forced Critical SQG

SQG+f

∂tθ + (R⊥θ) · ∇θ + Λθ = f


H1(T2) = θ |

T2θdx = 0,

T2|∇θ|2dx <∞



limt→∞

distH1 (S(t)θ0,X ) = 0,


Periodic case

I Poisson summation gives kernel for Riesz transform

I Poisson summation gives kernel for Λ

I Nonlinear max principle depends on L∞ aloneI Nonlinear fractional Poincare gives absorbing ball in L∞

Lemma(C-Glatt-Holtz-Vicol) There exists a constant C such that for everyp ≥ 2 even, and every φ,

T2 φ = 0 it holds that

T2φp−1Λφdx ≥ 1

p‖Λ 1

2 (φp2 )‖2

L2 + C‖θ‖pLp

Forgetting initial data: long time L∞ bound depends on forces only.Nonlinear max principle gives (small α) absorbing ball in Cα with sizedepending on forces only. Nonlinear max principle based on Cα

gives H1 absorbing ball (higher regularity as well)

Periodic case

I Poisson summation gives kernel for Riesz transformI Poisson summation gives kernel for Λ





p‖Λ 1

2 (φp2 )‖2

L2 + C‖θ‖pLp



Periodic case


I Nonlinear max principle depends on L∞ alone

I Nonlinear fractional Poincare gives absorbing ball in L∞




p‖Λ 1

2 (φp2 )‖2

L2 + C‖θ‖pLp



Periodic case






p‖Λ 1

2 (φp2 )‖2

L2 + C‖θ‖pLp



Periodic case






p‖Λ 1

2 (φp2 )‖2

L2 + C‖θ‖pLp

Forgetting initial data: long time L∞ bound depends on forces only.

Nonlinear max principle gives (small α) absorbing ball in Cα with sizedepending on forces only. Nonlinear max principle based on Cα


Periodic case






p‖Λ 1

2 (φp2 )‖2

L2 + C‖θ‖pLp

Forgetting initial data: long time L∞ bound depends on forces only.Nonlinear max principle gives (small α) absorbing ball in Cα with sizedepending on forces only.

Nonlinear max principle based on Cα


Periodic case






p‖Λ 1

2 (φp2 )‖2

L2 + C‖θ‖pLp



In this talk: bounded domains

I Lower bounds for Dirichlet Λ in bounded domains (nonlinear maxprinciple) (Ignatova, C)

I Global solutions for a model of electroconvection (Elgindi,Ignatova, Vicol, C)

I Global Holder bounds for critical SQG in bounded domains(Ignatova, C)

Basics in bounded domainsΩ ⊂ Rd , smooth boundary.

Normalized igenfunctions,

Ω

w2j dx = 1

with homogeneous Dirichlet BC:

−∆wj = λjwj ,

Well-known:0 < λ1 ≤ ... ≤ λj →∞

−∆ is a positive selfadjoint operator in L2(Ω) with domainD (−∆) = H2(Ω) ∩ H1

0 (Ω). The ground state w1 is positive and

c0d(x) ≤ w1(x) ≤ C0d(x),

whered(x) = dist(x , ∂Ω)

Basics in bounded domainsΩ ⊂ Rd , smooth boundary. Normalized igenfunctions,

Ω

w2j dx = 1


−∆wj = λjwj ,

Well-known:0 < λ1 ≤ ... ≤ λj →∞



c0d(x) ≤ w1(x) ≤ C0d(x),



Ω

w2j dx = 1


−∆wj = λjwj ,

Well-known:0 < λ1 ≤ ... ≤ λj →∞


0 (Ω).

The ground state w1 is positive and

c0d(x) ≤ w1(x) ≤ C0d(x),



Ω

w2j dx = 1


−∆wj = λjwj ,

Well-known:0 < λ1 ≤ ... ≤ λj →∞



c0d(x) ≤ w1(x) ≤ C0d(x),


Functional Calculus

(−∆)α f =∞∑j=1

λαj fjwj

withfj =

Ω

f (y)wj (y)dy

and for f ∈ D ((−∆)α) = f | (λαj fj ) ∈ `2(N). We denote by

ΛD = (−∆)12

It is well-known and easy to show that (Kato conjecture, trivial case)

D (ΛD) = H10 (Ω).

Indeed, for f ∈ D (−∆) we have

‖∇f‖2L2(Ω) =

Ω

f (−∆) fdx = ‖ΛDf‖2L2(Ω).

Fractional powers in terms of heat kernel

No explicit formulas for kernels.

Identity

λα = cα ∞

0(1− e−tλ)t−1−αdt ,

with1 = cα

∞0

(1− e−s)s−1−αds,

valid for 0 ≤ α < 1. Representation

(Λ2αD f )(x) = ((−∆)α f ) (x) = cα

∞0

[f (x)− et∆f (x)

]t−1−αdt

for f ∈ D ((−∆)α) = D(

(−ΛD)2α)

.


No explicit formulas for kernels. Identity

λα = cα ∞

0(1− e−tλ)t−1−αdt ,

with1 = cα

∞0

(1− e−s)s−1−αds,

valid for 0 ≤ α < 1.

Representation

(Λ2αD f )(x) = ((−∆)α f ) (x) = cα

∞0

[f (x)− et∆f (x)

]t−1−αdt

for f ∈ D ((−∆)α) = D(

(−ΛD)2α)

.


No explicit formulas for kernels. Identity

λα = cα ∞

0(1− e−tλ)t−1−αdt ,

with1 = cα

∞0

(1− e−s)s−1−αds,

valid for 0 ≤ α < 1. Representation

(Λ2αD f )(x) = ((−∆)α f ) (x) = cα

∞0

[f (x)− et∆f (x)

]t−1−αdt

for f ∈ D ((−∆)α) = D(

(−ΛD)2α)

.

Electroconvection

Electric field determined by charge density:∇× E = 0,∇ · E = ρ,

in Q ⊂ R3. Boundary conditions at ∂Q. Charge density ρ confined todomain Ω ⊂ Q:

ρ = 2qδΩ

carried by a flow in Ω with ε anisotropic permittivity (an operator).Conducting fluid confined to domain Ω:

∂tq +∇ · (uq + εE) = 0,∂tu + u · ∇u − ν∆u +∇p = qεE , ∇ · u = 0.

Ω ⊂ R2 × 0 (thin film): Fractional Laplacian emerges.

Electroconvection



ρ = 2qδΩ

carried by a flow in Ω with ε anisotropic permittivity (an operator).

Conducting fluid confined to domain Ω:∂tq +∇ · (uq + εE) = 0,∂tu + u · ∇u − ν∆u +∇p = qεE , ∇ · u = 0.


Electroconvection



ρ = 2qδΩ

carried by a flow in Ω with ε anisotropic permittivity (an operator).Conducting fluid confined to domain Ω:

∂tq +∇ · (uq + εE) = 0,∂tu + u · ∇u − ν∆u +∇p = qεE , ∇ · u = 0.


Electroconvection example

Thin film of fluid = 2DNS in fluid region Ω ⊂ R2 × 0.

Electrodsshare boundaries with ∂Ω. Two connected components of ∂Ω kept attwo different voltages, V and 0. Electric field

E = −∇Φ

defined in Q = Ω×R with inhomogeneous boundary conditions for Φ.

−∆3Φ = 2qδΩ, Φ∂Q = V , 0.

Solved by

Φ(x , z) = Φ0(x) +

e−zΛD Λ−1

D q, z > 0,ezΛD Λ−1

D q, z < 0

PermittivityεE = (−∂1Φ,−∂2Φ,0)|Ω


Thin film of fluid = 2DNS in fluid region Ω ⊂ R2 × 0. Electrodsshare boundaries with ∂Ω.

Two connected components of ∂Ω kept attwo different voltages, V and 0. Electric field

E = −∇Φ


−∆3Φ = 2qδΩ, Φ∂Q = V , 0.

Solved by

Φ(x , z) = Φ0(x) +

e−zΛD Λ−1


D q, z < 0



Thin film of fluid = 2DNS in fluid region Ω ⊂ R2 × 0. Electrodsshare boundaries with ∂Ω. Two connected components of ∂Ω kept attwo different voltages, V and 0. Electric field

E = −∇Φ


−∆3Φ = 2qδΩ, Φ∂Q = V , 0.

Solved by

Φ(x , z) = Φ0(x) +

e−zΛD Λ−1


D q, z < 0


Global Regularity for Electroconvection in 2D BoundedDomains

Theorem(C, Elgindi, Ignatova, Vicol) Let Ω ⊂ R2 open, bounded, with smoothboundary. Let u0 ∈ [H1

0 (Ω) ∩ H2(Ω)]2 be divergence-free. Letq0 ∈ H1

0 (Ω) ∩ H2(Ω). Then the electroconvection system ∂tu + u · ∇u +∇p = ν∆u − q∇Λ−1D q − q∇Φ0,

divu = 0,∂tq + u · ∇q + ΛDq = 0

with homogeneous Dirichlet boundary conditions for both u and qhas global unique strong solutions,

u ∈ L∞(0,T ; [H10 (Ω) ∩ H2(Ω)]2) ∩ L2(0,T ; H

52 (Ω)2),

q ∈ L∞(0,T ; W 1,40 (Ω) ∩ H2(Ω)) ∩ L2(0,T ; H

52 (Ω)),

Critical SQG in bounded domains

∂tθ + (R⊥D θ) · ∇θ + ΛDθ = 0

with RD = ∇Λ−1D . Ω ⊂ Rd bounded domain with smooth boundary.

Global weak solutions:

Theorem(C, Ignatova) Let θ0 ∈ L2(Ω) and let T > 0. There exists a weaksolution of critical SQG,

θ ∈ L∞(0,T ; L2(Ω)) ∩ L2(

(0,T ;D(

Λ12D

))satisfying limt→0θ(t) = θ0 weakly in L2(Ω).Local existence of smooth solutions: OK as well.


∂tθ + (R⊥D θ) · ∇θ + ΛDθ = 0

with RD = ∇Λ−1D .

Ω ⊂ Rd bounded domain with smooth boundary.Global weak solutions:


θ ∈ L∞(0,T ; L2(Ω)) ∩ L2(

(0,T ;D(

Λ12D



∂tθ + (R⊥D θ) · ∇θ + ΛDθ = 0




θ ∈ L∞(0,T ; L2(Ω)) ∩ L2(

(0,T ;D(

Λ12D



∂tθ + (R⊥D θ) · ∇θ + ΛDθ = 0




θ ∈ L∞(0,T ; L2(Ω)) ∩ L2(

(0,T ;D(

Λ12D

))satisfying limt→0θ(t) = θ0 weakly in L2(Ω).

Local existence of smooth solutions: OK as well.


∂tθ + (R⊥D θ) · ∇θ + ΛDθ = 0




θ ∈ L∞(0,T ; L2(Ω)) ∩ L2(

(0,T ;D(

Λ12D


Holder bounds for critical SQG in bounded domains

Theorem(C, Ignatova) Let θ(x , t) be a smooth solution of

∂tθ + (R⊥D θ) · ∇θ + ΛDθ = 0

in the smooth bounded domain Ω. There exists 0 < α < 1 dependingonly on ‖θ0‖L∞(Ω), and a constant Γ > 0 depending on the domain Ωsuch that, for any ` > 0 sufficiently small

supd(x)≥`, |h|≤ `

16 , t≥0

|θ(x + h, t)− θ(x , t)||h|α

≤ ‖θ0‖Cα + Γ`−α‖θ0‖L∞(Ω)

holds.recall: d(x) = dist(x , ∂Ω). Proof based on adequate nonlinear maxbounds and commutators.Higher interior regularity and global existence follow.



∂tθ + (R⊥D θ) · ∇θ + ΛDθ = 0


supd(x)≥`, |h|≤ `

16 , t≥0

|θ(x + h, t)− θ(x , t)||h|α

≤ ‖θ0‖Cα + Γ`−α‖θ0‖L∞(Ω)

holds.

recall: d(x) = dist(x , ∂Ω). Proof based on adequate nonlinear maxbounds and commutators.Higher interior regularity and global existence follow.



∂tθ + (R⊥D θ) · ∇θ + ΛDθ = 0


supd(x)≥`, |h|≤ `

16 , t≥0

|θ(x + h, t)− θ(x , t)||h|α

≤ ‖θ0‖Cα + Γ`−α‖θ0‖L∞(Ω)

holds.recall: d(x) = dist(x , ∂Ω).

Proof based on adequate nonlinear maxbounds and commutators.Higher interior regularity and global existence follow.



∂tθ + (R⊥D θ) · ∇θ + ΛDθ = 0


supd(x)≥`, |h|≤ `

16 , t≥0

|θ(x + h, t)− θ(x , t)||h|α

≤ ‖θ0‖Cα + Γ`−α‖θ0‖L∞(Ω)

holds.recall: d(x) = dist(x , ∂Ω). Proof based on adequate nonlinear maxbounds and commutators.

Higher interior regularity and global existence follow.



∂tθ + (R⊥D θ) · ∇θ + ΛDθ = 0


supd(x)≥`, |h|≤ `

16 , t≥0

|θ(x + h, t)− θ(x , t)||h|α

≤ ‖θ0‖Cα + Γ`−α‖θ0‖L∞(Ω)

holds.recall: d(x) = dist(x , ∂Ω). Proof based on adequate nonlinear maxbounds and commutators.Higher interior regularity and global existence follow.

Elements of the proof

I Gaussian bounds for heat kernel; cancellation due to translationinvariance effective for small time.

I Nonlinear maximum principle (lower bound for ΛD) givingsmoothing and a strong boundary repulsion damping effect.

I Good cutoff χ and bound for the commutator [δh,ΛD] away fromboundary; (the most expensive item, fighting boundary repulsion)

I Finite difference bounds for Riesz transforms using the nonlinearmax principle bound in its finite difference variant.

Bounds for heat kernelWe use precise upper and lower bounds for the kernel HD(t , x , y) ofthe heat operator,

(et∆f )(x) =

Ω

HD(t , x , y)f (y)dy .

These are as follows (Davies, Q.S Zhang): There exists a timeT > 0 depending on the domain Ω and constants c, C, k , K ,depending on T and Ω such that

c min(

w1(x)|x−y| ,1

)min

(w1(y)|x−y| ,1

)t−

d2 e−

|x−y|2kt ≤

HD(t , x , y) ≤ C min(

w1(x)|x−y| ,1

)min

(w1(y)|x−y| ,1

)t−

d2 e−

|x−y|2Kt

holds for all 0 ≤ t ≤ T . Moreover

|∇xHD(t , x , y)|HD(t , x , y)

≤ C

1d(x) , if

√t ≥ d(x),

1√t

(1 + |x−y|√

t

), if√

t ≤ d(x)

holds for all 0 ≤ t ≤ T .


(et∆f )(x) =

Ω


These are as follows (Davies, Q.S Zhang):

There exists a timeT > 0 depending on the domain Ω and constants c, C, k , K ,depending on T and Ω such that

c min(

w1(x)|x−y| ,1

)min

(w1(y)|x−y| ,1

)t−

d2 e−

|x−y|2kt ≤


w1(x)|x−y| ,1

)min

(w1(y)|x−y| ,1

)t−

d2 e−

|x−y|2Kt


|∇xHD(t , x , y)|HD(t , x , y)

≤ C

1d(x) , if

√t ≥ d(x),

1√t

(1 + |x−y|√

t

), if√

t ≤ d(x)



(et∆f )(x) =

Ω



c min(

w1(x)|x−y| ,1

)min

(w1(y)|x−y| ,1

)t−

d2 e−

|x−y|2kt ≤


w1(x)|x−y| ,1

)min

(w1(y)|x−y| ,1

)t−

d2 e−

|x−y|2Kt


Moreover

|∇xHD(t , x , y)|HD(t , x , y)

≤ C

1d(x) , if

√t ≥ d(x),

1√t

(1 + |x−y|√

t

), if√

t ≤ d(x)



(et∆f )(x) =

Ω



c min(

w1(x)|x−y| ,1

)min

(w1(y)|x−y| ,1

)t−

d2 e−

|x−y|2kt ≤


w1(x)|x−y| ,1

)min

(w1(y)|x−y| ,1

)t−

d2 e−

|x−y|2Kt


|∇xHD(t , x , y)|HD(t , x , y)

≤ C

1d(x) , if

√t ≥ d(x),

1√t

(1 + |x−y|√

t

), if√

t ≤ d(x)


Symmetry of bounds for the gradientNote that, in view of

HD(t , x , y) =∞∑j=1

e−tλj wj (x)wj (y)

elliptic regularity estimates and Sobolev embedding which implyuniform absolute convergence of the series (if ∂Ω is smooth enough),we have that

∂β1 HD(t , y , x) = ∂β2 HD(t , x , y) =∞∑j=1

e−tλj∂βx wj (y)wj (x)

for positive t , where we denoted by ∂β1 and ∂β2 derivatives with respectto the first spatial variables, respectively the second spatial variables.

Therefore, the previous gradient bounds result in

|∇y HD(t , x , y)|HD(t , x , y)

≤ C

1d(y) , if

√t ≥ d(y),

1√t

(1 + |x−y|√

t

), if√

t ≤ d(y)

Symmetry of bounds for the gradientNote that, in view of

HD(t , x , y) =∞∑j=1

e−tλj wj (x)wj (y)

elliptic regularity estimates and Sobolev embedding which implyuniform absolute convergence of the series (if ∂Ω is smooth enough),we have that

∂β1 HD(t , y , x) = ∂β2 HD(t , x , y) =∞∑j=1

e−tλj∂βx wj (y)wj (x)

for positive t , where we denoted by ∂β1 and ∂β2 derivatives with respectto the first spatial variables, respectively the second spatial variables.Therefore, the previous gradient bounds result in

|∇y HD(t , x , y)|HD(t , x , y)

≤ C

1d(y) , if

√t ≥ d(y),

1√t

(1 + |x−y|√

t

), if√

t ≤ d(y)

Additional bounds; translation invariance effect

|∇x∇xHD(x , y , t)| ≤ Ct−1− d2 e−

|x−y|2

Kt

holds for t ≤ cd(x)2 and 0 < t ≤ T .

This follows from previousbounds. Important additional bounds we need are

I1(x , t) =

Ω

|(∇x +∇y )HD(x , y , t)|dy ≤ Ct−12 e−

d(x)2

Kt

and

I2(x , t) =

Ω

|∇x (∇x +∇y )HD(x , y , t)|dy ≤ Ct−1e−d(x)2

Kt

valid for t ≤ cd(x)2. These bounds reflect the fact that translationinvariance is remembered in the solution of the heat equation withDirichlet boundary data for short time, away from the boundary. Theyimply that T

0t−

k2 Ij (x , t)dt ≤ d(x)2−j−k

for j = 1,2 and k ≥ 0.


|∇x∇xHD(x , y , t)| ≤ Ct−1− d2 e−

|x−y|2

Kt

holds for t ≤ cd(x)2 and 0 < t ≤ T . This follows from previousbounds.

Important additional bounds we need are

I1(x , t) =

Ω

|(∇x +∇y )HD(x , y , t)|dy ≤ Ct−12 e−

d(x)2

Kt

and

I2(x , t) =

Ω


Kt


0t−


for j = 1,2 and k ≥ 0.


|∇x∇xHD(x , y , t)| ≤ Ct−1− d2 e−

|x−y|2

Kt

holds for t ≤ cd(x)2 and 0 < t ≤ T . This follows from previousbounds. Important additional bounds we need are

I1(x , t) =

Ω

|(∇x +∇y )HD(x , y , t)|dy ≤ Ct−12 e−

d(x)2

Kt

and

I2(x , t) =

Ω


Kt


0t−


for j = 1,2 and k ≥ 0.


|∇x∇xHD(x , y , t)| ≤ Ct−1− d2 e−

|x−y|2

Kt


I1(x , t) =

Ω

|(∇x +∇y )HD(x , y , t)|dy ≤ Ct−12 e−

d(x)2

Kt

and

I2(x , t) =

Ω


Kt

valid for t ≤ cd(x)2.

These bounds reflect the fact that translationinvariance is remembered in the solution of the heat equation withDirichlet boundary data for short time, away from the boundary. Theyimply that T

0t−


for j = 1,2 and k ≥ 0.


|∇x∇xHD(x , y , t)| ≤ Ct−1− d2 e−

|x−y|2

Kt


I1(x , t) =

Ω

|(∇x +∇y )HD(x , y , t)|dy ≤ Ct−12 e−

d(x)2

Kt

and

I2(x , t) =

Ω


Kt

valid for t ≤ cd(x)2. These bounds reflect the fact that translationinvariance is remembered in the solution of the heat equation withDirichlet boundary data for short time, away from the boundary.

Theyimply that T

0t−


for j = 1,2 and k ≥ 0.

The convex damping inequality

Proposition(C, Ignatova) Let Ω be a bounded domain with smooth boundary, let0 < s < 2. There exists a constant C depending on the domain andon s such that for every Φ, a C2 convex function satisfying Φ(0) = 0,and every f ∈ C∞0 (Ω)

Φ′(f )ΛsDf − Λs

D(Φ(f )) ≥ Cd(x)s (f (x)Φ′(f (x))− Φ(f (x)))

holds pointwise in Ω.This generalizes the Cordoba-Cordoba inequality from Rd

(d(x) =∞). The proof follows from approximation, convexity andthe fact that Θ = et∆1 obeys 0 ≤ Θ ≤ 1 and

ΛsD1 =

∞0

t−1− s2 (1−Θ(x , t))dt ≥ cd(x)−s

Dramatically different from Rd !





holds pointwise in Ω.

This generalizes the Cordoba-Cordoba inequality from Rd


ΛsD1 =

∞0

t−1− s2 (1−Θ(x , t))dt ≥ cd(x)−s







(d(x) =∞).

The proof follows from approximation, convexity andthe fact that Θ = et∆1 obeys 0 ≤ Θ ≤ 1 and

ΛsD1 =

∞0

t−1− s2 (1−Θ(x , t))dt ≥ cd(x)−s







(d(x) =∞). The proof follows from approximation, convexity andthe fact that

Θ = et∆1 obeys 0 ≤ Θ ≤ 1 and

ΛsD1 =

∞0

t−1− s2 (1−Θ(x , t))dt ≥ cd(x)−s








ΛsD1 =

∞0

t−1− s2 (1−Θ(x , t))dt ≥ cd(x)−s


The nonlinear bound for derivatives

Theorem(C, Ignatova) Let f ∈ L∞(Ω) ∩ D(Λs

D), 0 ≤ s < 2. Assume that f = ∂qwith q ∈ L∞(Ω) and ∂ a first order derivative. Then there existconstants c, C depending on Ω and s such that

f ΛsDf − 1

2Λs

Df 2 ≥ c‖q‖−sL∞ |fd |

2+s

holds pointwise in Ω, with

|fd (x)| =

|f (x)| if |f (x)| ≥ C‖q‖L∞(Ω)

1d(x) ,

0 if |f (x)| ≤ C‖q‖L∞(Ω)1

d(x) ,

Proof: nontrivial, uses precise bounds on the heat kernel and

f ΛsDf − 1

2Λs

Df 2 ≥ cs

2

∞0

t−1− s2 dt

Ω

HD(t , x , y)(f (x)− f (y))2dy

Good cutoff

LemmaLet Ω be a bounded domain with C2 boundary. For ` > 0 smallenough (depending on Ω) there exist cutoff functions χ with theproperties: 0 ≤ χ ≤ 1, χ(y) = 0 if d(y) ≤ `

4 , χ(y) = 1 for d(y) ≥ `2 ,

|∇kχ| ≤ C`−k with C independent of ` and

Ω

(1− χ(y))

|x − y |d+j dy ≤ C1

d(x)j

and Ω

|∇χ(y)| 1|x − y |d

≤ C1

d(x)

hold for j ≥ 0 and d(x) ≥ `. We will refer to such χ as a “good cutoff”.

Nonlinear bound, finite differencesTheoremLet Ω be a bounded domain with smooth boundary. There exists aconstant C such that, for every f ∈ C∞0 (Ω)

D(f ) = f ΛDf − 12

ΛDf 2 ≥ Cd(x)

f 2(x)

holds for all x ∈ Ω. Let χ ∈ C∞0 (Ω) be a good cutoff with scale ` > 0and let

f (x) = χ(x)(δhq(x)) = χ(x)(q(x + h)− q(x)).

Then

(f ΛDf )(x)− 12

(ΛDf 2)(x) ≥ γ1|h|−1 |fd (x)|3

‖q‖L∞+ γ1

f 2(x)

d(x)

holds pointwise in Ω when |h| ≤ `16 , and d(x) ≥ ` with

|fd (x)| = |f (x)|, if |f (x)| ≥ M‖q‖L∞(Ω)|h|

d(x).

Commutator

Let χ be a good cutoff.

LemmaThere exists a constant Γ0 such that the commutator

Ch(θ) = χδhΛDθ − ΛD(χδhθ)

obeys

|Ch(θ)(x)| ≤ Γ0|h|

d(x)2 ‖θ‖L∞(Ω)

for d(x) ≥ `, |h| ≤ `16 .

Finite difference of Riesz transform

LemmaLet χ be a good cutoff, and let u be defined by

u = R⊥D θ.

Then

|δhu(x)| ≤ C(√

ρD(f )(x) + ‖θ‖L∞

(|h|

d(x)+|h|ρ

)+ |δhθ(x)|

)holds for d(x) ≥ `, ρ ≤ cd(x), f = χδhθ and with C a constant

depending on Ω.

This gives a bound on |h|−1|δhu(x)| which costs D(f ).

Finite difference of Riesz transform

LemmaLet χ be a good cutoff, and let u be defined by

u = R⊥D θ.

Then

|δhu(x)| ≤ C(√

ρD(f )(x) + ‖θ‖L∞

(|h|

d(x)+|h|ρ

)+ |δhθ(x)|

)holds for d(x) ≥ `, ρ ≤ cd(x), f = χδhθ and with C a constant

depending on Ω.This gives a bound on |h|−1|δhu(x)| which costs D(f ).

Idea of proof of Holder bound

Good cutoff, and equation for δhθ imply:

12

Lχ (δhθ)2 + D(f ) + (δhθ)Ch(θ) = 0

withLχg = ∂tg + u · ∇xg + δhu · ∇hg + ΛD(χ2g).

andD(f ) ≥ γ1|h|−1‖θ‖−1

L∞ |(δhθ)d |3 + γ1(d(x))−1|δhθ|2

Multiply by |h|−2α with ε = α‖θ0‖L∞ small. Obtain:

Lχ

(δhθ(x)2

|h|2α

)+

γ1

4d(x)

(δhθ(x)2

|h|2α− Γ1`

−2α‖θ‖2L∞

)≤ 0.



12

Lχ (δhθ)2 + D(f ) + (δhθ)Ch(θ) = 0


andD(f ) ≥ γ1|h|−1‖θ‖−1

L∞ |(δhθ)d |3 + γ1(d(x))−1|δhθ|2

Multiply by |h|−2α with ε = α‖θ0‖L∞ small.

Obtain:

Lχ

(δhθ(x)2

|h|2α

)+

γ1

4d(x)

(δhθ(x)2

|h|2α− Γ1`

−2α‖θ‖2L∞

)≤ 0.



12

Lχ (δhθ)2 + D(f ) + (δhθ)Ch(θ) = 0


andD(f ) ≥ γ1|h|−1‖θ‖−1

L∞ |(δhθ)d |3 + γ1(d(x))−1|δhθ|2

Multiply by |h|−2α with ε = α‖θ0‖L∞ small. Obtain:

Lχ

(δhθ(x)2

|h|2α

)+

γ1

4d(x)

(δhθ(x)2

|h|2α− Γ1`

−2α‖θ‖2L∞

)≤ 0.

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