Joint work with Arturo de Pablo, Ana Rodr´ıguez and Juan Luis V ´azquez

Applications of two new functional inequalities to fractional diffusion I

Fernando Quir ´os

Universidad Aut ´onoma de Madrid

Joint work with Arturo de Pablo, Ana Rodr´ıguez and Juan Luis V ´azquez

Nonlinear PDEs and functional inequalities U. Aut ´onoma de Madrid

September 19th, 2011

Fractional porous medium equation

(FPME) u t + (−∆) ^σ/2 (|u| ^m−1 u) = 0, x ∈ Ω, t > 0

I D OMAIN : Ω = R ^N or Ω bounded

I P ARAMETERS : 0 < σ < 2, m > 0 I I NITIAL DATA : u(·, 0) = f ∈ L ¹ (Ω)

I B OUNDARY DATA (Ω BOUNDED ): u = 0, x ∈ ∂Ω, t > 0

(−∆) ^σ/2

A non-negative, self-adjoint, linear operator

A ^α u(x) = 1 Γ(−α)

Z _∞

0

¡ e ^−tA u(x) − u(x) ¢ dt t ^1+α

• λ ^α = 1 Γ(−α)

Z _∞

0

³

e ^−tλ − 1

´ dt

t ^1+α , λ > 0

(−∆) ^σ/2 u(x) = 1 Γ(− ^σ ₂ )

Z _∞

0

¡ e ^t∆ u(x) − u(x) ¢ dt

t ¹⁺

Non-local operator Ω = R ^N

I F ¡

(−∆) ^σ/2 u ¢

(ξ) = |ξ| ^σ F(u)(ξ) I (−∆) ^σ/2 u(x) = C _N,σ P.V.

Z

R

u(x + y) − u(x)

|y| ^N+σ dy

Ω bounded, homogeneous Dirichlet B. C.

I u = X ∞

k=1

u k ϕ _k ⇒ (−∆) ^σ/2 u = X ∞

k=1

λ ^σ/2 _k u k ϕ _k

 



−∆ϕ _k = λ _k ϕ _k , x ∈ Ω,

ϕ _k = 0, x ∈ ∂Ω

Motivation

• Non-linear generalization of the fractional heat equation (FHE) u t + (−∆) ^σ/2 u = 0

• Non-local generalization of the porous medium equation (PME) u t − ∆u ^m = 0

(Other possible generalizations [Caffarelli-V ´azquez, 2009])

• Hydrodynamic limit of zero range processes [Jara, 2009]

Main results, m > m _∗ ≡ (N − σ) ₊ /N

T HEOREM :

f ∈ L ¹ (R ^N ) ⇒ Exists a unique weak solution

I S OME PROPERTIES :

• ∂ _t u ∈ L ^∞ ((τ, ∞) : L ¹ (R ^N )) for every τ > 0

• Conservation of mass

• L ¹ -L ^∞ smoothing effect

• Positivity for f ≥ 0

• H ¨older continuity if either m ≥ 1 or f ≥ 0

• Continuous dependence on the parameters and initial data

Supercritical region: m > m _∗ ≡ (N − σ) ₊ /N

1 0

1 2

m σ

0 1

1 2

m σ

0 1

1 2

m σ

N ≥ 3 N = 2 N = 1

Main results, m ≤ m _∗ ≡ (N − σ) ₊ /N

T HEOREM :

f ∈ L ¹ (R ^N ) ∩ L ^p (R ^N ), p > p _∗ (m) = (1 − m)N/σ

⇓

Exists a unique strong solution

I S OME PROPERTIES :

• Conservation of mass if m = m ∗ , extinction in finite time if m < m ∗

• L ^p -L ^∞ smoothing effect

• Positivity up to the extinction time for f ≥ 0

• H ¨older continuity if f ≥ 0 up to the extinction time

Existence region of strong solutions

0 1

1

m p

m *

N/ σ

Mild solutions [Crandall-Pierre, 1982]

T HEOREM :

m > 0, σ ∈ (0, 2), f ∈ L ¹ (R ^N )

⇓

Exists a unique mild solution (ITD)

I Abstract construction:

• Not enough information to prove that mild ⇒ weak

• No estimates ⇒ no further properties

I However [dPQRV, Preprint]: T HEOREM : mild ⇒ very weak

Integration by parts / Functional space

I Z

R

(−∆) ^σ/2 ψ ϕ = Z

R

|ξ| ^σ ψ ˆ ϕ ˆ = Z

R

|ξ| ^σ/2 ψ|ξ| ˆ ^σ/2 ϕ ˆ

= Z

R

(−∆) ^σ/4 ψ (−∆) ^σ/4 ϕ

I kψk _{H ˙}

= µZ

R

|ξ| ^σ | ψ| ˆ ² dξ

¶ _1/2

= k(−∆) ^σ/4 ψk ₂

Weak/strong solutions Weak (L ¹ -energy) solution

• u ∈ C([0, ∞) : L ¹ (R ^N )), |u| ^m−1 u ∈ L ² _loc ((0, ∞) : ˙ H ^σ/2 (R ^N ))

• Z _∞

0

Z

R

u ∂ϕ

∂t dxds − Z _∞

0

Z

R

(−∆) ^σ/4 (|u| ^m−1 u)(−∆) ^σ/4 ϕ dxds = 0,

∀ϕ ∈ C ¹ _c (R ^N × (0, ∞))

• u(·, 0) = f a.e.

Strong solution

• Weak (L ¹ -energy) solution

• ∂ _t u ∈ L ^∞ ((τ, ∞) : L ¹ (R ^N )) for every τ > 0

Difficulties

I (−∆) ^σ/4 (ϕψ) = ? I (−∆) ^σ/4 (ϕ ◦ ψ) = ?

I (−∆) ^σ/4 ϕ not compactly supported even when ϕ is

• (−∆) ^σ/4 non-local operator

σ-harmonic extension [Caffarelli-Silvestre, 2007]

v = E (g) :

 



L _σ v ≡ div(y ^1−σ ∇v) = 0, R ^N+1 ₊ = {x ∈ R ^N , y > 0}, v(x, 0) = g(x), x ∈ R ^N .

I v(x, y) = Z

R

P(x − ξ, y)g(ξ) dξ, P(x, y) = d N,σ y ^σ

(|x| ² + |y| ² ) ^(N+σ)/2 I lim

y→0

y ^1−σ ∂v

∂y = σ lim

y→0

v(x, y) − v(x, 0) y ^σ

= σ lim

y→0

Z

R

P(x − ξ, y)

y ^σ (g(ξ) − g(x)) dξ = − σd _N,σ

C _N,σ (−∆) ^σ/2 g

∂v

∂y ^σ ≡ µ _σ lim

y→0

y ^1−σ ∂v

∂y = −(−∆) ^σ/2 g

An equivalent local problem I w = E (|u| ^m−1 u), u = | Tr(w)|

⁻¹ Tr(w)

 

 

 



L _σ w = 0, (x, y) ∈ R ^N+1 ₊ , t > 0,

∂w

∂y ^σ − ∂|w|

⁻¹ w

∂t = 0, x ∈ R ^N , y = 0, t > 0, w = |f | ^m−1 f , x ∈ R ^N , y = 0, t = 0.

I Some proofs (e.g. existence) are easier in this formulation!

I Dynamical boundary conditions (Amann, Escher, Fila, Vitillaro, ...) I [Athanasopoulos-Caffarelli, 2009]:

 

 m > 1 

 

⇒

Weak solution (local problem)

Weak (L ¹ -energy) solution

• u ∈ C([0, ∞) : L ¹ (R ^N )), w ∈ L ² _loc ((0, ∞) : X ^σ (R ^N+1 ₊ ));

• Z _∞

0

Z

R

u ∂ϕ

∂t dxds − µ _σ Z _∞

0

Z

R

y ^1−σ h∇w, ∇ϕi dxdyds = 0,

∀ϕ ∈ C ¹ ₀

³

R ^N+1 ₊ × (0, ∞)

´

;

• u(·, 0) = f a.e.

I kvk _X

= Ã

µ _σ Z

R

y ^1−σ |∇v| ² dxdy

! _1/2

Extension / trace operators

I E : ˙ H ^σ/2 (R ^N ) → X ^σ (R ^N+1 ₊ ) isometry [Caffarelli-Silvestre, 2007]

I µ _σ Z

R

y ^1−σ h∇ E (ψ), ∇ E (ϕ)i = Z

R

(−∆) ^σ/4 ψ (−∆) ^σ/4 ϕ

I Tr(Φ ₁ ) = Tr(Φ ₂ ) Z ⇓

R

y ^1−σ h∇ E (ψ), ∇Φ ₁ i = Z

R

y ^1−σ h∇ E (ψ), ∇Φ ₂ i

I Tr : X ^σ (R ^N+1 ₊ ) → H ˙ ^σ/2 (R ^N ) surjective and continuous

Smoothing effect

T HEOREM .

f ∈ L ¹ (R ^N ) ∩ L ^∞ (R ^N ), u strong solution

I m > 0, p > max{1, p ∗ (m)}, p ∗ (m) = (1 − m)N/σ sup

x∈R

|u(x, t)| ≤ C t −N/(N(m−1)+σp) kf k σp/(N(m−1)+σp) p

I m > m ∗

sup

x∈R

|u(x, t)| ≤ C t −N/(N(m−1)+σ))γ kf k σ/(N(m−1)+σ)

1

Hardy-Littlewood-Sobolev’s inequality

I Hardy-Littlewood-Sobolev’s inequality: 1 < r < N/γ, 0 < γ < 2

kvk _r

≤ Ck(−∆) ^γ/2 vk _r , r ₁ = Nr N − γ r

I r = 2, γ = σ/2, σ < N ⇒ H ˙ ^σ/2 (R ^N ) , → L

(R ^N )

I N = 1 ≤ σ < 2?

Stroock-Varopoulos inequality (local case)

I q > 1:

Z

R

|v| ^q−2 v(−∆)v = Z

R

h∇(|v| ^q−2 v), ∇vi

= 4(q − 1) q ²

Z

R

|∇|v| ^q/2 | ²

= 4(q − 1) q ²

Z

R

¯ ¯

¯(−∆) ^1/2 |v| ^q/2

¯ ¯

¯ ²

Stroock-Varopoulos inequality T HEOREM :

Z

R

(|v| ^q−2 v)(−∆) ^σ/2 v ≥ 4(q − 1) q ²

Z

R

¯ ¯

¯(−∆) ^σ/4 |v| ^q/2

¯ ¯

¯ ² , q > 1

Z

R

(|v| ^q−2 v)(−∆) ^σ/2 v = Z

R

(−∆) ^σ/4 (|v| ^q−2 v)(−∆) ^σ/4 v

= µ _σ Z

R

y ^1−σ h∇ E (|v| ^q−2 v), ∇ E (v)i

= µ _σ Z

R

y ^1−σ h∇(| E (v)| ^q−2 E (v)), ∇ E (v)i

= µ _σ 4(q − 1) q ²

Z

R

y ^1−σ |∇(| E (v)| ^q/2 )| ²

Z ¯ ¯

Smoothing effect, σ < N: proof (Moser’s iteration)

I Multiply by |u| ^p

⁻² u, integrate on R ^N × (t _k , t _k+1 ), t _k = (1 − 2 ^−k )t:

Z

R

|u| ^p

(·, t k ) = Z _t

t

Z

R

(−∆) ^σ/2 (|u| ^m−1 u)|u| ^p

⁻² u + Z

R

|u| ^p

(·, t k+1 )

≥ C Z _t

t

k(−∆) ^σ/4 |u|

(·, τ )k ² ₂ dτ (Stroock-Varopoulos)

≥ C Z _t

t

ku(·, τ )k ^p

^+m−1

N−σ

dτ (Hardy-Littlewood-Sobolev)

≥ C2 ^−(k+1) tku(·, t _k+1 )k ^p

^+m−1

N−σ

(L ^p -decay)

Smoothing effect, σ < N: proof (Moser’s iteration)

I ku(·, t _k+1 )k

≤

³ c t

´

pk+m−1

2

k+1

pk+m−1

ku(·, t _k )k

pk pk+m−1

p

I p _k+1 ≡ N(p _k + m − 1)

N − σ > p _k if p ₀ = p > (1 − m)N

σ = p _∗ (m)

I Iteration

A Nash-Gagliardo-Nirenberg type inequality

T HEOREM :

p ≥ 1, r > 1, 0 < γ < min{N, 2}

kvk ^α+1 _r

≤ Ck(−∆) ^γ/2 vk _r kvk ^α _p , r ₂ = N(rp + r − p)

r(N − γ) , α = p(r − 1) r

Proof. Stroock-Varopoulos + Hardy-Littlewood-Sobolev + H ¨older

I r = 2, γ = σ/2, σ < 2N ⇒ H ˙ ^σ/2 (R ^N ) ∩ L ^p (R ^N ) , → L

N(p+2) 2N−σ

(R ^N )

Smoothing effect: proof (Moser’s iteration) I Multiply by |u| ^p

⁻² u, integrate on R ^N × (t _k , t _k+1 ), t _k = (1 − 2 ^−k )t:

Z

R

|u| ^p

(·, t _k ) = Z _t

t

Z

R

(−∆) ^σ/2 (|u| ^m−1 u)|u| ^p

⁻² u + Z

R

|u| ^p

(·, t _k+1 )

≥ C Z _t

t

k(−∆) ^σ/4 |u|

(·, τ )k ² ₂ dτ (Stroock-Varopoulos)

≥ C

ku(·, t k )k ^p _p

Z _t

t

ku(·, τ )k ^p _p

k(−∆) ^σ/4 |u|

(·, τ )k ² ₂ dτ (L ^p -decay)

≥ C

ku(·, t _k )k ^p _p

Z _t

t

ku(·, τ )k ^2p

^+m−1

2N−σ

dτ (Nash-Gagliardo-Nirenberg)

≥ C2 ^−(k+1) t

ku(·, t )k ^2p

^+m−1 (L ^p -decay)

Smoothing effect: proof (Moser’s iteration)

I ku(·, t _k+1 )k

≤

³ c t

´

2pk+m−1

2

k+1

2pk+m−1

ku(·, t _k )k

2pk 2pk+m−1

p

I p _k+1 ≡ N(2p _k + m − 1)

2N − σ > p _k if p ₀ = p > (1 − m)N

σ = p _∗ (m)

I Iteration

References

Athanasopoulos, I.; Caffarelli, L. A.

Continuity of the temperature in boundary heat control problems.

Adv. Math. 224 (2010), no. 1, 293–315.

Caffarelli, L.; Silvestre, L.

An extension problem related to the fractional Laplacian.

Comm. Partial Differential Equations 32 (2007), no. 7-9, 1245–1260.

Caffarelli, L. A.; V ´azquez, J. L.

Nonlinear porous medium flow with fractional potential pressure.

ArXiv:1001.0410 [math.AP], to appear in Arch. Rational Mech. Anal.

Crandall, M. G.; Pierre, M.

Regularizing effects forut+Aϕ(u) =0inL¹. J. Funct. Anal. 45 (1982), no. 2, 191–212.

Jara, M.

Hydrodynamic limit of particle systems with long jumps.

arXiv:0805.1326v2 [math.PR]

de Pablo, A.; Quir ´os, F.; Rodriguez, A.; V ´azquez, J. L.

A fractional porous medium equation.

Adv. Math. 226 (2011), no. 2, 1378–1409.

de Pablo, A.; Quir ´os, F.; Rodriguez, A.; V ´azquez, J. L.

A general fractional porous medium equation.