Theory of fractional nonlinear diffusion equations.

(1)

The Fractional Porous Medium / Fast Diffusion Equation onRd Theory of very weak solutions for FDE/PME Existence and uniqueness of initial traces

Theory of fractional nonlinear diffusion equations.

Estimates and traces.

Matteo Bonforte

Departamento de Matemáticas, Universidad Autónoma de Madrid,

Campus de Cantoblanco 28049 Madrid, Spain

[email protected] http://www.uam.es/matteo.bonforte

Joint work withJ. L. Vázquez (UAM)

Analysis Seminar, Department of Mathematics University of Texas at Austin, TX, USA

April 3, 2013

(2)

Outline

1 The Fractional Porous Medium / Fast Diffusion Equation onR^d The setup of the problem

References

Several definition of solutions

Properties of weak solutions I,m>mc[DPQRV1,DPQRV2]

Properties of weak solutions II,0<m≤mc[DPQRV1,DPQRV2]

2 Theory of very weak solutions for FDE/PME WeightedL¹Estimates. The case0<m<1.

Existence and uniqueness of very weak solutions when0<m<1 Positivity estimates in the Good Fast Diffusion range

Positivity estimates in the Very Fast Diffusion range Positivity estimates in the Porous Medium range

3 Existence and uniqueness of initial traces Understanding the problem

Existence and uniqueness of initial trace for0<m<1.

Existence and uniqueness of initial trace form>1.

(3)

The setup of the problem

The Cauchy Problem for The Fractional Porous Medium / Fast Diffusion Equation inR^d

(CP)







∂tu(t,x) +(−∆)^su^m(t,x) =0, ∀(t,x)∈(0,+∞)×R^d

u(0,x) =u0(x), ∀x∈R^d where 0<s≤1 , m>0 and d≥1.

m>1 is PME, m=1 is HE, 0<m<1 is FDE.

Definition(s) of the fractional Laplacian

Several equivalent definitions of the nonlocal operator(−∆)^σ/2(Laplacian of orderσ):

(1) Fourier transform ((−∆)^σ/2f)^b(x) = (2π|x|)^σˆf(x), can be used for positive and negative values ofσ (2) Hypersingular Kernel (−∆)^σ/2g(x) =cd,σP.V.

Z

R^d

g(x)−g(z)

|x−z|^d+σ dz,

can be used for 0< σ <2 , wherecd,σ=2σ−1σΓ((d+σ)/2)

πd/2Γ(1−σ/2) is a normalization constant.

(3) Associated semigroup (−∆)^σ/2g(x) = 1 Γ(−^σ

2) Z ∞

0

e^t∆g(x)−g(x) dt

t¹⁺^σ2

.

•In this talk we will always let σ=2s, 0<s≤1.

(4)

The setup of the problem

The Cauchy Problem for The Fractional Porous Medium / Fast Diffusion Equation inR^d

(CP)







∂tu(t,x) +(−∆)^su^m(t,x) =0, ∀(t,x)∈(0,+∞)×R^d

u(0,x) =u0(x), ∀x∈R^d where 0<s≤1 , m>0 and d≥1.

m>1 is PME, m=1 is HE, 0<m<1 is FDE.

Definition(s) of the fractional Laplacian

Several equivalent definitions of the nonlocal operator(−∆)^σ/2(Laplacian of orderσ):

(1) Fourier transform ((−∆)^σ/2f)^b(x) = (2π|x|)^σˆf(x), can be used for positive and negative values ofσ (2) Hypersingular Kernel (−∆)^σ/2g(x) =cd,σP.V.

Z

R^d

g(x)−g(z)

|x−z|^d+σ dz,

can be used for 0< σ <2 , wherecd,σ=2σ−1σΓ((d+σ)/2)

πd/2Γ(1−σ/2) is a normalization constant.

(3) Associated semigroup (−∆)^σ/2g(x) = 1 Γ(−^σ

2) Z ∞

0

e^t∆g(x)−g(x) dt

t¹⁺^σ2

.

•In this talk we will always let σ=2s, 0<s≤1.

(5)

References

References:

[BV] M. BONFORTE, J. L. VÁZQUEZ, Quantitative Local and Global A Priori Estimates for Fractional Nonlinear Diffusion Equations,Preprint(2012).

http://arxiv.org/abs/1210.2594

[DPQRV1] A.DEPABLO, F. QUIRÓS, A. RODRIGUEZ, J. L. VÁZQUEZ, A fractional porous medium equationAdv. Math. 226(2011), no. 2, 1378−1409.

[DPQRV2] A.DEPABLO, F. QUIRÓS, A. RODRIGUEZ, J. L. VÁZQUEZ, A general fractional porous medium equation,Comm. Pure Appl. Math.,65(2012), 1242−1284.

[V] J. L. VÁZQUEZ, Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type,Preprint,(2012).

http://arxiv.org/abs/1205.6332v2

More:

[AC] I. ATHANASOPOULOS, L. A. CAFFARELLI, Continuity of the temperature in boundary heat control problems.Adv. Math.224(2010), no. 1, 293−315.

[CS] L. CAFFARELLI, L. SILVESTRE, An extension problem related to the fractional Laplacian.Comm. Partial Differential Equations32(2007), no. 7-9, 1245−1260.

[VBk] J. L. VÁZQUEZ,The porous medium equation. Mathematical theory.Oxford

Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007.

(6)

References

References:

[BV] M. BONFORTE, J. L. VÁZQUEZ, Quantitative Local and Global A Priori Estimates for Fractional Nonlinear Diffusion Equations,Preprint(2012).

http://arxiv.org/abs/1210.2594

[DPQRV1] A.DEPABLO, F. QUIRÓS, A. RODRIGUEZ, J. L. VÁZQUEZ, A fractional porous medium equationAdv. Math. 226(2011), no. 2, 1378−1409.

[DPQRV2] A.DEPABLO, F. QUIRÓS, A. RODRIGUEZ, J. L. VÁZQUEZ, A general fractional porous medium equation,Comm. Pure Appl. Math.,65(2012), 1242−1284.

[V] J. L. VÁZQUEZ, Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type,Preprint,(2012).

http://arxiv.org/abs/1205.6332v2

More:

[AC] I. ATHANASOPOULOS, L. A. CAFFARELLI, Continuity of the temperature in boundary heat control problems.Adv. Math.224(2010), no. 1, 293−315.

[CS] L. CAFFARELLI, L. SILVESTRE, An extension problem related to the fractional Laplacian.Comm. Partial Differential Equations32(2007), no. 7-9, 1245−1260.

[VBk] J. L. VÁZQUEZ,The porous medium equation. Mathematical theory.Oxford

Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007.

(7)

Consider the Fractional Porous Medium / Fast Diffusion Equation inR^d: (EQ) ∂tu(t,x) + (−∆)^su^m(t,x) =0. Definition of Weak and Strong solutions, [DPQRV1, DPQRV2]

A functionuis aweaksolution to (EQ)if

(i) u∈C((0,∞) :L¹(R^d)),|u|^m−1u∈L²_loc((0,∞) : ˙H^s(R^d));

(ii) The following identity holds for everyϕ∈C¹0(R^d×(0,∞)):

Z ∞ 0

Z

R^d

u∂ϕ

∂t dxdt− Z ∞

0

Z

R^d

(−∆)^s/2(|u|^m−1u)(−∆)^s/2ϕ dxdt=0.

Aweaksolution to the Cauchy Problem (CP)is a weak solution to Equation (EQ) such that moreoveru∈C([0,∞) :L¹(R^d))andu(0,·) =u0∈L¹(R^d).

We say that a weak solutionuto the Cauchy Problem (CP) is astrongsolution if moreover∂tu∈L^∞((τ,∞) :L¹(R^d)), for everyτ >0.

The fractional Sobolev spaceH˙^s(R^d)is the completion ofC₀^∞(R^d)with the norm

kψkH˙^s= Z

R^d

|ξ|^σ|ψ|ˆ²dξ 1/2

=k(−∆)^s/2ψk2.

In [DPQRV2] there are other concepts of solutions, such as Mild solutions, Weak solutions via the so-called “Caffarelli-Silvestre extension”...

(8)

Consider the Fractional Porous Medium / Fast Diffusion Equation inR^d: (EQ) ∂tu(t,x) + (−∆)^su^m(t,x) =0. Definition of Weak and Strong solutions, [DPQRV1, DPQRV2]

A functionuis aweaksolution to (EQ)if

(i) u∈C((0,∞) :L¹(R^d)),|u|^m−1u∈L²_loc((0,∞) : ˙H^s(R^d));

(ii) The following identity holds for everyϕ∈C¹0(R^d×(0,∞)):

Z ∞ 0

Z

R^d

u∂ϕ

∂t dxdt− Z ∞

0

Z

R^d

(−∆)^s/2(|u|^m−1u)(−∆)^s/2ϕ dxdt=0.

Aweaksolution to the Cauchy Problem (CP)is a weak solution to Equation (EQ) such that moreoveru∈C([0,∞) :L¹(R^d))andu(0,·) =u0∈L¹(R^d).

We say that a weak solutionuto the Cauchy Problem (CP) is astrongsolution if moreover∂tu∈L^∞((τ,∞) :L¹(R^d)), for everyτ >0.

The fractional Sobolev spaceH˙^s(R^d)is the completion ofC₀^∞(R^d)with the norm

kψkH˙^s= Z

R^d

|ξ|^σ|ψ|ˆ²dξ 1/2

=k(−∆)^s/2ψk2.

In [DPQRV2] there are other concepts of solutions, such as Mild solutions, Weak solutions via the so-called “Caffarelli-Silvestre extension”...

(9)

Consider the Fractional Porous Medium / Fast Diffusion Equation inR^d: (EQ) ∂tu(t,x) + (−∆)^su^m(t,x) =0.

Definition of Very Weak solutions, [BV]

A functionuis avery weaksolution to (EQ)if

(i) u∈C((0,∞) :L¹_loc(R^d)),|u|^m−1u∈L¹_loc (0,∞) :L¹ R^d,(1+|x|)^−(d+2s)dx

; (ii) The following identity holds for everyϕ∈C^∞c ([0,T]×R^d):

Z∞ 0

Z

R^d

u∂ϕ

∂t dxdt− Z∞

0

Z

R^d

|u|^m−1u(−∆)^sϕ dxdt=0. Avery weaksolution to the Cauchy Problem (CP)is very weak sol. to (EQ) such that moreoveru∈C([0,∞) :L¹_loc(R^d))andu(0,·) =u0∈L¹_loc(R^d).

Notice thatu(t,x) =constant is a very weak solution to (EQ).

At least in the Fast Diffusion case0<m<1 , there exist very weak solutions corresponding to initial data that can grow at infinity like|x|^2s^m^−ε, for allε >0.

Such solutions are moreover unique, strictly positive and they exist globally in time. Therefore, by [AC], they are regular.

(10)

Consider the Fractional Porous Medium / Fast Diffusion Equation inR^d: (EQ) ∂tu(t,x) + (−∆)^su^m(t,x) =0.

Definition of Very Weak solutions, [BV]

A functionuis avery weaksolution to (EQ)if

(i) u∈C((0,∞) :L¹_loc(R^d)),|u|^m−1u∈L¹_loc (0,∞) :L¹ R^d,(1+|x|)^−(d+2s)dx

; (ii) The following identity holds for everyϕ∈C^∞c ([0,T]×R^d):

Z∞ 0

Z

R^d

u∂ϕ

∂t dxdt− Z∞

0

Z

R^d

|u|^m−1u(−∆)^sϕ dxdt=0. Avery weaksolution to the Cauchy Problem (CP)is very weak sol. to (EQ) such that moreoveru∈C([0,∞) :L¹_loc(R^d))andu(0,·) =u0∈L¹_loc(R^d).

Notice thatu(t,x) =constant is a very weak solution to (EQ).

At least in the Fast Diffusion case0<m<1 , there exist very weak solutions corresponding to initial data that can grow at infinity like|x|^2s^m^−ε, for allε >0.

Such solutions are moreover unique, strictly positive and they exist globally in time. Therefore, by [AC], they are regular.

(11)

Consider a weak solutionu(t,x)to the Cauchy problem (CP) and let mc= d−2s d . The casem>mc:PME + HE + Good FDE

EXISTENCE AND UNIQUENESS FORL¹ DATA. Letm > mc, then for every u0∈L¹(R^d)there exists aunique strong solutionof Problem (CP).

[The linear casem=1included]

WELLPOSEDNESS. The solution depends continuously on the parameterss∈ (0,1),m>mc, andu0∈L¹(R^d)in the norm of the spaceC([0,∞) : L¹(R^d)).

SMOOTHINGEFFECTS. For allp≥1, ku(t,·)k_L^∞(R^d)≤K∞

t^dϑ^pku0k^2spϑ_Lp(R^p^d), with

ϑp= _2sp+d(m−1)¹ >0 K∞=K∞(m,p,d,s)>0 L¹-CONTRACTION ANDCOMPARISON. Letu0,v0∈L¹(R^d)then

Z

R^d

(u−v)+(x,t)dx≤ Z

R^d

(u0−v0)+(x)dx. MASSCONSERVATION.R

R^du(x,t)dx=R

R^du0(x)dxfor allt≥0.

POSITIVITY. Ifu0≥0 , thenu(t,x)>0 , for allx∈R^dand allt>0.

C^αREGULARITY. If eitherm≥1 oru0 ≥0, thenu∈C^α((0,∞)×R^d)for some 0< α <1.

REMARK: Non-negative initial data produce solutions with the same mass, which are positive andC^αglobally in time.[Mass Conservation + Positivity + Regularity]

(12)

Z

R^d

R^du(x,t)dx=R

(13)

Z

R^d

R^du(x,t)dx=R

(14)

Z

R^d

R^du(x,t)dx=R

(15)

Z

R^d

R^du(x,t)dx=R

(16)

Let 0<m≤mc, mc=d−2s

d . and pc=d(1−m) 2s The case0<m≤mc:Very Fast FDE

EXISTENCE AND UNIQUENESS FORL^pDATA.

Let0<m≤mcandu0∈L¹(R^d)∩L^p(R^d)withp>pc. There exists aunique strong solutionof Problem (CP).

[Well posedness, i.e., continuous dependence on “m,s” not known in this range] SMOOTHINGEFFECTS. For allp>pc,

ku(t,·)k_L∞(R^d)≤K∞

ϑp= _2sp+d(m−1)¹ >0 K∞=K∞(m,p,d,s)>0 L¹-CONTRACTION ANDCOMPARISON. As before, it holds for allm>0 . LOSS OFMASS.The mass is conserved only whenm=mc.

When 0<m<mcthere is a finite timeT =T(u0)>0,calledExtinction Time, such that u(x,t)≡0 for allt≥Tandx∈R^d .

POSITIVITY. Ifu0≥0 , then the solution is positive up to the extinction timeT, i.e.,u(t,x)>0 , for allx∈R^dand allt∈(0,T).

C^αREGULARITY. Letu0≥0 andTbe the extinction time.

Thenu∈C^α((0,T)×R^d)for some 0< α <1.

REMARK: Non-negative initial data produce solutions which lose mass, are positive andC^αup to the extinction time.[Loss of Mass + Positivity + Regularity]

(17)

(18)

(19)

(20)

(21)

(22)

WeightedL¹Estimates. The case0<m<1.

Class of weightsWα. We say thatϕ∈C²(R^d)belongs to the classWαif:

(i)ϕis positive, radially symmetric and decreasing outsideB1(0), i.e. for all|x| ≥1 . (ii)ϕ(x)≤ |x|^−αand|D²ϕ(x)| ≤c0|x|^−α−2, for somec0>0 and for any|x|>>1.

Example. ϕ(x) =

( 1, if|x| ≤1,

1+ (|x|²−1)⁴⁻^α₈

, if|x| ≥1. Lemma. Estimates for the weightsWα

Letϕ∈ Wα, forα >0. Then, for all|x| ≥ |x0|>>1 we have

|(−∆)^sϕ(x)| ≤











c1|x|^−(α+2s), ifα <d, c2 log|x| |x|^−(d+2s), ifα=d, c3|x|^−(d+2s), ifα >d. Moreover, forα >dthe reverse estimate holds from below ifϕ≥0:

|(−∆)^sϕ(x)| ≥ c4

|x|^d+2s for all|x| ≥ |x0|>>1 . The positive constantsc1,c2,c3,c4>0 depend only onα,s,dandkϕk_C2(R^d). Remark.Note that ifϕ∈ Wαis compactly supported, then|(−∆)^sϕ(x)|is no more compactly supported, and it behaves like|x|^−(d+2s)for any|x|>>1.

(23)

Example. ϕ(x) =

( 1, if|x| ≤1, 1+ (|x|²−1)⁴⁻^α₈

|(−∆)^sϕ(x)| ≤











|(−∆)^sϕ(x)| ≥ c4

(24)

Example. ϕ(x) =

( 1, if|x| ≤1, 1+ (|x|²−1)⁴⁻^α₈

|(−∆)^sϕ(x)| ≤











|(−∆)^sϕ(x)| ≥ c4

(25)

Theorem 1. WeightedL¹estimates for0<m<1, [BV]

Letu≥vbe two ordered very weak solutions to the equation (EQ), with 0<m<1.

LetϕR(x) =ϕ(x/R)whereR>0 andϕ∈ Wαwith d− 2s

1−m < α <d+2s m.

Then, there existsC1>0 that depends only onα,m,d, such that∀0≤τ,t<∞ : Z

R^d

u−v

(t,x)ϕR(x)dx 1−m

≤ Z

R^d

u−v

(τ,x)ϕR(x)dx 1−m

+ C1|t−τ|

R^{2s−d(1−m)}. The estimate holds for changing sign solutions and also fors=1.

The estimate holds both forτ <tand forτ >t.[Quantitative loss of mass] Whenmc<m<1:Conservation of mass,by lettingR→ ∞.

When 0<m<mc: ifu0∈L¹(R^d)∩L^p(R^d)withp≥pc= ^d(1−m)_2s , then sols extinguish in finite timeT>0 , see [DPQRV2] .

Lower bound for the extinction time,[letτ=Tandt=0] 1

C1R^{d(1−m)−2s} Z

R^d

u0ϕRdx 1−m

≤T

Ifu0is such that the limit asR→+∞of the right-hand side diverges to+∞, then the corresponding solutionu(t,x)exists (and is positive) globally in time.

(26)

1−m < α <d+2s m.

R^d

u−v

(t,x)ϕR(x)dx 1−m

≤ Z

R^d

u−v

+ C1|t−τ|

C1R^{d(1−m)−2s} Z

R^d

u0ϕRdx 1−m

≤T

(27)

1−m < α <d+2s m.

R^d

u−v

(t,x)ϕR(x)dx 1−m

≤ Z

R^d

u−v

+ C1|t−τ|

C1R^{d(1−m)−2s} Z

R^d

u0ϕRdx 1−m

≤T

(28)

Proof. •STEP1.A differential inequality for the weightedL¹-norm.

Ifψis a smooth and sufficiently decaying function we have

d dt

Z

R^d

u(t,x)−v(t,x) ψ(x)dx

= Z

R^d

((−∆)^su^m−(−∆)^sv^m)ψdx

=(a)

Z

R^d

(u^m−v^m) (−∆)^sψdx

≤(b)2^1−m Z

R^d

(u−v)^m |(−∆)^sψ| dx

≤(c)2 Z

R^d

(u−v)ψdx m Z

R^d

|(−∆)^sψ|^1−m¹

ψ^1−m^m dx

!1−m

=C^1−m_ψ Z

R^d

(u−v)ψdx m

.

Notice that in(a)we have used the fact that(−∆)^sis a symmetric operator, while in (b)we have used that(u^m−v^m)≤2^1−m(u−v)^m, whereu^m=|u|^m−1uas mentioned.

In(c)we have used Hölder inequality withq=1/m>1 andq⁰=1/(1−m).

IfCψis finite, integrating the above differential inequality on(τ,t)gives:

Z

R^d

u−v

(t,x)ψ(x)dx 1−m

− Z

R^d

u−v

(τ,x)ψ(x)dx 1−m

≤(1−m)C^1−m_ψ |t−τ|

•STEP2. Estimating the constant Cψ. Chooseψ(x) =ϕR(x) :=ϕ(x/R) =ϕ(y), withϕ ∈ Wα. Use estimates forWαweights. Finally, we get thatCψ is finite if d−_1−m^2s < α <d+^2s_m.

(29)

Proof. •STEP1.A differential inequality for the weightedL¹-norm.

Ifψis a smooth and sufficiently decaying function we have

d dt

Z

R^d

u(t,x)−v(t,x) ψ(x)dx

= Z

R^d

((−∆)^su^m−(−∆)^sv^m)ψdx

=(a)

Z

R^d

(u^m−v^m) (−∆)^sψdx

≤(b)2^1−m Z

R^d

(u−v)^m |(−∆)^sψ| dx

≤(c)2 Z

R^d

(u−v)ψdx m Z

R^d

|(−∆)^sψ|^1−m¹

ψ^1−m^m dx

!1−m

=C^1−m_ψ Z

R^d

(u−v)ψdx m

.

Notice that in(a)we have used the fact that(−∆)^sis a symmetric operator, while in (b)we have used that(u^m−v^m)≤2^1−m(u−v)^m, whereu^m=|u|^m−1uas mentioned.

In(c)we have used Hölder inequality withq=1/m>1 andq⁰=1/(1−m).

IfCψis finite, integrating the above differential inequality on(τ,t)gives:

Z

R^d

u−v

(t,x)ψ(x)dx 1−m

− Z

R^d

u−v

(τ,x)ψ(x)dx 1−m

≤(1−m)C^1−m_ψ |t−τ|

•STEP2. Estimating the constant Cψ. Chooseψ(x) =ϕR(x) :=ϕ(x/R) =ϕ(y), withϕ ∈ Wα. Use estimates forWαweights. Finally, we get thatCψ is finite if d−_1−m^2s < α <d+^2s_m.