Fractional Nonlinear Diffusion Equation: Numerical
Analysis and Large-time Behavior
Goro Akagi (Tohoku University), Hélène Hivert (Géosciences Rennes),
Florian Salin (Tohoku University, Institut Camille Jordan)

French-Japanese Laboratory of Mathematics and its Interactions

florian.salin@ec-lyon.fr

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Outline

1

Large-time behaviour of fractional nonlinear diffusion equation
Derivation of the equation
Decay estimates
Convergence to asymptotic profiles with energy method

2

Numerical analysis of fractional nonlinear diffusion equation
Discretization of the equation
Analysis of the numerical scheme
Numerical illustrations of the large-time behaviour

florian.salin@ec-lyon.fr

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Large-time behaviour of fractional nonlinear diffusion equation1

Large-time behaviour of fractional nonlinear diffusion equation
Derivation of the equation
Decay estimates
Convergence to asymptotic profiles with energy method

2

Numerical analysis of fractional nonlinear diffusion equation
Discretization of the equation
Analysis of the numerical scheme
Numerical illustrations of the large-time behaviour

florian.salin@ec-lyon.fr

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Large-time behaviour of fractional nonlinear diffusion equation

Derivation of the equation

Nonlocal diffusion equation with general interaction kernelu(t, x)J(|x − y |)dxdy
u(t, x)dx••u(t, y )dyRdu(t, y )J(|y − x|)dy dxu(t, x) : density, J(|x − y |) : interaction kernel.
Z
Z
J(|x − y |)dyu(t, y )J(|x − y |)dy − u(t, x)∂t u(t, x) =Rd
Rd Z
= P.V.
(u(t, y ) − u(t, x))J(|x − y |)dy
RdZ
for singularkernel J:= lim(u(t, y ) − u(t, x))J(|x − y |)dy
r →0 Rd \Br (x)florian.salin@ec-lyon.fr

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Large-time behaviour of fractional nonlinear diffusion equation

Derivation of the equation

A particular choice of kernel: the fractional diffusion
equation
Choice of kernel: J α2 (|x − y |) = |x−y1|d+α ,α ∈ (0, 2).

Definition (Fractional Laplacian)u(x) − u(y )dy ,d+α
Rd |x − y |
α2α−1 Γ α+d2
Cd,α := d/2 2−α .
π Γ 2Z(−∆) u(x) := Cd,α P.V.
α2αF[(−∆) 2 u](ξ) = |ξ|α F[u](ξ),
J α2 has a non-integrable singularity, and is heavy-tailed.
αFractional diffusion equation: ∂t u + (−∆) 2 u = 0.
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Large-time behaviour of fractional nonlinear diffusion equation

Derivation of the equation

Fractional nonlinear diffusion equation on bounded domainα∂t u + (−∆) 2 u m = 0.
m > 1: fractional porous medium equation,
0 < m < 1: fractional fast diffusion equation.
αChange of variable: q := 1/m + 1, v = u m ⇒ ∂t (v q−1 ) + (−∆) 2 v = 0.
Fractional nonlinear diffusion equation on a bounded domain Ω q−1α+ (−∆) 2 v = 0 in Ω × (0, +∞),
∂t v
v = 0 in (Rd \ Ω) × (0, +∞),

v (·, 0) = v0 in Ω.

florian.salin@ec-lyon.fr

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Large-time behaviour of fractional nonlinear diffusion equation

Derivation of the equation

Purposeα∂t (v q−1 ) + (−∆) 2 v = 0.
q ∈ (1, 2): fractional porous medium equation,q ∈ (2, +∞): fractional fast diffusion equation.
Purpose:Energy decay estimates,
Fast diffusion case: Near extinction asymptotics,
Numerical scheme preserving energy decay estimates.

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Large-time behaviour of fractional nonlinear diffusion equation

Decay estimates

Energy decay estimates: porous medium case

Proposition ([Bonforte, Vazquez], [Akagi, S.])
Assume q ∈ (1, 2). Let v be an energy solution of (CDP).
There exist c, C > 0 such that, for any t > 0,
∥v 0 ∥q−2
Lq (Ω) + ct

florian.salin@ec-lyon.fr

 1

q−2

≤ ∥v (t)∥Lq (Ω) ≤∥v 0 ∥q−2
Lq (Ω) + Ct

Fractional Nonlinear Diffusion Equation

 1

q−2

.

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Large-time behaviour of fractional nonlinear diffusion equation

Decay estimates

Energy decay estimates: fast diffusion case
Proposition ([Bonforte, Ibarrondo, Ispizua], [Akagi, S.])
Assume q ∈ (2, 2∗α ]. Let v be an energy solution of (CDP).
There exist c, C > 0 such that, for any t > 0,

∥v0 ∥q−2
Lq (Ω) − ct

 1

q−2

+


 1
q−2
≤ ∥v (t)∥Lq (Ω) ≤ ∥v0 ∥q−2−
Ct.
Lq (Ω)+

In particular, u extincts at a time t∗ ≤ T∗ :=∥v 0 ∥q−2
q d

L (R )

C.

Moreover, for any t > 0,11c(t∗ − t)+q−2 ≤ ∥v (t)∥Lq (Ω) ≤ C (t∗ − t)+q−2 ,and the same is true when ∥ · ∥Lq (Ω) is replaced by ∥ · ∥H α2 (Rd ) .
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Large-time behaviour of fractional nonlinear diffusion equation

Decay estimates

Idea of proof
Lemma (Fractional Sobolev inequality)2d, and q ∈ (1, 2∗α ]. There exists K > 0 such that,Let 2∗α := (d−α)+∥u∥Lq (Rd ) ≤ K [u]H α2 (Rd ) = Krα(−∆) 2 u, u
L2 (Rd ),α

for any u ∈ H 2 (Rd ) with u ≡ 0 in Rd \ Ω.

1

Obtain energy identity from the variational form of the equation,

2

Use the fractional Sobolev inequality, or monotonicity of Rayleighquotient, to obtain an ordinary differential inequality,

3

Integrate the ordinary differential inequality.
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Large-time behaviour of fractional nonlinear diffusion equation

Convergence to asymptotic profiles with energy method

Assume 2 < q < 2∗α . Then11c(t∗ − t)+q−2 ≤ ∥v (t)∥H α2 (Rd ) ≤ C (t∗ − t)+q−2 .
Rescaled solution: w (s) := (t∗ − t)αThen ∂s w q−1 + (−∆) 2 w =

−1
q−2


v (t),

s := logt∗t∗ − t.

q − 1 q−1w
, and c < ∥w (s)∥H α2 (Rd ) < C
q−2

Does w (s) converges as s → ∞ ?
Non-fractional case:
Berryman, Holland (’80): convergence along subsequences,
Feireisl, Simondon (’00): convergence along the full sequence,
Bonforte, Grillo, Vázquez (’12): convergence in relative error,
Bonforte, Figalli (’20), Jin, Xiong (’23), Akagi (’23), Choi, McCann,
Seis (’23): Sharp rate of convergence.
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Large-time behaviour of fractional nonlinear diffusion equation

Convergence to asymptotic profiles with energy method

Rescaled solution and asymptotic profiles

α∂s w q−1 + (−∆) 2 w =q − 1 q−1w
, and c < ∥w (s)∥H α2 (Rd ) < C .
q−2

Proposition ([Akagi, S.])For any sn → +∞, there exists a subsequence (still denoted by (sn )), andαϕ ∈ H 2 (Rd ) \ {0}, with ϕ = 0 in Rd \ Ω, such thatw (sn ) → ϕααstrongly in H 2 (Rd ),(−∆) 2 ϕ = λq ϕq−1

in Ω,with λq := q−1
q−2 > 0.

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Large-time behaviour of fractional nonlinear diffusion equation

Convergence to asymptotic profiles with energy method

Generalized gradient flow structure for nonlinear diffusion
DefineαX α2 (Ω) := {u ∈ H 2 (Rd ) : u = 0 a.e. in Rd \ Ω},λq
1J(w ) := ∥w ∥2 α2 d −∥w ∥qLq (Ω)
H
(R
)
2
q

for w ∈ Xθ (Ω).

Then the rescaled solution s > 0 7→ w (s) ∈ Xθ (Ω) solves∂s w q−1 (s) = −J ′ (w (s)),

for a.e. s > 0.

Therefore it holds
24d∂s (|w |(q−2)/2 w )(s) 2 + J(w (s)) ≤ 0,
′
qq
ds
L (Ω)

for a.e. s > 0,

and J(w (·)) is non-increasing.
florian.salin@ec-lyon.fr

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Large-time behaviour of fractional nonlinear diffusion equation

Convergence to asymptotic profiles with energy method

Lojasiewicz inequality
Theorem ([Lojasiewicz])
Let U ⊂ Rd open and f : U → R a real-analytic function. Let x0 ∈ U suchthat ∇f (x0 ) = 0. Then there exists a neighborhood V of x0 , ω > 0 andθ ∈ (0, 1/2] such that|f (x) − f (x0 )|1−θ ≤ ω|∇f (x)|,x ∈ V.

Proof for d = 1: ∃N ≥ 2 s.t. f (N) (x0 ) ̸= 0 and
f (N) (x0 ) N
h + o(hN ),N!
f (N) (x0 ) N−1
f ′ (x0 + h) =h+ o(hN−1 )(N − 1)!
N−1
1⇒ (f (x0 + h) − f (x0 )) N =f ′ (x0 + h) + o(hN−1 )| {z }
Nf (N) (x0 )1/N<
∼ f ′ (x0 +h)f (x0 + h) − f (x0 ) =florian.salin@ec-lyon.fr

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Large-time behaviour of fractional nonlinear diffusion equation

Convergence to asymptotic profiles with energy method

Lojasiewicz inequality implies full convergence of gradientflows
Lemma ([Lojasiewicz])
Let x ∈ ACloc (0, ∞; Rd ) be a solution to ẋ(t) = −∇f (x(t)) for a.e. t > 0.
Assume ∃ (tn )n s.t. tn ↗ ∞ and x(tn ) → x∞ with ∇f (x∞ ) = 0.
Then x(t) → x∞ as t → +∞.
Sketch of proof:d(f (x(t)) − f (x∞ )) = −|∇f (x(t))||ẋ(t)| ≤ −ω(f (x(t)) − f (x∞ ))1−θ |ẋ(t)|dt
1 d⇒ |ẋ(t)| ≤ −(f (x(t)) − f (x∞ ))θ
ω dt
Z ∞
1
⇒
|ẋ(t)|dt < (f (x(0)) − f (x∞ ))θ < ∞ω0florian.salin@ec-lyon.fr

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Large-time behaviour of fractional nonlinear diffusion equation

Convergence to asymptotic profiles with energy method

PDE case: Lojasiewicz-Simon inequality

Simon (’83) extended Lojasiewicz inequality to certain analytic functionals.
Applicable to semilinear parabolic equation with analytic nonlinearities.
Feiresl and Simondon (’00) extended Lojasiewicz-Simon inequality to thestandard Laplacian and non-analytic nonlinearities.
Akagi, Schimperna, Segatti (’19) extended Lojasiewicz-Simon inequality to
the fractional Laplacian and to non-analytic nonlinearities.

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Large-time behaviour of fractional nonlinear diffusion equation

Convergence to asymptotic profiles with energy method

Lojasiewicz-Simon inequality for fractional LaplacianI (w ) :=1∥w ∥2H α2 (Rd ) +2

Z Z w (x)g (s)dsdxΩ

α

for w ∈ H 2 (Rd ), ĝ (w (·)) ∈ L1 (Ω).

0

(H0) g ∈ C 1 (R) and g (0) = 0,
(H1) g ∈ C ∞ ((0, ∞)), and for all β ∈ (0, ∞), there exist C , M ≥ 0 such that,|g (n) (s)| ≤ C

M n n!
,|s|n∀s ∈ (0, β), n ∈ N(H2) there exists 0 ≤ p < ∞ with p ≤ 2∗α − 1 such that|g ′ (s)| ≤ C (|s|p−1 + 1)

for all s ∈ R.

Lemma ([Akagi, Schimperna, Segatti])αAssume (H0), (H1), (H2), and let ψ ∈ H 2 (Rd ) ∩ L∞ (Ω) such that I ′ (ψ) = 0 andψ > 0. Then, there exists θ ∈ (0, 1/2] and ω, δ > 0 s.t.
|I (w ) − I (ψ)|1−θ ≤ ω∥I ′ (w )∥H − α2 (Ω) ,florian.salin@ec-lyon.fr

if ∥w − ψ∥H α2 (Rd ) < δ.

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Large-time behaviour of fractional nonlinear diffusion equation

Convergence to asymptotic profiles with energy method

Full convergence to the asymptotic profile
Proposition ([Akagi, S.])αLet 2 < q < 2∗α . Assume that ϕ > 0 and w (sn ) → ϕ strongly in H 2 (Rd )for some sequence of times (sn )n∈N such that sn → +∞. Thenw (s) → ϕαstrongly in H 2 (Rd ) as s → +∞.

Idea of proof: Use the Lojasiewicz-Simon inequality with the functionnal
λq
1J(w ) := ∥w ∥2 α2 d −∥w ∥qLq (Ω) .
H
(R
)
2
q
Open problems:Estimation of the extinction time,
Rate of convergence to the asymptotic profile.
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Numerical analysis of fractional nonlinear diffusion equation1

Large-time behaviour of fractional nonlinear diffusion equation
Derivation of the equation
Decay estimates
Convergence to asymptotic profiles with energy method

2

Numerical analysis of fractional nonlinear diffusion equation
Discretization of the equation
Analysis of the numerical scheme
Numerical illustrations of the large-time behaviour

florian.salin@ec-lyon.fr

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Numerical analysis of fractional nonlinear diffusion equation

Discretization of the equation

Notation
We restrict to dimension d = 1.
Ω = (−L, L), space step h = L/(Nx + 1):|−L· · · x−Nx −2

0
h
L
|
|
|
|
|
x−Nx · · · x−2 x−1 x0 x1 x2 · · · xNx
xNx +2
|

|

|

···

time step ∆t > 0.
For u a fonction over [0, +∞) × R,
uin := u(n∆t, ih),X
∥u n ∥qlq (R) :=|ui |q h.
hflorian.salin@ec-lyon.fri∈Z

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Numerical analysis of fractional nonlinear diffusion equation

Discretization of the equation

Discretization of the fractional Laplacianα2(−∆) u(xi ) := Cd,α!
Zui − u(y )ui − u(y )
dy +
dy
d+α
d+α
|xi −y |<h |xi − y ||x −y |>h |xi − y |
{z
} | i
{z
}

Z
P.V.
|

singular parttail partsingular part: u(y ) replaced by Taylor expansion,
tail part: u(y ) replaced by piecewise quadratic interpolation.
Then integrating explicitly yields, for some weights (γjh )j∈Z ,α(−∆) 2 u(xi ) ≈X

γjh (ui − ui−j ).

j∈ZConvergence result: For u ∈ C 4 , the error is in O(h3−α ).
➢ Y. Huang and A. Oberman. “Numerical Methods for the Fractional
Laplacian: A Finite Difference-Quadrature Approach”. In: (2014)
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Numerical analysis of fractional nonlinear diffusion equation

Discretization of the equation

Convolution structure of the discrete fractional Laplacianhα(−∆)h2 u

i
i

:=

X

γjh (ui − ui−j )

j∈Z

x
y
α
2

Z

(−∆) u(xi ) = P.V.

C1,α(u(xi ) − u(xi − z))dz.
1+α
Rd |z|

Theorem ([Ayi, Herda, Hivert, Tristani, 2022])
There exists positive constants bα and Bα independent of h such thatbαBαh ≤ γjh ≤
h.
1+α
|jh|
|jh|1+α

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Numerical analysis of fractional nonlinear diffusion equation

Discretization of the equation

Convolution structure of the discrete fractional Laplacian
For u a Schwartz function,
Xh
α i
1 X
(−∆)h2 u ui =hγjh |ui − ui−j |2 ∼ [u]2 α2 ,
H (R)
2
i
i∈Z
i∈Z,j∈Z{z}|=:[u]2 α

H 2 (R)h

Lemma ([Hivert, S.], Discrete fractional Sobolev inequality)
For q ≤ 2∗α , there exists K > 0 independent of h,∥u∥l q (R) ≤ K [u]hαHh2 (R),for u ∈ ZN with u ≡ 0 outside Ω.
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Numerical analysis of fractional nonlinear diffusion equation

Discretization of the equation

Numerical scheme for fractional nonlinear diffusion
equation

Implicit scheme for CDP:
 n+1 q−1iα(ui )− (uin )q−1 h

2 n+1


+
(−∆)
u
= 0,

h
∆t
i
uin = 0,

ui0 = (u 0 )i

|i| ≤ Nx and n ≥ 0,|i| ≥ Nx + 1 and n ≥ 0,|i| ≤ Nx .

Does the scheme has the same property as the continuous equation (decayrate, extinction) ?

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Numerical analysis of fractional nonlinear diffusion equation

Analysis of the numerical scheme

Discrete decay estimates: porous medium case

Proposition ([Hivert, S.])
Assume q ∈ (1, 2). There exists (βn∆t )n≥0 , independent of h, such that

 −1
2−q
∥u n ∥l q (R) ≤ ∥u 0 ∥q−2+
Cn∆t
+ βn∆t ,ql (R)hfor any n ≥ 0.

hMoreover,sup βn∆t → 0,

as ∆t → 0.

n≥0florian.salin@ec-lyon.fr

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Numerical analysis of fractional nonlinear diffusion equation

Analysis of the numerical scheme

Numerical results: energy decay for PME
Figure: Energy decay for q = 1.5, α = 0.5, h = 0.04, ∆t = 0.03, L = 5.
1.5

∥u n ∥q∥u 0 ∥q−2+ ctnq

1/(q−2)∥u 0 ∥q−2+ Ctnq1/(q−2)1012+ βn∆t

1

0.5

0
0

2

4

6

8

14

tn
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Numerical analysis of fractional nonlinear diffusion equation

Analysis of the numerical scheme

Discrete decay estimates: fast diffusion case
Proposition ([Hivert, S.])
Assume q ∈ (2, 2∗α ].
Decay estimate: There exists (βn∆t )n≥0 , independent of h, such that

 1
q−2
+ βn∆t ,∥u n ∥l q (R) ≤ ∥u 0 ∥q−2
−
Cn∆t
q
l (R)
h

for any n ≥ 0.

+hMoreover,sup βn∆t → 0,

as ∆t → 0.

n≥0

Extinction estimate:n0∥u ∥l q (R) ≤ ∥u ∥l q (R)hflorian.salin@ec-lyon.fr

h

T∗
n∆t

n/2
,

Fractional Nonlinear Diffusion Equation

for any n ≥ 0.

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Numerical analysis of fractional nonlinear diffusion equation

Analysis of the numerical scheme

Numerical results: energy decay for FDE
Figure: Energy decay for q = 2.4, α = 0.5, h = 0.04, ∆t = 0.03, L = 5.
1.2

∥u n ∥q1/(q−2)∥u 0 ∥q−2− ctn +q
1/(q−2)∥u 0 ∥q−2− Ctn +
+ βn∆t
q

1

0.8

0.6

0.4

0.2

0
0

2

4

6

8

10

tn
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Numerical analysis of fractional nonlinear diffusion equation

Analysis of the numerical scheme

Numerical results: energy decay for FDE
Figure: Energy decay for q = 3.5, α = 1.5, h = 0.04, ∆t = 0.03, L = 5.
1.2

∥u n ∥q1/(q−2)∥u 0 ∥q−2− ctnq+1/(q−2)+ βn∆t∥u 0 ∥q−2
− Ctn
q

1

+

0.8

0.6

0.4

0.2

0
0

2

4

6

8

10

tn
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Numerical analysis of fractional nonlinear diffusion equation

Analysis of the numerical scheme

Idea of proof

1

Obtain energy inequality,
1 ∥u
q′

2

∆t

+ ∥u n+1 ∥2 α2≤ 0.

Hh (R)

Use discrete fractional Sobolev inequality to obtain a discretization of
the ordinary differential inequality,
1 ∥u
q′

3

n+1 ∥q− ∥u n ∥qlq (R)lhq (R)hn+1 ∥q− ∥u n ∥qlq (R)lhq (R)h∆t+ K −2 ∥u n+1 ∥2l q (R) ≤ 0.
h

Sum in time and use convexity inequalities.

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Numerical analysis of fractional nonlinear diffusion equation

Numerical illustrations of the large-time behaviour

Computation of the extinction time when q > 2
u : solution to the continuous problem with initial data u 0t∗ (u 0 ): extinction time of u
−1

Let t̃ > 0 and w (s) := (t̃ − t) q−2 u(t), s := log (t̃/(t̃ − t)). Then−1
q − 1 q−1w
, w (0) = t̃ q−2 u(0),q−2• ∥w (s)∥q → ∞ as s → ∞ if t̃ > t∗ (u 0 ),α

• ∂s w q−1 + (−∆) 2 w =• w extinct in finite time if t̃ < t∗ (u 0 ).
To compute t∗ (u 0 ) we proceed by dichotomy using the scheme
 n+1
h
i
q−1 − (w n )q−1
α

q − 1 n+1 q−1 (wi )i+ (−∆)h2 w n+1 =
(w
) ,
∆s
q−2
i


wi0 = t̃ −1/(q−2) (u 0 )i .
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Numerical analysis of fractional nonlinear diffusion equation

Numerical illustrations of the large-time behaviour

Figure: q = 2.4, α = 0.5, h = 0.04, ∆t = 0.03, L = 5t∗ computed by dichotomy with ∆s = 0.001∥u n ∥q

0.13279∆s 0.92667

1
10−2

0.5

t∗ = 5.56
0
0

2

4

6

8

10

10−3

10−2

tn

∆s

(a) Energy decay of the non-rescaled
solutionflorian.salin@ec-lyon.fr

10−1(b) Convergence of the computation of theextinction time as ∆s → 0

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Numerical analysis of fractional nonlinear diffusion equation

Numerical illustrations of the large-time behaviour

Figure: w (t) for α = 0.5, q = 2.4 < 2∗α , h = 0.01, ∆s = 0.01t∗ computed by dichotomy with ∆s = 0.01
w ∞ : theoretical asymptotic profile

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Numerical analysis of fractional nonlinear diffusion equation

Numerical illustrations of the large-time behaviour

Figure: q = 2.4 < 2∗α , h = 0.01, ∆s = 0.01t∗ computed by dichotomy with ∆s = 0.01
w ∞ : theoretical asymptotic profile

100

log(∥w n − w∞ ∥q )

10−1
10−2
10−3
10−4

0

2

4

6

8

10

12

14

sn
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Conclusion

Summary
We showed convergence to asymptotic profiles in fast diffusion case.
We introduced a numerical scheme having same decay energy as thefractional nonlinear diffusion equation, and a method for the computationof the extinction time.
Extensions:
Rates of convergence to asymptotic profiles,
Numerical analysis in dimension d ≥ 1,
Better convergence results for the numerical scheme.
Thank you for your attention!

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