On nonlinear conservation laws with nonlocal diffusion term Franz - - PowerPoint PPT Presentation

On nonlinear conservation laws with nonlocal diffusion term

Franz Achleitner1 Sabine Hittmeir1 Christian Schmeiser2

1 Vienna University of Technology 2 University of Vienna

Padova, June 2012 financial support by the Austrian Science Fund (FWF)

• utline

1 examples in physics 2 nonlocal operator 3 partial integro-differential equations 4 traveling wave solutions

existence of traveling wave solutions asymptotic stability of traveling wave solutions

5 outlook 6 References

shallow water flow

boundary conditions at free surface incompressible Navier-Stokes equations no-slip boundary conditions at rigid bottom Assumptions (Kluwick, Cox, Exner and Grinschgl (2010))

1 Froude number 1 <

U0 √gH << 2

2 length scales H << L 3 Reynolds number 1 << Re = L√gH

ν H2 L2

⇒ viscous effects only important in boundary layer

incompressible Navier-Stokes equation

ux + vy = 0 , ut + uux + vuy = −px + 1 Re H2 L2 uxx + uyy

• ,

vt + uvx + vvy = − L2 H2 (py + 1) + 1 Re H2 L2 vxx + vyy

• .

x, y ...coordinates, u, v ...velocity components, t ...time, p ...pressure no-slip boundary condition y = s(x, t) : u = 0 , v = st . boundary conditions at free surface y = h(x, t) : ht + uhx − v = 0 , p = −Thxx

• 1 + H2

L2 (hx)2 −3/2 .

triple-deck structure: lower deck

matching conditions lim

Y →∞ U(X, Y ) = Y + A ,

lim

X→−∞ U(X, Y ) = Y .

no-slip b.c. at Y = 0: U = 0 , V = 0 . governing equations for (U, V , P) with X ∈ R, Y ∈ R+ and t ∈ R+ ∂XU + ∂Y V = 0 , U∂XU + V ∂Y U = −∂XP + ∂2

Y U ,

∂tP + ∂XP − P∂XP = K1∂XA + K2∂3

XP .

linear flow response

interaction equation for pressure P ∂tP + ∂XP = K1∂XD1/3P + K2∂3

XP,

X ∈ R, t ∈ R+, with constants K1 and K2 related to K1 streamline curvature and boundary displacement effects K2 detuning and surface tension and a nonlocal operator Dα with 0 < α < 1 defined as (DαP)(X) := 1 Γ(1 − α) X

−∞

P′(ξ) (X − ξ)α dξ .

Fowler equation: model for dune formation

∂tu + 1 2∂xu2 = −∂xD1/3u + ∂2

xu,

x ∈ R, t ∈ R+, where u(x, t) represents dune amplitude. no maximum principle (Alibaud, Azerad and Is` ebe (2010)) bore-like traveling wave solutions

nonlocal operator ∂xDα

For 0 < α < 1, the operator (∂xDαu)(x) = ∂x 1 Γ(1 − α) x

−∞

u′(y) (x − y)α dy is a Fourier multiplier operator F(∂xDαu)(ξ) = 1 √ 2π

• R

e−iξx(∂xDαu)(x) dx = Λ(ξ)Fu(ξ) with u ∈ S(R) and symbol Λ(ξ) = −

• sin
• απ

2

• − i sgn(ξ) cos
• απ

2

• |ξ|α+1 ,

ξ ∈ R .

Fourier multiplier operator (FTf )(ξ) = −ψ(−ξ)(Ff )(ξ)

Id −H −Id H ∂x −∂xH −∂x ∂xDαu ∂2

x

−H∂2

x

∂3

x

1 − α 1 + α a θ 2 1 −1 3 2 1 Riesz-Feller operator T ψ(ξ) = |ξ|a exp

• i sgn(ξ)θπ

2

• real-valued parameters

a index of stability 0 < a ≤ 2 θ skewness parameter |θ| ≤ min(a, 2 − a) Hilbert transform Hf := p.v. 1 π ∞

−∞

f (y) x − y dy

Fourier multiplier ψa,θ(ξ) := |ξ|a exp

• i sgn(ξ)θ π

2

• −1

i sgn(ξ) 1 −i sgn(ξ) iξ |ξ| −iξ −(iξ)a |ξ|2 −(iξ)3 1 − α 1 + α a θ 2 1 −1 3 2 1 real-valued parameters a index of stability 0 < a ≤ 2 θ skewness parameter |θ| ≤ min(a, 2 − a)

fractional diffusion equation

∂tu = ∂xDαu , x ∈ R , t ∈ R+ , for some fixed α with 0 < α < 1. strongly continuous, convolution semigroup Tt : Lp(R) → Lp(R) , u0 → Ttu0 = u(t, x) = K(t, ·) ∗ u0 , with 1 ≤ p < ∞ and kernel K(t, x) = F−1(exp(Λ(.)t))(x). Properties of K(t, x): for all x ∈ R, t > 0 and m ∈ N, non-negative K(t, x) ≥ 0 integrable K(t, .)L1(R) = 1 scaling K(t, x) = t−

1 1+α K(1, xt− 1 1+α )

smooth K(t, x) is C∞ smooth bounded there exists Bm ∈ R+ such that |∂m

x K|(t, x) ≤ t− 1+m

1+α

Bm 1 + t−

2 1+α |x|2

L´ evy strictly stable distributions on R

L C H N 1 − α 1 + α a θ 2 1 −1 3 2 1 random variable X E[exp(iξX)] = exp(−ψ(ξ)) ψ(ξ) = |ξ|a exp

• i sgn(ξ)θ π

2

• distributions

L L´ evy-Smirnov PDF x−3/2

2√π exp

• − 1

4x

• ,

x > 0. C Cauchy(-Lorentz) PDF 1

π 1 1+x2

H Holtsmark N Normal (Gaussian) PDF

1 √ 2πσ2 exp

• − x2

2σ2

approximate identity

Theorem (Stein, Weiss)

Suppose φ ∈ L1(Rn) with

• Rn φ(x)dx = 1 and for ǫ > 0 let

φǫ(x) = ǫ−nφ(x/ǫ). If f ∈ Lp(Rn), 1 ≤ p < ∞, or f ∈ C0 ⊂ L∞(Rn), then f ∗ φǫ − f p → 0 as ǫ → 0.

Theorem (Lieb, Loss)

Let j be in L1(Rn) with

• Rn j = 1. For ǫ > 0, define jǫ(x) := ǫ−nj(x/ǫ), so

that

• Rn jǫ = 1 and jǫ1 = j1. Let f ∈ Lp(Rn) for some 1 ≤ p < ∞ and

define the convolution fǫ := jǫ ∗ f . Then fǫ ∈ Lp(Rn) and fǫp ≤ j1f p . fǫ → f strongly in Lp(Rn) as ǫ → 0 . If j ∈ C ∞

c (Rn), then fǫ ∈ C ∞(Rn) and Dαfǫ = (Dαjǫ) ∗ f .

Theorem (L´ evy process)

For every 0 < α < 1, there exists a L´ evy process {Xt | t ≥ 0} such that the probability distribution µt of Xt has probability density function K(t, x). Moreover, the associated transition semigroup {Pt} is a strongly continuous semigroup on C0(R) with Pt = 1. The infinitesimal generator Lα of {Pt} is given for f ∈ C 2

0 (R) by

Lαf (x) = cos

• (1 − α)π

2 ∞ f (x + y) − f (x) − yf ′(x) y2+α dy and has core C ∞

c (R).

transition function Pt(x, B) = µt(B − x) for t ≥ 0, x ∈ R, B ∈ B(R) transition semigroup Define for f ∈ C0(R) (Ptf )(x) =

• R

Pt(x, dy)f (y) =

• R

f (x + y)K(t, y) dy = E[f (x + Xt)] , then Ptf ∈ C0(R) by the Lebesgue convergence theorem.

scalar conservation law with nonlocal diffusion

∂tu + ∂xf (u) = ∂xDαu, x ∈ R, t ∈ R+, (1) where f (u) is a smooth flux function.

Theorem (Droniou, Gallou¨ et and Vovelle (2003))

The Cauchy problem of (1) with initial datum u0 ∈ L∞(R) has a unique global solution u(t, x), in the sense that u ∈ L∞((0, ∞) × R) satisfies u(t, x) = (K(t, .) ∗ u0)(x) − t

• K(t − τ, .) ∗ ∂xf (u(τ, .))
• (x)dτ

(2) almost everywhere. Moreover,

1 u ∈ C ∞((0, ∞) × R) and u ∈ C ∞

b ((t0, ∞) × R) for all t0 > 0.

2 u satisfies equation (1) in the classical sense. 3 for all t > 0, u(t, .)∞ ≤ u0∞ and, in fact, u takes its values

between the essential lower and upper bounds of u0.

4 u(t) t→0

− − → u0 in L∞(R) weak-∗ and in Lp

loc(R) for all p ∈ [1, ∞).

sketch of proof

Droniou, Gallou¨ et and Vovelle established the result in case of a nonlocal diffusion operator with symbol −|ξ|1+α for 0 < α < 1. existence Suppose u0 ∈ C ∞

c (R).

Construct approximate solution uδ, δ > 0, by a splitting method: Define uδ(0, .) = u0 Define uδ(t, x) on (t, x) ∈ (2nδ, (2n + 1)δ] × R, n ∈ N0, as the solution of ∂tu = 2∂xDαu with initial condition uδ(2nδ, .). Define uδ(t, x) on (t, x) ∈ ((2n + 1)δ, (2n + 2)δ] × R, n ∈ N0, as the solution of ∂tu + 2∂xf (u) = 0 with initial condition uδ((2n + 1)δ, .). For δ0 > 0 sufficiently small, any compact set Q ⊂ R and T > 0, {uδ | δ ∈ (0, δ0]} is relatively compact in C([0, T]; L1(Q)). limit function u ∈ C([0, T]; L1(R)) satisfies mild formulation (2).

scalar conservation law with vanishing nonlocal diffusion

∂tuǫ + ∂xf (uǫ) = ǫ∂xDαuǫ, x ∈ R, t ∈ R+, (3) where f (u) is a smooth flux function.

Theorem (Droniou (2003))

Suppose 0 < α ≤ 1 and u0 ∈ L∞(R). The solution uǫ of ∂tuǫ + ∂xf (uǫ) = ǫ∂xDαuǫ , uǫ(0, x) = u0(x) , converges as ǫ → 0 in C([0, T]; L1

loc(Rn)) for all T > 0 to the entropy

solution of the Cauchy problem ∂tu + ∂xf (u) = 0 , u(0, x) = u0(x) . Moreover, if u0 ∈ L∞(R) ∩ L1(R) ∩ BV (R), then uǫ − uC([0,T];L1(R)) = O

• ǫ1/(1+α)

.

traveling wave solutions of equation (1)

Consider wave speed s ∈ R and traveling wave variable ξ := x − st.

Definition

A traveling wave solution of (1) is a solution of the form u(t, x) = ¯ u(ξ), for some function ¯ u that connects different endstates limξ→±∞ ¯ u(ξ) = u±. traveling wave equation h(u) := f (u) − su −

• f (u−) − su−
• = Dαu =

1 Γ(1 − α) x

−∞

u′(y) (x − y)α dy (4) properties of specific traveling wave solutions Rankine-Hugoniot condition f (u+) − f (u−) = s(u+ − u−) For convex flux function f (u) and a monotone solution, standard entropy condition u− > u+.

existence of traveling wave solution

Theorem (A., Hittmeir and Schmeiser (2011))

Suppose f ∈ C ∞(R) is a convex function and (u−, u+, s) satisfy the Rankine-Hugoniot condition as well as the entropy condition u− > u+. Then there exists a traveling wave solution of (1), which is decreasing and unique (up to translations) among all functions u ∈ {u− + v | v ∈ H2(−∞, 0) ∩ C 1

b (R)}.

traveling wave equation linearized at u = u− h′(u−)v(ξ) = Dαv(ξ) := 1 Γ(1 − α) ξ

−∞

v′(y) (ξ − y)α dy, ξ ∈ R−, has solutions of the form v(ξ) := b exp(λξ) with λ := h′(u−)1/α and b ∈ R.

local existence + uniqueness

Lemma

For every sufficiently small ǫ > 0, ξǫ := log ǫ

λ

and Iǫ = (−∞, ξǫ], there exists an ǫ-independent δ > 0, such that the equation (4) has solutions uup,ǫ, udown,ǫ ∈ u− + H2(Iǫ), which satisfy uup,ǫ(ξǫ) = u− + ǫ , udown,ǫ(ξǫ) = u− − ǫ (5) and are unique among all functions {u | u − u−H2(Iǫ) ≤ δ}. Moreover, uup,ǫ − u− − eλξH2(Iǫ) ≤ Cǫ2 , udown,ǫ − u− + eλξH2(Iǫ) ≤ Cǫ2 holds with an ǫ-independent constant C and λ := h′(u−)1/α. proof Consider ¯ u = udown,ǫ(ξ) − u− + exp(λξ) which satisfies a BVP

• Dα−h′(u−)
• ¯

u = h(u−−exp(λξ)+¯ u)+h′(u−)

• exp(λξ)−¯

u

• ,

¯ u(ξǫ) = 0. Cast BVP as a fixed point problem + use Banach’s fixed point theorem.

local monotonicity

Lemma

For all ξ ∈ Iǫ, the solution udown,ǫ(ξ) is bounded (udown,ǫ(ξ) < u−) and monotone (u′down,ǫ(ξ) < 0). proof Due to Sobolev imbedding H2(Iǫ) ֒ → C 1

b (Iǫ),

|udown,ǫ − u− − eλξ| ≤ Cǫ2 for all ξ < ξǫ . There exists ξ⋆ < ξǫ, such that udown,ǫ(ξ⋆) = u− − 2Cǫ2 and the solution is bounded from above by u− for all ξ ∈ (ξ⋆, ξǫ). For ǫ2 = 2Cǫ2

1 with ǫ1 = ǫ,

the translated function udown,ǫ(ξ − ξǫ2 + ξ⋆) is the unique solution udown,ǫ2. Iteration of the argument produces a sequence {ǫn}, determined by ǫn+1 = 2Cǫ2

n, such that the unique solution satisfies udown,ǫ(ξ) < u− for

all ξ ∈ (ξǫn, ξǫ) and n ∈ N. Similar argument proves monotonicity.

continuation of solution

Lemma

Let u ∈ C 1

b (−∞, ξ0] be a solution of (4). For sufficiently small δ > 0, the

solution has a unique continuation in the function space C 1

b (−∞, ξ0 + δ).

proof For monotone functions u ∈ C 1

b (R), equation (4) is equivalent to

u(ξ) − u− = 1 Γ(α) ξ

−∞

h(u(y)) (ξ − y)1−α dy . (6) Rewrite equation (6) as a Volterra integral equation on a bounded interval u(ξ) = f (ξ) + 1 Γ(α) ξ

ξ0

h(u(y)) (ξ − y)1−α dy , where f (ξ) = u− + 1 Γ(α) ξ0

−∞

h(u(y)) (ξ − y)1−α dy . Local existence of a smooth solution is a standard result.

global properties

Lemma

Let u ∈ C 1

b (−∞, ξ0] be (a continuation of) the solution udown(ξ) of (4).

Then for all ξ ∈ (−∞, ξ0], u(ξ) is nonincreasing and bounded by u+ < u(ξ) < u−.

Proof of Theorem.

global existence boundedness+local continuation ⇒ The solution u(ξ) exists for all ξ ∈ R and satisfies limξ→+∞ u(ξ) = u+. global uniqueness Let u ∈ {u− + v | v ∈ H2(−∞, 0) ∩ C 1

b (R)} be a

solution of (4). Then the restriction of the solution to an interval (−∞, ξ0] is, up to a shift in ξ, the continuation of uup or udown, or the constant function u ≡ u−.

asymptotic stability of traveling wave φ

Theorem (A., Hittmeir and Schmeiser (2011))

Suppose f is a convex function, φ is a traveling wave solution of (4), and u0 is such that W0(ξ) = ξ

−∞(u0(η) − φ(η)) dη satisfies W0 ∈ H2(R).

If W0H2 is small enough, then the Cauchy problem for ∂tu + ∂ξ(f (u) − su) = ∂ξDαu with initial datum u0 has a unique global solution converging to the traveling wave solution φ in the sense that lim

t→∞

∞

t

u(τ, ·) − φH1 dτ = 0 .

idea of proof

Perturbation U = u − φ satisfies ∂tU + ∂ξ

• f (u) − f (φ) − sU
• = ∂ξDαU .

The energy estimate 1 2 d dt U2

L2+1

2

• R

f ′′(φ)φ′U2 dξ−1 2

• R

f ′′(φ+ϑU)U2∂ξU dξ = −aαU2

˙ H

1+α 2

holds for some 0 < ϑ < 1 and leads to 1 2 d dt U2

L2 − C0U2 L2 − L(UL∞)UL∞U2 H1 ≤ −aαU2 ˙ H

1+α 2

, for a positive nondecreasing function L and positive constants C0 and aα.

The primitive of the perturbation

W (t, ξ) := ξ

−∞

U(t, η) dη satisfies for some 0 < ϑ < 1 ∂tW + (f ′(φ) − s)∂ξW + 1 2f ′′(φ + ϑU)(∂ξW )2 = ∂ξDαW (7) and the energy estimate 1 2 d dt W L2 − L(UL∞)W L∞∂ξW 2

L2 ≤ −aαW 2 ˙ H

1+α 2

.

Lemma

Suppose W0 ∈ H2(R). Then there exists T > 0 such that the Cauchy problem for (7) with initial data W0 has a unique solution W (t) ∈ H2(R) for all t ∈ [0, T).

Lyapunov functional

J(t) = 1 2(W 2

L2 + γ1U2 L2 + γ2∂ξU2 L2)

with positive constants γ1 and γ2. Linear combination of energy estimates yields d dt J + aα

• W 2

˙ H

1+α 2

+ γ1W 2

˙ H

3+α 2

+ γ2W 2

˙ H

5+α 2

• − γ1C0U2

L2 − γ2C1U2 H1 − L(W H2)W H2U2 H(5+α)/4 ≤ 0

Choose γ1, γ2 > 0 such that γ1C0U2

L2 +γ2C1U2 H1 ≤ aα

2

• W 2

˙ H

1+α 2

+ γ1W 2

˙ H

3+α 2

+ γ2W 2

˙ H

5+α 2

• and get the final estimate

d dt J ≤ −γ∗ aα 2 − L γ∗ W H2 W 2

˙ H

1+α 2

+ γ1W 2

˙ H

3+α 2

+ γ2W 2

˙ H

5+α 2

fractional Korteweg-de Vries-Burgers equation

∂tu + u∂xu = ǫ∂xDαu + δ∂3

xu,

x ∈ R, t ∈ R+, (8) for some fixed α with 0 < α < 1 and ǫ, δ ∈ R.

Theorem (Molinet and Ribaud (2001))

global well-posedness of the Cauchy problem for the Kortweg-de Vries-Burgers equation with fractional Laplacian and initial datum in Hs(R) for s > − 3

4.

existence of traveling wave solutions

Theorem (A., Cuesta and Hittmeir (2012))

Suppose (u−, u+, s) satisfy the Rankine-Hugoniot condition as well as the entropy condition u− > u+. Then there exists a traveling wave solution ¯ u ∈ C 2

b (R) of (8), such that limξ→−∞ ¯

u(ξ) = u−.

references

• F. Achleitner, S. Hittmeir and Ch. Schmeiser. On nonlinear conservation

laws with nonlocal diffusion term J. Diff. Equ., 250: 2177–2196, 2011.

• J. Droniou, T. Gallou¨

et and J. Vovelle. Global solution and smoothing effect for a non-local regularization of a hyperbolic equation.

• J. Evol.

Equ., 3: 499–521, 2003.

• J. Droniou Vanishing non-local regularization of a scalar conservation law.

Electronic J. Diff. Equ., 117: 1–20, 2003.

• A. Kluwick, E. A. Cox, A. Exner and C. Grinschgl On the internal

structure of weakly nonlinear bores in laminar high Reynolds number flow. Acta Mech., 210: 135–157, 2010.

Thank you for your attention.

Riemann-Liouville fractional derivative

For a finite interval [a, b] ⊂ R, α ∈ C\N0 with ℜα ≥ 0 and n = ⌊ℜα⌋ + 1 (Dα

a+f )(x) =

1 Γ(n − α) dn dxn x

a

f (y) (x − y)α−n+1 dy , x ∈ [a, b] properties: For α = n ∈ N0 (Dn

a+f )(x) = f (n)(x)

For α, β ∈ C with ℜα ≥ 0 and ℜβ > 0 (Dα

a+(. − a)β−1)(x) =

• Γ(β)

Γ(β−α)(x − a)β−α−1

for α − β / ∈ N0 for α − β ∈ N For α ∈ C\N0 with ℜα ≥ 0 and f ∈ AC n[a, b] (Dα

a+f )(x) = n−1

• k=0

f (k)(a) Γ(1 + k − α)(x−a)k−α+ 1 Γ(n − α) x

a

f (n)(y) dy (x − y)α−n+1

Caputo fractional derivative on finite interval

For a finite interval [a, b] ⊂ R, α ∈ C\N0 with ℜα ≥ 0 and n = ⌊ℜα⌋ + 1 (CDα

a+f )(x) = Dα a+

• f (.) −

n−1

• k=0

f (k)(a) k! (. − a)k

• (x) ,

x ∈ [a, b] properties: alternative representation (CDα

a+f )(x) =

• 1

Γ(n−α)

x

a f (n)(y) (x−y)α−n+1 dy

for α / ∈ N0 f (n)(x) for α ∈ N0 For α, β ∈ C with ℜα ≥ 0 and ℜβ > 0 (CDα

a+(. − a)β−1)(x) =

• Γ(β)

Γ(β−α)(x − a)β−1

for α − β / ∈ N0 for α − β ∈ N For α / ∈ N0 and f ∈ C n[a, b] (CDα

a+f )(a) = 0

Liouville fractional derivative

For α ∈ C\N0 with ℜα ≥ 0 and n = ⌊ℜα⌋ + 1 (Dα

+f )(x) =

1 Γ(n − α) dn dxn x

−∞

f (y) (x − y)α−n+1 dy , x ∈ R properties: For α = n ∈ N0 (Dn

+f )(x) = f (n)(x)

For α, λ ∈ C with ℜα ≥ 0 and ℜλ > 0 (Dα

+ exp(λ.))(x) = λα exp(λx)

For α ∈ C with ℜα > 0 and f ∈ S(R) (FDα

+f )(ξ) = (−iξ)α(Ff )(ξ)

where (−iξ)α = exp(−απi sgn(ξ)/2).

Caputo fractional derivative on R

For α ∈ C\N0 with ℜα > 0 and n = ⌊ℜα⌋ + 1 (CDα

+f )(x) =

1 Γ(n − α) x

−∞

f (n)(y) (x − y)α−n+1 dy , x ∈ R properties: For α = n ∈ N0 (CDα

+f )(x) = f (n)(x)

For α ∈ C with ℜα > 0 and λ > 0 (CDα

+ exp(λ.))(x) = λα exp(λx)

For α ∈ C with ℜα > 0 and f ∈ S(R) (FCDα

+f )(ξ) = (−iξ)α(Ff )(ξ)

where (−iξ)α = exp(−απi sgn(ξ)/2).

Cauchy problem with fractional derivative

For a finite interval [a, b] ⊂ R and α ∈ C\N0 with ℜα ≥ 0 (Dα

a+y)(x) = f (x, y(x))

with n = ⌊ℜα⌋ + 1 initial conditions (Dα−k

a+ y)(a+) = bk ,

bk ∈ C , k = 1, . . . , n

Theorem (Kilbas et al (2006))

For an open set G ⊂ C and a function f : (a, b] × G → C with f (x, y) integrable w.r.t. x for all y ∈ G and f (x, y) Lipschitz continuous in y uniformly w.r.t. x ∈ (a, b]. Then there exists a unique solution y(x) for Cauchy type problem in the space { y ∈ L1(a, b) | Dα

a+y ∈ L1(a, b) }.

Step 1: Cauchy type problem is equivalent to Volterra integral equation y(x) =

n

• j=1

bj Γ(α − j + 1)(x − a)α−j + 1 Γ(α) x

a

f (t, y(t)) (x − t)1−α dt , x ∈ [a, b] Step 2: Banach fixed point theorem

Cauchy problem with Caputo fractional derivative

For a finite interval [a, b] ⊂ R and α ∈ R\N0 with α > 0 (CDα

a+y)(x) = f (x, y(x))

with n = ⌊α⌋ + 1 initial conditions y(k)(a+) = bk , bk ∈ C , k = 0, . . . , n − 1

Theorem (Kilbas et al (2006))

For an open set G ⊂ C and a function f : (a, b] × G → C with f (x, y) continuous w.r.t. x for all y ∈ G and f (x, y) Lipschitz continuous in y uniformly w.r.t. x ∈ (a, b]. Then there exists a unique solution y(x) for Cauchy type problem in the space { y ∈ C ⌊α⌋[a, b] | CDα

a+y ∈ C[a, b] }.

Step 1: Cauchy type problem is equivalent to Volterra integral equation y(x) =

n−1

• j=0

bj j! (x − a)j + 1 Γ(α) x

a

f (t, y(t)) (x − t)1−α dt , x ∈ [a, b] Step 2: Banach fixed point theorem

L´ evy strictly α-stable distributions on R

A random variable is said to be strictly stable (or to have a strictly stable distribution), if it has the property that linear combinations of two independent copies of the variable have the same distribution, up to a scaling. random variable X with L´ evy strictly stable distribution characteristic function E[exp(iξX)] = exp(ψ(ξ)) characteristic exponent ψ(ξ) := −c0|ξ|α exp

• − i sgn(ξ)θα π

2

• parameters

α index of stability 0 < α ≤ 2 θ skewness parameter |θ| ≤ min 2−α

α , 1

• c0 scaling parameter c0 > 0

L´ evy α-stable distributions on R

A random variable is said to be stable (or to have a stable distribution), if it has the property that linear combinations of two independent copies of the variable have the same distribution, up to location and scale parameters. random variable X with L´ evy stable distribution characteristic function E[exp(iξX)] = exp(ψ(ξ)) characteristic exponent ψ(ξ) =

• −c|ξ|α

1 − iβ(sgn ξ) tan απ

2

• + iτξ

for α = 1 , −c|ξ|

• 1 − iβ 2

π(sgn ξ) log |ξ|

• + iτξ

for α = 1 . parameters α index of stability 0 < α ≤ 2 β skewness parameter −1 ≤ β ≤ 1 c scaling parameter 0 < c τ location parameter τ ∈ R

L´ evy operator

Theorem (Sato (1999) Theorem 31.5)

Suppose {Xt} is a L´ evy process on Rd with generating triplet (A, ν, γ), whereat A = (Ajk) ∈ Rd×d and γ = (γj) ∈ Rd. The associated family of

• perators {Pt | t ≥ 0} is a strongly continuous semigroup on C0(Rd) with

norm Pt = 1. Let L be its infinitesimal generator. Then C ∞

c (Rd) is a

core of L, C 2

0 (Rd) ⊂ D(L), and

Lf (x) = 1 2

d

• j,k=1

Ajk ∂2f ∂xj∂xk (x) +

d

• j=1

γj ∂f ∂xj (x)+ +

• Rd
• f (x + y) − f (x) −

d

• j=1

yj ∂f ∂xj (x)1D(y)

• ν( dy)

(9) for f ∈ C 2

0 (Rd) and D = {x ∈ Rd | |x| ≤ 1}.

L´ evy operator: examples for d = 1

1 Any non-trivial α-stable distribution with 0 < α < 2 has absolutely

continuous L´ evy measure ν(dx) =

• c1x−1−α
• n (0, ∞) ,

c2|x|−1−α

• n (−∞, 0) ,

with c1 ≥ 0, c2 ≥ 0, c1 + c2 > 0.

2 If the L´

evy measure ν is non-singular, then

• R
• f (x + y) − f (x) − y ∂f

∂x (x)1D(y)

• ν( dy) = (K ∗ f − µf )(x)

for some K ∈ L1(R) and µ ∈ R.