Unique Continuation Property of Solutions to Anomalous Diffusion - - PowerPoint PPT Presentation

Unique Continuation Property of Solutions to Anomalous Diffusion Equations

Ching-Lung Lin

cllin2@mail.ncku.edu.tw Department of Mathematics, National Cheng Kung University, Taiwan Joint work with Gen Nakamura

24th Annual Workshop on Differential Equations

Ching-Lung Lin (National Cheng Kung Univ.) Unique Continuation Property of Solutions to Anomalous Diffusion Equations 24th Annual Workshop on Differential Equations / 16

Outline of my talk

· Background of this study · Main results · Ingredients of proofs of main results · Conclusion and some future studies

Ching-Lung Lin (National Cheng Kung Univ.) Unique Continuation Property of Solutions to Anomalous Diffusion Equations 24th Annual Workshop on Differential Equations / 16

Background

material science: anomalous slow diffusion on fractals such as some amorphous semiconductors or strongly porous materials (R. Metzler, J. Klafter, Phys. Rep., 339 (2000)) enviromental science: spread of pollution in soils is uncorrectly modeled by the usual diffusion equation (Y. Hatano, N. Hatano, Water Resour. Res., 134 (1998)). ⇒ anomalous slow diffusion equation: ∂α

t u−∆u+(l.o.t. in space derivatives) = (source term) with 0 < α < 1,

where ∂α

t u(t, ·) :=

1 Γ(1 − α) t (t − s)−α∂su(x, ·) ds is the fractional derivative in the Caputo sense.

Ching-Lung Lin (National Cheng Kung Univ.) Unique Continuation Property of Solutions to Anomalous Diffusion Equations 24th Annual Workshop on Differential Equations / 16

The exponent α describes the long time behavior of the mean square displacement < x2(t) >∼ positive const. tα of an anomalous diffusive particle x(t).

Known results (incomplete list)

well-posedness of initial boundary value problem: eigenfunction expansion approach: K. Sakamoto, M. Yamamoto, JMAA, 382 (2011) layer potential approach: J. Kemppainen, K. Rustsainen, Integr. Equ.

• Oper. Theory, 64 (2009)

Carleman estimate and UCP: α = 1/2 ⇒ transform to that of ∂t − ∆2 = (∂1/2 + ∆)(∂1/2 − ∆) C-L. Lin-G. Nakamura, JDE, 254 (2013)

Ching-Lung Lin (National Cheng Kung Univ.) Unique Continuation Property of Solutions to Anomalous Diffusion Equations 24th Annual Workshop on Differential Equations / 16

inverse problems: identifying α or source (with numerics)

• G. Li, D. Zhang, X. Jia, M. Yamamoto, Inverse Problems (2013)
• M. Kirane, S.A. Malik, M.A. Al-Gwaig, MMA Sci., 36 (2013)

UCP for general α has been missing which is important to give stability estimate and reconstruction schemes such as linear sampling method (LSM) and dynamical probe method (DPM). Formulation of UCP Even in the case discussing UCP of solution satisfying zero Cauchy data on a small part Γ of the C2 boundary ∂Ω of a domain Ω ⊂ Rn over some time interval, we can always consider the 0 extension of the solution outside Ω in a neighborhood of Γ.

Ching-Lung Lin (National Cheng Kung Univ.) Unique Continuation Property of Solutions to Anomalous Diffusion Equations 24th Annual Workshop on Differential Equations / 16

Assuming 0 ∈ ∂Ω without loss of generality, consider UCP (i.e. extending 0 across yn = 0 near the origin) of solution u(t, y) ∈ Hα,2(Rn+1) solving      ∂α

t u(t, y) − ∆yu(t, y) = l1(t, y; ∇y)u(t, y) (|t| < T, y ∈ ω− ∪ ω+)

u(t, y) = 0 (t ≤ 0), u(t, y) = 0 (y ∈ ω−, 0 < t < T), (1) where l1(t, y; ∇y) is a linear differential operator of order 1 with C∞(Rn+1

t,y ) coefficients,

ω± = {(y1, · · · , yn) : |yj − ˆ yj| < ℓ (1 ≤ j ≤ n − 1), 0 ≤ ±yn < ℓ} ⊂ Rn with a fixed constant ℓ > 0, ˆ y = (ˆ y1, · · · , ˆ yn−1, 0) ∈ Rn, and Hm,s(Rn+1) = {u(t, y) ∈ S′(Rn+1) : u2

Hm,s :=

(1 + |η|s + |τ|m)2|ˆ u(τ, η)|2dτ dη < ∞}.

Ching-Lung Lin (National Cheng Kung Univ.) Unique Continuation Property of Solutions to Anomalous Diffusion Equations 24th Annual Workshop on Differential Equations / 16

Main results

Concerning the Carleman estimate, let P(t, x, Dt, Dx) be the operator

• btained by transforming eτ0t(∂α

t − ∆y) with fixed τ0 < 0 by a Holmgren

type transformation: (t, y) → (t, x) x′ = y′ − ˆ y′, xn = yn + |y′ − ˆ y′|2 + X T (t − T), t = t, where x′ = (x1, · · · , xn−1), y′ = (y1, · · · , yn−1), ˆ y′ = (ˆ y1, · · · , ˆ yn−1) and X, T are small positive constants. Take the weight function ψ: ψ = 1 2(xn − X)2. Then, we have the following Carleman estimate for P(t, x, Dt, Dx).

Ching-Lung Lin (National Cheng Kung Univ.) Unique Continuation Property of Solutions to Anomalous Diffusion Equations 24th Annual Workshop on Differential Equations / 16

Theorem 1 There exist a small open neighborhood U ⊂ Rn+1 of the origin and sufficiently large constant β1 depending on n such that for all v(t, x) ∈ C∞

0 (U) supported on t ≥ 0 and β ≥ β1, we have that

• |γ|≤1 β3−2|γ|

e2βψ(x)|Dγ

xv|2dtdx

• e2βψ(x)|P(t, x; Dt, Dx)v|2dtdx.

(2) By applying this Carleman estimate, we can have the following UCP for the solution u of (1). Theorem 2 Let u ∈ Hα,2(R1+n) satisfy (1). Then u will be zero across yn = 0 near the origin of Rn+1.

Ching-Lung Lin (National Cheng Kung Univ.) Unique Continuation Property of Solutions to Anomalous Diffusion Equations 24th Annual Workshop on Differential Equations / 16

Ingredients of Proofs

Ingredients of the proof of Theorem 2 The proof of Theorem 2 is just an application of the Carleman estimate and it is quite standard. (omitted) Ingredients of the proof of Theorem 1 Notations Define a pseudo-differential operator Λm

α (Dt, Dx) by

Λm

α (Dt, Dx)· := F−1(Λm α (τ, ξ)ˆ

·), where F−1 is the inverse Fourier transform and Λm

α (τ, ξ) = ((1 + |ξ|2)1/α + iτ)mα/2

for m ∈ R.

Ching-Lung Lin (National Cheng Kung Univ.) Unique Continuation Property of Solutions to Anomalous Diffusion Equations 24th Annual Workshop on Differential Equations / 16

Let Pψ(x, Dt, Dx, Dz) = P(x, Dt, Dx + i|Dz|∇ψ) be a pseudo-differential

• perator defined by

Pψv(z, t, x) =

• ei(x·ξ+tτ+zσ)P(x; τ, ξ+i|σ|∇ψ)ˆ

v(σ, ξ, τ)dσdτdξ. (3) for any compactly supported distribution v in Rn+1 × Rz. The ”principal symbol” ˜ pψ of Pψ can be given by ˜ pψ = (i(τ + iτ0))α + |ξ′|2 + 4gξn + fξ2

n − f|σ|2(xn − X)2

+i4g(xn − X)|σ| + i2fξn(xn − X)|σ|, where g = g(x′, ξ′) = Σn−1

j=1 xjξj and f = f(x′) = 1 + 4|x′|2.

Ching-Lung Lin (National Cheng Kung Univ.) Unique Continuation Property of Solutions to Anomalous Diffusion Equations 24th Annual Workshop on Differential Equations / 16

The Carleman estimate will be derived from the following subelliptic estimate for the operator Pψ. Lemma 3 There exists a sufficiently small constant z0 such that for all u(t, x, z) ∈ C∞

0 (U × [−z0, z0]) ∩ ˙

S( ¯ R1+n+1

+

), we have that Σk+s<2||h(Dz)2−k−sΛs

αDk zu|| ||Pψu||,

(4) where h(Dz) = (1 + D2

z)1/4 and U is a small open neighborhood of the

• rigin in R1+n.

Ching-Lung Lin (National Cheng Kung Univ.) Unique Continuation Property of Solutions to Anomalous Diffusion Equations 24th Annual Workshop on Differential Equations / 16

Let’s fix a non-zero function g ∈ C∞

0 ((−z0, z0)).

Since h(β) ≃ β1/2, for any f(t, x) ∈ ˙ S( ¯ R1+n

+

), we have the following by the subelliptic estimate

• |γ|≤1 β3−2|γ|

|Dγ

xf|2dtdx Σk+s<2h(β)2(2−k−s)β2k||Λs αf||2

(by Plancherel thm. and absorbing argument) Σk+s<2||h(Dz)2−k−sΛs

αDk z(eiβzf(t, x)g(z))||2

(by subelliptic estimate) ||Pψ(eiβzf(t, x)g(z))||2 (5) On the other hand by ”Treve’s trick”, we have ||Pψ(eiβzf(t, x)g(z))||

• ||P(t, x, Dt, Dx + i|β|∇ψ)f|| + Σk+|γ|<2βk · ||Dγ

xf||.

(6)

Ching-Lung Lin (National Cheng Kung Univ.) Unique Continuation Property of Solutions to Anomalous Diffusion Equations 24th Annual Workshop on Differential Equations / 16

By going back to what we had

• |γ|≤1 β3−2|γ|

|Dγ

xf|2dtdx

||P(t, x, Dt, Dx + i|β|∇ψ)f||2 + Σk+|γ|<2β2k||Dγ

xf||2

(7) and let β large enough, we have

• |γ|≤1 β3−2|γ|

|Dγ

xf|2dtdx

||P(t, x, Dt, Dx + i|β|∇ψ)f||2. (8) By letting f = eβψv in (8), we immediately have the Carleman estimate:

• |γ|≤1 β3−2|γ|

e2βψ(x)|Dγ

xv|2dtdx

• e2βψ(x)|P(t, x; Dt, Dx)v|2dtdx.

Ching-Lung Lin (National Cheng Kung Univ.) Unique Continuation Property of Solutions to Anomalous Diffusion Equations 24th Annual Workshop on Differential Equations / 16

Conclusion and some future studies

Conclusion We have obtained a Carleman estimate and UCP for anomalous slow diffusion equations which were missing for developing a stability estimate and reconstruction schemes such as linear sampling method (LSM) and dynamical probe method (DPM) for inverse boundary value problem to identify unknown cavities and inclusions inside an anomalous diffusive medium. Some future studies (i) Develop the reconstruction schemes LSM and DPM. (ii) Study the fast diffusion case i.e. 1 < α < 2 which appears in fractional

• rder viscoelastic equations. (ex. springpot model for human tissues)

Ching-Lung Lin (National Cheng Kung Univ.) Unique Continuation Property of Solutions to Anomalous Diffusion Equations 24th Annual Workshop on Differential Equations / 16

Some ingredients for the proof of subelliptic estimate

Let |x′|2 ≤ X/4 with X << 1. If Re ˜ pψ = 0, we have {Re ˜ pψ, Im ˜ pψ} (|τ|α + |ξ|2 + σ2)3/2. (9) This enables to handle the commutator [P ∗

ψ, Pψ] in

||Pψu||2

L2 = (Pψu, Pψu) = (P ∗ ψPψu, u)

= (PψP ∗

ψu, u) + ([P ∗ ψ, Pψ]u, u)

(10) for u as in the lemma with Fourier transform supported away from σ = 0. If |σ|/(|ξ|2 + σ2 + |τ|α)1/2 is small, we have |˜ pψ| |ξ|2 + σ2 + |τ|α, (11) which enables to have an elliptic estimate in this case.

Ching-Lung Lin (National Cheng Kung Univ.) Unique Continuation Property of Solutions to Anomalous Diffusion Equations 24th Annual Workshop on Differential Equations / 16

Thank you for your attention

Ching-Lung Lin (National Cheng Kung Univ.) Unique Continuation Property of Solutions to Anomalous Diffusion Equations 24th Annual Workshop on Differential Equations / 16