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

Outline of my talk · Background of this study · Main results · Ingredients of proofs of main results · Conclusion and some future studies 24th Annual Workshop on Differential Equations Ching-Lung Lin (National Cheng Kung Univ.) Unique Continuation Property of Solutions to Anomalous Diffusion 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 � t 1 ∂ α ( t − s ) − α ∂ s u ( x, · ) ds t u ( t, · ) := Γ(1 − α ) 0 is the fractional derivative in the Caputo sense. 24th Annual Workshop on Differential Equations Ching-Lung Lin (National Cheng Kung Univ.) Unique Continuation Property of Solutions to Anomalous Diffusion Equations / 16

The exponent α describes the long time behavior of the mean square displacement < x 2 ( t ) > ∼ positive const. t α 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) 24th Annual Workshop on Differential Equations Ching-Lung Lin (National Cheng Kung Univ.) Unique Continuation Property of Solutions to Anomalous Diffusion 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 C 2 boundary ∂ Ω of a domain Ω ⊂ R n over some time interval, we can always consider the 0 extension of the solution outside Ω in a neighborhood of Γ . 24th Annual Workshop on Differential Equations Ching-Lung Lin (National Cheng Kung Univ.) Unique Continuation Property of Solutions to Anomalous Diffusion Equations / 16

Assuming 0 ∈ ∂ Ω without loss of generality, consider UCP (i.e. extending 0 across y n = 0 near the origin) of solution u ( t, y ) ∈ H α, 2 ( R n +1 ) solving  ∂ α t u ( t, y ) − ∆ y u ( t, y ) = l 1 ( 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 l 1 ( t, y ; ∇ y ) is a linear differential operator of order 1 with C ∞ ( R n +1 t,y ) coefficients, y j | < ℓ (1 ≤ j ≤ n − 1) , 0 ≤ ± y n < ℓ } ⊂ R n ω ± = { ( y 1 , · · · , y n ) : | y j − ˆ y n − 1 , 0) ∈ R n , and with a fixed constant ℓ > 0 , ˆ y = (ˆ y 1 , · · · , ˆ H m,s ( R n +1 ) = { u ( t, y ) ∈ S ′ ( R n +1 ) : (1 + | η | s + | τ | m ) 2 | ˆ � u � 2 u ( τ, η ) | 2 dτ dη < ∞} . H m,s := � � 24th Annual Workshop on Differential Equations Ching-Lung Lin (National Cheng Kung Univ.) Unique Continuation Property of Solutions to Anomalous Diffusion Equations / 16

Main results Concerning the Carleman estimate, let P ( t, x, D t , D x ) be the operator obtained by transforming e τ 0 t ( ∂ α t − ∆ y ) with fixed τ 0 < 0 by a Holmgren type transformation: ( t, y ) �→ ( t, x ) y ′ | 2 + X x ′ = y ′ − ˆ y ′ , x n = y n + | y ′ − ˆ T ( t − T ) , t = t, where x ′ = ( x 1 , · · · , x n − 1 ) , y ′ = ( y 1 , · · · , y n − 1 ) , ˆ y ′ = (ˆ y 1 , · · · , ˆ y n − 1 ) and X, T are small positive constants. Take the weight function ψ : ψ = 1 2( x n − X ) 2 . Then, we have the following Carleman estimate for P ( t, x, D t , D x ) . 24th Annual Workshop on Differential Equations Ching-Lung Lin (National Cheng Kung Univ.) Unique Continuation Property of Solutions to Anomalous Diffusion Equations / 16

Theorem 1 There exist a small open neighborhood U ⊂ R n +1 of the origin and sufficiently large constant β 1 depending on n such that for all v ( t, x ) ∈ C ∞ 0 ( U ) supported on t ≥ 0 and β ≥ β 1 , we have that e 2 βψ ( x ) | D γ | γ |≤ 1 β 3 − 2 | γ | � x v | 2 dtdx � e 2 βψ ( x ) | P ( t, x ; D t , D x ) v | 2 dtdx. � � (2) By applying this Carleman estimate, we can have the following UCP for the solution u of (1). Theorem 2 Let u ∈ H α, 2 ( R 1+ n ) satisfy (1) . Then u will be zero across y n = 0 near the origin of R n +1 . 24th Annual Workshop on Differential Equations Ching-Lung Lin (National Cheng Kung Univ.) Unique Continuation Property of Solutions to Anomalous Diffusion 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 Λ m α ( D t , D x ) by Λ m α ( D t , D x ) · := F − 1 (Λ m α ( τ, ξ )ˆ · ) , where F − 1 is the inverse Fourier transform and α ( τ, ξ ) = ((1 + | ξ | 2 ) 1 /α + iτ ) mα/ 2 Λ m for m ∈ R . 24th Annual Workshop on Differential Equations Ching-Lung Lin (National Cheng Kung Univ.) Unique Continuation Property of Solutions to Anomalous Diffusion Equations / 16

Let P ψ ( x, D t , D x , D z ) = P ( x, D t , D x + i | D z |∇ ψ ) be a pseudo-differential operator defined by e i ( x · ξ + tτ + zσ ) P ( x ; τ, ξ + i | σ |∇ ψ )ˆ P ψ v ( z, t, x ) = � v ( σ, ξ, τ ) dσdτdξ. (3) for any compactly supported distribution v in R n +1 × R z . The ”principal symbol” ˜ p ψ of P ψ can be given by ( i ( τ + iτ 0 )) α + | ξ ′ | 2 + 4 gξ n + fξ 2 n − f | σ | 2 ( x n − X ) 2 p ψ = ˜ + i 4 g ( x n − X ) | σ | + i 2 fξ n ( x n − X ) | σ | , where g = g ( x ′ , ξ ′ ) = Σ n − 1 j =1 x j ξ j and f = f ( x ′ ) = 1 + 4 | x ′ | 2 . 24th Annual Workshop on Differential Equations Ching-Lung Lin (National Cheng Kung Univ.) Unique Continuation Property of Solutions to Anomalous Diffusion Equations / 16

The Carleman estimate will be derived from the following subelliptic estimate for the operator P ψ . Lemma 3 There exists a sufficiently small constant z 0 such that for all 0 ( U × [ − z 0 , z 0 ]) ∩ ˙ S ( ¯ R 1+ n +1 u ( t, x, z ) ∈ C ∞ ) , we have that + Σ k + s< 2 || h ( D z ) 2 − k − s Λ s α D k z u || � || P ψ u || , (4) z ) 1 / 4 and U is a small open neighborhood of the where h ( D z ) = (1 + D 2 origin in R 1+ n . 24th Annual Workshop on Differential Equations Ching-Lung Lin (National Cheng Kung Univ.) Unique Continuation Property of Solutions to Anomalous Diffusion Equations / 16

Let’s fix a non-zero function g ∈ C ∞ 0 (( − z 0 , z 0 )) . Since h ( β ) ≃ β 1 / 2 , for any f ( t, x ) ∈ ˙ S ( ¯ R 1+ n ) , we have the + following by the subelliptic estimate | D γ | γ |≤ 1 β 3 − 2 | γ | � x f | 2 dtdx � Σ k + s< 2 h ( β ) 2(2 − k − s ) β 2 k || Λ s α f || 2 � (by Plancherel thm. and absorbing argument) � Σ k + s< 2 || h ( D z ) 2 − k − s Λ s α D k z ( e iβz f ( t, x ) g ( z )) || 2 (5) (by subelliptic estimate) � || P ψ ( e iβz f ( t, x ) g ( z )) || 2 On the other hand by ”Treve’s trick”, we have || P ψ ( e iβz f ( t, x ) g ( z )) || (6) || P ( t, x, D t , D x + i | β |∇ ψ ) f || + Σ k + | γ | < 2 β k · || D γ � x f || . 24th Annual Workshop on Differential Equations Ching-Lung Lin (National Cheng Kung Univ.) Unique Continuation Property of Solutions to Anomalous Diffusion Equations / 16

By going back to what we had | D γ | γ |≤ 1 β 3 − 2 | γ | � x f | 2 dtdx � � (7) || P ( t, x, D t , D x + i | β |∇ ψ ) f || 2 + Σ k + | γ | < 2 β 2 k || D γ x f || 2 and let β large enough, we have | D γ | γ |≤ 1 β 3 − 2 | γ | � x f | 2 dtdx � || P ( t, x, D t , D x + i | β |∇ ψ ) f || 2 . � (8) By letting f = e βψ v in (8), we immediately have the Carleman estimate: e 2 βψ ( x ) | D γ | γ |≤ 1 β 3 − 2 | γ | � x v | 2 dtdx � e 2 βψ ( x ) | P ( t, x ; D t , D x ) v | 2 dtdx. � � 24th Annual Workshop on Differential Equations Ching-Lung Lin (National Cheng Kung Univ.) Unique Continuation Property of Solutions to Anomalous Diffusion Equations / 16