Analyticity of solutions to parabolic equations and observability Coron60: Conference in honor of Jean-Michel Coron. Santiago Montaner University of the Basque Country (UPV/EHU) June, 21st 2016 A joint work with Luis Escauriaza (UPV/EHU) and Can Zhang (UPMC Paris 6) S. Montaner (UPV/EHU) Analyticity June, 21st 2016 1 / 18
Interior observability inequality over open sets The interior null-controllability property for the Heat equation is equivalent to the interior observability , i.e., there exists a constant N = N ( ω, Ω , T ) s.t. the solution to ∂ t v − ∆ v = 0 , in Ω × (0 , T ] , v = 0 , on ∂ Ω × (0 , T ] , v (0) = v 0 . in Ω , satisfies the observability inequality � v ( T ) � L 2 (Ω) ≤ N � v � L 2 ( ω × (0 , T )) . The null-controllability property for the Heat equation and other second-order parabolic equations was obtained by Fattorini-Russell (1971), Imanuvilov, Lebeau-Robbiano (1995). Also some results for 4th-order parabolic equations by Le Rousseau-Robbiano (2015). S. Montaner (UPV/EHU) Analyticity June, 21st 2016 2 / 18
An interior observability inequality over measurable sets Theorem (J. Apraiz, L. Escauriaza, G. Wang, C. Zhang, 2014) Let 0 < T < 1, D ⊂ Ω × (0 , T ) ( ∂ Ω Lipschitz) be a measurable set, |D| > 0. Then ∃ N = N ( D , Ω , T ) s.t. � � u ( T ) � L 2 (Ω) ≤ N | u ( x , t ) | dxdt D holds for all solutions to ∂ t u − ∆ u = 0 , in Ω × (0 , T ] , on ∂ Ω × (0 , T ] , u = 0 u 0 ∈ L 2 (Ω) . u (0) = u 0 , S. Montaner (UPV/EHU) Analyticity June, 21st 2016 3 / 18
Null-controllability of a parabolic equations from measurable sets Corollary (J. Apraiz, L. Escauriaza, G. Wang, C. Zhang, 2014) Let 0 < T < 1 and D ⊆ Ω × (0 , T ) ( ∂ Ω Lipschitz) be a measurable set, |D| > 0. Then for each u 0 ∈ L 2 (Ω) exists f ∈ L ∞ (Ω × (0 , T )) s.t. � f � L ∞ ( D ) ≤ N ( D , Ω , T ) � u 0 � L 2 (Ω) and the solution to ∂ t u − ∆ u = χ D f , in Ω × (0 , T ] , u = 0 , on ∂ Ω × (0 , T ] , u (0) = u 0 . in Ω , satisfies u ( T ) ≡ 0. S. Montaner (UPV/EHU) Analyticity June, 21st 2016 4 / 18
In Observation from measurable sets for parabolic analytic evolutions and applications (Escauriaza, Montaner, Zhang (2015)), these results are extended to some equations and systems with real-analytic coefficients not depending on time such as: higher-order parabolic evolutions, strongly coupled second-order systems with a possibly non-symmetric structure, one-component control of a weakly coupled system of two equations, In this work, the real-analyticity of coefficients is quantified as: | ∂ γ x a α ( x ) | ≤ ρ 0 − 1 −| γ | | γ | ! in Ω × [0 , T ] . S. Montaner (UPV/EHU) Analyticity June, 21st 2016 5 / 18
The proof of these results rely on: An inequality of propagation of smallness from measurable sets by S. Vessella (1999). S. Montaner (UPV/EHU) Analyticity June, 21st 2016 6 / 18
The proof of these results rely on: An inequality of propagation of smallness from measurable sets by S. Vessella (1999). New quantitative estimates of space-time analyticity of the form t u ( x , t ) | ≤ e 1 /ρ t 1 / (2 m − 1) ρ −| γ |− p | γ | ! p ! t − p � u 0 � L 2 (Ω) , | ∂ γ x ∂ p 0 < t ≤ 1, γ ∈ N n , p ≥ 0 and 2 m is the order of the parabolic problem solved by u . These estimates are obtained quantifying each step of a reasoning developed by Landis and Oleinik (1974) which reduces the strong UCP within characteristic hyperplanes of parabolic equations to its elliptic counterpart and is based on a spectral representation of solutions. S. Montaner (UPV/EHU) Analyticity June, 21st 2016 6 / 18
The proof of these results rely on: An inequality of propagation of smallness from measurable sets by S. Vessella (1999). New quantitative estimates of space-time analyticity of the form t u ( x , t ) | ≤ e 1 /ρ t 1 / (2 m − 1) ρ −| γ |− p | γ | ! p ! t − p � u 0 � L 2 (Ω) , | ∂ γ x ∂ p 0 < t ≤ 1, γ ∈ N n , p ≥ 0 and 2 m is the order of the parabolic problem solved by u . These estimates are obtained quantifying each step of a reasoning developed by Landis and Oleinik (1974) which reduces the strong UCP within characteristic hyperplanes of parabolic equations to its elliptic counterpart and is based on a spectral representation of solutions. The so-called telescoping series method (L. Miller; K. D. Phung, G. Wang). S. Montaner (UPV/EHU) Analyticity June, 21st 2016 6 / 18
S. Vessella. A continuous dependence result in the analytic continuation problem. Forum Math. 11 , 6 (1999), 695–703. Lemma. (Propagation of smallness from measurable sets) Let ω ⊂ B R be a measurable set | ω | > 0. Let f be a real-analytic function in B 2 R s.t. there exist numbers M and ρ for which | ∂ γ x f ( x ) | ≤ M ( ρ R ) −| γ | | γ | ! holds when x ∈ B 2 R and γ ∈ N n . Then, there are N = N ( B R , ρ, | ω | ) and θ = θ ( B R , ρ, | ω | ), 0 < θ < 1, such that � 1 � θ � � f � L ∞ ( B R ) ≤ NM 1 − θ | f | dx . | ω | ω S. Montaner (UPV/EHU) Analyticity June, 21st 2016 7 / 18
Some remarks on the quantitative estimates The quantitative estimate of space-time real-analyticity 1 2 m − 1 ρ − 1 −| γ |− p t − p | γ | ! p ! � u 0 � L 2 (Ω) t u ( x , t ) | ≤ e t − | ∂ γ x ∂ p S. Montaner (UPV/EHU) Analyticity June, 21st 2016 8 / 18
Some remarks on the quantitative estimates The quantitative estimate of space-time real-analyticity 1 2 m − 1 ρ − 1 −| γ |− p t − p | γ | ! p ! � u 0 � L 2 (Ω) t u ( x , t ) | ≤ e t − | ∂ γ x ∂ p yields a positive lower bound ρ for the radius of convergence of the Taylor series in the spatial variables independent of t , S. Montaner (UPV/EHU) Analyticity June, 21st 2016 8 / 18
Some remarks on the quantitative estimates The quantitative estimate of space-time real-analyticity 1 2 m − 1 ρ − 1 −| γ |− p t − p | γ | ! p ! � u 0 � L 2 (Ω) t u ( x , t ) | ≤ e t − | ∂ γ x ∂ p yields a positive lower bound ρ for the radius of convergence of the Taylor series in the spatial variables independent of t , 1 2 m − 1 when t → 0 + . if p = 0, it blows up like e t − S. Montaner (UPV/EHU) Analyticity June, 21st 2016 8 / 18
Some remarks on the quantitative estimates The quantitative estimate of space-time real-analyticity 1 2 m − 1 ρ − 1 −| γ |− p t − p | γ | ! p ! � u 0 � L 2 (Ω) t u ( x , t ) | ≤ e t − | ∂ γ x ∂ p yields a positive lower bound ρ for the radius of convergence of the Taylor series in the spatial variables independent of t , 1 2 m − 1 when t → 0 + . if p = 0, it blows up like e t − These features of the quantitative estimates of analyticity are essential in the proof of the interior observability estimate over measurable sets. S. Montaner (UPV/EHU) Analyticity June, 21st 2016 8 / 18
Parabolic operators with time dependent coefficients In order to deal with time-dependent coefficients, we cannot adapt the reasoning by Landis and Oleinik! S. Montaner (UPV/EHU) Analyticity June, 21st 2016 9 / 18
Parabolic operators with time dependent coefficients In order to deal with time-dependent coefficients, we cannot adapt the reasoning by Landis and Oleinik! Consider the 2 m -th order operator � a α ( x , t ) ∂ α � ∂ α x ( A αβ ( x , t ) ∂ β � A γ ( x , t ) ∂ γ L = x = x ) + x , | α |≤ 2 m | α | , | β |≤ m | γ |≤ m assume that for some ρ 0 , 0 < ρ 0 < 1 A α,β ( x , t ) ξ α + β ≥ ρ 0 | ξ | 2 m ∀ ξ ∈ R n , in Ω × [0 , T ] , � | α | = | β | = m S. Montaner (UPV/EHU) Analyticity June, 21st 2016 9 / 18
Parabolic operators with time dependent coefficients In order to deal with time-dependent coefficients, we cannot adapt the reasoning by Landis and Oleinik! Consider the 2 m -th order operator � a α ( x , t ) ∂ α � ∂ α x ( A αβ ( x , t ) ∂ β � A γ ( x , t ) ∂ γ L = x = x ) + x , | α |≤ 2 m | α | , | β |≤ m | γ |≤ m assume that for some ρ 0 , 0 < ρ 0 < 1 A α,β ( x , t ) ξ α + β ≥ ρ 0 | ξ | 2 m ∀ ξ ∈ R n , in Ω × [0 , T ] , � | α | = | β | = m x ∂ p | ∂ γ t a α ( x , t ) | ≤ ρ 0 − 1 −| γ |− p | γ | ! p ! in Ω × [0 , T ] . S. Montaner (UPV/EHU) Analyticity June, 21st 2016 9 / 18
As far as we know, the best estimate that follows from the works of S. D. Eidelman, A. Friedman, D. Kinderlehrer, L. Nirenberg, G. Komatsu and H. Tanabe is: Theorem There is 0 < ρ ≤ 1 , ρ = ρ ( ρ 0 , n , ∂ Ω) such that ∀ α ∈ N n , p ∈ N t u ( x , t ) | ≤ ρ − 1 − | γ | 2 m − p | γ | ! p ! t − | γ | 2 m − p − n | ∂ γ x ∂ p 4 m � u 0 � L 2 (Ω) , in Ω × (0 , T ] when u solves ∂ t u + ( − 1) m Lu = 0 , in Ω × (0 , T ] , u = Du = . . . = D m − 1 u = 0 , in ∂ Ω × (0 , T ] , u 0 ∈ L 2 (Ω) . u ( · , 0) = u 0 , and ∂ Ω is a real-analytic hypersurface. S. Montaner (UPV/EHU) Analyticity June, 21st 2016 10 / 18
If u satisfies t u ( x , t ) | ≤ ρ − 1 − | γ | 2 m − p | γ | ! p ! t − | γ | 2 m − p − n | ∂ γ x ∂ p 4 m � u 0 � L 2 (Ω) , ∀ γ ∈ N n , p ∈ N , we observe that: the space analyticity estimate blows up as t tends to zero, which is unavoidable if u 0 is an arbitrary L 2 (Ω) function; S. Montaner (UPV/EHU) Analyticity June, 21st 2016 11 / 18
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