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Time-periodic parabolic equations CEMRACS 2019 Jean-Jérôme Casanova

Introduction to analytic semigroups 1 The periodic problem 2 Application to a fluid–structure interaction problem 3 Jean-Jérôme Casanova Time-periodic parabolic equations CEMRACS 2019 2 / 18

Abstract parabolic evolution equation: � y ′ ( t ) = Ay ( t ) + f ( t ) , t > 0 , (1.1) y (0) = y 0 . Hypothesis: Hilbertian framework H . A is the infinitesimal generator of an analytic semigroup of operators S ( t ). The resolvent of A is compact. ( σ ( A ) = σ p ( A )) Definition of a semigroup of operators S ( t ) ∈ L ( H ), t ≥ 0: (i) S (0) = Id on H (ii) S ( t + s ) = S ( t ) ◦ S ( s ) for every t , s ≥ 0. A trivial example: y ′ ( t ) = ay ( t ) ⇒ y ( t ) = e at y 0 = S ( t ) y 0 . Jean-Jérôme Casanova Time-periodic parabolic equations CEMRACS 2019 3 / 18

Analytic semigroups ( C 0 -) Analytic semigroup: S (0) = I , z → 0 , z ∈ ∆ S ( z ) x = x for all x ∈ H . lim z �→ S ( z ) is analytic in a sector ∆. S ( z 1 + z 2 ) = S ( z 1 ) ◦ S ( z 2 ) for all z 1 , z 2 ∈ ∆. i R • z 1 ∆ R • z 2 Jean-Jérôme Casanova Time-periodic parabolic equations CEMRACS 2019 4 / 18

Another definition/property A is the infinitesimal generator of an analytic semigroup ⇔ The resolvent set ρ ( A ) contains a sector Σ = { λ ∈ C | λ � = ω and | arg( λ − ω ) | < θ } with ω ∈ R and θ > π 2 . M � R ( λ, A ) � L ( H ) ≤ | λ − ω | , ∀ λ ∈ Σ with M > 0. Dunford integral: 1 � e tA := S ( t ) = e λ t R ( λ, A ) d λ, t > 0 . 2 i π γ Jean-Jérôme Casanova Time-periodic parabolic equations CEMRACS 2019 5 / 18

Why are analytic semigroups so important? Formally, in Fourier: y ( k ) + ˆ ik ˆ y ( k ) = A ˆ f ( k ) , 1 y ( k ) = R ( ik , A )ˆ ˆ f ( k ) , 2 y ( k ) | ≤ ( M + 1) | ˆ ⇒ | A ˆ f ( k ) | , 3 Using that: AR ( λ, A ) = − Id + λ R ( λ, A ). y ′ , Ay and f have the same regularity ⇒ “Maximal regularity property” Theorem 1 Assume that S is an analytic semigroup, then for each T > 0 , the map � L 2 (0 , T ; D ( A )) ∩ H 1 (0 , T ; H ) → L 2 (0 , T ; H ) × [ D ( A ) , H ] 1 / 2 Iso : y �→ ( y ′ − Ay , y (0)) is an isomorphism. Jean-Jérôme Casanova Time-periodic parabolic equations CEMRACS 2019 6 / 18

A concrete example Consider the heat equation: � y ′ ( t ) − ∆ y ( t ) = f ( t ) , t > 0 , (1.2) y (0) = y 0 with : Ω a smooth bounded domain. D (∆) = H 2 (Ω) ∩ H 1 0 (Ω) (Dirichlet boundary condition). f ∈ L 2 (0 , + ∞ ; L 2 (Ω)). y 0 ∈ [ H 2 (Ω) ∩ H 1 0 (Ω) , L 2 (Ω)] 1 / 2 = H 1 0 (Ω). Theorem 1: ⇒ ∃ ! y ∈ L 2 (0 , + ∞ ; H 2 (Ω) ∩ H 1 0 (Ω)) ∩ H 1 (0 , + ∞ ; L 2 (Ω)) solution to (1.2) . And with a nonlinear term y ∆ y ? Jean-Jérôme Casanova Time-periodic parabolic equations CEMRACS 2019 7 / 18

The periodic problem Periodic evolution equation: � y ′ ( t ) = Ay ( t ) + f ( t ) , for all t ∈ [0 , T ] , (2.1) y (0) = y ( T ) . � T From the Duhamel formula: y (0) = y ( T ) = S ( T ) y (0) + S ( T − s ) f ( s ) ds . 0 Existence of time-periodic solutions ⇐ ⇒ Existence of a solution z to � T (2.2) ( I − S ( T )) z = S ( T − s ) f ( s ) ds . 0 We need some spectral assumptions on ( A , T ) to invert ( I − S ( T )). Jean-Jérôme Casanova Time-periodic parabolic equations CEMRACS 2019 8 / 18

The periodic problem Periodic evolution equation: � y ′ ( t ) = Ay ( t ) + f ( t ) , for all t ∈ [0 , T ] , (2.1) y (0) = y ( T ) . � T From the Duhamel formula: y (0) = y ( T ) = S ( T ) y (0) + S ( T − s ) f ( s ) ds . 0 Existence of time-periodic solutions ⇐ ⇒ Existence of a solution z to � T (2.2) ( I − S ( T )) z = S ( T − s ) f ( s ) ds . 0 We need some spectral assumptions on ( A , T ) to invert ( I − S ( T )). Jean-Jérôme Casanova Time-periodic parabolic equations CEMRACS 2019 8 / 18

The periodic problem Periodic evolution equation: � y ′ ( t ) = Ay ( t ) + f ( t ) , for all t ∈ [0 , T ] , (2.1) y (0) = y ( T ) . � T From the Duhamel formula: y (0) = y ( T ) = S ( T ) y (0) + S ( T − s ) f ( s ) ds . 0 Existence of time-periodic solutions ⇐ ⇒ Existence of a solution z to � T (2.2) ( I − S ( T )) z = S ( T − s ) f ( s ) ds . 0 We need some spectral assumptions on ( A , T ) to invert ( I − S ( T )). Jean-Jérôme Casanova Time-periodic parabolic equations CEMRACS 2019 8 / 18

Assumptions on A : A is the infinitesimal generator of an analytic semigroup and its resolvent is compact. Spectral theorem: σ p ( S ( T )) = e T σ p ( A ) . 1 ∈ σ p ( S ( T )) ⇔ 0 ∈ σ p ( A ) or A has a complex eigenvalue 2 ik π with k ∈ Z ∗ . T Jean-Jérôme Casanova Time-periodic parabolic equations CEMRACS 2019 9 / 18

Assumptions on A : A is the infinitesimal generator of an analytic semigroup and its resolvent is compact. Spectral theorem: σ p ( S ( T )) = e T σ p ( A ) . 1 ∈ σ p ( S ( T )) ⇔ 0 ∈ σ p ( A ) or A has a complex eigenvalue 2 ik π with k ∈ Z ∗ . T i R R Jean-Jérôme Casanova Time-periodic parabolic equations CEMRACS 2019 9 / 18

Assumptions on A : A is the infinitesimal generator of an analytic semigroup and its resolvent is compact. Spectral theorem: σ p ( S ( T )) = e T σ p ( A ) . 1 ∈ σ p ( S ( T )) ⇔ 0 ∈ σ p ( A ) or A has a complex eigenvalue 2 ik π with k ∈ Z ∗ . T i R R Jean-Jérôme Casanova Time-periodic parabolic equations CEMRACS 2019 9 / 18

Assumptions on A : A is the infinitesimal generator of an analytic semigroup and its resolvent is compact. Spectral theorem: σ p ( S ( T )) = e T σ p ( A ) . 1 ∈ σ p ( S ( T )) ⇔ 0 ∈ σ p ( A ) or A has a complex eigenvalue 2 ik π with k ∈ Z ∗ . T i R • • • R • • • • • • • Jean-Jérôme Casanova Time-periodic parabolic equations CEMRACS 2019 9 / 18

Assumptions on A : A is the infinitesimal generator of an analytic semigroup and its resolvent is compact. Spectral theorem: σ p ( S ( T )) = e T σ p ( A ) . 1 ∈ σ p ( S ( T )) ⇔ 0 ∈ σ p ( A ) or A has a complex eigenvalue 2 ik π with k ∈ Z ∗ . T i R • • • R • • • • • • • Jean-Jérôme Casanova Time-periodic parabolic equations CEMRACS 2019 9 / 18

Denote by { ib j } 0 ≤ j ≤ N A the (finite) number of eigenvalue of A on the imaginary axis i R ( • in the previous example). Assumption on the period T : T ∈ R + \ { 2 k π (2.3) | k ∈ Z , 0 ≤ j ≤ N A } b j Under the previous assumptions on ( A , T ) we have � T y (0) = ( I − S ( T )) − 1 S ( T − s ) f ( s ) ds ∈ [ D ( A ) , H ] 1 / 2 and we obtain: 0 Theorem 2 For f ∈ L 2 (0 , T ; H ) , the periodic evolution equation (2.1) admits a unique strict solution y ∈ L 2 (0 , T ; D ( A )) ∩ H 1 ♯ (0 , T ; H ) in L 2 (0 , T ; H ) . The following estimate holds � y � L 2 (0 , T ; D ( A )) ∩ H 1 ♯ (0 , T ; H ) ≤ C � f � L 2 (0 , T ; H ) . Jean-Jérôme Casanova Time-periodic parabolic equations CEMRACS 2019 10 / 18

Hölder regularity in time When the source term f is Hölder continuous in time: Theorem 3 For f ∈ C ρ ♯ ([0 , T ]; H ) with ρ ∈ (0 , 1) the periodic evolution equation (2.1) admits a unique strict solution y in C ([0 , T ]; H ) . Moreover y ∈ C ρ ([0 , T ]; D ( A )) ∩ C ρ +1 ([0 , T ]; H ) , and the following estimate holds (2.4) � y � C ρ ([0 , T ]; D ( A )) ∩C ρ +1 ([0 , T ]; H ) ≤ C � f � C ρ ([0 , T ]; H ) . Remark 4 Very specific result for parabolic equation ⇒ Not true in the non-periodic framework. Jean-Jérôme Casanova Time-periodic parabolic equations CEMRACS 2019 11 / 18

Simplified model of blood flow through arteries Structure Fluid Incompressible fluid, viscous, Newtonian : Incompressible Navier–Stokes equations. Viscoelastic structure : Damped Euler–Bernoulli beam equation. Jean-Jérôme Casanova Time-periodic parabolic equations CEMRACS 2019 12 / 18

Γ η ( t ) η ( x , t ) 1 Γ s Γ i Γ o Ω η ( t ) Γ b 0 L Eulerian-Lagrangian formulation. Structure displacement η : Γ s × (0 , T ) → ( − 1 , + ∞ ). Fluid domain Ω η ( t ) : Unknown of the problem. Jean-Jérôme Casanova Time-periodic parabolic equations CEMRACS 2019 13 / 18

Fluid–structure interaction system Fluid : 2 D Incompressible Navier–Stokes equation u t + ( u · ∇ ) u − ν ∆ u + ∇ p = 0 et div u = 0 in Ω η ( t ) , t > 0 . Structure : Damped Euler–Bernoulli beam equation η tt − βη xx − γη txx + αη xxxx = F ( u , p , η ) on Γ s , t > 0 . Kinematic coupling : u = η t e 2 on Γ η ( t ) , t > 0 . Boundary conditions and time-periodic forcing term : u = ω 1 on Γ i , u 2 = 0 and p + (1 / 2) | u | 2 = ω 2 on Σ o , u = 0 on Γ b , η (0 , t ) = η ( L , t ) = η x (0 , t ) = η x ( L , t ) = 0 , t > 0 . Periodic solutions: ( u (0) , η (0) , η t (0)) = ( u ( T ) , η ( T ) , η t ( T )). Jean-Jérôme Casanova Time-periodic parabolic equations CEMRACS 2019 14 / 18