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Time-periodic parabolic equations

CEMRACS 2019 Jean-Jérôme Casanova

1

Introduction to analytic semigroups

2

The periodic problem

3

Application to a fluid–structure interaction problem

Jean-Jérôme Casanova Time-periodic parabolic equations CEMRACS 2019 2 / 18

Abstract parabolic evolution equation: (1.1)

• y ′(t) = Ay(t) + f (t), t > 0,

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) = σp(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) = eaty 0 = S(t)y 0.

Jean-Jérôme Casanova Time-periodic parabolic equations CEMRACS 2019 3 / 18

Analytic semigroups

(C0-) Analytic semigroup: S(0) = I, lim

z→0,z∈∆ S(z)x = x for all x ∈ H.

z → S(z) is analytic in a sector ∆. S(z1 + z2) = S(z1) ◦ S(z2) for all z1, z2 ∈ ∆. iR R

• z1

z2 ∆

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 .

R(λ, A)L(H) ≤ M |λ − ω|, ∀λ ∈ Σ with M > 0. Dunford integral: etA := S(t) = 1 2iπ

• γ

eλtR(λ, A)dλ, t > 0.

Jean-Jérôme Casanova Time-periodic parabolic equations CEMRACS 2019 5 / 18

Why are analytic semigroups so important?

Formally, in Fourier:

1

ikˆ y(k) = Aˆ y(k) + ˆ f (k),

2

ˆ y(k) = R(ik, A)ˆ f (k),

3

⇒ |Aˆ y(k)| ≤ (M + 1)|ˆ f (k)|, Using that: AR(λ, A) = −Id + λR(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 Iso :

• L2(0, T; D(A)) ∩ H1(0, T; H) → L2(0, T; H) × [D(A), H]1/2

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: (1.2)

• y ′(t) − ∆y(t) = f (t), t > 0,

y(0) = y 0 with : Ω a smooth bounded domain. D(∆) = H2(Ω) ∩ H1

0(Ω) (Dirichlet boundary condition).

f ∈ L2(0, +∞; L2(Ω)). y 0 ∈ [H2(Ω) ∩ H1

0(Ω), L2(Ω)]1/2 = H1 0(Ω).

Theorem 1: ⇒ ∃!y ∈ L2(0, +∞; H2(Ω) ∩ H1

0(Ω)) ∩ H1(0, +∞; L2(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: (2.1)

• y ′(t) = Ay(t) + f (t) , for all t ∈ [0, T],

y(0) = y(T). From the Duhamel formula: y(0) = y(T) = S(T)y(0) + T S(T − s)f (s)ds. Existence of time-periodic solutions ⇐ ⇒ Existence of a solution z to (2.2) (I − S(T))z = T S(T − s)f (s)ds. 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: (2.1)

• y ′(t) = Ay(t) + f (t) , for all t ∈ [0, T],

y(0) = y(T). From the Duhamel formula: y(0) = y(T) = S(T)y(0) + T S(T − s)f (s)ds. Existence of time-periodic solutions ⇐ ⇒ Existence of a solution z to (2.2) (I − S(T))z = T S(T − s)f (s)ds. 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: (2.1)

• y ′(t) = Ay(t) + f (t) , for all t ∈ [0, T],

y(0) = y(T). From the Duhamel formula: y(0) = y(T) = S(T)y(0) + T S(T − s)f (s)ds. Existence of time-periodic solutions ⇐ ⇒ Existence of a solution z to (2.2) (I − S(T))z = T S(T − s)f (s)ds. 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: σp(S(T)) = eTσp(A). 1 ∈ σp(S(T)) ⇔ 0 ∈ σp(A) or A has a complex eigenvalue 2ikπ T with k ∈ Z∗.

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: σp(S(T)) = eTσp(A). 1 ∈ σp(S(T)) ⇔ 0 ∈ σp(A) or A has a complex eigenvalue 2ikπ T with k ∈ Z∗. iR 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: σp(S(T)) = eTσp(A). 1 ∈ σp(S(T)) ⇔ 0 ∈ σp(A) or A has a complex eigenvalue 2ikπ T with k ∈ Z∗. iR 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: σp(S(T)) = eTσp(A). 1 ∈ σp(S(T)) ⇔ 0 ∈ σp(A) or A has a complex eigenvalue 2ikπ T with k ∈ Z∗. iR 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: σp(S(T)) = eTσp(A). 1 ∈ σp(S(T)) ⇔ 0 ∈ σp(A) or A has a complex eigenvalue 2ikπ T with k ∈ Z∗. iR R

• Jean-Jérôme Casanova

Time-periodic parabolic equations CEMRACS 2019 9 / 18

Denote by {ibj}0≤j≤NA the (finite) number of eigenvalue of A on the imaginary axis iR (• in the previous example). Assumption on the period T: (2.3) T ∈ R+ \ {2kπ bj | k ∈ Z, 0 ≤ j ≤ NA} Under the previous assumptions on (A, T) we have y(0) = (I − S(T))−1 T S(T − s)f (s)ds ∈ [D(A), H]1/2 and we obtain:

Theorem 2

For f ∈ L2(0, T; H), the periodic evolution equation (2.1) admits a unique strict solution y ∈ L2(0, T; D(A)) ∩ H1

♯ (0, T; H) in L2(0, T; H). The following estimate

holds yL2(0,T;D(A))∩H1

♯(0,T;H) ≤ C f L2(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 ∈ Cρ

♯ ([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 ∈ Cρ([0, T]; D(A)) ∩ Cρ+1([0, T]; H), and the following estimate holds (2.4) yCρ([0,T];D(A))∩Cρ+1([0,T];H) ≤ C f Cρ([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

Γb L 1 Γo Γi Γs Γη(t) Ωη(t) η(x, t) Eulerian-Lagrangian formulation. Structure displacement η : Γs × (0, T) → (−1, +∞). Fluid domain Ωη(t): Unknown of the problem.

Jean-Jérôme Casanova Time-periodic parabolic equations CEMRACS 2019 13 / 18

Fluid–structure interaction system

Fluid : 2D Incompressible Navier–Stokes equation ut + (u · ∇)u − ν∆u + ∇p = 0 et div u = 0 in Ωη(t), t > 0. Structure : Damped Euler–Bernoulli beam equation ηtt − βηxx − γηtxx + αηxxxx = F(u, p, η) on Γs, t > 0. Kinematic coupling : u = ηte2 on Γη(t), t > 0. Boundary conditions and time-periodic forcing term : u = ω1 on Γi, u2 = 0 and p + (1/2)|u|2 = ω2 on Σo, u = 0 on Γb, η(0, t) = η(L, t) = ηx(0, t) = ηx(L, t) = 0, t > 0. Periodic solutions: (u(0), η(0), ηt(0)) = (u(T), η(T), ηt(T)).

Jean-Jérôme Casanova Time-periodic parabolic equations CEMRACS 2019 14 / 18

Theorem 5 (C, 19)

Fix θ ∈ (0, 1) and T > 0. There exists R > 0 such that, for all T-periodic source terms (ω1, ω2) ∈

• Cθ

♯ ([0, T]; H3/2

(Γi)) ∩ C1+θ

♯

([0, T]; H−1/2(Γi))

• ×Cθ

♯ ([0, T]; H1/2(Γo)),

satisfying ω1Cθ

♯ ([0,T];H3/2

(Γi))∩C1+θ

♯

([0,T];H−1/2(Γi)) + ω2Cθ

♯ ([0,T];H1/2(Γo)) ≤ R,

the fluid–structure system admits a T-periodic strict solution (u, p, η) belonging to (after a change of variables mapping Ωη(t) into Ω) u ∈ Cθ

♯ ([0, T]; H2(Ω)) ∩ C1+θ ♯

([0, T]; L2(Ω)). p ∈ Cθ

♯ ([0, T]; H1(Ω)).

η ∈ Cθ

♯ ([0, T]; H4(Γs) ∩ H2 0(Γs)) ∩ C1+θ ♯

([0, T]; H2

0(Γs)) ∩ C2+θ ♯

([0, T]; L2(Γs)).

Jean-Jérôme Casanova Time-periodic parabolic equations CEMRACS 2019 15 / 18

Main steps

Change of variables to fix the domain Ωη(t) → Ω. Linearization. Introduce the Leray projector Π adapted to the mixed boundary conditions. Use Π to remove the pressure in the fluid equations. The pressure appears in the right-hand side of the beam equation (in the term F(u, p, η)). Apply (I − Π) to the fluid equations to express p in terms of Πu and η. Rewrite the linear system associated to (FS) as an evolution equation on (Πu, η, ηt), (3.1) d dt   Πu η ηt   = A   Πu η ηt   + f ,   Πu(0) η(0) ηt(0)   =   Πu(T) η(T) ηt(T)   .

Jean-Jérôme Casanova Time-periodic parabolic equations CEMRACS 2019 16 / 18

Here: (3.2) A =   I I (I + Ns)−1     As −AsΠL1 I Nv Aα,β γ∆s   , Energy identity for the eigenvalue of A: λ

• Ω

|u|2 +

• Γs

|η2|2

• + λ
• β
• Γs

|η1,x|2 + α

• Γs

|η1,xx|2

• + ν
• Ω

|∇u|2 + γ

• Γs

|η2,x|2 = 0. Re λ < 0 ⇒ no restriction on the period T.

Jean-Jérôme Casanova Time-periodic parabolic equations CEMRACS 2019 17 / 18

1

Introduction to analytic semigroups

2

The periodic problem

3

Application to a fluid–structure interaction problem

Jean-Jérôme Casanova Time-periodic parabolic equations CEMRACS 2019 18 / 18