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Chaotic behavior of an analog of the 2D Kuramoto-Sivashinsky equation Jiao He Universit e Lyon 1 Institut Camille-Jordan August 9, 2019 CEMRACS Outline 1D Kuramoto-Sivashinsky equation 1 Some PDE models Dynamical systems An analog of

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Chaotic behavior of an analog of the 2D Kuramoto-Sivashinsky equation

Jiao He

Universit´ e Lyon 1 Institut Camille-Jordan

August 9, 2019 CEMRACS

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Outline

1

1D Kuramoto-Sivashinsky equation Some PDE models Dynamical systems

2

An analog of 2D Kuramoto-Sivashinsky equation Global existence and Absorbing set Analyticity and Attractor Bound of the number of spatial oscillations

3

Numerical simulations

(Institut Camille-Jordan) Chaotic behavior August 9, 2019 CEMRACS 2 / 20

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1D Kuramoto-Sivashinsky equation

In mathematics, the Kuramoto–Sivashinskys equation is a fourth-order nonlinear partial differential equation, named after Yoshiki Kuramoto and Gregory Sivashinsky, who derived the equation to model the diffusive instabilities in a laminar flame front in the late 1970s. The Kuramoto-Sivashinsky equation is writen as : ut + uux = −uxx − uxxxx (1) The KS equations bridge the gap between infinite-dimensional behavior of PDEs and the finite-dimensional behavior in dynamical

  • systems. The solutions to the KS equation reveal a complex interplay
  • f simple spatial patterns and low fractal dimensional chaos.

(Institut Camille-Jordan) Chaotic behavior August 9, 2019 CEMRACS 3 / 20

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Some PDEs

In one variable, we consider the Heat equation : ut − uxx = 0, u(0, x) = f (x) (2) with general solution u(x, t) =

  • Φ(x − y, t)f (y)dy

where Φ(x, t) is the fundamental solution.

(Institut Camille-Jordan) Chaotic behavior August 9, 2019 CEMRACS 4 / 20

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Some PDEs

In one variable, we consider the Heat equation : ut − uxx = 0, u(0, x) = f (x) (2) with general solution u(x, t) =

  • Φ(x − y, t)f (y)dy

where Φ(x, t) is the fundamental solution. The Burgers’ equation : ut + uux = uxx (3) We can solve it by means of the Cole-Hopf transformation, which transform Burgers’ equation to the linear diffusion equation by a nonlinear transformation.

(Institut Camille-Jordan) Chaotic behavior August 9, 2019 CEMRACS 4 / 20

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Some PDEs

KdV equation : ut + uux + uxxx = 0 (4) with solution −1 2 c sech2 √c 2 (x − c t − a)

  • where sech stands

for the hyperbolic secant and a is an arbitrary constant. This describes a right-moving soliton.

(Institut Camille-Jordan) Chaotic behavior August 9, 2019 CEMRACS 5 / 20

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1D Kuramoto-Sivashinsky equation

Kuramoto-Sivashinsky equation : ηt + ηηx + ηxx + ηxxxx = 0 (5) In this equation, the large scale dynamics are dominated by a destabilising ‘diffusion’ ηxx, whereas small scale dynamics are dominated by stabilising hyperdiffusion ηxxxx, and a nonlinear advective term ηηx stabilises the system by transferring energy from the large unstable modes to the small stable modes ηt + ηηx = −ηxx − ηxxxx ↓ ↓ ↓ non-linearity instability dissipation

(Institut Camille-Jordan) Chaotic behavior August 9, 2019 CEMRACS 6 / 20

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1D Kuramoto-Sivashinsky equation

Applying the Fourier transfor- mation to the linear part, ∂t ˆ η(k) = (k2 − k4)ˆ η(k), it results in the stability

  • f high frequencies (|k| > 1)

and instability of low frequencies (0 < |k| < 1). Specifically, the term ηxx leads to instability at large scales; the dissipative term ηxxxx is responsible for damping at small scales. When the nonlinear term ηηx is added, stabilization occurs as this term transfers energy from the long wavelengths to the short wavelengths and balances the exponential growth due to the linear parts. This interaction between the unstable linear parts and a nonlinearity makes the solution to develop chaotic dynamics.

(Institut Camille-Jordan) Chaotic behavior August 9, 2019 CEMRACS 7 / 20

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Dynamical systems

The focus of dynamical systems is to understand the qualitative behavior of the solutions. Typical questions include: What is the long-time asymptotic behavior of general solutions? Do solutions behave chaotically? What about attactors ? etc...

(Institut Camille-Jordan) Chaotic behavior August 9, 2019 CEMRACS 8 / 20

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Dynamical systems

Abstract form of an evolution equation : ut = Au(t) + f (u(t)), u(0) = u0 (6) where A is an infinitesimal generator of a semi-group S(t) = eAt. Due to Duhamel’s formula, u(t) = eAtu0 + t eA(t−s)f (s, u(s))ds

(Institut Camille-Jordan) Chaotic behavior August 9, 2019 CEMRACS 9 / 20

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Compact global attractor : A nonlinear semigroup S(t) has a compact global attractor A if (i) A is a compact subset of X; (ii) A is an invariant set, i.e., S(t)A = A, ∀t ≥ 0; (iii) A attract every bounded set B of X, i.e. dist (S(t)B, A) → 0 as t → ∞.

(Institut Camille-Jordan) Chaotic behavior August 9, 2019 CEMRACS 10 / 20

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Compact global attractor : A nonlinear semigroup S(t) has a compact global attractor A if (i) A is a compact subset of X; (ii) A is an invariant set, i.e., S(t)A = A, ∀t ≥ 0; (iii) A attract every bounded set B of X, i.e. dist (S(t)B, A) → 0 as t → ∞. A dynamic system with a chaotic attractor is locally unstable but globally stable: once some sequences have entered the attractor, nearby points diverge from one another but never depart from the attractor.

(Institut Camille-Jordan) Chaotic behavior August 9, 2019 CEMRACS 10 / 20

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Compact global attractor : A nonlinear semigroup S(t) has a compact global attractor A if (i) A is a compact subset of X; (ii) A is an invariant set, i.e., S(t)A = A, ∀t ≥ 0; (iii) A attract every bounded set B of X, i.e. dist (S(t)B, A) → 0 as t → ∞. A dynamic system with a chaotic attractor is locally unstable but globally stable: once some sequences have entered the attractor, nearby points diverge from one another but never depart from the attractor. The K-S equation: a bridge between PDE’s and dynamical systems : The existence of a compact global attractor with finite fractal dimension shows that the asymptotic behavior of solutions of equations is essentially finite-dimensional.

(Institut Camille-Jordan) Chaotic behavior August 9, 2019 CEMRACS 10 / 20

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A small spoiler

(Institut Camille-Jordan) Chaotic behavior August 9, 2019 CEMRACS 11 / 20

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A small spoiler Simulation of 1D Kuramoto-Sivashinsky equation

(Institut Camille-Jordan) Chaotic behavior August 9, 2019 CEMRACS 11 / 20

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An analog of the 2D KS equation

We consider the following canonical equation for overlying electrified films, which was derived by Tomlin, Papageorgiou & Pavliotis [1]: ηt + ηηx + (β − 1)ηxx − ηyy − γΛ3η + ∆2η = 0 (7) where β > 0 is the Reynolds number, 0 ≤ γ ≤ 2 measures the electric field strength and Λ is a non-local operator corresponding to the electric field effect given on the Fourier variables as

  • Λη = |ξ|ˆ

η(ξ) = (ξ2

1 + ξ2 2)

1 2 ˆ

η(ξ). We observe that the term corresponding to the electric field, −γΛ3(η), always has a destabilizing effect,

  • R. Tomlin, D. Papageorgiou, and G. Pavliotis.

Three-dimensional wave evolution on electrified falling films. Journal of Fluid Mechanics, 822:54–79, 2017.

(Institut Camille-Jordan) Chaotic behavior August 9, 2019 CEMRACS 12 / 20

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Physical model

A Newtonian liquid of constant density ρ and viscosity µ, flows under gravity along an infinitely long flat plate which is inclined at an angle β to the horizontal. A coordinate system (x, z) is adopted with x measuring distance down and along the plate and z distance perpendicular to it. The film thickness is z = h(x, t) and its unperturbed value is h0 .

(Institut Camille-Jordan) Chaotic behavior August 9, 2019 CEMRACS 13 / 20

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An analog of the 2D KS equation

We will study the initial value problem for nonlocal 2D Kuramoto-Sivashinsky-type equation

  • ηt + ηηx + (β − 1)ηxx − ηyy − δΛ3(η) + ǫ∆2η = 0,

η(x, y, 0) = η0(x, y), (x, y) ∈ T2. (8) with periodic boundary conditions and initial data with zero mean L L η0(x, y)dxdy = 0. Step 1 : Global existence Step 2 : Existence of an absorbing set Step 3 : Analyticity Step 4 : Existence of an attractor. Step 5 : Oscillations

(Institut Camille-Jordan) Chaotic behavior August 9, 2019 CEMRACS 14 / 20

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Global existence (R. Granero-Belinch´

  • n, J. He, 19):

If η0 ∈ H2(T2), then for every 0 < T < ∞ the initial value problem has a unique solution η ∈ C([0, T]; H2(T2)) ∩ L2(0, T; H4(T2)). By Picard’s theorem and some a priori estimates

(Institut Camille-Jordan) Chaotic behavior August 9, 2019 CEMRACS 15 / 20

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Global existence (R. Granero-Belinch´

  • n, J. He, 19):

If η0 ∈ H2(T2), then for every 0 < T < ∞ the initial value problem has a unique solution η ∈ C([0, T]; H2(T2)) ∩ L2(0, T; H4(T2)). By Picard’s theorem and some a priori estimates Absorbing set (R. Granero-Belinch´

  • n, J. He, 19)

Let η0 ∈ H2(T2) be the zero mean initial data. Then the solution η

  • f the initial-value problem (8) satisfies

lim sup

t→∞ η(t)L2(T2) ≤ Rǫ,δ.

Based on the construction of a Lyapunov functional

(Institut Camille-Jordan) Chaotic behavior August 9, 2019 CEMRACS 15 / 20

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Analyticity (R. Granero-Belinch´

  • n, J. He, 19)

Let η0 be given in H2(T2). Then, there exists T0 depending on η0, ǫ, β, δ such that the solution of (8) satisfies eσ(t)Λη(t)2

L2 ≤ 1 + 2C 2 ǫ,δ,η0, ∀ t ≥ 0

where σ(t) = min{tanh(t), tanh T0

2

  • }. In particular, it becomes

analytic for t > 0. Based on a priori estimates in a Gevrey class.

(Institut Camille-Jordan) Chaotic behavior August 9, 2019 CEMRACS 16 / 20

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Analyticity (R. Granero-Belinch´

  • n, J. He, 19)

Let η0 be given in H2(T2). Then, there exists T0 depending on η0, ǫ, β, δ such that the solution of (8) satisfies eσ(t)Λη(t)2

L2 ≤ 1 + 2C 2 ǫ,δ,η0, ∀ t ≥ 0

where σ(t) = min{tanh(t), tanh T0

2

  • }. In particular, it becomes

analytic for t > 0. Based on a priori estimates in a Gevrey class. Attractor (R. Granero-Belinch´

  • n, J. He, 19)

The system has a maximal, connected, compact attractor in the space H2(T2). By showing that S(·)η0 ∈ C ([0, T]; H2 (T2)) defines a compact semiflow in H2 (T2).

(Institut Camille-Jordan) Chaotic behavior August 9, 2019 CEMRACS 16 / 20

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Spatial oscillations (R. Granero-Belinch´

  • n, J. He, 19)

Let η be a solution of system for initial data η0 ∈ H2(T2). Then, T = I ∪ R, where I is a union of at most [

4π tanh( T0

2 )] open intervals in T

and the following estimates hold for t ≥ T0

2 (T0 is explicit),

|∂xη(x, y, t)| ≤ 1, for all x ∈ I, y ∈ T and card{x ∈ R : |∇η(x, y, t)| = 0} ≤ 4π log 2 log Cǫ,δ,η0 tanh T0

2

  • where Cǫ,δ,η0 is a constant depending on ǫ, δ, η0.

(Institut Camille-Jordan) Chaotic behavior August 9, 2019 CEMRACS 17 / 20

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Spatial oscillations (R. Granero-Belinch´

  • n, J. He, 19)

Let η be a solution corresponding to the initial data η0 ∈ H2(T2), then for t ≥ T0

2 , the number of peaks for η can be bounded as

card {peaks for η} ≤ 4π log 2 log Cǫ,δ,η0 tanh T0

2

  • where Cǫ,δ,η0 depends on ǫ, δ, η0 and T0 is defined as before.

(Institut Camille-Jordan) Chaotic behavior August 9, 2019 CEMRACS 18 / 20

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Numerical simulations of usual 2D K-S equation

(Institut Camille-Jordan) Chaotic behavior August 9, 2019 CEMRACS 19 / 20

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Thanks for your attentions!

(Institut Camille-Jordan) Chaotic behavior August 9, 2019 CEMRACS 20 / 20