Euler 2D and coupled systems: Coherent structures, solutions and stable measures

Transcription

1 Euler 2D and coupled systems: Coherent structures, solutions and stable measures R. Vilela Mendes Centro de Matemática, Aplicações Fundamentais e Investigação Operacional, Universidade de Lisboa PSPDE - V, November 28-30, 2016 PSPDE - V, November 28-30,

2 Contents 1 2D problems: Geophysics and plasmas 2 Persistent coherent structures: Euler 2D and Scrape-o layer equation in Toamas 3 Noise-stable invariant measures: The Euler 2D equation 4 A two- eld model of the SOL: Travelling waves. PSPDE - V, November 28-30,

3 Persistent structures in (quasi) 2-dimensional uid motions Jupiter red spot PSPDE - V, November 28-30,

4 Persistent structures in (quasi) 2-dimensional uid motions Jupiter red spot PSPDE - V, November 28-30,

5 Persistent structures in (quasi) 2-dimensional uid motions Large-scale vortices in the oceans PSPDE - V, November 28-30,

6 Persistent structures in (quasi) 2-dimensional uid motions Large-scale vortices in the oceans PSPDE - V, November 28-30,

7 Persistent structures in (quasi) 2-dimensional uid motions The SOL (scrape-o layer): Jets and blobs PSPDE - V, November 28-30,

8 Persistent structures in (quasi) 2-dimensional uid motions The SOL (scrappe-o layer): Jets and blobs PSPDE - V, November 28-30,

9 Persistent structures in (quasi) 2-dimensional uid motions The SOL (scrappe-o layer): Jets and blobs PSPDE - V, November 28-30,

10 The equations The 2D Euler equation v t = (v r)v rp div v = 0 (Periodic boundary conditions) Since div v = 0 there is a function ψ(x, t) such that and the Euler equation becomes v = r? ψ = ( x2 ψ, x1 ψ) t ψ=-r? ψ r ψ Solutions on the 2-dimensional torus T 2 = [0, 2π] [0, 2π] subjected to periodic boundary conditions ψ(0, x 2, t) = ψ(2π, x 2, t), ψ(x 1, 0, t) = ψ(x 1, 2π, t) Let e (x) = 1 2π ei x, 2 Z 2 be the eigenfunctions for with eigenvalues 2 = Are a complete set in L2 (T 2 ). PSPDE - V, November 28-30,

11 The equations Since ψ is a real function assume R ψdx = 0, then ω T 2 = ω ψ(x, t) = ω (t)e (x), 2Z 2 + Z 2 + denotes the set f 2 Z 2 : 1 > 0, 2 2 Z or 1 = 0, 2 > 0g. ψ = fω g 2Z 2 and introducing the operator + B(ω) = fb (ω)g 2Z 2 + = B (ω) ω B (ω) = 1 2π 2? h h 2 ω h ω h6= h,2z 2 + where? = ( 2, 1 ), the 2D Euler equation becomes the following in nite-dimensional ordinary di erential equations system d dt ω =B (ω) 2 Z 2 + h B ω = 0 PSPDE - V, November 28-30,

12 The equations The SOL equations (Two- eld model) L t = rφ r? L g 2 (L φ) + D r 2 L + jrlj 2 σ e (Λ φ 4φ t = rφ r? 4φ g 2 L + ν4 2 φ + σ h1 e (Λ φ)i, Conservative component L t 4φ t = rφ r? L g 2 (L φ) = rφ r? 4φ g 2 L PSPDE - V, November 28-30,

13 Two equations, two problems 2D Euler equation: Persistent large-scale structures 2- eld SOL equation: Coherent structures travelling from the core to the wall Solutions? 2D Euler equation: Stochastic stability of invariant measures 2- eld SOL equation: Travelling wave solutions PSPDE - V, November 28-30,

14 Stochastic stability of invariant measures Physical measure (or SBR measure) and stochastically stable measure are closely related notions. M a state space, f : M! M a smooth dynamical system and µ a positive Borel measure on M such that n 1 1 lim n! n ϕ f j (x) Z! ϕdµ j=0 M for a positive measure set A of initial points x and any continuous function ϕ : M! R. (Means that time averages are given by µ spatial averages, at least for a large set of initial states x). For uniformly hyperbolic systems there is a complete theory concerning existence and uniqueness of physical measures and partial results for non-uniformly hyperbolic and partially hyperbolic systems. PSPDE - V, November 28-30,

15 Stochastic stability of invariant measures Consider the stochastic process f ε obtained by adding a small random noise to the deterministic system f. Under general conditions, there exists a stationary probability measure µ ε such that, almost surely, n 1 1 lim n! n ϕ f j (x) Z! j=0 ϕdµ ε Stochastic stability of the measure means that µ ε converges to the physical measure µ when the noise level ε goes to zero. (uniformly hyperbolic maps, Lorenz and Hénon strange attractors and also results for partially hyperbolic systems). Existence and uniqueness of the invariant measure µ ε under general conditions provides a powerful tool to obtain the physical measure of f, by randomly perturbing it and letting the noise level ε! 0. That this might also be useful for in nite-dimensional systems is the source of inspiration for this wor. It might contribute to the understanding of the large-scale structures in geophysical phenomena. PSPDE - V, November 28-30,

16 In nitesimally invariant measures of PDE s φ t the ow of a partial di erential equation and φt the push-forward semigroup acting on measures. A measure µ is invariant if and in nitesimally invariant if Z φ t (µ) = µ B ϕdµ = 0 B being the generator of the ow φ t. Equivalently B 1 = 0. Let the generator B be a rst or second order di erential operator on a discrete set of coordinates ω = fω i g, B = u ij (ω) i,j 2 and consider a measure of the form ω i ω j + i dµ = R (ω) dω i i b i (ω) ω i (1) PSPDE - V, November 28-30,

17 In nitesimally invariant measures of PDE s To obtain the invariance condition R (B ϕ) R (ω) i dω i = 0, compute the adjoint of B obtaining ( ) B 1 = R 2 (Rb i ) (Ru ij ) ω i ω i ω j Theorem + i ( i b i + 1 R j i,j ω j [R (u ij + u ji )] ) + ω i 2 u ij i,j ω i ω j Therefore, to have B 1 = 0, the rst term should vanish leading to A generator B of the form in Eq.(1), u ij and b i being di erentiable functions, has dµ = R (ω) i dω i as an in nitesimally invariant measure i i ω i (Rb i ) i,j 2 ω i ω j (Ru ij ) = 0 PSPDE - V, November 28-30,

18 Invariant measures of the 2D Euler equation With u ij (ω) = 0, the invariance condition is simply or B i i i ω i R = i ω i (RB i ) = 0 d dt ω i ω i R = d dt R = 0 In conclusion: any constant of motion of the Euler equation generates an (in nitesimally) invariant measure. Among them the energy E = ω 2 and the enstrophy S = ω 2 However, the Poisson structure of the Euler 2D equation being degenerate, there is an in nite set of Casimir invariants, Z C f = f (4ψ) d 2 x f being an arbitrary di erentiable function. Therefore there are in nitely many invariant measures for the 2D Euler equation. The enstrophy is the Casimir invariant for f (x) = x 2. PSPDE - V, November 28-30,

19 Stochastic stability of 2D Euler invariant measures Consider the following in nite dimensional Ornstein-Uhlenbec operator εq a (ω) εqf (ω) = ε f (ω) + σ (ω) 2 ω ω 2 f (ω) Lf (ω) = εqf (ω) + B (ω) f (ω) ω is the in nitesimal generator for a stochastically perturbed Euler ow. 1 With W (t) = jj b (t)e a normalized brownian motion, b (t) being independent copies of a complex brownian motion, the following perturbed Euler equation is obtained X (t) = X (0) + R t 0 fb (X (s)) + εa (X (s))g ds + R t 0 p 2εσ (X (s))db (s), 8 2 Z 2 + PSPDE - V, November 28-30,

20 Stochastic stability of 2D Euler invariant measures Theorem If dµ = R (ω) i dω i is an invariant measure for the unperturbed Euler equation, then this is also an invariant measure for the perturbed equation if a (ω) and σ (ω) in satisfy a 2 σ R a + R ω ω ω 2 σ ω 2 2 R σ ω 2 = 0 As an example, the Gaussian measure constructed from the enstrophy dµ S = e ω 2 j dω j remains invariant if a = 2 ω and σ = 1 2 However, we are not only adding noise but also modifying the deterministic part, actually adding noise to a Navier-Stoes equation t ψ = r? ψ r ψ + ε4 2 ψ v t = (v r)v + ε4v rp These are NOT the stochastically stable measures of 2D Euler. PSPDE - V, November 28-30,

21 Stochastic stability of 2D Euler invariant measures Instead, consider noise perturbations without changing the deterministic part (a (ω) = 0). Use Galerin approximations of arbitrary order N B N (ω) = 1 2π 2? h h 2 ω h ω h 0<jhjN 0<j hjn The equation for the density R (ω) of the invariant measure becomes B N (ω) 2 R εσ ω ω 2 R = 0 Two cases are of physical interest, σ = 1 and σ = 1. However, by the 2 change of variables z = jj ω and B N 0 (ω) = jj B N (ω) the second case becomes identical to the rst one and we have to deal with B N 0 (z) R z ε 2 z 2 R = 0 PSPDE - V, November 28-30,

22 Stochastic stability of 2D Euler invariant measures This is an elliptic regularization of a rst order Hamilton-Jacobi equation (ε = 0) which has at least as many solutions as the number of constants of motion of the N Galerin approximation of the Euler equation. Hence, existence and uniqueness of a stochastically-stable solution for R is equivalent to the establishment of a viscosity solution for this Hamilton-Jacobi problem, in its vanishing viscosity modality. Associated to the uniformly elliptic equation there is a di usion process X t with di usion coe cient p ε and drift B N 0 (z). In each bounded domain D of z space ( R N ) the drift, being a quadratic polynomial, is uniformly Lipschitz continuous. Therefore the Dirichlet problem has a unique solution with stochastic representation R (z)j D = E z ff (X (τ))g f being the boundary condition at D and τ the rst exit time from D For the unbounded z space we may consider a sequence of nesting open domains fd n g, with boundary functions f n j Dn. Then in each D n nd n 1 domain one has a unique solution. PSPDE - V, November 28-30,

23 Stochastic stability of 2D Euler invariant measures For a bounded smooth boundary condition the solution R ε is bounded and continuous on compact subsets of D. Then, when ε! 0 R ε converges locally uniformly to a function R. This function is not necessarily a classical solution of B N 0 (z) z R = 0, but a standard construction shows that it is a viscosity solution, in the sense that, given a C function g, if R g has a local maximum at a point z 0 then B N (z 0) z g (z 0 ) 0 and if it is a local minimum B N (z 0) z g (z 0 ) 0. Hence Theorem For each choice of boundary conditions in z space and noise level (ε), one has a unique invariant measure density R ε (z), solution of (??). Furthermore, in the ε! 0 limit, R ε converges to a viscosity solution of B N 0 (z) z R = 0. PSPDE - V, November 28-30,

24 Stochastic stability of 2D Euler invariant measures So far we have dealt with N-dimensional Galerin approximations of the 2D Euler equation. When N! several modi cations are needed. It maes no sense to de ne R (ω) as a density of the non-existent at measure in in nite dimensions. Instead, R (ω) should be de ned as the Radon-Nyodim derivative for some other measure, for example the Gaussian enstrophy measure. Then the equation for the density R (ω) would be B (ω) ω 4 ω B (ω) R (ω) = 0 an Hamilton-Jacobi equation in in nite dimensions. Such equations have been extensively studied and given the appropriate boundary condition, for example R (ω)! 1 when jωj!, the construction of the density as a limiting viscosity solution of B (ω) R (ω) = 0 4 ω B (ω) ε 2 ω ω 2 would follow similar steps as in the nite dimensional case. PSPDE - V, November 28-30,

25 Travelling wave solutions of the SOL equations With the transformation becomes el t 4eφ t L (x 1, x 2, t) = el (x 1, x 2, t) gx 1 φ (x 1, x 2, t) = eφ (x 1, x 2, t) gx 1 = reφ r? el = reφ r? 4eφ g 2 el + g 2 4eφ. Loo for travelling-wave solutions to this system el (x 1, x 2, t) = el (x 1 v 1 t, x 2 v 2 t) eφ (x 1, x 2, t) = eφ (x 1 v 1 t, x 2 v 2 t) PSPDE - V, November 28-30,

26 Travelling wave solutions of the SOL equations Several classes of travelling wave solutions were constructed: L (x 1, x 2, t) = α [φ (x 1, x 2, t) v 2 (x 1 v 1 t) + v 1 (x 2 v 2 t)] φ (x 1, x 2, t) = + (α 1) gx 1 v 2 + g + αg 2 (x 1 v 1 t) v 1 (x 2 v 2 t) gx 1 +A cos [ 1 (x 1 v 1 t) + 2 (x 2 v 2 t)] + B sin [ 1 (x 1 v 1 t) + 2 (x 2 v 2 t)] +C 0 + C 1 cos [ (x 1 v 1 t)] +C 2 sin [ (x 1 v 1 t)]. PSPDE - V, November 28-30,

27 Travelling wave solutions of the SOL equations L (x 1, x 2, t) = α [φ (x 1, x 2, t) v 2 (x 1 v 1 t) + v 1 (x 2 v 2 t)] φ (x 1, x 2, t) = + (α 1) gx 1 αg v 2 + g 2 (x 1 v 1 t) v 1 (x 2 v 2 t) gx 1 +Ae [ 1(x 1 v 1 t)+ 2 (x 2 v 2 t)] + C 0 +C 1 e (x 1 v 1 t) + C 2 e (x 1 v 1 t) L (x 1, x 2, t) = γae γ[(x 1 v 1 t) 2 +(x 2 v 2 t) 2 ]/2 n h 2 γ (x 1 v 1 t) 2 + (x 2 v 2 t) 2io gx 1 φ (x 1, x 2, t) = Ae γ[(x 1 v 1 t) 2 +(x 2 v 2 t) 2 ]/2 + v 2 (x 1 v 1 t) v 1 (x 2 v 2 t) gx 1 PSPDE - V, November 28-30,

28 Travelling wave solutions of the SOL equations PSPDE - V, November 28-30,

29 Travelling wave solutions of the SOL equations PSPDE - V, November 28-30,

30 References and J. P. Bizarro; Analytical study of growth estimates, control of uctuations and conservatives structures in a two- eld model of the scrape-o layer, Phys. of Plasmas, to appear., F. Cipriano and H. Ouerdiane; Stochastic stability of invariant measures: The 2D Euler equation, in preparation. PSPDE - V, November 28-30,