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On 2–d Euler equations with partial damping and some related model problems.

Wenqing Hu. 1

1. Department of Mathematics and Statistics, Missouri S&T.

The 2–d Navier–Stokes equation on a torus.

I ut + u∇u +∇p − ν∆u = f ,

divu = 0 , u|t=0 = u0 .

I u = u(x , t) = (u1(x , t), u2(x , t)),

I p = p(x , t),

I x ∈ T2 = R2/2πZ2.

The 2–d Navier–Stokes equation on a torus.

I Solution u(x , t) can be written as a Fourier series

u(x , t) = 1

2π

∑ k∈Z2

û(k , t)e ikx .

I The Laplace operator ∆ is a “damping” term if we view it from the Fourier space

(∆̂u)(k) = −|k |2û(k) for all k ∈ Z2 .

Modification of the 2–d Navier–Stokes equation on a torus.

I Introduce the “partial damping” operator Y :

(Ŷu)(k) =

{ −|k |2û(k) , k 6∈ K , 0 , k ∈ K .

I We will assume that K is symmetric (i.e. invariant under k → −k) to keep the solution real–valued.

I Ultimate goal is to study ut + u∇u +∇p − νYu = f ,

divu = 0 , u|t=0 = u0 .

Background.

I Canonical (conjectural) picture due to Kraichnan 2 : the energy and enstrophy cascades which spread the excitations to other Fourier modes through the nonlinearity.

I Nonlinear interactions should tend to distribute energy uniformly between all degrees of freedom, and hence a system for which some of the degrees of freedom are forced while some other degrees of freedom are damped should still reach some kind of dynamical equilibrium. 3

I Kraichnan’s downward cascade of energy : if in the case K = {k0,−k0} where k0 is one of the lowest non–trivial frequencies, the solution will presumably not stay bounded generically, even if the forcing acts “far away” in the Fourier space.

2. Kraichnan, R.H., Inertial ranges in two dimensional turbulence, Physics of Fluids, 10(7), pp. 1417–1423, 1967.

3. At least if there is enough interaction between the damped and forced parts of the system.

Background.

I The simplest way to establish “dynamical equilibrium” is to assume that f is random, and validate the fact that the system admits a unique invariant measure. 4

I “Asymptotic Strong Feller” property.

I It does not work in the case of partial damping.

4. Hairer, M., Mattingly, J.C., Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing, Annals of Mathematics, (2) 164 (2006), no. 3., pp. 993–1032.

Our problem.

I We study the modest case ut + u∇u +∇p − νYu = 0 ,

divu = 0 , u|t=0 = u0 .

I Case 1 : K is finite (remove damping from finitely many modes) ;

I Case 2 : K is co–finite (i.e. Z2\K is finite, leave damping at finitely many modes) ;

I Case 3 : Both K and Z2\K are infinite.

General Result.

I Write

(Ẑu)(k) =

{ −|k|2û(k) , k ∈ K , 0 , k 6∈ K .

I Y + Z = ∆.

I “Regularization”.

General Result.

I Theorem. Let K be any symmetric subset of Z2. For each divergence–free vector field u0 ∈ L2 on the torus T2 the initial–value problem

(uε)t + uε∇uε +∇pε − νYuε − εZuε = 0 , divuε = 0 ,

uε|t=0 = u0

has a unique solution uε ∈ C ([0,∞), L2x) ∩ L2t Ḣ2x . The solution uε is smooth in T2 × (0,∞) and satisfies the energy identity∫

T2

1

2 |uε(x , t)|2dx +

∫ t 0

∫ T2

(−ν(Yuε)uε − ε(Zuε)uε)dxdt ′

=

∫ T2

1

2 |u0(x)|2dx

for each t ≥ 0.

General Result.

I Theorem. (continued) Moreover, if ω0 = curlu0 ∈ L2, then uε ∈ L2t Ḣ2x and satisfies∫

T2

1

2 ω2ε(x , t)dx +

∫ t 0

∫ T2

(−ν(Yuε)uε − ε(Zuε)uε)dxdt ′

=

∫ T2

1

2 ω20(x)dx

for each t ≥ 0.

General Result.

I Weak Solution.

u ∈ C ([0,∞), L2) ∩ L∞t Ḣ1x ∩ {v ,Yv ∈ L2t L2x} .

I Corollary. For any symmetric K ⊂ Z2 and any initial datum u0 ∈ H1 our problem has at least one weak solution.

I Leray–Hopf argument.

Case 1. Finitely many undamped frequencies.

I Let κ = max k∈K |k|.

I ‖∇v‖2 ≤ −(Yv , v) + κ2‖v‖2L2x .

I Uniform control of the solutions uε in L ∞ t L

2 x ∩ L2t Ḣ1x .

I Standard embedding argument gives existence and uniqueness of solutions to the initial–value problem with u0 ∈ L2 in the class C ([0,∞), L2) ∩ L2t Ḣ1x .

Case 1. Finitely many undamped frequencies.

I Long–time behavior :

EK ,E ,I =

v : T 2 → R2,

v solves the 2d incompressible Euler equation and, in addition

v̂(k) = 0 for each k 6∈ K ,∫ T2 |v |

2 = 2E , ∫ T2 |curlv |

2 = I .

. I Krasovskii–LaSalle principle.

I Question : What is the structure of the solution of 2–d Euler on T2 that is supported on finitely many Fourier modes ?

I Answer : Independent of time and is supported either on a line passing through the origin or on a circle centered at the origin. (Will come back to this later.)

I So in the case finitely many undamped modes solutions converge to steady state !

Case 2. Finitely many damped frequencies.

I Vorticity formulation : ωt + (u · ∇)ω = νYω . I Bound ‖Yω‖L∞ ≤ cK‖ω‖L2 due to the finiteness of Z2\K . I ‖ω(t)‖L2 is not increasing.

I Thus d

dt ‖ω(t)‖L∞ ≤ cK‖ω0‖L2 .

I We arrive at the a–priori estimate

‖ω(t)‖L∞ ≤ ‖ω0‖L∞ + cK‖ω0‖L2t , t > 0 .

I If ω0 ∈ L∞, uniqueness follows from Yudovich theory.

The structure of Euler solution supported on finitely many Fourier modes.

I Question : What is the structure of the solution of 2–d Euler on T2 that is supported on finitely many Fourier modes ?

I Answer : Independent of time and is supported either on a line passing through the origin or on a circle centered at the origin.

I It must be steady state !

The structure of Euler solution supported on finitely many Fourier modes.

I 2–d Euler can be written in Fourier components as

d

dt ω̂(m, t) = − 1

4π

∑ k+l=m

(k1l2−k2l1) (

1

|k |2 − 1 |l |2

) ω̂(k, t)ω̂(l , t) .

I Suppose the support is a finite set of Fourier modes S .

The structure of Euler solution supported on finitely many Fourier modes.

I An (unordered) pair {k , l} of two distinct points k = (k1, k2), l = (l1, l2) ∈ Z2\{(0, 0)} is called degenerate if either k , l lie on the same circle centered at the origin (i.e. k21 + k

2 2 = l

2 1 + l

2 2 ), or k, l lie on the same line passing through

the origin (i. e. k1l2 − k2l1 = 0). In the former case we call the pair to be c–degenerate, and in the latter case we call the pair to be l–degenerate. A pair which is not degenerate is called non–degenerate.

I If the set S is not degenerate, then there exists a non–zero element m ∈ Z2\S such that m = k + l for exactly one non–degenerate (unordered) pair {k , l} ∈ S .

I S “wakes up” those modes that are not in S .

Figure: Proof that S is symmetric w.r.t. O (if not a line).

Figure: Proof that S is degenerate.

Reference.

Elgindi, T., Hu, W., Sverak, V., On 2d incompressible Euler equations with partial damping. Communications in Mathematical Physics, 355, Issue 1, October 2017, pp. 145-159.

Model problem : AB–model.

I We consider here a model problem{ dxt = −xtytdt , dyt = x

2 t dt .

I A phase picture is shown in the next Figure.

I We see that the whole line OyA contains stable equilibriums and the whole line OyB contains unstable equilibriums. This is different from the cases considered in the classical Freidlin-Wentzell theory.

I In this case we can understand the symmetry of our model in a more rough way : the stable and unstable equilibriums are symmetric with respect to shifts in the directions of OyA and OyB , respectively.

I Our model preserves the energy E (x , y) = x2 + y 2. The driving vector field b(x , y) = (−xy , x2) is degenerate on x = 0.

Figure: The AB model.

Randomly perturbed AB–model.

I We add friction and random perturbation to this system{ dX εt = −X εt Yεt dt − εX εt dt +

√ εdW 1t , X ε0 = x0 ,

dYεt = (X εt )2dt − εYεt dt + √ εdW 2t , Yε0 = y0 .

I Here W 1t and W 2 t are two independent standard

1–dimensional Brownian motions ;

I The small parameter ε > 0 is the intensity of the friction, and the small parameter

√ ε > 0 is the intensity of the noise.

Randomly perturbed AB–model.

I Question : What is the long–time behavior of the system (X εt ,Yεt ) as t →∞ and ε ↓ 0 ?

Randomly perturbed AB–model : Background.

I Our model problem here differs from the set–up in the classical Frei

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