Long time behavior of solutions of Vlasov-like Equations

Transcription

1 Long time behavior of solutions of Vlasov-like Equations Emanuele Caglioti "Sapienza" Università di Roma (Mathematics) Journée Claude Bardos - Laboratoire Jacques Louis Lions

2

3 Equations Vlasov Poisson Equation and related models 2D Euler

4 Equations Vlasov Poisson Equation and related models 2D Euler Stationary stable solutions

5 Equations Vlasov Poisson Equation and related models 2D Euler Stationary stable solutions BGK solutions and Kirchhoff Ellipses

6 Equations Vlasov Poisson Equation and related models 2D Euler Stationary stable solutions BGK solutions and Kirchhoff Ellipses Landau Damping

7 Equations Vlasov Poisson Equation and related models 2D Euler Stationary stable solutions BGK solutions and Kirchhoff Ellipses Landau Damping Other possible limiting behaviors

8 Equations Vlasov Poisson Equation and related models 2D Euler Stationary stable solutions BGK solutions and Kirchhoff Ellipses Landau Damping Other possible limiting behaviors What can we say in general

9 Equations Vlasov Poisson Equation and related models 2D Euler Stationary stable solutions BGK solutions and Kirchhoff Ellipses Landau Damping Other possible limiting behaviors What can we say in general Conclusions

10 Vlasov-Poisson 2D Euler The Vlasov Equation t f + v x f + F f v f = 0, f (x, v) : S 1 R is the phase space density ρ(x) = dvf (x, v) is the space density F f (x) = dyf(x y) is the force.

11 Vlasov-Poisson 2D Euler Vlasov-Poisson Equation - VPE Vlasov-Poisson Equation F = x V V = ( xx ) 1δ for x [0, 2π) : F(x) = 1 2 x 2π Hamiltonian Mean Field Model - HMF model V = cos(x y) In this case F = C(t) cos x + S(t) sin x

12 Vlasov-Poisson 2D Euler 2D Euler Equation t ω + u ω = 0 ω : D R is the vorticity u(x, t), the velocity field, is given by Some cases: D = R 2, D = T 2, D = R [0, h] u = ( ) 1 ω

13 Vlasov-Poisson 2D Euler For VPE the density f (x, v, t) is transported along the trajectories of an Hamiltonian system: ẋ = v, v = F f (x) The hamiltonian is a functional of the density f itself: self-consistent force field. Therefore the area of the level stes of f is conseved. Same for 2D Euler: the vorticity is transported along the trajectories of an Hamiltonian system.

14 Stationary Solutions Time periodic soutions Landau Damping Stable Stationary Solutions o VPE is a stationary solution of VPE f = g(v) If g is not increasing and f (1 + v 2 )dxdv < then f is a stationary stable solution of VPE Marchioro and Pulvirenti (1986).

15 Stationary Solutions Time periodic soutions Landau Damping BGK waves Bernstein, Greene and Kruskal (1957) discovered the existence of inhomogeneous traveling wave solution of VPE f = f 0 (x u 0 t, v u 0 ) BGK waves are time-periodic solutions of VPE. They solve f (x, v) = G(H(x, v)) = G( 1 2 v 2 + V (x)) where G is a generic C 2 function, H(x, v) is the 2D Hamiltonian, and V (x) is the potential energy. V must be compatible with its operative definition: V xx = dvg( 1 2 v 2 + V (x)) 1

16 Stationary Solutions Time periodic soutions Landau Damping Stable Stationary Solutions of 2D Euler Any radial vorticity ω = g(ρ), ρ = x 2 + y 2 is a stationary solution of 2D Euler If g is not increasing and ω(1 + ρ 2 )dxdv < then f is a stationary stable solution of VPE Marchioro and Pulvirenti. In particular the circular vortex patch is stable (Pulvirenti and Wan).

17 Stationary Solutions Time periodic soutions Landau Damping Kirchoff Ellipse Kirchhoff (1876) showed that elliptical patches are rotating solutions of 2D Euler. The patch rotates with angular velocity w The patch is stable in shape for a < 3 b. ab a 2 +b 2

18 Stationary Solutions Time periodic soutions Landau Damping Landau Damping Landau, on the basis of the analysis of the VPE linearized around an equilibrium conjectured that for initial data close to euilibrium f 0 = f (v) + ɛg(x, v) asymptotically the electric field will vanish and the phase space density will become homogeneous.

19 Stationary Solutions Time periodic soutions Landau Damping Landau Damping Landau, on the basis of the analysis of the VPE linearized around an equilibrium conjectured that for initial data close to euilibrium f 0 = f (v) + ɛg(x, v) asymptotically the electric field will vanish and the phase space density will become homogeneous. The linear case has been fully characterized (see for instance Maslov and Fedoryuk)

20 Stationary Solutions Time periodic soutions Landau Damping Landau Damping Landau, on the basis of the analysis of the VPE linearized around an equilibrium conjectured that for initial data close to euilibrium f 0 = f (v) + ɛg(x, v) asymptotically the electric field will vanish and the phase space density will become homogeneous. The linear case has been fully characterized (see for instance Maslov and Fedoryuk) Existence of a class of damped analytic solutions has been proved with a scattering approach by C. and Maffei (1998). Also non small solutions allowed. The result has been extended to close to equilibrium solutions by Hwang and Velasquez (2009). Initial data cannot be characterized.

21 Stationary Solutions Time periodic soutions Landau Damping Landau Damping Finally Mouhot and Villani (2009) proved that close to equlibrium initial data are exponentially damped (Landau Damping). The result is proved in an analytic framework (also Gevray type regularity).

22 Stationary Solutions Time periodic soutions Landau Damping Landau Damping Finally Mouhot and Villani (2009) proved that close to equlibrium initial data are exponentially damped (Landau Damping). The result is proved in an analytic framework (also Gevray type regularity). Recently Lin and Zeng have shown that close to equilibrium BGK waves exist for small regularity: W s,p : s < p Therefore No Landau Damping for small regularity.

23 Stationary Solutions Time periodic soutions Landau Damping Possible limiting behaviors: figures t f + v x f = 0

24 Stationary Solutions Time periodic soutions Landau Damping Possible limiting behaviors: figures t f + v x f = 0

25 Stationary Solutions Time periodic soutions Landau Damping Possible limiting behaviors: figures t f + v x f = 0

26 Stationary Solutions Time periodic soutions Landau Damping Possible limiting behaviors: figures t f + v x f = 0

27 In the case of Landau Damping, weakly, f (x, v, t) Stationary Solutions Time periodic soutions Landau Damping w t f + (v) and more precisely: it exists f + such that, strongly, where f (x, v, t) f + (x vt, v) < f + > x = f + (v) For any concave functional S(f ), for instance the entropy, unless f 0 = f + S(f ) = f log f So entropy increases. S(f + ) > S(f 0 )

28 Stationary Solutions Time periodic soutions Landau Damping Possible limiting behaviors: simulations

29 Stationary Solutions Time periodic soutions Landau Damping Possible limiting behaviors: state of the art

30 Stationary Solutions Time periodic soutions Landau Damping Possible limiting behaviors: state of the art stationary (stable) solution for VPE and for 2D Euler

31 Stationary Solutions Time periodic soutions Landau Damping Possible limiting behaviors: state of the art stationary (stable) solution for VPE and for 2D Euler time-periodic solutions BGK for VPE and Kirkhhoff Ellipses for 2D Euler notice that BGK solution for VPE and Kirkhhof ellipses are stationary solutions in a translating frame and in rotating frame respectively,

32 Stationary Solutions Time periodic soutions Landau Damping Possible limiting behaviors: state of the art stationary (stable) solution for VPE and for 2D Euler time-periodic solutions BGK for VPE and Kirkhhoff Ellipses for 2D Euler notice that BGK solution for VPE and Kirkhhof ellipses are stationary solutions in a translating frame and in rotating frame respectively, Landau Damping: asymptotic convergence to analytic stationary stable solutions

33 Stationary Solutions Time periodic soutions Landau Damping Possible limiting behaviors: state of the art stationary (stable) solution for VPE and for 2D Euler time-periodic solutions BGK for VPE and Kirkhhoff Ellipses for 2D Euler notice that BGK solution for VPE and Kirkhhof ellipses are stationary solutions in a translating frame and in rotating frame respectively, Landau Damping: asymptotic convergence to analytic stationary stable solutions No such result for 2D Euler. May be the case of the channel is the "easiest".

34 Stationary Solutions Time periodic soutions Landau Damping Possible limiting behaviors: state of the art stationary (stable) solution for VPE and for 2D Euler time-periodic solutions BGK for VPE and Kirkhhoff Ellipses for 2D Euler notice that BGK solution for VPE and Kirkhhof ellipses are stationary solutions in a translating frame and in rotating frame respectively, Landau Damping: asymptotic convergence to analytic stationary stable solutions No such result for 2D Euler. May be the case of the channel is the "easiest". Is it possible to prove damping to BGK solutions?

35 Shnirelman construction Related conjectures What we can say in general: Shnirelman construction Mixing operators in L 2 K φ(x) dyk(x, y)φ(y) where x : M k(x, y)dy = 1 and y : bistochastic operators Examples: k(x, y) = δ(x y) k is the heat kernel M M k(x, y)dx = 1 K = {K } defines a partial order between vorticity fields.

36 Shnirelman construction Related conjectures Definition. Let V s = {u H s : u = 0, u n M admissible velocity field Definition. If u 1, u 2 V 1 then u 1 u 2 iff curl u 1 curl u 2 Definition. For any u 0 V 1 define = 0} the set of Ω u0 = {u H 1 : u u 0, u L 2 = u 0 L 2}

37 Shnirelman construction Related conjectures If u(t) is the solution of the Euler Equation with initial data u 0, and O(u 0 ) = {u(t) : t R}is the orbit of u 0 then Ō(u 0 ) Ω u0. Definition. Minimal Elements of Ω u0 with respect to the order relation are called minimal flows.. It is possible to prove that minimal flows are stationary stable solutions of 2D Euler.

38 Shnirelman construction Related conjectures A first conjecture. The set of minimal flows is an attractor for the 2D Euler flow. Motivation: if the fluid does not go to stationary solutions level lines of vorticity are stretched and stretched and therefore the solution reacesh a minimal element. The conjecture is probably wrong because more complicate behaviors are expected form simulations

39 Shnirelman construction Related conjectures Generalized minimal flows (Shnirelman). A vector field is called a GMF if for any v Ō(u) : curl v L 2 = curl u L 2 Now the conjecture is that (for the Navier Stokes Equation with Random Forcing in the null viscosity limit) the attractor is concentrated on Generalized Minimal Flows

40 In the case of Landau Damping, weakly, f (x, v, t) Shnirelman construction Related conjectures w t f + (v) and more precisely: it exists f + such that, strongly, where f (x, v, t) f + (x vt, v) < f + > x = f + (v) For any concave functional S(f ), for instance the entropy S(f ) = f log f Equality holds only if f + = f 0 S(f + ) S(f 0 )

41 Shnirelman construction Related conjectures A strictly related conjecture essentially rephrasing it in the Vlasov and deterministic case Given f 0 define Ω(f 0 ) = {weak limit points of f (x, v, t) : t + } Then it is reasonable that generically

42 Shnirelman construction Related conjectures A strictly related conjecture essentially rephrasing it in the Vlasov and deterministic case Given f 0 define Ω(f 0 ) = {weak limit points of f (x, v, t) : t + } Then it is reasonable that generically For any g Ω(f 0 ) : S(g) S(f 0 ) (and generically S(g) > S(f 0 ) unless f 0 is not already on the attractor)

43 Shnirelman construction Related conjectures A strictly related conjecture essentially rephrasing it in the Vlasov and deterministic case Given f 0 define Ω(f 0 ) = {weak limit points of f (x, v, t) : t + } Then it is reasonable that generically For any g Ω(f 0 ) : S(g) S(f 0 ) (and generically S(g) > S(f 0 ) unless f 0 is not already on the attractor) If g 1 and g 2 are in Ω(f 0 ) then S(g 1 ) = S(g 2 ) The first is quasi obvious, the second it is not

44 Shnirelman construction Related conjectures Counterexample Let T0 t the evolution with the free motion Let f 0 (x, v) not homogeneous:

45 Shnirelman construction Related conjectures Counterexample Let T0 t the evolution with the free motion Let f 0 (x, v) not homogeneous: evolve for a time 1 with T

46 Shnirelman construction Related conjectures Counterexample Let T0 t the evolution with the free motion Let f 0 (x, v) not homogeneous: evolve for a time 1 with T then evolve for a time 1 with T 1

47 Shnirelman construction Related conjectures Counterexample Let T0 t the evolution with the free motion Let f 0 (x, v) not homogeneous: evolve for a time 1 with T then evolve for a time 1 with T 1 then evolve for a time 2 with T.

48 Shnirelman construction Related conjectures Counterexample Let T0 t the evolution with the free motion Let f 0 (x, v) not homogeneous: evolve for a time 1 with T then evolve for a time 1 with T 1 then evolve for a time 2 with T....

49 Shnirelman construction Related conjectures Counterexample Let T0 t the evolution with the free motion Let f 0 (x, v) not homogeneous: evolve for a time 1 with T then evolve for a time 1 with T 1 then evolve for a time 2 with T.... then evolve for a time n with T

50 Shnirelman construction Related conjectures Counterexample Let T0 t the evolution with the free motion Let f 0 (x, v) not homogeneous: evolve for a time 1 with T then evolve for a time 1 with T 1 then evolve for a time 2 with T.... then evolve for a time n with T... There are two limit points of f, one is f 0 and the other is lim n T n f 0 =< f 0 > x and their entropy is not the same.

51 Shnirelman construction Related conjectures Limiting behavior: Lagrangian-chaotic motion In this picture nothing hosts the fact that the asymptotic motion is Lagrangian-chaotic if, for instance, it is Eulerian-periodic. That is, for some T > 0, f (x, v, t) = f (x, vt + T ), and therefore H(x, v, t) = H(x, v, t + T ),

52 Shnirelman construction Related conjectures Limiting behavior: Lagrangian-chaotic motion where f (x, v, t) = lim n f (x, v, t + nt ), and where H = v V f In this case it is reasonable that f is constant on the invariant sets of the flow Φ T induced by H in the time interval [0, T ].

53 Shnirelman construction Related conjectures Limiting behavior: Lagrangian-chaotic motion A new interesting behavior has been discovered numerically by H. Morita and K. Kaneco, PRL (2006) (see also A. Antoniazzi, D. Fanelli, J. Barrè, P. H. Chavanis, T. Dauxois, S. Ruffo) They consider the HMF model V f (x) dydv f (x, v) cos(x y) with a far from equilibrium initial condition: f 0 = e v 2 2T 0 e M 0 cos x Teq (M 0 ) where M 0 is the magnetization, T e q(m 0 ) is the corresponding equilibrium temperature, and where T 0 T eq is a (temperature) parameter.

54 Shnirelman construction Related conjectures Limiting behavior: Lagrangian-chaotic motion The asymptotic state, depending on the parameter T 0,seems to be stationary solution periodic solution quasi-periodic solution

55 Shnirelman construction Related conjectures Explanation - Poincaré Section

56 Shnirelman construction Related conjectures Explanation - Poincaré Section

57 Shnirelman construction Related conjectures Conclusions It seems reasonable that the VPE flow (and the 2D Euler flow) cannot be to complicated asymptotically. Because, if too complicate, mixing would lead to simple behavior. Reasonable to expect asymptotically to reach a simple motion: may be even chaotic but with a few degrees of freedom involved.