Mean Field Limits for Ginzburg-Landau Vortices

Transcription

1 Mean Field Limits for Ginzburg-Landau Vortices Sylvia Serfaty Courant Institute, NYU & LJLL, Université Pierre et Marie Curie Conference in honor of Yann Brenier, January 13, 2017

2 Fields Institute, 2003

3 The Ginzburg-Landau equations u : Ω R 2 C u = u ε 2 (1 u 2 ) Ginzburg-Landau equation (GL) t u = u + u ε 2 (1 u 2 ) parabolic GL equation (PGL) i t u = u + u ε 2 (1 u 2 ) Gross-Pitaevskii equation (GP) Associated energy E ε (u) = 1 2 Ω u 2 + (1 u 2 ) 2 2ε 2 Models: superconductivity, superfluidity, Bose-Einstein condensates, nonlinear optics

4 Vortices in general u 1, u 1 = superconducting/superfluid phase, u 0 = normal phase u has zeroes with nonzero degrees = vortices u = ρe iϕ, characteristic length scale of {ρ < 1} is ε = vortex core size degree of the vortex at x 0 : 1 ϕ 2π B(x 0,r) τ = d Z In the limit ε 0 vortices become points, (or curves in dimension 3).

5 Solutions of (GL), bounded number N of vortices [Bethuel-Brezis-Hélein 94] u ε minimizing E ε has vortices all of degree +1 (or all 1) which converge to a minimizer of W ((x 1, d 1 ),..., (x N, d N )) = π i j d i d j log x i x j +boundary terms... renormalized energy", Kirchhoff-Onsager energy (in the whole plane) minimal energy min E ε = πn log ε + min W + o(1) as ε 0 Some boundary condition needed to obtain nontrivial minimizers nonminimizing solutions: u ε has vortices which converge to a critical point of W : i W ({x i }) = 0 i = 1, N [Bethuel-Brezis-Hélein 94] stable solutions converge to stable critical points of W [S. 05]

6 Solutions of (GL), bounded number N of vortices [Bethuel-Brezis-Hélein 94] u ε minimizing E ε has vortices all of degree +1 (or all 1) which converge to a minimizer of W ((x 1, d 1 ),..., (x N, d N )) = π i j d i d j log x i x j +boundary terms... renormalized energy", Kirchhoff-Onsager energy (in the whole plane) minimal energy min E ε = πn log ε + min W + o(1) as ε 0 Some boundary condition needed to obtain nontrivial minimizers nonminimizing solutions: u ε has vortices which converge to a critical point of W : i W ({x i }) = 0 i = 1, N [Bethuel-Brezis-Hélein 94] stable solutions converge to stable critical points of W [S. 05]

7 Dynamics, bounded number N of vortices For well-prepared initial data, d i = ±1, solutions to (PGL) have vortices which converge (after some time-rescaling) to solutions to dx i dt = iw (x 1,..., x N ) [Lin 96, Jerrard-Soner 98, Lin-Xin 99, Spirn 02, Sandier-S 04] For well-prepared initial data, d i = ±1, solutions to (GP) dx i dt = i W (x 1,..., x N ) = ( 2, 1 ) [Colliander-Jerrard 98, Spirn 03, Bethuel-Jerrard-Smets 08] All these hold up to collision time For (PGL), extensions beyond collision time and for ill-prepared data [Bethuel-Orlandi-Smets 05-07, S. 07]

8 Vorticity In the case N ε, describe the vortices via the vorticity : supercurrent j ε := iu ε, u ε a, b := 1 2 (a b + āb) vorticity µ ε := curl j ε vorticity in fluids, but quantized: µ ε 2π i d iδ a ε i µε 2πN ε µ signed measure, or probability measure,

9 Mean-field limit for stationary solutions If u ε is a solution to (GL) and N ε 1 then µ ε /N ε µ solution to µ h = 0 h = 1 µ in a suitable weak sense ( Delort): T µ := h h h 2 δ j i Weak relation is div T µ = 0 in finite parts" [Sandier-S 04] h is constant on the support of µ c 1 c 2 Ω

10 Dynamics in the case N ε 1 Back to N ε log ε tu = u + u ε 2 (1 u 2 ) in R 2 (PGL) in ε t u = u + u ε 2 (1 u 2 ) in R 2 (GP) For (GP), by Madelung transform, the limit dynamics is expected to be the 2D incompressible Euler equation. Vorticity form t µ div (µ h) = 0 h = 1 µ (EV) For (PGL), formal model proposed by [Chapman-Rubinstein-Schatzman 96], [E 95]: if µ 0 t µ div (µ h) = 0 h = 1 µ (CRSE)

11 Dynamics in the case N ε 1 Back to N ε log ε tu = u + u ε 2 (1 u 2 ) in R 2 (PGL) in ε t u = u + u ε 2 (1 u 2 ) in R 2 (GP) For (GP), by Madelung transform, the limit dynamics is expected to be the 2D incompressible Euler equation. Vorticity form t µ div (µ h) = 0 h = 1 µ (EV) For (PGL), formal model proposed by [Chapman-Rubinstein-Schatzman 96], [E 95]: if µ 0 t µ div (µ h) = 0 h = 1 µ (CRSE)

12 Study of the Chapman-Rubinstein-Schatzman-E equation [Lin-Zhang 00, Du-Zhang 03, Masmoudi-Zhang 05] existence of weak solutions (à la Delort) by vortex approximation method, existence and uniqueness of L solutions, which decay in 1/t (uses pseudo-differential operators) [Ambrosio-S 08] variational approach in the setting of a bounded domain. The equation is formally the gradient flow of F (µ) = 1 2 Ω 1 µ 2 for the 2-Wasserstein metric (à la [Otto, Ambrosio-Gigli-Savaré]). Existence of weak solutions for bounded energy initial data (i.e. µ H 1 ) via the minimizing movement scheme of De Giorgi, uniqueness in the class L, propagation of L p regularity. Takes into account possible entrance / exit of mass via the boundary ( Ω µ not preserved). Extension by [Ambrosio-Mainini-S 11] for signed measures. [S-Vazquez 13] PDE approach in all dimension. Existence via limits in fractional diffusion t µ + div (µ s µ) when s 1, uniqueness in the class L, propagation of regularity, asymptotic self-similar profile µ(t) = 1 πt 1 B t

13 Study of the Chapman-Rubinstein-Schatzman-E equation [Lin-Zhang 00, Du-Zhang 03, Masmoudi-Zhang 05] existence of weak solutions (à la Delort) by vortex approximation method, existence and uniqueness of L solutions, which decay in 1/t (uses pseudo-differential operators) [Ambrosio-S 08] variational approach in the setting of a bounded domain. The equation is formally the gradient flow of F (µ) = 1 2 Ω 1 µ 2 for the 2-Wasserstein metric (à la [Otto, Ambrosio-Gigli-Savaré]). Existence of weak solutions for bounded energy initial data (i.e. µ H 1 ) via the minimizing movement scheme of De Giorgi, uniqueness in the class L, propagation of L p regularity. Takes into account possible entrance / exit of mass via the boundary ( Ω µ not preserved). Extension by [Ambrosio-Mainini-S 11] for signed measures. [S-Vazquez 13] PDE approach in all dimension. Existence via limits in fractional diffusion t µ + div (µ s µ) when s 1, uniqueness in the class L, propagation of regularity, asymptotic self-similar profile µ(t) = 1 πt 1 B t

14 Study of the Chapman-Rubinstein-Schatzman-E equation [Lin-Zhang 00, Du-Zhang 03, Masmoudi-Zhang 05] existence of weak solutions (à la Delort) by vortex approximation method, existence and uniqueness of L solutions, which decay in 1/t (uses pseudo-differential operators) [Ambrosio-S 08] variational approach in the setting of a bounded domain. The equation is formally the gradient flow of F (µ) = 1 2 Ω 1 µ 2 for the 2-Wasserstein metric (à la [Otto, Ambrosio-Gigli-Savaré]). Existence of weak solutions for bounded energy initial data (i.e. µ H 1 ) via the minimizing movement scheme of De Giorgi, uniqueness in the class L, propagation of L p regularity. Takes into account possible entrance / exit of mass via the boundary ( Ω µ not preserved). Extension by [Ambrosio-Mainini-S 11] for signed measures. [S-Vazquez 13] PDE approach in all dimension. Existence via limits in fractional diffusion t µ + div (µ s µ) when s 1, uniqueness in the class L, propagation of regularity, asymptotic self-similar profile µ(t) = 1 πt 1 B t

15 Study of the Chapman-Rubinstein-Schatzman-E equation [Lin-Zhang 00, Du-Zhang 03, Masmoudi-Zhang 05] existence of weak solutions (à la Delort) by vortex approximation method, existence and uniqueness of L solutions, which decay in 1/t (uses pseudo-differential operators) [Ambrosio-S 08] variational approach in the setting of a bounded domain. The equation is formally the gradient flow of F (µ) = 1 2 Ω 1 µ 2 for the 2-Wasserstein metric (à la [Otto, Ambrosio-Gigli-Savaré]). Existence of weak solutions for bounded energy initial data (i.e. µ H 1 ) via the minimizing movement scheme of De Giorgi, uniqueness in the class L, propagation of L p regularity. Takes into account possible entrance / exit of mass via the boundary ( Ω µ not preserved). Extension by [Ambrosio-Mainini-S 11] for signed measures. [S-Vazquez 13] PDE approach in all dimension. Existence via limits in fractional diffusion t µ + div (µ s µ) when s 1, uniqueness in the class L, propagation of regularity, asymptotic self-similar profile µ(t) = 1 πt 1 B t

16 Study of the Chapman-Rubinstein-Schatzman-E equation [Lin-Zhang 00, Du-Zhang 03, Masmoudi-Zhang 05] existence of weak solutions (à la Delort) by vortex approximation method, existence and uniqueness of L solutions, which decay in 1/t (uses pseudo-differential operators) [Ambrosio-S 08] variational approach in the setting of a bounded domain. The equation is formally the gradient flow of F (µ) = 1 2 Ω 1 µ 2 for the 2-Wasserstein metric (à la [Otto, Ambrosio-Gigli-Savaré]). Existence of weak solutions for bounded energy initial data (i.e. µ H 1 ) via the minimizing movement scheme of De Giorgi, uniqueness in the class L, propagation of L p regularity. Takes into account possible entrance / exit of mass via the boundary ( Ω µ not preserved). Extension by [Ambrosio-Mainini-S 11] for signed measures. [S-Vazquez 13] PDE approach in all dimension. Existence via limits in fractional diffusion t µ + div (µ s µ) when s 1, uniqueness in the class L, propagation of regularity, asymptotic self-similar profile µ(t) = 1 πt 1 B t

17 Previous rigorous convergence results (PGL) case : [Kurzke-Spirn 14] convergence of µ ε /(2πN ε ) to µ solving (CRSE) under assumption N ε (log log log ε ) 1/4 + well-preparedness (GP) case: [Jerrard-Spirn 15] convergence to µ solving (EV) under assumption N ε (log log ε ) 1/2 + well-preparedness both proofs push" the fixed N proof (taking limits in the evolution of the energy density) by making it more quantitative difficult to go beyond these dilute regimes without controlling distance between vortices, possible collisions, etc

18 Previous rigorous convergence results (PGL) case : [Kurzke-Spirn 14] convergence of µ ε /(2πN ε ) to µ solving (CRSE) under assumption N ε (log log log ε ) 1/4 + well-preparedness (GP) case: [Jerrard-Spirn 15] convergence to µ solving (EV) under assumption N ε (log log ε ) 1/2 + well-preparedness both proofs push" the fixed N proof (taking limits in the evolution of the energy density) by making it more quantitative difficult to go beyond these dilute regimes without controlling distance between vortices, possible collisions, etc

19 Alternative method: the modulated energy" Exploits the regularity and stability of the solution to the limit equation Works for dissipative as well as conservative equations Works for gauged model as well Let v(t) be the expected limiting velocity field (such that 1 N ε u ε, iu ε v and curl v = 2πµ). Define the modulated energy E ε (u, t) = 1 2 R 2 u iun ε v(t) 2 + (1 u 2 ) 2 ) 2ε 2, modelled on the Ginzburg-Landau energy. Analogy with relative entropy" and modulated entropy" methods [Dafermos 79] [DiPerna 79] [Yau 91] [Brenier 00]...

20 Alternative method: the modulated energy" Exploits the regularity and stability of the solution to the limit equation Works for dissipative as well as conservative equations Works for gauged model as well Let v(t) be the expected limiting velocity field (such that 1 N ε u ε, iu ε v and curl v = 2πµ). Define the modulated energy E ε (u, t) = 1 2 R 2 u iun ε v(t) 2 + (1 u 2 ) 2 ) 2ε 2, modelled on the Ginzburg-Landau energy. Analogy with relative entropy" and modulated entropy" methods [Dafermos 79] [DiPerna 79] [Yau 91] [Brenier 00]...

21 Alternative method: the modulated energy" Exploits the regularity and stability of the solution to the limit equation Works for dissipative as well as conservative equations Works for gauged model as well Let v(t) be the expected limiting velocity field (such that 1 N ε u ε, iu ε v and curl v = 2πµ). Define the modulated energy E ε (u, t) = 1 2 R 2 u iun ε v(t) 2 + (1 u 2 ) 2 ) 2ε 2, modelled on the Ginzburg-Landau energy. Analogy with relative entropy" and modulated entropy" methods [Dafermos 79] [DiPerna 79] [Yau 91] [Brenier 00]...

22 Main result: Gross-Pitaevskii case Theorem (S. 15) Assume u ε solves (GP) and let N ε be such that log ε N ε 1 ε. Let v be a L (R +, C 0,1 ) solution to the incompressible Euler equation { t v = 2v curl v + p in R 2 div v = 0 in R 2, (IE) with curl v L (L 1 ). Let {u ε } ε>0 be solutions associated to initial conditions u 0 ε, with E ε (u 0 ε, 0) o(n 2 ε ). Then, for every t 0, we have 1 N ε u ε, iu ε v in L 1 loc(r 2 ). Implies of course the convergence of the vorticity µ ε /N ε curl v Works in 3D

23 Main result: parabolic case Theorem (S. 15) Assume u ε solves (PGL) and let N ε be such that 1 N ε O( log ε ). Let v be a L ([0, T ], C 1,γ ) solution to if N ε log ε { t v = 2vcurl v + p in R 2 div v = 0 in R 2, (L1) if N ε λ log ε t v = 1 λ div v 2vcurl v in R2. (L2) Assume E ε (u 0 ε, 0) πn ε log ε + o(n 2 ε ) and curl v(0) 0. Then t T we have 1 N ε u ε, iu ε v in L 1 loc(r 2 ). Taking the curl of the equation yields back the (CRSE) equation if N ε log ε, but not if N ε log ε! Long time existence proven by [Duerinckx 16].

24 Proof method Go around the question of minimal vortex distances by using instead the modulated energy and showing a Gronwall inequality on E. the proof relies on algebraic simplifications in computing d dt E ε(u ε (t)) which reveal only quadratic terms Uses the regularity of v to bound corresponding terms An insight is to think of v as a spatial gauge vector and div v (resp. p) as a temporal gauge

25 The disordered case In real superconductors one wants to flow currents and prevent the vortices from moving because that dissipates energy Model pinning and applied current by pinning potential 0 < a(x) 1 and force F equation reduces to (α+i log ε β) t u ε = u ε + au ε ε 2 (1 u ε 2 )+ a a u ε+i log ε F u ε +fu ε competition between vortex interaction, pinning force h := log a and applied force F Case of finite number of vortices treated in [Tice 10], [S-Tice 11], [Kurzke-Marzuola-Spirn 15]

26 The disordered case In real superconductors one wants to flow currents and prevent the vortices from moving because that dissipates energy Model pinning and applied current by pinning potential 0 < a(x) 1 and force F equation reduces to (α+i log ε β) t u ε = u ε + au ε ε 2 (1 u ε 2 )+ a a u ε+i log ε F u ε +fu ε competition between vortex interaction, pinning force h := log a and applied force F Case of finite number of vortices treated in [Tice 10], [S-Tice 11], [Kurzke-Marzuola-Spirn 15]

27 The disordered case In real superconductors one wants to flow currents and prevent the vortices from moving because that dissipates energy Model pinning and applied current by pinning potential 0 < a(x) 1 and force F equation reduces to (α+i log ε β) t u ε = u ε + au ε ε 2 (1 u ε 2 )+ a a u ε+i log ε F u ε +fu ε competition between vortex interaction, pinning force h := log a and applied force F Case of finite number of vortices treated in [Tice 10], [S-Tice 11], [Kurzke-Marzuola-Spirn 15]

28 Convergence to fluid-like equations Gross-Pitaevskii case Theorem (Duerinckx-S) In the regime log ε N ε 1 ε, convergence of j ε/n ε to solutions of { t v = p + ( F + 2v )curl v in R 2 div (av) = 0 in R 2,

29 Parabolic case Theorem (Duerinckx-S) N ε log ε, λ ε := j ε /N ε converges to N ε log ε, F ε = λ ε F, a ε = a λε (h ε = λ ε h) { t v = p + ( h F 2v)curl v in R 2 div v = 0 in R 2, j ε /N ε converges to N ε = λ log ε (λ > 0) t v = 1 λ (1 a div (av)) + ( h F 2v)curl v in R 2. vorticity evolves by t µ = div (Γµ) with Γ = pinning + applied force + interaction

30 Homogenization questions we want to consider rapidly oscillating pinning force η ε h(x, x η ε ) η ε 1 and scale η ε with ε too difficult to take the diagonal limit η ε 0 directly from GL eq. Instead homogenize the limiting equations t µ = div (Γµ) Γ = h F 2v homogenization of nonlinear transport equations. easier when interaction is negligible Γ independent of µ, washboard model Understand depinning current and velocity law (in F F c ) Understand thermal effects by adding noise to such systems creep, elastic effects

31 Homogenization questions we want to consider rapidly oscillating pinning force η ε h(x, x η ε ) η ε 1 and scale η ε with ε too difficult to take the diagonal limit η ε 0 directly from GL eq. Instead homogenize the limiting equations t µ = div (Γµ) Γ = h F 2v homogenization of nonlinear transport equations. easier when interaction is negligible Γ independent of µ, washboard model Understand depinning current and velocity law (in F F c ) Understand thermal effects by adding noise to such systems creep, elastic effects

32 Homogenization questions we want to consider rapidly oscillating pinning force η ε h(x, x η ε ) η ε 1 and scale η ε with ε too difficult to take the diagonal limit η ε 0 directly from GL eq. Instead homogenize the limiting equations t µ = div (Γµ) Γ = h F 2v homogenization of nonlinear transport equations. easier when interaction is negligible Γ independent of µ, washboard model Understand depinning current and velocity law (in F F c ) Understand thermal effects by adding noise to such systems creep, elastic effects

33 Sketch of proof: quantities and identities E ε (u, t) = 1 2 u iun ε v(t) 2 + (1 u 2 ) 2 ) R 2ε 2 2 (modulated energy) j ε = iu ε, u ε curl j ε = µ ε (supercurrent and vorticity) V ε = 2 i t u ε, u ε (vortex velocity) t j ε = iu ε, t u ε + V ε t curl j ε = t µ ε = curl V ε (V ε transports the vorticity). S ε := k u ε, l u ε 1 2 ( u ε ) 2ε 2 (1 u ε 2 ) 2 δ kl (stress-energy tensor) S ε = k u ε iu ε N ε v k, l u ε iu ε N ε v l 1 ( u ε iu ε N ε v ) 2 2ε 2 (1 u ε 2 ) 2 δ kl modulated stress tensor"

34 Sketch of proof: quantities and identities E ε (u, t) = 1 2 u iun ε v(t) 2 + (1 u 2 ) 2 ) R 2ε 2 2 (modulated energy) j ε = iu ε, u ε curl j ε = µ ε (supercurrent and vorticity) V ε = 2 i t u ε, u ε (vortex velocity) t j ε = iu ε, t u ε + V ε t curl j ε = t µ ε = curl V ε (V ε transports the vorticity). S ε := k u ε, l u ε 1 2 ( u ε ) 2ε 2 (1 u ε 2 ) 2 δ kl (stress-energy tensor) S ε = k u ε iu ε N ε v k, l u ε iu ε N ε v l 1 ( u ε iu ε N ε v ) 2 2ε 2 (1 u ε 2 ) 2 δ kl modulated stress tensor"

35 The Gross-Pitaevskii case Time-derivative of the energy (if u ε solves (GP) and v solves (IE)) de ε (u ε (t), t)) = N ε (N ε v j ε ) t v N dt R }{{}}{{} ε V ε v 2 2v curl v+ p linear term linear term a priori controlled by E unsufficient But div S ε = N ε (N ε v j ε ) curl v N ε v µ ε N εv ε Multiply by 2v 2v div S ε = N ε (N ε v j ε ) 2v curl v + N ε V ε v R 2 R 2 de ε dt = R 2 2 S ε }{{} controlled by E ε : v }{{} bounded Gronwall OK: if E ε (u ε (0)) o(n 2 ε ) it remains true (vortex energy is πn ε log ε N 2 ε in the regime N ε log ε )

36 The Gross-Pitaevskii case Time-derivative of the energy (if u ε solves (GP) and v solves (IE)) de ε (u ε (t), t)) = N ε (N ε v j ε ) t v N dt R }{{}}{{} ε V ε v 2 2v curl v+ p linear term linear term a priori controlled by E unsufficient But div S ε = N ε (N ε v j ε ) curl v N ε v µ ε N εv ε Multiply by 2v 2v div S ε = N ε (N ε v j ε ) 2v curl v + N ε V ε v R 2 R 2 de ε dt = R 2 2 S ε }{{} controlled by E ε : v }{{} bounded Gronwall OK: if E ε (u ε (0)) o(n 2 ε ) it remains true (vortex energy is πn ε log ε N 2 ε in the regime N ε log ε )

37 The Gross-Pitaevskii case Time-derivative of the energy (if u ε solves (GP) and v solves (IE)) de ε (u ε (t), t)) = N ε (N ε v j ε ) t v N dt R }{{}}{{} ε V ε v 2 2v curl v+ p linear term linear term a priori controlled by E unsufficient But div S ε = N ε (N ε v j ε ) curl v N ε v µ ε N εv ε Multiply by 2v 2v div S ε = N ε (N ε v j ε ) 2v curl v + N ε V ε v R 2 R 2 de ε dt = R 2 2 S ε }{{} controlled by E ε : v }{{} bounded Gronwall OK: if E ε (u ε (0)) o(n 2 ε ) it remains true (vortex energy is πn ε log ε N 2 ε in the regime N ε log ε )

38 The parabolic case If u ε solves (PGL) and v solves (L1) or (L2) de ε (u ε (t), t)) N ε = dt R log ε tu ε 2 + (N ε (N ε v j ε ) t v N ε V ε v) 2 R 2 div S ε = N ε log ε tu ε iu ε N ε φ, u ε iu ε N ε v + N ε (N ε v j ε ) curl v N ε v µ ε. φ = p if N ε log ε φ = λ div v if not Multiply by v and insert: de ε = 2 S ε : v N ε V ε v 2N ε v 2 µ ε dt R 2 N ε log ε tu ε iu ε N ε φ 2 +2v N ε log ε tu ε iu ε N ε φ, u ε iu ε N ε v. R 2

39 The parabolic case If u ε solves (PGL) and v solves (L1) or (L2) de ε (u ε (t), t)) N ε = dt R log ε tu ε 2 + (N ε (N ε v j ε ) t v N ε V ε v) 2 R 2 div S ε = N ε log ε tu ε iu ε N ε φ, u ε iu ε N ε v + N ε (N ε v j ε ) curl v N ε v µ ε. φ = p if N ε log ε φ = λ div v if not Multiply by v and insert: de ε = 2 S ε : v N ε V ε v 2N ε v 2 µ ε dt R 2 N ε log ε tu ε iu ε N ε φ 2 +2v N ε log ε tu ε iu ε N ε φ, u ε iu ε N ε v. R 2

40 The vortex energy πn ε log ε is no longer negligible with respect to N 2 ε. We now need to prove de ε dt Need all the tools on vortex analysis: C(E ε πn ε log ε ) + o(n 2 ε ). vortex ball construction [Sandier 98, Jerrard 99, Sandier-S 00, S-Tice 08]: allows to bound the energy of the vortices from below in disjoint vortex balls B i by π d i log ε and deduce that the energy outside of i B i is controlled by the excess energy E ε πn ε log ε product estimate" of [Sandier-S 04] allows to control the velocity: V ε v 2 ( ) 1 t u ε iu ε N ε φ 2 ( u ε iu ε N ε v) v 2 2 log ε 1 ( ) 1 t u ε iu ε N ε φ ( u ε iu ε N ε v) v 2 log ε 2

41 de ε = dt R 2 de ε dt N ε R 2 2 S ε }{{} C(E ε πn ε log ε ) : v }{{} N ε V ε v }{{} 2N ε v 2 µ ε bounded controlled by prod. estimate N ε log ε tu ε iu ε N ε φ 2 +2v log ε tu ε iu ε N ε φ, u ε iu ε N ε v. }{{} bounded by Cauchy-Schwarz C(E ε πn ε log ε ) + R 2 N ε log ε ( ) tu ε iu ε N ε φ 2 + 2N ε ( u ε iu ε N ε v) v 2 + ( u ε iu ε N ε v) v 2 2N ε v 2 µ ε log ε R 2 R 2 = C(E ε πn ε log ε )+ 2N ε u ε iu ε N ε v 2 v 2 2N ε v 2 µ ε log ε R 2 R }{{ 2 } bounded by C(E ε πn ε log ε ) by ball construction estimates Gronwall OK

42 de ε = dt R 2 de ε dt N ε R 2 2 S ε }{{} C(E ε πn ε log ε ) : v }{{} N ε V ε v }{{} 2N ε v 2 µ ε bounded controlled by prod. estimate N ε log ε tu ε iu ε N ε φ 2 +2v log ε tu ε iu ε N ε φ, u ε iu ε N ε v. }{{} bounded by Cauchy-Schwarz C(E ε πn ε log ε ) + R 2 N ε log ε ( ) tu ε iu ε N ε φ 2 + 2N ε ( u ε iu ε N ε v) v 2 + ( u ε iu ε N ε v) v 2 2N ε v 2 µ ε log ε R 2 R 2 = C(E ε πn ε log ε )+ 2N ε u ε iu ε N ε v 2 v 2 2N ε v 2 µ ε log ε R 2 R }{{ 2 } bounded by C(E ε πn ε log ε ) by ball construction estimates Gronwall OK

43 Joyeux anniversaire, Yann!