A scaling limit from Euler to Navier-Stokes equations with random perturbation

Transcription

1 A scaling limit from Euler to Navier-Stokes equations with random perturbation Franco Flandoli, Scuola Normale Superiore of Pisa Newton Institute, October 208 Newton Institute, October 208 /

2 Subject of the talk The talk deals with a special scaling limit connecting 2D Euler and Navier-Stokes equations, both with noise, but of different type: t ω + u ω = ξ ω t ω + u ω = ω + η Euler, multiplicative noise Navier-Stokes, additive noise where u = velocity, div u = 0 ω = u = vorticity ξ, η = space-dependent noise. Newton Institute, October /

3 Overview of some directions in stochastic fluid dynamics Due to the interdisciplinary character of the meeting, let me spend a few introductory words on stochastic fluid dynamics. The most studied equation is the stochastic Navier-Stokes equation with additive noise: t u + u u + p = ν u + W From the time of Kolmogorov it is indicated as a potential model to explain some feature of turbulence. For instance, the formal average energy balance t 2 E u t 2 dx + ν E u s 2 dxds = 2 E u 0 2 dx + t Trace (Q) 0 put the basis for investigating dissipation for small ν: energy input is under control, and dissipation is constant. Much work has been devoted to the attempt to prove well posedness in 3D, outstanding open problem. Newton Institute, October /

4 Overview of some directions in stochastic fluid dynamics For Euler equation, a more natural noise is of transport type t u + u u + p = ξ u. It preserves certain physical quantities (energy, in the form above, enstrophy in the 2D model t ω + u ω = ξ ω). Intuitively, it may correspond to the Lagrangian motion of small scales, acting on larger scales. Similarly to additive noise for 3D Navier-Stokes equation, an open problem is whether this noise "regularizes", namely it produces better results of well posedness. For instance, it prevents point vortex collision (F.-Gubinelli-Priola ). Newton Institute, October /

5 Back to the subject of the talk Thus the two models t ω + u ω = ξ ω Euler, multiplicative noise t ω + u ω = ω + η Navier-Stokes, additive noise have some physical and mathematical motivation. (Here they are written in vorticity form) The purpose of this talk is to discuss a special scaling limit which connects the two, based on a computation similar in spirit to a renormalization. Newton Institute, October /

6 Disclaimer There is an obvious misunderstanding that we have to declare. In deterministic fluid mechanics, one of the most important open problems is the vanishing viscosity limit. Is it true that the solution ω ɛ of t ω ɛ + u ɛ ω ɛ = ɛ ω ɛ Navier-Stokes converges to a solution ω of t ω + u ω = 0 Euler as ɛ 0? This is not the limit investigated in this talk. Newton Institute, October /

7 From Euler to Navier-Stokes Somewhat opposite, we aim to prove that solutions ω ɛ of t ω N + u N ω N = ξ N ω N Euler converge to a solution ω of t ω + u ω = ω + η Navier-Stokes as N. In other words, in a suitable scaling limit, a transport type noise ξ N ω N gives rise to ω + η. Newton Institute, October /

8 The stochastic Euler equations The equations will be considered on the torus T 2 = R 2 /Z 2. The noise ξ N in t ω N + u N ω N = ξ N ω N Euler has the form where ξ N (x, t) = ɛ N σ k (x) d β k (t) dt k Z 2 = Z 2 \ {0} σ k (x) : T 2 R 2 are divergence free fields β k are independent scalar BM s. Newton Institute, October /

9 Coeffi cients on the noise t ω N + u N ω N = ξ N ω N d β ξ N = ɛ N σ k k dt We assume σ k (x) : T 2 R 2, divergence free fields, of the form where (k, k 2 ) = (k 2, k ), σ k (x) = k k 2 e k (x) e k (x) = { sin (2πk x) for k Z 2 + cos (2πk x) for k Z 2. Newton Institute, October /

10 The case of infinite dimensional noise Consider the divergence free noise (infinite series) parametrized by γ. ξ (γ) k (x, t) = k γ e k (x) d β k (t) dt k Z 2 The case γ = is the divergence free analog of space-time white noise. Completely untractable as a transport noise. The case γ = 2 is the divergence free analog of Gaussian free field noise. It is our case but with truncated series. The infinite series (as a transport noise) is an open problem. Only the case γ > 2 (infinite series) can been treated by standard methods. Newton Institute, October /

11 Itô-Stratonovich The threshold γ = 2 appears when rewriting Stratonovich into Itô: ξ (γ) k = d β k γ e k k dt k Z 2 ξ (γ) ω ξ (γ) ω = 2 k (e ) k k k γ (e ) k k γ ω. k Z 2 and it turns out that the second order differential operator is well defined only for γ > 2. Let us see the details. Newton Institute, October 208 /

12 The Itô-Stratonovich corrector 2 k (e ) k k k γ (e ) k k γ ω k Z 2 ( k e k =0 = 2 k e 2 ) ( k ) k k γ k γ ω k Z 2 ( sin 2 + cos 2 = = 2 k ) ( k ) k γ k γ ω k Z 2 + = 2 k Z 2 + = 2 k 2γ 2 i,j= k Z 2 k 2γ 2 + k i k j i j ω ω finite only for γ > 2. Newton Institute, October /

13 The Itô-Stratonovich corrector The previous computation ( shows that, for γ > 2, the Itô-Stratonovich corrector in 2 k Z 2 + ω. k 2γ 2 ) For γ = 2 the coeffi cient diverges. But, for γ = 2, the Itô-Stratonovich corrector of the noise is 2 ξ N (x, t) = ɛ N σ k (x) d β k (t) dt ( ɛ 2 N k 2 ) ω which converges if we take ɛ N / log N. Newton Institute, October /

14 Summary For the Euler equation d β t ω N + u N ω N = ξ N ω N, ξ N = ɛ N σ k k dt with σ k (x) = k k 2 e k (x) and ɛ N / log N we can write t ω N + u N ω N = ξ N ω }{{ N + ν } N ω N Itô with ν N ν R +. In the case σ k (x) = k k γ e k (x) with γ > 2 we get the same result without rescaling with ɛ N. What about the martingale term? Newton Institute, October /

15 The martingale term The martingale M N (t, x) := ɛ N t σ k (x) ω N (s, x) d β k (s) 0 in general depends strongly on the solution ω N (s, x) and preserves this dependence in the limit N. This dependence spoils the dissipativity of ω N : the sum of the martingale and corrector is the Stratonovich term, which is not dissipative. For γ > 2, not rescaled by ɛ N, this remains true in the limit of the infinite dimensional noise: we get the stochastic Euler equation t ω + u ω = ξ ω, k ξ = d β k γ e k k dt. k Z 2 Newton Institute, October /

16 The martingale term In the case γ = 2, the martingale M N (t, x) := t log N 0 k k 2 e k (x) ω N (s, x) d β k (s) behaves in a special way for special solutions ω N (described below) and converges to a Gaussian process, precisely of the form W (t, x) where W (t, x) is a divergence free space-time white noise. When true, the limit of the 2D Euler equation will be t ω + u ω = ν ω + W (t, x). It remains to describe the special solutions when this happens. Newton Institute, October /

17 Again To avoid misunderstanding, let us insist on the fact that usually the Laplacian in the Itô-Stratonovich reformulation t ω N + u N ω N = ξ N ω }{{ N + ν } N ω N Itô is a fake Laplacian, it does not dissipate. The sum of Itô + corrector is just Stratonovich. Thus, in general, both the approximating and the limit equation are of hyperbolic type. But we have identified a special regime where the Laplacian (the corrector) converges to a Laplacian, and the Itô term converges to a Gaussian process. In this case the limit equation changes nature, becomes parabolic. Newton Institute, October /

18 White noise solutions to 2D Euler equations Consider, on T 2 = R 2 /Z 2, the equation where ξ (γ) (x, t) = k Z 2 Theorem (F.-Luo 8) t ω + u ω = ξ (γ) ω ω = u, div u = 0 k k γ e k (x) d β k (t) dt. If γ > 2, there exists a solution, weak in the probabilistic and analytic sense, stationary process with trajectories of class C ( [0, T ] ; H ), such ω (t) is a white noise in space, for every t 0. In spite of many attempts, we do not know how to extend this theorem to γ = 2. ranco Flandoli, Scuola Normale Superiore of Pisa From Euler () to Navier-Stokes Newton Institute, October /

19 Remarks For γ > 3, solutions in a more classical sense, precisely of class ω L, have been obtained by Brzezniak-F.-Maurelli, ARMA 206. In the class ω L, also uniqueness is know, as a generalization of the deterministic result of Yudovich. Below ω L, determinisc 2D Euler equations lack uniqueness and our hope is that this kind of transport noise may restore it. Therefore we investigate all possible ranges of noise, since perhaps those which may regularize are very rough. The range γ > 2 is covered by the previous result. γ = 2 is the threshold we do not know how to approach (with infinite dimensional noise). Newton Institute, October /

20 The approximating system Consider the equation with Similarly to above, Lemma (F.-Luo 8) t ω N + u N ω N = ξ N ω N ξ N (x, t) = ɛ N k k 2 e k (x) d β k (t). dt There exists a solution (called below White Noise solution), weak in the probabilistic and analytic sense, stationary process with trajectories of class C ( [0, T ] ; H ), such ω N (t) is a white noise in space, for every t 0. ranco Flandoli, Scuola Normale Superiore of Pisa From Euler () to Navier-Stokes Newton Institute, October /

21 Main result Theorem (F.-Luo 8) For every N N, let ω N be a White Noise solution of t ω N + u N ω N = ξ N ω N, ξ N = ɛ N given by the previous lemma. Assume ɛ N = log N. k k 2 e k Then ω N converges weakly to ω, unique solution of the stochastic Navier-Stokes equation with space-time noise t ω + u ω = ω + W. d β k dt ranco Flandoli, Scuola Normale Superiore of Pisa From Euler () to Navier-Stokes Newton Institute, October /

22 Remark on the Navier-Stokes equations 2D Navier-Stoke equations with space-time white noise t u + u u + p = u + W have been studied by Da Prato-Debussche, Albeverio-Ferrario and others. It is well posed also in the strong sense (for general initial conditions). The vorticity formulation, ω = u, in the equation above: t ω + u ω = ω + W. It has a White Noise solution as a stationary solution. This is the regime considered by the theorem above. Newton Institute, October /

23 Scheme of the proof Tightness of the laws of solutions of the approximating Euler equations can be proved in C ( [0, T ] ; H ), taking advantage of their stationarity in time and knowledge of the time-marginals. The identification of the limit requires standard work for the nonlinear term (based on recent works on White Noise solutions of 2D Euler) and for the Laplacian arising as a corrector. The diffi cult part is the convergence of the martingale term M N (t, x) := t log N 0 k k 2 e k (x) ω N (s, x) d β k (s) to the Gaussian process W (t, x). Newton Institute, October /

24 Quadratic variation We have to prove, for M N (t, x) := that the joint variation t log N 0 k k 2 e k (x) ω N (s, x) d β k (s) converges to [ M N, e l, M N, e m ] t t [ W, e l, W, e m ]t = e l, e m ds = t l 2 δ l,m. 0 Newton Institute, October /

25 Quadratic variation From M N (t, x) = log N M N (t, x) M N (t, x) := t 0 k k 2 e k (x) ω N (s, x) d β k (s) one can show, similarly to the corrector above, that [ ] M N, e l, M N, e m diverges. Precisely, [ ] M N, e l, M N, e m = ( k 4 k l t ) ( k m t ) t ω, e k e l ω, e k e m ds 0 Newton Institute, October /

26 Quadratic variation [ ] M N, e l, M N, e m = ( k 4 k l t ) ( k m ) t ω, e k e l ω, e k e m ds 0 [ ] E M N, e l, M N, e m ω(t) WN = t ( k 2 ) k 4 t l 2 δ l,m t k l 2 δ l,m and the coeffi cient k 2 diverges. Newton Institute, October /

27 Analogy: renormalized energy In the theory of 2D Euler with the enstrophy measure there is a well-known renormalization: the interaction kinetic energy. The average kinetic energy of the fluid is (u k = Fourier components of u) E [ 2 ] u (x) 2 dx = 2 [ E u k 2] ω= u=wn = k Z 2 2 k Z 2 k 2 = + but the renormalized partial sums (corresponding to interaction energy in the case of point vortices) [ ] u k 2 E u k 2 converge in L 2 µ (µ=law of White Noise on H ) to a well defined random variable : E : called renormalized kinetic energy. Newton Institute, October /

28 Renormalizing quadratic variation For [ M N, e l, M N, e m ] t it is similar. Without the term log N, we have seen above that [ ] E M N, e l, M N, e m diverges. t Moreover, it happens that [ [ ] M N, e l, M N, e m ]t E M N, e l, M N, e m converges in L 2 µ (µ=law of White Noise on H ) to a well defined random variable. t Newton Institute, October /

29 Renormalizing quadratic variation As a consequence, including the term log N, L 2 µ lim N ([ M N, e l, M N, e m ] t E [ M N, e l, M N, e m ] t ) = 0. And E [ M N, e l, M N, e m ] t = = ( log N ) k 2 t l 2 δ l,m t l 2 δ l,m ] [ W, e l, W, e m t. Newton Institute, October /

30 Discussion: extensions, interpretation, motivations There are heuristic reasons to believe that part of the scaling limit discussed above is a more general fact; in particular, the presence of the Laplacian in the limit equation, with a true parabolic character. Maybe precisely additive white noise is special, due to the "white noise" solution regime. [Private discussions years ago with Weber and Hauray.] A common space-dep noise with very short correlation range acting on particles is similar to independent BMs and should give rise to a Laplacian in the mean field limit. With Dejun Luo we spent much time trying to prove that multiplicative noise restores uniqueness in 2D Euler equations and had the heuristic impression that γ = 2 was relevant (but it is not admissible). In a sense we have studied here the case γ = 2, and the limit equation, stochastic Navier-Stokes, has uniqueness! Newton Institute, October 208 /