Regularization by noise in infinite dimensions

Transcription

1 Regularization by noise in infinite dimensions Franco Flandoli, University of Pisa King s College 2017 Franco Flandoli, University of Pisa () Regularization by noise King s College / 33

2 Plan of the talk Regularization by noise in finite dimensions and the attempts on 3D Navier-Stokes equations with additive noise Inviscid models with multiplicative noise Franco Flandoli, University of Pisa () Regularization by noise King s College / 33

3 Introduction on regularization by noise in finite dimensions Consider the equation dx t = b (t, X t ) dt + σdw t, X 0 = x 0 with b : [0, T ] R d R d and (W t ) t 0 a Brownian motion on a probability space (Ω, F, P). Theorem i) (Veretennikov 1981) If b is bounded measurable, then there is strong existence and pathwise uniqueness. ii) (Krylov-Röckner 2005) If b L q ( 0, T ; L p ( R d, R d )) with d p + 2 q < 1, the same result holds. A modern proof is based on the following idea: consider the associated (backward, vector valued) Kolmogorov equation U t U + b U = b + λu, U t=t = 0 which has solutions twice more regular (in space) than b. ranco Flandoli, University of Pisa () Regularization by noise King s College / 33

4 Introduction on regularization by noise in finite dimensions (b = b (x) to simplify notations) 1 2 U + b U = b + λu, dx t = b (X t ) dt + σdw t Itô formula: du (X t ) = U (X t ) dx t + σ2 2 U (X t) dt ( ) 1 = U + b U (X t ) dt + U (X t ) σdw t 2 = b (X t ) dt + λu (X t ) dt + U (X t ) σdw t and therefore t b (X s ) ds = U (x 0 ) U (X t ) + 0 X t = x 0 + U (x 0 ) U (X t ) + t 0 t 0 t λu (X s ) ds + λu (X s ) ds + t 0 0 U (X s ) σdw s (I + U (X s )) σdw s. Franco Flandoli, University of Pisa () Regularization by noise King s College / 33

5 Extensions to Hilbert spaces dx t = AX t dt + B (X t ) dt + QdW t, X 0 = x 0 where A : D (A) H H is a negative selfadjoint linear operator, H Hilbert space, B : H H is bounded measurable, (A, Q) satisfy proper conditions. Theorem (Da Prato, F., Priola, Röckner 2015) There is strong existence and pathwise uniqueness. Extended with Veretennikov to more general B, which includes V. However, these results are restricted to applications in space-dimension 1 (due to the assumptions on (A, Q)), and B cannot be unbounded. ranco Flandoli, University of Pisa () Regularization by noise King s College / 33

6 3D Navier-Stokes equations: some extensions of deterministic results Consider the 3D Navier-Stokes equations perturbed by noise: du + (u u + p) dt = udt + noise div u = 0 in a bounded regular domain, with non-slip boundary condition and initial condition u 0 L 2 σ. Two main examples of noise: additive noise (generic form of a forcing term, of small fluctuations, of hidden variables): noise = σ k (x) dwt k k=1 under the typical assumption that σ k (x) are orthogonal functions with k σ k 2 L 2 < Franco Flandoli, University of Pisa () Regularization by noise King s College / 33

7 3D Navier-Stokes equations: some extensions of deterministic results transport type noise (more comments below for Euler): noise = σ k (x) u (t, x) dwt k k=1 (Stratonovich form). In both cases, see F.-Gatarek, PTRF 1996 and several other papers: Theorem (Existence of weak solutions) In d = 2, 3, given u 0 L 2 σ, there exists a weak martingale solution with the property [ ] E sup u (t) 2 L + 2 t [0,T ] T It is strong and pathwise unique in d = 2. 0 u (t) 2 W 1,2 dt <. ranco Flandoli, University of Pisa () Regularization by noise King s College / 33

8 The dream for 3D Navier-Stokes equations Due to the special well posedness results for stochastic equations with irregular drift, some people devoted an enormous effort to investigate uniqueness of weak solutions for the stochastic 3D Navier-Stokes equations with additive noise du + (u u + p) dt = udt + div u = 0. k=1 σ k dw k t Unfortunately all techniques failed, including one based on Kolmogorov equations (developed by Da Prato and Debussche, but with solutions U (t, x) not suffi ciently regular). Franco Flandoli, University of Pisa () Regularization by noise King s College / 33

9 A special result for 3D Navier-Stokes equations The best result that has been obtained (collecting a series of paper by Da Prato - Debussche around 2005 and F. - Romito around 2008) is the following one. There exists selections of weak solutions with the Markov property (this holds true for all kind of noise, also without noise): probability kernels P (t, u 0, A), u 0 L 2 σ, A B ( L 2 ) σ satisfying Chapman-Kolmogorov equation P (t + s, u 0, A) = P (s, v, A) P (t, u 0, dv). L 2 σ (the existence of a solution flow Φ (t, ) : L 2 σ L 2 σ with the semi-flow property Φ (s, Φ (t, u 0 )) = Φ (t + s, u 0 ) is an open problem in the deterministic theory) Franco Flandoli, University of Pisa () Regularization by noise King s College / 33

10 A special result for 3D Navier-Stokes equations Under proper non-degeneracy of the sequence (σ i ) i N, every Markov selection is Strong Feller in H 1 σ, namely it has the following special continuity property in the initial conditions, globally in time, fully unknown in the deterministic case: if u n 0 u 0 in H 1 σ, then for every t 0 P (t, u n 0, ) P (t, u 0, ) in the topology of bounded variations of probability measures, on L 2 σ. Each selection, hence, is a very strong object. But we cannot exclude the possibility that there are several different selections. Franco Flandoli, University of Pisa () Regularization by noise King s College / 33

11 Do I still have a dream? After many years I do not know if classical stochastic method may help. I suspect that special quantities like the vorticity production t 0 ( ) ω T u + u T ω dxdt could have better properties in the stochastic case, similarly to the properties of integrals of the form t 0 as seen above. b (X s ) ds = U (x 0 ) U (X t ) + t 0 t λu (X s ) ds + U (X s ) σdw s 0 Franco Flandoli, University of Pisa () Regularization by noise King s College / 33

12 Recent variant for inviscid problems In the case of Euler equations, additive noise does not look so natural and promising. It is more natural to consider a transport type noise, in Stratonovich form (div σ i = 0): Motivations: du + (u u + p) dt = σ k u dwt k k=1 div u = 0. energy conservation geometric variational principle (Darryl Holm 2015) the results of F.-Gubinelli-Priola, of well posedness of the transport equations with non regular drift b (Invent. Math. 2010) du + b u dt + u dw t = 0 and of no-collapse of point vortices (SPA 2014). Franco Flandoli, University of Pisa () Regularization by noise King s College / 33

13 Results in 2D Let us first list a few generalizations of deterministic results. existence of solutions when vorticity is L p (Brzezniak-Peszat). uniqueness of solutions when vorticity is L (Brzezniak-F.-Maurelli, ARMA 2016). The equation is dω + u ωdt + k=1 σ k ω dw k t = 0 on a torus, with div σ i = 0. If ω 0 L, then there exists a unique bounded measurable solution. The proof is based on a reformulation by means of flows (not Yudovich proof). Franco Flandoli, University of Pisa () Regularization by noise King s College / 33

14 Method for uniqueness in 2D dω + u ωdt + k=1 σ k ω dw k t = 0 ω (t, x) = ω 0 ( ϕ 1 t (x) ) d ϕ t (x) = u (t, ϕ t (x)) dt + k=1 σ k (ϕ t (x)) dw k t. A fixed point argument with ω L and measure preserving flows works. An intriguing problem is a deeper understanding of the flow equation t ϕ t (x) = x + 0 K (ϕ t (x) ϕ t (y)) ω 0 (dy) + k=1 σ k (ϕ t (x)) dw k t compared to the recent results of flows for equations with irregular drift. Franco Flandoli, University of Pisa () Regularization by noise King s College / 33

15 3D theory In 3D, the generalization of deterministic results is more diffi cult. Let me mention a work in progress with Dan Crisan and Darryl Holm. We have considered the equation dω + L u ω dt + k=1 L σk ω dw k t = 0, ω t=0 = ω 0 proposed by Holm 2015; the Lie derivative of vector fields is defined as L A B = [A, B] = A B B A. The equations above is stated in Stratonovich form; the corresponding Itô form is dω + L u ω dt + k=1 L σk ω dw k t = 1 2 L 2 σ k ω dt. k=1 Franco Flandoli, University of Pisa () Regularization by noise King s College / 33

16 3D theory We want to prove a priori estimates in high order Sobolev spaces. The deterministic classical result proves well posedness in the space ω (t) W 3/2+ɛ,2 ( R 3 ; R 3), for some ɛ > 0, when ω 0 belongs to the same space. Here we simplify (due to a number of new very non-trivial facts) and work in the space ω (t) W 2,2 ( R 3 ; R 3). This way we may examine ω (t) (to avoid fractional derivatives) and investigate existence and uniqueness in the class of regularity ω (t) L 2 ( R 3 ; R 3). Franco Flandoli, University of Pisa () Regularization by noise King s College / 33

17 3D theory Rigorously we work on a regularized problem of the form dω ν + f R ( u ν L ) L u ω dt + = ν 3 ω ν dt but I will omit this detail in the sequel. k=1 L σk ω ν dw k t L 2 σ k ω ν dt, ω ν t=0 = ω 0 k=1 Franco Flandoli, University of Pisa () Regularization by noise King s College / 33

18 3D theory; deterministic case We look for a priori estimates on R ω (t, x) 2 dx. In the deterministic 3 case L u ω, ω (arising from d dt R ω (t, x) 2 dx) is equal to the sum 3 of several terms. Using Sobolev embedding theorems W 2,2 ( R 3) ( C b R 3 ), W 2,2 ( R 3) W 1,4 ( R 3), one can estimate all terms by C ω 2 W 2,2 except for the term with higher order derivatives R 3 (u ω) ω dx but this term is equal to zero, being equal to 1 (u ) ω 2 dx = 1 ω 2 div u dx = 0. 2 R 3 2 R 3 Franco Flandoli, University of Pisa () Regularization by noise King s College / 33

19 3D theory; diffi culty in the stochastic case In the stochastic case we have many more terms, coming from two sources: 1 the term 1 2 k L 2 σk ω, ω, which contains 6 derivatives, opposite to the deterministic term L u ω, ω ; 2 the Itô correction term 1 2 k L σk ω, L σk ω dt in Itô formula for d R 3 ω (t, x) 2 dx. In their sum the highest order terms cancel each other. However, a priori there is a large amount of terms with 5 derivatives, hence not bounded by ω 2 W 2,2. Franco Flandoli, University of Pisa () Regularization by noise King s College / 33

20 3D theory A careful algebraic manipulation of differential operators, their commutators and adjoint operators, finally shows that also all terms with 5 derivatives cancel each other. One has L 2 σk ω, ω + L σk ω, L σk ω C k ω 2 W 2,2 At the end we estimate again by ω 2 W 2,2 and obtain the a priori estimates. The final result is: Theorem Assume ω 0 W 2,2 ( R 3 ; R 3). Then there exists a random interval [0, τ R ], with P (τ R > 0) = 1, such that the stochastic 3D Euler equations have a unique continuous adapted solution on [0, τ R ], with paths of class C ( [0, τ R ] ; W 2,2 ( R 3 ; R 3)). ranco Flandoli, University of Pisa () Regularization by noise King s College / 33

21 Could transport noise improve the 3D theory? For the true Euler equations in vorticity form we do not know. We have a result for a much simplified model: a linear vector advection equation in 3D, with non-regular coeffi cients. It is a vector-valued generalization of the results with Gubinelli and Priola mentioned above for the transport equation. I will describe the result of F. - Maurelli - Neklyudov, JMFM Franco Flandoli, University of Pisa () Regularization by noise King s College / 33

22 Linear vector advection equation The equation is db + L u B dt + k=1 L σk B dw k t = 0 for a passive vector field B (let us think to a magnetic field), driven by a deterministic field u and noise. Its Itô formulation is db + L u B dt + k=1 L σk B dw k t = 1 2 k L 2 σ k B dt. Franco Flandoli, University of Pisa () Regularization by noise King s College / 33

23 What is diffi cult Although being a linear model, blow-up of the passive field B may occur for non-regular driving vector field u (without noise). Our aim is to show that, for the same class of non-regular driving vector field u, the addition of the transport noise leads to better solutions, prevents blow-up. Obviously this does not mean that the same should happen for a nonlinear model. Franco Flandoli, University of Pisa () Regularization by noise King s College / 33

24 Example of infinite stretching In cylindrical coordinates, an example is u = u θ e θ, u θ (r) = r α u θ = u θ (r) for some α (0, 1). The planar "velocity" field u is not Lipschitz, only C α. One can solve explicitly the equation for B and see that, for generic smooth bounded initial conditions B 0, B(t, r, θ, z) C t, t > 0. r 1 α The "magnetic" field B blows up around the z-axis. In the next pictures we shall see the planar Lagrangian dynamics. Franco Flandoli, University of Pisa () Regularization by noise King s College / 33

25 y y Planar Lagrangian dynamics Lagrangian trajectories for α = 1 (Lipschitz velocity field) and α = 0.3 (only C α velocity field) x x Franco Flandoli, University of Pisa () Regularization by noise King s College / 33

26 y Planar Lagrangian dynamics The picture shows the image at time t = 1 of the ideal line initially equal to the x axis (α = 0.3). This line is infinitely stretched near the origin x Franco Flandoli, University of Pisa () Regularization by noise King s College / 33

27 The stochastic case: absence of blow-up We have shown that, without noise, there exists u C α b (R3 ; R 3 ) and B 0 C (R 3 ; R 3 ) such that sup x 1 B (t, x) = + for all t > 0. Theorem (noise prevents infinite stretching) With the transport noise, for all u C ([0, T ]; C α b (R3 ; R 3 )) and B 0 C (R 3 ; R 3 ) one has B C ([0, T ] R 3 ; R 3 ), with probability one. This result has been proved by F. - Maurelli - Neklyudov, JMFM 2015 for the equation db + L u B dt + k=1 L σk B dw k t = 0 in the case of finitely many noise terms; it has been extended by F. - Olivera, on arxiv, to infinitely many terms and to the assumption for some p > 3. u L ( 0, T ; L p ( R 3 ; R 3)) ranco Flandoli, University of Pisa () Regularization by noise King s College / 33

28 y y Simulations x x Franco Flandoli, University of Pisa () Regularization by noise King s College / 33

29 y y Other samples x x Franco Flandoli, University of Pisa () Regularization by noise King s College / 33

30 y y Other samples x x Franco Flandoli, University of Pisa () Regularization by noise King s College / 33

31 Intuitive digression The emergence of singularity seems to require a certain degree of organization of the fluid structures and perhaps this organization is lost, broken, under the influence of randomness. With further degree of speculation, one could even think that a turbulent regime may contain the necessary degree of randomness to prevent blow-up; if so, singularities of the vorticity could more likely be associated to strong transient phases, instead of established turbulent ones. Franco Flandoli, University of Pisa () Regularization by noise King s College / 33

32 Idea of proof The stochastic characteristics in R 3 associated to the linear vector valued SPDE are dx t = u(t, X t )dt + σ k (X t ) dwt k, X 0 = x. k In F. - Gubinelli - Priola, IM 2010, it is proven: Theorem If u C ([0, T ]; C α b (R3 ; R 3 )) for some α (0, 1), then for every x R 3, there exists a unique strong solution X t and there exists a stochastic flow Φ t (x) of C 1,α diffeomorphisms, for every α < α. Using this result one can prove: Lemma If B 0 C (R 3 ; R 3 ) then B(t, Φ t (x)) = DΦ t (x)b 0 (x). ranco Flandoli, University of Pisa () Regularization by noise King s College / 33

33 Thank you for your attention Franco Flandoli, University of Pisa () Regularization by noise King s College / 33