Zdzislaw Brzeźniak. Department of Mathematics University of York. joint works with Mauro Mariani (Roma 1) and Gaurav Dhariwal (York)

Navier-Stokes equations with constrained L 2 energy of the solution Zdzislaw Brzeźniak Department of Mathematics University of York joint works with Mauro Mariani (Roma 1) and Gaurav Dhariwal (York) Stochastic Dynamical Systems and Ergodicity Loughborough, December 5-9, 216 Zdzislaw Brzeźniak (York) Constrained NSEs December 6, 216 1 / 18

Motivation Caglioti et.al [5] studied 2D NSEs in R 2 with constraints E(ω) = ψ(x) ω(x) dx = u(x) 2 dx = a, R 2 R 2 I(ω) = x 2 ω(x) dx = b, R 2 where ω = curl u, ψ = ( ) 1 ω. They proved that for a certain stationary solution ω MF of the Euler equation (in the vorticity form) with constraints a, b, for every initial data ω close enough ω MF with the same constraints a, b; ω(t) ω MF, as t, where ω(t) is the solution of the NSEs (in the vorticity form) with inital data ω and the same constraints. Zdzislaw Brzeźniak (York) Constrained NSEs December 6, 216 2 / 18

Motivation Rybka [7] and Caffarelli & Lin [4] studied heat equation with constraint u L 2 = 1. (1) The heat equation is given by u = A u, (2) t where A u = u is a self adjoint operator on H. We define a Hilbert manifold M = {u H : u H = 1}. (3) Note that A u / T um for u M but Π u( A u) T um for every u M, where Π u : H x x x, u H u T um = {y H : u, y H = } (4) is the orthogonal projection. Since Π u( A u) = A u + A 1/2 u 2 Hu, we get u t = A u + A1/2 2 Hu. (5) Zdzislaw Brzeźniak (York) Constrained NSEs December 6, 216 3 / 18

Constrained Heat equation A special case of heat equation with Dirichlet boundary condition u t = u + u 2 L 2u (6) u() = u Note that the heat equation (2) can be seen as an L 2 gradient flow of energy E(u) = 1 u(x) 2 dx, (7) 2 as formally L 2E(u) = u. Similarly, the constrained heat equation (6) can be seen as the gradient flow of E restricted to the manifold M with L 2 metric on the tangent bundle. In fact one can prove that the solution of (6) with u H 1 (O) M satisfies E(u(t)) + from which one can deduce the global existence. O u(s) + u 2 L 2u(s) 2 L2 ds = E(u()) (8) An essential step in proving the global existence is to establish the invariance of M. Zdzislaw Brzeźniak (York) Constrained NSEs December 6, 216 4 / 18

Navier-Stokes equations We consider NSEs u + A u + B(u, u) = t (9) u() = u which is an abstract form of u ν u + u u + p =, t div u =, u(, ) = u ( ). Here B(u, u) = Π(u u) (1) where Π : L 2 (O) H is the orthogonal projection. H = {u L 2 (O): div u = and u O n = (Dirichlet b.c.) or u(x) dx = (Torus)} (11) O Zdzislaw Brzeźniak (York) Constrained NSEs December 6, 216 5 / 18

Constrained Navier-Stokes equations We put M = {u H : u L 2 = 1}. The projected version of (9) can be found in a similar way as before. Note So we get Π u(b(u, u)) = B(u, u) B(u, u), u H u = B(u, u). (12) }{{} = u t + A u + B(u, u) = u 2 L 2u u() = u V M, where V = H 1,2 M or H 1,2 M. We can show existence of a local maximal solution u(t), t < τ which lies on M. However to prove the global existence one needs to assume that we deal with periodic boundary conditions (or torus), because then A u, B(u, u) H =. (13) Zdzislaw Brzeźniak (York) Constrained NSEs December 6, 216 6 / 18

Global existence for Constrained NSEs Since L 2 norm of u(t) doesn t explode, in order to show that u(t) 2 V doesn t explode, it suffices to show that u(t) L 2 neither does (as u 2 V = u 2 H + u 2 L 2 = u 2 H + 2E(u)). Formally, we have 1 d 2 dt u(t) 2 L 2 = u, A u L 2 = A u B(u, u) + u 2 L 2u, u L 2 = A u 2 L 2 + u 2 L2. (14) But recall Thus ME(u) = Π u( ue(u)) = Π u(a u) = Au u 2 L2u. (15) ME(u) 2 L 2 = A u 2 + u 4 L 2 u 2 L }{{ 2 2 u 2 L } 2 u, A u L }{{ 2 } = 1 = u 2 L 2 = A u 2 L 2 u 4 L2. (16) Hence A u 2 L 2 u 4 L 2 and 1 2 u(t) 2 L 2 + ME(u(s)) 2 L 2 ds = 1 2 u 2 L2, t [, T ). (17) Zdzislaw Brzeźniak (York) Constrained NSEs December 6, 216 7 / 18

Deterministic constrained NSEs - Main Theorem Thus we can summarise our results in the following theorem : Theorem 1 Let X T = C([, T ]; V) L 2 (, T; D(A)). Then for every u V M there exists a unique global solution u X T of (13). Zdzislaw Brzeźniak (York) Constrained NSEs December 6, 216 8 / 18

Stochastic Constrained NSEs We assume that W = (W 1,, W m) is R m valued Wiener process, c 1, c m and Ĉ 1,, Ĉm are respectively vector fields and assosciated linear operators given by Ĉ ju = c j(x) u, : div c j =, j {1,, m}. Since C ju = ΠĈju, j {1,, m}, is skew symmetric in H, these operators don t produce any correction term when projected on T um. Thus the stochastic NSE m m du + [A u + B(u, u)] dt = C ju dw j = C ju dw j + 1 m Cj 2 u dt 2 }{{} Stratonovich = Itô + correction under the constraint is given by du + [ A u + B(u, u) u 2 L 2 u] dt = 1 m m Cj 2 u dt + C ju dw j. (18) 2 Zdzislaw Brzeźniak (York) Constrained NSEs December 6, 216 9 / 18

Martingale solution Definition 2 We say that there exists a martingale solution of (18) iff there exist a stochastic basis (ˆΩ, ˆF, ˆF, ˆP) with filtration ˆF { } = ˆFt, an R m -valued ˆF-Wiener process Ŵ, and an ˆF-progressively measurable process u : [, T ] ˆΩ V with ˆP-a.e. paths u(, ω) C([, T ]; V w) L 2 (, T; D(A)), such that for all t [, T ] and all v D(A): u(t), v + + Au(s), v ds + u(s) 2 L 2 u(s), v ds + 1 2 t B(u(s)), v ds = u, v m C 2 j u(s), v ds + m C ju(s), v dw j, (19) the identity hold ˆP-a.s. Zdzislaw Brzeźniak (York) Constrained NSEs December 6, 216 1 / 18

Existence of a martingale solution Theorem 3 (Assume that our domain is the 2-d torus) Then for every u V M, there exists a martingale solution to (18). Sketch of the proof : Galerkin approximation : Let {e j} be ONB of H and eigenvectors of A. H n := lin{e 1,, e n} is the finite dimensional Hilbert space P n : H H n be the orthogonal projection operator given by P nu = n u, e i e i. { dun = [ ] P nau n + P nb(u n) u n 2 L 2un dt + m PnCjun dwj, t [, T ], u n() = Pnu P nu L 2 for n large enough (2) We fix T >. (2) is a Stochastic ODE on a manifold M n = {u H n : u L 2 = 1}. It has a.s. continuous paths and q 2 In particular u n(t) M, t [, T ]. E T u n(t) q H dt <. Zdzislaw Brzeźniak (York) Constrained NSEs December 6, 216 11 / 18 i=1

A priori estimates These depend deeply on the property that B(u), Au H =, u D(A). Lemma 4 For every p [1, ) there exists positive constants C 1(p), C 2(p) and C 3 such that ( ) sup E n 1 sup u n(r) 2p V r [,T ] C 1(p), (21) and Moreover T sup E n 1 u n(s) 2(p 1) V Au n(s) u n(s) 2 L 2un(s) 2 H ds C 2(p). (22) T sup E u n(s) 2 D(A) ds C 2(1) + C 1(2)T =: C 3. (23) n 1 We put Z T = C([, T ]; H) L 2 w(, T; D(A)) L 2 (, T; V) C([, T]; V w), and T T the corresponding topology. Zdzislaw Brzeźniak (York) Constrained NSEs December 6, 216 12 / 18

Aldous condition In order to prove that the laws of u n are tight on Z T. Apart from a priori estimates we also need one additional property to be satisfied : Lemma 5 (Aldous condition in H) ε >, η > δ > : for every stopping time τ n : Ω [, T ] sup n N sup P ( u n(τ n + θ) u n(τ n) H η) < ε. (24) θ δ Lemma 5 can be proved using Lemma 4 and (2). Corollary 6 Laws of (u n) are tight on Z T, i.e. ε > K ε Z T compact, such that P (u n K ε) 1 ε, n N. Zdzislaw Brzeźniak (York) Constrained NSEs December 6, 216 13 / 18

Skorokhod theorem By the application of the Prokhorov and Jakubowski-Skorokhod theorem (since Z T is not Polish we need Jakubowski) we deduce that there exists a subsequence, a probability space (ˆΩ, ˆF, ˆP), Z T valued random variables ũ n such that Law(ũ n) = Law(u n), and there exists ũ: ˆΩ Z T random variable such that ũ n ũ in, ˆP a.s. Then, using Kuratowski Theorem, we can deduce that the sequence ũ n satisfies the same a priori estimates as u n. In particular p [1, ) ( ) sup E n 1 sup ũ n(r) 2p V r [,T ] sup E n 1 T C 1(p), (25) ũ n(s) 2 D(A) ds C 3. (26) Zdzislaw Brzeźniak (York) Constrained NSEs December 6, 216 14 / 18

Convergence The choice of Z T allows to deduce that ψ H(orV) and s, t [, T ]: (a) lim n ũ n(t), P nψ = ũ(t), ψ, P-a.s., (b) lim n Aũn(σ), Pnψ dσ = Aũ(σ), ψ dσ, P-a.s., s s (c) lim n B(ũn(σ), ũn(σ)), Pnψ dσ = B(ũ(σ), ũ(σ)), ψ dσ, P-a.s., s s (d) lim n s ũn(σ) 2 L 2 ũn(σ), Pnψ dσ = s ũ(σ) 2 L2 ũ(σ), ψ dσ, P-a.s. (e) lim n s C2 j ũ n(σ), P nψ dσ = s C2 j ũ(σ), ψ dσ, P-a.s. We are close to conclude the proof of Theorem 3. We are just left to deal with the Itô integral. Zdzislaw Brzeźniak (York) Constrained NSEs December 6, 216 15 / 18

Itô integral Define m M n(t) = M n is a martingale on (Ω, F, P). Moreover P nc ju n(s) dw j(s). M n(t) = u n(t) P nu n() + u n(s) 2 L 2un(s) ds 1 2 P nau n(s) ds + m P nb(u n(s)) ds (P nc j) 2 u n(s) ds (27) The equation (27) can also be used on (ˆΩ, ˆF, ˆF, ˆP) to define M n(t). Using earlier results on convergence and (25), (26), we can prove that M n(t) M(t) = ũ(t) ũ() + 1 2 m C 2 j ũ(s) ds. Aũ(s) ds + B(ũ(s)) ds ũ(s) 2 L2ũ(s) ds (28) Zdzislaw Brzeźniak (York) Constrained NSEs December 6, 216 16 / 18

Martingale representation theorem From (28) one can show that (i) M is F martingale. (ii) Cov( M n) Cov( M) = m Cjũ(s) (Cjũ(s)) ds. This allows to use the martingale representation theorem to deduce that there exists a bigger probability space ( Ω, F, F, P) and a Wiener process W on the same probability space such that Hence we proved Theorem 3. M(t) = m C jū(s) d W j(s). Zdzislaw Brzeźniak (York) Constrained NSEs December 6, 216 17 / 18

References Z. Brzeźniak, G. Dhariwal and M. Mariani, 2D Constrained Navier-Stokes equations, arxiv:166.836v2 (216) (Submitted). Z. Brzeźniak and G. Dhariwal, 2D Constrained Stochastic Navier-Stokes equations, arxiv:171.1385 (217) Z. Brzeźniak and E. Motyl, Existence of a martingale solution of the stochastic Navier-Stokes equations in unbounded 2D and 3D-domains, J. Differential Equations, 254(4), 1627-1685 (213). L. Caffarelli and F. Lin, Nonlocal heat flows preserving the L 2 energy, Discrete and Continuous Dynamical Systems, 23(1&2), 49-64 (29). E. Caglioti, M. Pulvirenti and F. Rousset, On a constrained 2D Navier-Stokes Equation, Communications in Mathematical Physics, 29, 651-677 (29). A. Jakubowski, The almost sure Skorokhod representation for subsequences in nonmetric spaces, Teor. Veroyatn. Primen. 42(1), 29-216 (1997); translation in Theory Probab. Appl. 42(1), 167-174 (1998). P. Rybka, Convergence of a heat flow on a Hilbert manifold, Proceedings of the Royal Society of Edinburgh, 136A, 851-862 (26). Zdzislaw Brzeźniak (York) Constrained NSEs December 6, 216 18 / 18