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) LMS EPSRC Durham Symposium Stochastic Analysis Durham, July 1-2, 217 Zdzislaw Brzeźniak (York) Constrained NSEs July 1, 217 1 / 19

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 July 1, 217 2 / 19

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 July 1, 217 3 / 19

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 July 1, 217 4 / 19

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 July 1, 217 5 / 19

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 July 1, 217 6 / 19

Global existence for Constrained NSEs Since u 2 V = u 2 H + u 2 L 2 = u 2 H + 2E(u) and the 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. 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, A u L 2 = A u 2 L 2 + u 4 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) Zdzislaw Brzeźniak (York) Constrained NSEs July 1, 217 7 / 19

Deterministic constrained NSEs - Main Theorem 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) Thus we can summarise our results in the following theorem : Theorem 1 For every u V M there exists a unique global solution u of the constrained NSEs (13) such that u X T for all T >. Here X T = C([, T ]; V) L 2 (, T; D(A)). Zdzislaw Brzeźniak (York) Constrained NSEs July 1, 217 8 / 19

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 July 1, 217 9 / 19

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 M 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 July 1, 217 1 / 19

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 the stochastic constrained NSEs (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 n P n : H H n be the orthogonal projection operator given by P nu = u, e i e i. We consider the following projection of onto H n: { dun = [ ] P nau n + P nb(u n) u n 2 L 2un dt + m PnCjun dwj, t, u n() = Pnu, for n large enough P nu L 2 (2) We fix T >. Equation (2) is a stochastic ODE on a finite dimensional compact manifold M n = {u H n : u L 2 = 1}. Hence it has a unique M-valued solution (with continuous paths). Moreover, q 2 E T u n(t) q H dt <. Zdzislaw Brzeźniak (York) Constrained NSEs July 1, 217 11 / 19 i=1

A priori estimates These depend deeply on the property that B(u), Au H =, and a very specific assumption We assume c 1, c m are constant vector fields. Let K c = max j 1,,m c j R 2. u D(A). Lemma 4 [ ) Let p 1, 1 + 1 and ρ >. Then there exist positive constants C Kc 1(p, ρ), C 2(p, ρ) 2 and C 3(ρ) such that if u V ρ, then ( ) sup E n 1 sup E n 1 sup u n(r) 2p V r [,T ] T C 1(p, ρ), (21) u n(s) 2(p 1) V Au n(s) u n(s) 2 L 2un(s) 2 H ds C 2(p, ρ), (22) and T sup E u n(s) 2 D(A) ds C 2(1) + C 1(2)T =: C 3(ρ). (23) n 1 Zdzislaw Brzeźniak (York) Constrained NSEs July 1, 217 12 / 19

Aldous condition 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. 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 by applying Lemma 4 to equations (2). Corollary 6 The 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 July 1, 217 13 / 19

Skorokhod theorem By the application of the Prokhorov and the Jakubowski-Skorokhod Theorems (since Z T is not a Polish space, 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, 1 + 1 ) Kc 2 ( ) 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 July 1, 217 14 / 19

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. Since ũ n ũ in C([, T ]; H) and u n(t) M for every t [, T ], we infer that ũ(t) M, t [, T ]. (27) 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 July 1, 217 15 / 19

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 (28) The equation (28) can also be used on (ˆΩ, ˆF, ˆF, ˆP) to define a process M n, i.e. M n(t) = ũ n(t) P nũ n() + ũ n(s) 2 L 2ũn(s) ds 1 2 P naũ n(s) ds + m P nb(ũ n(s)) ds (P nc j) 2 ũ n(s) ds (29) Zdzislaw Brzeźniak (York) Constrained NSEs July 1, 217 16 / 19

Martingale representation theorem Using the earlier convergence results and a priori estimates (25), (26), we can prove that M n(t) M(t) := ũ(t) ũ() + ũ(s) 2 L 2ũ(s) ds 1 2 From equality (3) one can deduce that (i) M is F martingale. Aũ(s) ds + m B(ũ(s)) ds C 2 j ũ(s) ds. (3) (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 July 1, 217 17 / 19

Pathwise Uniqueness Theorem 7 Pathwise Uniqueness holds for the the stochastic constrained NSEs (18). Theorem 8 The stochastic constrained NSEs (18) have a unique strong solution for each u V M. Moreover, the paths of this solution belong to the space X T for all T >. In particular, the paths are V-valued continuous (strongly and not only weakly). Zdzislaw Brzeźniak (York) Constrained NSEs July 1, 217 18 / 19

References Z. Brzeźniak, G. Dhariwal and M. Mariani, 2D Constrained Navier-Stokes equations, arxiv:166.836v2 (216) (Submitted). Z. Brzeźniak and G. Dhariwal, Stochastic Constrained Navier-Stokes equations on T 2, 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 July 1, 217 19 / 19