Stochastic Shear Thickening Fluids: Strong Convergence of the Galerkin Approximation and the Energy Equality 1

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1 Stochastic Shear Thickening Fluids: Strong Convergence of the Galerkin Approximation and the Energy Equality Nobuo Yoshida Contents The stochastic power law fluids. Terminology from hydrodynamics Function spaces The noise The SPDE The Galerkin approximation Proof of Theorem.4.(outline) The stochastic shear thickening fluids 4. Strong convergence of the Galerkin approximation The energy equality Proof of Theorem..(outline) The stochastic power law fluids We consider viscous, incompressible fluid whose motion is subject to a random perturbation.. Terminology from hydrodynamics T d = (R/Z) d = [, ] d (container of the fluid) v : T d R d (velocity field of the fluid) Π : T d R (pressure field of the fluid) e(v), τ(v) : T d R d R d, ( ) i v j + j v i e(v) = Then, where div τ = (rate of strain), τ(v) = ν( + e(v) ) p e(v) (extra stress), ν >, p >. }{{} viscosity the force exerted on the fluid per volume = Π + div τ(v) : T d R d, (.) ( d ) d j= jτ ij i=. The fluid is said shear thinning if p <, Newtonian if p =, shear thickening if p >. Notes for 9 minutes talk on February 4,, at Tohoku University. Division of Mathematics, Graduate School of Science, Kyoto University, Kyoto 66-85, Japan. nobuo@math.kyoto-u.ac.jp URL: nobuo/ (.)

2 Remark: Note that:. Function spaces p = τ(v) = νe(v) (Stokes law) div τ(v) = ν v. (.3) V = {v : T d R d ; trigonometric polynomial, div v = } (test functions). v p p,α = ( ) α/ v p, v V, p [, ), α R. T d V p,α = p,α -completion of V (spaces of the solutions). Note that each v V p,α is a vector fieled on T d such that div v = in the distribution sense..3 The noise Γ : V, V,, trace class, self-adjoint,, Γ = Γ. (.4) W = (W t ) t : BM(V,, Γ) (Brownian motion on V, with the covariance operator Γ) : a process such that W and: ( E [exp (i φ, W t W s ) (W u ) u s ] = exp t s ) φ, Γφ, a.s. (.5) for each φ V, and s < t..4 The SPDE Given an initial velocity X = ξ V,, the (random) time evolution of the velocity field X = (X t ) t and the pressure field Π = (Π t ) t is described by the following SPDE: for t >, X t V, V p,, (.6) t X t + (X t )X t = Π t + div τ(x t ) + t W t. (.7) Here, and in what follows, v = d j= v j j. Remark: By (.3), the SPDE (.6) (.7) for p = is the stochastic Navier-Stokes equation. As in the case of (stochastic) Navier-Stokes equation, we will reformulate the problem (.6) (.7) into the one which does not involve the pressure. Let: P : L (T d R d ) V, : orthogonal projection, b(v) = (v )v + div τ(v), v V, V p,. More precisely, b(v) is defined as the following linear functional on V via a formal integration by parts: φ, b(v) = v, (v )φ e(φ), τ(v), φ V. (.8) With these new notation, we formally have that: (.7) t X t = Π t + b(x t ) + t W t, = t X t = Pb(X t ) + t W t (since X t, W t V,, P ) X t = X + Pb(X s )ds + W t. (.9)

3 Roughly speaking, a weak solution to (SPLF) p (stochastic power law fluid) is defined as a pair (X, W ) of processes which satisfy (.6) and (.9). More precisely: Definition.4. Let (X, W ) be a pair of processes such that W is a BM(V,, Γ). (X, W ) is called a weak solution to (SPLF) p if: a) (.6) is satiefied in the sense that t X t belongs to: L p,loc (R + V p, ) L,loc (R + V, ) C(R + V p, β) for β >, (.) where p = p p. b) (.9) is satisfied as a linear functional on V. Here is the basic result for the existence and the uniqueness for (SPLF) p. We limit ourselves to d =, 3, 4 for simplicity. Theorem.4. [TY] In addition to (.4), suppose that: Then, Γ is of trace class; (.) ξ : (Ω, F, P ) V,, E[ ξ,] < ; (.) d =, 3, 4 and p 3d d +. (.3) a) weak solution (X, W ) to (SPLF) p on a probability space ( Ω, F, P ) with X law = ξ. b) p + d pathwise uniqueness, i.e., for any two weak solutions (X, W ), ( X, W ), X = X, W W = X X. Remarks: ) Related results: [FG95, Fl8](p = ), [MNRR96](non-random PDE). ) The solution (X, W ) in Theorem.4. is obtained on a different probability space ( Ω, F, P ) from (Ω, F, P ), where the initial data ξ is defined. Even if we realize the initial data ξ and the solution (X, W ) on a common probability space (e.g., (Ω, F, P ) ( Ω, F, P )), Theorem.4. does not guarantee narurally expected properties such as: X = ξ, X t σ(ξ, (W s ) s t ), t. (.4) When the pathwise uniqueness is valid, one may use the Yamada-Watanabe theorem to obtain the solution with the property (.4), where σ(ξ, W ) is replaced by the completion with respect to the law of ξ times the Wiener measure. Another method to obtain such solution is via the strong convergence of the Galerkin approximation (Theorem.. below). 3

4 .5 The Galerkin approximation We now discuss a finite dimensional approximation to (SPLF) p. V n = {v V, degree n}, so that dim V n <. P n : L (T d R d ) V n : orthogonal projection. Theorem.5. [TY] Suppose (.4) and that we are given: ξ : (Ω, F, P ) V,, with E[ ξ ] <, W : BM(V,, Γ) on (Ω, F, P ). Then, n, solution X n to the following SDE with values in V n, defined on (Ω, F, P ): (cf. (.9)). Moreover, If p d, in addition, then d+ sup E n X n t = P n ξ + [ P n b(x n s )ds + P n W t. (.5) X n t σ(p n ξ, (P n W s ) s t ), t. (.6) sup Xt n + t T.6 Proof of Theorem.4. (outline) T ] Xt n p p,dt C T <, (.7) a priori bounds for X n (e.g., (.7)) + compact embedding theorems = {X n } n is tight n(k) s.t. X n(k) X in law. { = n (k) s.t. Xn (k) X a.s. on ( Ω, F, P ), where X n (k) law = X n (k), X law = X (Skhrohod s representation theorem). The stochastic shear thickening fluids. Strong convergence of the Galerkin approximation For p [ + d, d, d ), the solution to (SPLF)p is well behaved and is well approximated by the Galerkin approximation: Theorem.. [Y] In addition to (.4), suppose (.) (.) and d =, 3, 4, and + d p < Then, weak solution (X, W ) to (SPLF) p on (Ω, F, P ) such that: Moreover, sup t T T d d. (.) X = ξ, X t σ(ξ, (W s ) s t ), t. (.) X t X n t,α X t X n t,+αdt n in probability for α <, (.3) n in probability for α >. (.4) 4

5 Remark: Related results: [DD, Kuk6](d = p = ). Here, we adopt rather different approach from the ones in these references. The good thing is that our method is more direct.. The energy equality The strong convergence of the Galerkin approximation proved in Theorem.. has the following application: Theorem.. [Y] Suppose the same assumptions as in Theorem... Then, the pathwise energy equality holds in the sense that martingale M such that: X t = X e(x s ), τ(x s ) ds + tr(γ)t + M t, t. (.5) }{{}}{{} () (3) } {{ } () (()=the energy dissipated by the friction, ()=the energy injected by the external force, (3)=the fluctuation of the energy). In particular, the mean energy equality holds: E [ ] X t = E [ [ ] t ] X E e(x s ), τ(x s ) ds + tr(γ)t, t. (.6) Proof: ) (.5) is valid if X t is replaced by X n t (Itô s formula). ) X n t X t strongly enough by Theorem... Remarks: )Although the mean energy equality is plausible from the physics viewpoint, its validity is a non-trivial mathematical issue and remains open in general (e.g., 3D Navier-Stokes equation). )The mean energy equality plays an important role in studying Kolmogorov type scaling law for turbulent fluids, see e.g. [FGHR8]..3 Proof of Theorem.. (outline) The core of the proof is that: sup Xt m t T X n t m,n in probability. (.7) We will prove this by a series of elementary bounds (mainly, Gronwall s inequality), instead of functional analytic method based on compact embedding as in [Kuk6]. Let: By Itô s formula, where A m,n t = A m,n t + M m,n t + Z m,n t = X m t X n t. t = (P m P n )ξ + tr(p m Γ P n Γ)t + M m,n t = (P m P n )Z m,n s, dw s. 5 Z m,n s, b(x m s ) b(x n s ) ds, (.8) (P m P n )Z m,n s, b(x n s ) ds

6 On the other hand, we have the following bound (a similar bound is used in Theorem.4.b)): Z m,n s, b(x m s ) b(x n s ) C X m s By (.8) (.9), we observe that for t [, T ]: t sup (A m,n t t T Thus, by Gronwall s lemma, Since sup t T t + M m,n t ) + C p p d p X m s s. (.9) p p d p s ds (.) sup (A m,n t + M m,n t ) exp t T C X m p p d s p ds. (.) }{{}}{{} =:S m,n =:I m T T p p( p + d ), we see from (.7) that: p d Therefore, the convergence (.7) follows if: {I m T } m are tight, and so are {exp(ci m T )} m. (.) S m,n T T m,n in probability. (.3) Unfortunately, the proof of (.3) requires series of rather involved statements and bounds (for about five pages). However, the good news is that no big tool is used there. References [DD] Da Prato, Giuseppe; Debussche, Arnaud: Two-dimensional Navier-Stokes equations driven by a spacetime white noise. J. Funct. Anal. 96 (), no., 8. [Fl8] Flandoli, Franco : An introduction to 3D stochastic fluid dynamics. SPDE in hydrodynamic: recent progress and prospects, 5 5, Lecture Notes in Math., 94, Springer, Berlin, 8. [FG95] Flandoli, Franco ; Gatarek, Dariusz: Martingale and stationary solutions for stochastic Navier-Stokes equations. Probab. Theory Related Fields (995), no. 3, [FGHR8] Flandoli, F.; Gubinelli, M.; Hairer, M.; Romito, M.: turbulent fluids. Comm. Math. Phys. 78 (8), no., 9. Rigorous remarks about scaling laws in [Kuk6] Kuksin, Sergei B.: Randomly forced nonlinear PDEs and statistical hydrodynamics in space dimensions. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zurich, 6. x+93 pp. ISBN: [MNRR96] Málek, J.; Ne cas, J.; Rokyta, M.; R u zi cka, M.: Weak and measure-valued solutions to evolutionary PDEs. Applied Mathematics and Mathematical Computation, 3. Chapman & Hall, London, 996. xii+37 pp. ISBN: X [Te79] Temam, Roger: Navier-Stokes Equations. North-Holland Publishing Company (979). [TY] Terasawa, Yutaka; Yoshida, Nobuo: Stochastic Power Law Fluids: the Existence and the Uniqueness of the Weak Solution, preprint. [Y] Yoshida, Nobuo: Stochastic Shear Thickening Fluids: Strong Convergence of the Galerkin Approximation and the Energy Equality, in preparation. 6