flow problems in two and three dimensional bounded, periodic as well as .... Open Problems 1: The solvability of strong as well as martingale solutions are.
Stochastic Navier-Stokes Equations: Solvability, Control and Filtering S.S.Sritharan Department of Mathematics University of Wyoming Laramie, WY 82071, USA
ABSTRACT: In this paper we describe the mathematical problem of filtering and control for the stochastic Navier-Stokes equation. Main issues discussed include martingale or weak solutions and strong solutions, solvability of infinite dimensional Hamilton-Jacobi equations of first and second order associated with feedback synthesis, infinite dimensional variational and quasi-variational inequalities associated with stopping time and impulse controls, Fujisaki-Kallianpur-Kunita and Zakai equations for nonlinear filtering.
1. Stochastic Navier-Stokes Equation: solvability Let us begin with the abstract evolution form of the controlled stochastic NavierStokes equation [9] in the divergence free subspace H of square integrable vector fields which are parallel to the boundary: du(t) + (νAu(t) + B(u(t)))dt = U (t)dt + dW (t).
(1)
Here ν is the coefficient of kinematic viscosity, A is the Stokes operator and B(·) is the nonlinear inertia term with well known properties. U (t) is a distributed control with possible local support and W (t) is an H-valued Wiener process with covariance operator Q. Here both the cases of degenerate noise (where Q is of trace class) and non-degenerate noise (where for example Q = I) are of importance. Moreover, flow problems in two and three dimensional bounded, periodic as well as unbounded physical regions are of interest. In this paper we will denote k · k for H-norm, k · k1 for the norm of the space V = D(A1/2 ) and k · k−1 for the norm of the dual space V 0 = D(A−1/2 ). 1
Let us first consider the solvability of the stochastic Navier-Stokes equations with degenerate noise. Theorem 1: Strong Solutions [5] Let (Ω, Σ, Σt , m) be a complete filtered probability space and W (t) be an H-valued Wiener process with trace class covariance. Let the control function U (·) ∈ L2 (Ω; L2 (0; T ; V 0 )) be adapted to Σt and the initial data be u0 ∈ L2 (Ω; H). Then there exists a unique strong solution u(·) ∈ C([0, T ]; H) ∩ L2 (0, T ; V ), a.s. and adapted to Σt such that "
E
2
sup ku(t)k + ν t∈(0,T )
Z
#
T
kA
1/2
0
"
2
2
u(t)k dt ≤ E ku0 k +
Z 0
T
#
kU (t)k2−1 dt
+ TrQ.T.
This solution satisfies the equation (1) in the generalized sense and also can be conveniently described as u(·) ∈ L2 (Ω; C([0, T ]; H)) ∩ L2 (Ω; L2 (0, T ; V )). The proof given in [5] relies on a local monotonicity property associated with the inertia term and Minty-Browder type technique and does not utilize the usual compactness arguments used in classical Navier-Stokes theory. Because of this reason this theorem holds in two dimensional bounded as well as unbounded regions including exterior domains and R2 . We will now turn to the concept of weak solutions. In this case we will also consider the multiplicative noise: du(t) + (νAu(t) + B(u(t)))dt = U (t)dt + g(u)dW (t),
(2)
where the map u → g(u) from H → L(H; H) satisfies (i) kg(u)k ≤ C1 kuk + C2 , ∀u ∈ H, (ii) kg(u1 ) − g(u2 )k ≤ C3 ku1 − u2 k, ∀u1 , u2 ∈ H. For the path space we will take the Lusin space Ω := L2 (0, T ; H) ∩ D([0, T ]; V 0 ) ∩ L2 (0, T ; V )σ ∩ L∞ (0, T ; H)w∗ . Here D([0, T ]; V 0 ) is the V 0 -valued Skorohod space of Cadlag (right continuous, left limit) functions endowed with J-topology [7] and σ and w∗ denote weak and weak-star topologies respectively. We will take the cannonical filtration Σt = σ{u(s), s ≤ t}. The martingale problem is to find a Radon measure P on the Borel algebra B(Ω) such that, Z t
Mt := u(t) +
(νAu(s) + B(u(s)) − U (s)) dt
0
is an H-valued, (Ω, B(Ω), Σt , P )-martingale (i.e. a Σt -adapted process such that E[Mt |Σs ] = Ms ) with quadratic variation process >t :=
Z
t
g(u(s))Qg ∗ (u(s))ds.
0
2
The following theorem is a simplified version of what is proved in the paper [10] which includes measure-valued relaxed controls. Theorem 2: Martingale solutions For three dimensional Navier-Stokes equation in bounded domains, there exists a martingale solution P which is carried by the subset of paths satisfying the following bounds: "
E
P
2
sup ku(t)k + ν
Z
#
T
kA
1/2
0
t∈(0,T )
"
2
2
u(t)k dt ≤ C1 E ku0 k +
Z 0
T
#
kU (t)k2−1 dt
+C2 TrQ.T.
Moreover, the martingale solution is unique for the two dimensional case. We note finally that for the above strong and weak solutions it is possible to get the following a-priori estimate for the higher order moments: "
E
P
2l
sup ku(t)k + ν
Z
#
T
ku(s)k
2l−2
kA
1/2
2
u(t)k dt
0
t∈(0,T )
" 2l
≤ C1 E ku0 k +
Z
T
0
#
kU (t)k2l −1 dt
+ C2 (TrQ.T ) .
Other a-priori estimates and continuous dependence theorems as well as discussions on earlier literature can be found in [5],[10]. Open Problems 1: The solvability of strong as well as martingale solutions are open for the non-degenerate case of Q = I. 2. Feedback Control and Infinite Dimensional Hamilton-Jacobi Equations Let us consider the control problem: J(t, u; U ) := E
(Z t
T
kA
1/2
)
1 u(r)k + kU (r)k2 dr + ku(T )k2 → inf . 2
2
We get insight in to the nature of cost functional by noting the well known equivalence of the integrand kA1/2 uk2 and the enstrophy or the total vorticity k Curl uk2 . We will take the state equation as du(t) + (νAu(t) + B(u(t)))dt = KU (t)dt + dW (t),
(3)
where K ∈ L(H; V ) and the control U (·) : [0, T ] × Ω → U will be taken from the set of control strategies Ut . The control set U = BH (0, R) ⊂ H is the ball of radius R in H. Let us define the value function as V(t, v) := inf J(t, u; U (·)), U (·)∈Ut
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for the initial data u(t) = v. Formally the value function satisfies the infinite dimensional second-order Hamilton-Jacobi (-Bellman) equation: 1 Vt + Tr QD2 V −(νAv+B(v), DV)+kA1/2 vk2 +H(K ∗ DV) = 0, for (t, v) ∈ (0, T )×D(A), 2
V(T, v) = kvk2 , for v ∈ H. Here H(·) : H → R is given by H(Z) := inf
1 (U, Z) + kU k2 . 2
U ∈U
More explicitly we can write 1 2 − 2 kZk
H(Z) =
for kZk ≤ R
−RkZk + 12 R2
for kZk > R.
Moreover, the optimal feedback control is given formally by U˜ (t) = Υ (K ∗ Dv V(t, u(t))) where Υ(Z) := DZ H(Z) =
−Z
for kZk ≤ R
−Z R kZk
for kZk > R.
Let us now state a rigorous result on viscosity solutions to the above Hamilton-Jacobi equation from the paper[4] where more general cost functionals involving polynomial growth in the V -norm also treated. In [3] a semigroup treatment of this problem with non-degenerate noise is given. Definition: Test Functions A function ψ is a test function of the above Hamilton-Jacobi equation if ψ = φ + δ(t)(1 + kvk21 )m , where (i) φ ∈ C 1,2 ((0, T ) × H), and φ, φt , Dφ, D2 φ are uniformly continuous on [, T − ] × H for every > 0, and (ii) δ ∈ C 1 ((0, T )) is such that δ > 0 on (0, T ) and m ≥ 1. Definition: Viscosity Solution A function V : (0, T ) × V → R that is weakly sequentially upper-semicontinuous (respectively lower-semicontinuous) on (0, T ) × V is called a viscosity subsolution (respectively, supersolution) of the above HamiltonJacobi equation if for every test function ψ, whenever V − ψ has a global maximum (respectively, V + ψ has a global minimum) over (0, T ) × V at (t, v) then we have v ∈ D(A) and 1 ψt + Tr QD2 ψ − (νAv + B(v), Dψ) + kA1/2 vk2 + H(K ∗ Dψ) ≥ 0, 2
4
( respectively 1 −ψt − Tr QD2 ψ + (νAv + B(v), Dψ) + kA1/2 vk2 + H(K ∗ (−Dψ)) ≤ 0.) 2
A function is a viscosity solution if it is both a viscosity subsolution and a supersolution. For the two dimensional stochastic Navier-Stokes equation on a periodic domain (or compact manifold) with H and V degeneracies on the noise (i.e. TrQ < ∞ and Tr(A1/2 QA1/2 ) < ∞) we can establish the following result: Theorem 3: Continuity of the Value Function For each r > 0, there exists a modulus of continuity ωr such that |V(t1 , v) − V(t2 , z)| ≤ ωr (|t1 − t2 | + kv − zk), for t1 , t2 ∈ [0, T ] and kvk1 , kzk1 ≤ r, and |V(t, v)| ≤ C(1 + kvk21 ). Theorem 4: Existence and Uniqueness The value function V is the unique viscosity solution for the Hamilton-Jacobi equations. Open Problems 2: Existence and uniqueness of viscosity solutions for the cases of arbitrary two dimensional domains (bounded and unbounded) as well as the nondegenerate noise cases (e.q. Q = I) are open. 3. Optimal Stopping and Infinite Dimensional Variational Inequality Optimal stopping problem for the stochastic Navier-Stokes equations in two dimensional bounded domains has been studied in [6] and in [2] by different methods. In this section we present a slightly simplified version of the results in [2]. Consider the optimal stopping problem of characterizing the value function: V(t, v) := inf E τ
Z
τ
kA1/2 u(s)k2 ds + k(u(τ ))ku(τ )k2 ,
t
with state equation du(t) + (νAu(t) + B(u(t)))dt = dW (t). Formally, the value function solves the following variational inequality: 1 Vt − Tr QD2 V + (νAv + B(v), DV) ≤ kA1/2 vk2 , for t > 0, v ∈ D(A), 2
V(t, v) ≤ k(v)kvk2 , for t ≥ 0, v ∈ H, 5
(4)
V(0, v) = φ0 (v), for v ∈ H, and in the (continuation) set n
o
(t, v) ∈ R+ × H; V(t, v) < k(v)kvk2 ,
we have equality: 1 Vt − Tr QD2 V + (νAv + B(v), DV) = kA1/2 vk2 , for t > 0, v ∈ D(A). 2
This problem can be viewed as a nonlinear evolution problem with multi-valued nonlinearity: Wt − N W + NK (W) 3 kA1/2 · k2 , t ∈ [0, T ], W(0) = φ0 . Here N is the generator of the stochastic Navier-Stokes process (infinitesimal generator of the transition semigroup P (t)) and NK is the normal cone to the closed convex subset K ⊂ L2 (H, µ), n
o
K = φ ∈ L2 (H; µ); φ ≤ k(·)k · k2 on H , where µ is an invariant measure for P (t). In fact NK is defined as
NK (φ) = η ∈ L2 (H; µ);
Z
η(v)(ψ(v) − φ(v))µ(dv) ≤ 0, ∀ψ ∈ K , φ ∈ K.
H
let us use the solvability theorem (Theorem 1) for strong solutions to define the transition semigroup P (t) : Cb (H) → Cb (H) by (P (t)ψ)(v) = Eψ(u(t, v)), v ∈ H, ∀t ≥ 0, ψ ∈ Cb (H), where u(t, v) is the strong solution with initial data v. Existence of invariant measure µ and its uniqueness for large ν are shown in [1]: Z
(P (t)ψ)(v)µ(dv) =
H
Z H
ψ(v)µ(dv), ψ ∈ Cb (H).
Then P (t) has an extension to a C0 -contraction semigroup on L2 (H; µ). We denote by N : D(N ) ⊂ L2 (H, µ) → L2 (H, µ) the infinitesimal generator of P (t) and let N0 ⊂ N be defined by 1 (N0 ψ)(v) = Tr QD2 ψ(v) − (νAv + B(v), Dψ(v)), ∀ψ ∈ εA (H), 2
where εA (H) is the linear span of all functions of the form φ(·) = exp (i(h, ·)), h ∈ D(A). It is shown in[1] that if ν ≥ C(kQkL(H;H) + TrQ) 6
is sufficiently large and if Tr[Aδ Q] < ∞ for δ > 2/3 then N0 is dissipative in L2 (H, µ) and its closure N¯0 in L2 (H, µ) coincides with N . Moreover, from the definition of the invariant measure, taking ψ(v) = kvk2 we have Z
(N ψ)(v)µ(dv) = 0
H
which implies the integrability of enstrophy kcurlvk2 = kA2 vk2 with respect to the invariant measure µ: Z 2ν kA1/2 vk2 µ(dv) = TrQ < ∞. H
We will now state a slightly simplified version of the solvability theorem from [2] for the the variational inequality (or the nonlinear evolution problem formulated above). The proof is based on nonlinear semigroup theory for the m-accretive operator A = −N + NK in L2 (H, µ). Theorem 5: Suppose k(v) such that G(v) = k(v)kvk2 satisfies G ∈ C 2 (H) and (N0 G)(v) ≤ 0, ∀v ∈ D(A). Then, for each φ0 ∈ D(N )∩K there exists a unique function φ ∈ W 1,∞ ([0, T ]; L2 (H, µ)) such that N φ ∈ L∞ (0, T ; L2 (H, µ)) and d φ(t) − N φ(t) + η(t) − kA1/2 · k2 = 0, a.e. t ∈ (0, T ), dt η(t) ∈ NK (φ(t)), a.e. t ∈ (0, T ), φ(0) = φ0 . Moreover, φ : [0, T ] → L2 (H, µ) is differentiable from the right and d+ φ(t) − N φ(t) − kA1/2 · k2 + PNK (φ(t)) (kA1/2 · k2 + N φ(t)) = 0, ∀t ∈ [0, T ), dt where PNK (φ) is the projection on the cone NK (φ). Remarks on Impulse Control In [6] impulse control problem is treated for the two dimensional stochastic NavierStokes equation with degenerate noise in bounded domains. This problem is of the form: du(t) + (νAu(t) + B(u(t)))dt =
X
Ui δ(t − τi )dt + dW (t),
(5)
i≥1
where the control consists of the set of random stopping times τi and the control decisions Ui : U := {(τ1 , U1 ); (τ2 , U2 ); , · · ·} . 7
The goal would be to find the optimal control such that a cost functional of the following form is minimized: J(v; U ) := E
(Z
)
∞
F (u(t))dt +
0
X
L(Ui ) .
i
In this case we end up with quasi-variational inequalities of the following form for the value function V: N V ≤ F, V ≤ M (V), and N V = F in the set {V < M (V)} . Here the nonlinear operator M is defined as M (V)(v) = inf {L(U ) + V(v)} . U
In [6] a “hybrid control” generalization of this problem is formulated and solvability is proved by designing a convergent sequence of stopping time problems (and a sequence of variational inequalities)to approximate the quasi-variational inequality. The proof involves a generalization of the generator N using the concept of weak generator and the resolvant operator of the Feller semigroup associated with the stochastic Navier-Stokes process. This is a technical construction so we omit the details and refer the interested readers to this paper. 4. Nonlinear Filtering of Stochastic Navier-Stokes Equations Let us now consider a flow field in which both the viscosity coefficient and noise may be unknown so we propose the situation to be modelled by equation (4). Let us also assume that we have sensors at specific locations measuring the flow characteristics in real time: Z t h(u(s))ds + W2 (t). z(t) = 0
Here W2 is a Wiener process representing uncertainty in measurements and h is called the observation vector. Depending on the type of measurement h could be finite or infinite dimensional. Moreover, the domain of h(·) will be H if we are making velocity measurements and D(A1/2 ) if we measure the vorticity. Let us assume that we have the back measurements {z(s), 0 ≤ s ≤ t} . How does the least square best estimate of a function of the velocity f (u(t)) evolve in time?. It is well known that the best estimator is the conditional expectation of f (u(t)) given the back measurements (or the sigma algebra Σzt generated by {z(s), 0 ≤ s ≤ t} .). Let us denote µzt [f ] := E[f (u(t))|Σzt ].
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Let us describe the special case of uncorrelated W and W2 from the more general correlated case developed in [8]. Using martingale methods we derive the equation of evolution for µzt [f ] called the Fujisaki-Kallianpur-Kunita equation: dµzt [f ] = µzt [N0 f ]dt + (µzt [hf ] − µzt [h]µzt [f ]) (dz(t) − µzt [h]dt) , If we set:
for f ∈ εA (H).
1Z t z 2 := · dz(s) − |µ [h]| ds . 2 0 s 0 We then get (using Ito formula) a linear equation called Duncan-Mortensen-Zakai equation: dϑzt [f ] = ϑzt [N0 f ]dt + ϑzt [hf ] · dz(t), for f ∈ εA (H). ϑzt [f ]
µzt [f ]. exp
Z
t
µzs [h]
Existence and uniqueness of measure-valued solutions to the above two evolution equations has been proven in [8] for the case of two-dimensional periodic domains with H and V degeneracies on the noise (i.e. TrQ < ∞ and Tr(A1/2 QA1/2 ) < ∞): Theorem 6 Let M(H) and P(H) respectively denote the class of positive Borel measures and Borel probability measures on H. Then there exists a unique P(H)valued random probability measure µzt and a unique M(H)-valued random measure ϑzt , both processes being adapted to the filtration Σzt such that the FujisakiKallianpur-Kunita and the Zakai equations are respectively satisfied for the class of functions from εA (H). The proof is based on the uniqueness theorem for the backward Kolmogorov equation. Acknowledgement. This research has been supported by the Army Research Office, Probability and Statistics Program.
References [1] V. Barbu, G. Da Prato and A. Debussche, “The Transition Semigroup of Stochastic Navier-Stokes Equations in 2-D”, Atti. Acad. Naz. Lincei. (to appear). [2] V. Barbu and S. S. Sritharan, “Optimal Stopping-Time Problem for Stochastic Navier-Stokes Equations and Infinite-Dimensional Variational Inequalities”, preprint, 2004. [3] G. Da Prato and A. Debussche, “Dynamic Programming of the Stochastic Navier-Stokes Equation”, Math. Model. Numer. Anal. 34(2), pp.459-475, (2000). [4] F. Gozzi, S. S. Sritharan, A. Swiech, “Bellman Equations Associated to the Optimal Feedback Control of Stochastic Navier-Stokes Equations”, to appear in Communications on Pure and Applied Mathematics, 2004/5.
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[5] J. L. Menaldi and S. S. Sritharan, “Stochastic 2-D Navier-Stokes Equation”, Applied Mathematics and Optimization, 46, pp. 31-52, (2002). [6] J. L. Menaldi and S. S. Sritharan, “Impulse Control of Stochastic Navier-Stokes Equations”, Nonlinear Analysis, 52, pp. 357-381, (2003). [7] M. Metivier, Stochastic Partial Differential Equations in Infinite Dimensional Spaces, Scuola Normale Superiore, PISA, (1988). [8] S. S. Sritharan, “Nonlinear Filtering of Stochastic Navier-Stokes Equations”, in T. Funaki and W. A. Woycznski, Editors, Nonlinear Stochastic PDEs: Burgers Turbulence and Hydrodynamic Limit, Springer-Verlag, 1994. [9] S. S. Sritharan, Editor, Optimal Control of Viscous Flow, SIAM, Philadelphia, (1998). [10] S. S. Sritharan, “Deterministic and Stochastic Control of Navier-Stokes Equation with Linear, Monotone and Hyperviscosities”, Applied Mathematics and Optimization, 41, pp. 255-308, (2000).
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