Stochastic Analysis of Backward Tidal Dynamics Equation Hong Yin

Communications on Stochastic Analysis

Volume 5 | Number 4 Article 9

12-1-2011 Stochastic analysis of backward tidal dynamics equation Hong Yin

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Recommended Citation Yin, Hong (2011) "Stochastic analysis of backward tidal dynamics equation," Communications on Stochastic Analysis: Vol. 5 : No. 4 , Article 9. DOI: 10.31390/cosa.5.4.09 Available at: https://digitalcommons.lsu.edu/cosa/vol5/iss4/9 Communications on Stochastic Analysis Serials Publications Vol. 5, No. 4 (2011) 745-768 www.serialspublications.com

STOCHASTIC ANALYSIS OF BACKWARD TIDAL DYNAMICS EQUATION

HONG YIN

Abstract. The backward stochastic tidal dynamics equations, a system of coupled backward stochastic diﬀerential equations, in bounded domains are studied in this paper. Un- der suitable projections and truncations, a priori estimates are obtained, which enable us to establish the uniformly boundedness of an adapted solution to the system. Such regularity does not usually hold for stochastic diﬀerential equations. The well-posedness of the projected system is given by means of the contraction property of the elevation component. The existence of solutions are proved by utilizing the Galerkin approximation scheme and the monotonicity properties for bounded terminal conditions. The uniqueness and continuity of solutions with respect to terminal conditions are also provided.

1. Introduction Tide, the alternate rising and falling of the sea levels, is the result of the combination of the rotation of the Earth, and the gravitational attraction exerted at different parts of the Earth by the Moon and the Sun. The study of ocean tides trace back to the early seventeenth century. The first attempt of a theoretical explanation of ocean tides was given by Galileo Galilei[6] in 1632. Although not very successful, it inspired many successors to further the studies in this field, including Johannes Kepler and Isaac Newton[25]. The latter established a scientific formulation in 1687, which pointed out the role of the lunar and solar gravitational effect on ocean tides. Later Maclaurin[20] used Newton’s theory of fluxion and took into account Earth’s rotational effects on motion, Euler discovered that the horizontal component of the tidal force, as opposed to the vertical component, is the main driving force of ocean tides that causes the wavelike progression of high tide, and Jean le Rond d’Alembert observed tidal equations for the atmosphere which did not include rotation. A major break through of the mathematical formulation of ocean tides is accredited to Laplace[14], who introduced a system of three linear partial differential equations for the horizontal components of ocean velocity, and the vertical displacement of the ocean surface in 1775. His work remains the basis of tidal computation to this day, and was followed up by Thomson and Tait[33], and Poincaré[28], among others. The former applied systematic harmonic analysis to tidal analysis and rewrote Laplace’s equations in terms of vorticity. In the first half of the nineteenth century, researchers further expanded the studies of ocean tides. Encouraging developments include Arctic tides, and geophysical tides such as tidal motion in the atmosphere and in the Earth’s molten core. In the second

Received 2011-7-26; Communicated by P. Sundar. 2000 Mathematics Subject Classification. Primary 60H15, 35Q30; Secondary 76D05, 35R60. Key words and phrases. Backward stochastic tidal dynamics equation, backward stochastic differential equations, coupled backward stochastic differential equations, the Itôformula, contraction mapping, method of monotonicity, infinite dimensional analysis, Galerkin approximation. 745 746 HONG YIN half of the nineteenth century, quantum advances in computers, satellite technology and numerical methods of solving PDEs made it possible to solve Laplace tidal equations with realistic boundary conditions and depth functions. In this article, we consider the tidal dynamics equations constructed by Marchuk and Kagan[21, 22]. The existence and uniqueness of the tide equations in forward case have been shown by Ipatova[9], Marchuk and Kagan[21], and Manna, Menaldi and Sritharan [23]. The backward version of stochastic tidal dynamics equations is, to our best knowl- edge, new. It appears as an inverse problem wherein the velocity profile and elevation component at a time T are observed and given, and the noise coefficient has to be ascer- tained from the given terminal data. Such a motivation arises naturally when one under- stands the importance of inverse problems in partial differential equations (see J. L. Lions [15, 16]). Since the problem of specifying the function of boundary condition on the liq- uid boundary, and the problem of specifying the tide-generating forces must be solved simultaneously with the tide theory equation system, Agoshkov[2], among other authors, has considered tidal dynamics models as inverse problems. Some studies of backward stochastic analysis on fluid dynamics has been put forth in our previous work [31]. Lin- ear backward stochastic differential equations were introduced by Bismut in 1973 ([3]), and the systematic study of general backward stochastic differential equations (BSDEs for short) were put forward first by Pardoux and Peng[27], Ma, Protter, Yong, Zhou, and several other authors in a finite-dimensional setting. Ma and Yong[19] have studied linear degenerate backward stochastic differential equations motivated by stochastic control theory. Later, Hu, Ma and Yong [8] considered the semi-linear equations as well. Backward stochastic partial differential equations were shown to arise naturally in stochastic ver- sions of the Black-Scholes formula by Ma, Protter and Yong [17, 18]. A nice introduction to backward stochastic differential equations is presented in the book by Yong and Zhou [34], with various applications. The usual method of proving existence and uniqueness of solutions by fixed point arguments do not apply to the stochastic system on hand since the drift coefficient in the backward stochastic tidal flow is nonlinear, non-Lipschitz and unbounded. However, the 4 1 drift coefficient is monotone on bounded L (G) balls in H0(G), which was first observed by Manna, Menaldi and Sritharan [23]. One may also refer to Menaldi and Sritharan [24] for more information. The Galerkin approximation scheme is employed in the proof of existence and uniqueness of solutions to the system. To this end, a priori estimates of finite-dimensional projected systems are studied, and uniformly boundedness of adapted solutions are established. Such regularity does not usually hold for stochastic differential equations. The well-posedness of the projected system is also given by means of the contraction property of the elevation component. In order to establish the monotonicity property of the drift term, a truncation of R(x), the depth of the calm sea, is introduced. Then the generalized Minty-Browder technique is used in this paper to prove the existence of solutions to the tidal dynamics system. The proof of the uniqueness and continuity of solutions are wrought by establishing the closeness of solutions of the system via monotonicity arguments. The structure of the paper is as follows. The functional setup of the paper is introduced and several frequently used inequalities are listed in section 2. Some a priori estimates for the solutions of the projected system are given underdifferent assumptions on the terminal conditions and external force in section 3. Section 4 is devoted to well-posedness of the BACKWARD STOCHASTIC TIDAL DYNAMICS EQUATION 747 projected system. The existence of solutions of the tidal dynamics equations under suitable assumptions is shown by Minty-Browder monotonicity argument in section 5. The uniqueness and continuity of the solution under the assumption that terminal condition is uniformly bounded in H1 sense are given in section 6.

2. Formulation of the Problem Let us consider the time interval [0, T], and let G, the horizontal ocean basin where tides are induced, be a bounded domain in R2 with smooth boundary conditions. The boundary contour ∂G is composed of two disconnected parts, a solid part of Γ1 coinciding with the edge of the continental and island shelves, and an open boundary Γ2. Let us assume that sea water is incompressible and the vertical velocities are small compared with the horizontal velocities. Thus we are able to exclude acoustic waves. Also long waves, including tidal waves, are stood out from the family of gravitational oscillations. Furthermore, to reduce computational difficulties, we assume that the Earth is absolutely rigid, and the gravitational field of the Earth is not affected by movements of ocean tides. Also the effect of the atmospherictides on the ocean tides and the effect of curvatureof the surface of the Earth on horizontal turbulent friction are ignored. Under these commonly used assumptions, we are able to adopt the following tide dynamics model: ∂w + lk w = g ζ r w w + κ w + g; ∂t × − ∇ − R | | h△ ∂ζ + (Rw) = 0;  ∂t ∇ (2.1) w = w0 on [0, T] ∂G;   0 × 0 w = 0 on Γ1, and w dΓ2 = 0, Γ2  where w, the horizontal transport vector, is the averaged integral of the velocity vector  over the vertical axis, l = 2ρ cos θ is the Coriolis parameter, where ρ is the angular velocity of the Earth rotation and θ is the colatitude, k is an unit vector oriented vertically upward, g is the free fall acceleration, r is the bottom friction factor, κh is the horizontal turbulent viscosity coefficient, g is the external force vector, and ζ is the displacement of the free surface with respect to the ocean floor. The function w0 is an known function 0 on the boundary. The restriction w Γ1 = 0 is the no-slip condition on the shoreline, and 0 | w dΓ2 = 0 follows from the mass conservation law. Here R is the vertical scale of mo- Γ2 tion, i.e., the depth of the calm sea. Let us assume that R is a continuously differentiable function of x, so that infx G R(x) C0 and supx G R(x)+ R(x) C1 for some positive ∈ { } ≥ ∈ { |∇ |} ≤ constants C0 and C1. In order to simplify the non-homogeneous boundary value problem to a homogeneous Dirichlet boundary value problem, we set u(t, x) = w(t, x) w0(t, x), − and t ξ(t, x) = ζ(t, x) + R(x)w0(s, x) ds. ∇ 0 Let us denote by A the matrix κ 2ρ cos θ A = − h△ − , 2ρ cos θ κh − △ 748 HONG YIN

, r and γ(x) R(x) . Thus we are able to rewrite the tide dynamics model as ∂u = Au γ u + w0 (u + w0) g ξ + f on [0, T] G; ∂t − − | | − ∇ × ∂ξ + (Ru) = 0;  ∂t ∇ (2.2) u = 0 on [0, T] ∂G;  × u = u and ξ = ξ at t = 0,  0 0  where   0 t f = g ∂w + g (Rw0)ds + κ w0 lk w0; − ∂t ∇ 0 ∇ h△ − × u (x) = w (x) w0(0, x);  0 0 − ξ (x) = ζ (x).  0 0 To unify the language, let us introduce the following definitions and notations.  Definition 2.1. Let A be an operator on a separable Hilbert space K with complete orthonormal system (CONS for short) e . If Ax, y = x, A y for any x, y K, then A { j}∞j=1 ∗ ∈ ∗ is called the adjoint of A. If A = A∗, then A is called self-adjoint. Definition 2.2. Let A be a linear operator from a separable Hilbert space K with CONS e ∞ to a separable Hilbert space H. { j} j=1 (a) We denote by L(K, H) the class of all bounded linear operators with the uniform operator norm L. 1 (b) If A = ∞ (A A) 2 e , e < , then A is called a trace class(nuclear) opera- L1 k=1 ∗ k kK ∞ tor. We denote by L1(K, H) the class of trace class operators equipped with norm . L1 (c) We also denote by L2(K, H) the class of Hilbert-Schmidt operators with norm 1 given by A =( ∞ Ae , Ae ) 2 . Sometimes is also denoted by L2 L2 k=1 k kH L2 H.S. (d) Let Q L1(K, K) be self-adjoint and positive definite. Let K0 be the Hilbert subspace∈ of K with inner product

1 1 f, g = Q− 2 f, Q− 2 g , K0 K and we denote LQ = L2(K0, H) with the inner product

F, G = tr(FQG∗) = tr(GQF∗), F, G L . LQ ∈ Q Deﬁnition 2.3. A stochastic process W(t) is called an H-valued Q-Wiener process, where Q is a trace class operator on H, if W(t) satisﬁes the following: (a) W(t) has continuous sample paths in H-norm with W(0) = 0. (b) (W(t), h) has stationary independent increments for all h H. (c) W(t) is a Gaussian process with mean zero and covariance∈ operator Q, i.e. E(W(t), g)(W(s), h) = (t s)(Qg, h) for all g, h H. ∧ ∈ 2 1 Let L (G) and H0(G) be standard Sobolev spaces with norms

u 2 , u 2dx L2 | | G BACKWARD STOCHASTIC TIDAL DYNAMICS EQUATION 749 and 2 , 2 u H1 u dx, 0 |∇ | G 1 1 respectively. Denote H− (G) the dual space of H0(G). Denote ( , ) the inner product of 2 1 1 L (G), ( , ) 1 the inner product of H (G), and , the duality paring between H (G) and H0 0 0 1 2 1 H− (G). Let L2 be the norm of L and 1 be the norm of H (G). Similarly, we can H0 0 2 1 1 define the norms, inner products of L (G), H0 (G) and H− (G). It is clear that 1 2 1 1 2 1 H (G) L (G) H− (G) and H (G) L (G) H− (G) 0 ⊂ ⊂ 0 ⊂ ⊂ are Gelfand triples, and for any x L2(G) and y H1(G), there exists x H 1(G), such ∈ ∈ 0 ′ ∈ − that (x, y) = x′, y . The mapping x x′ is linear, injective, compact and continuous. A 2 1 → 1 similar result holds for L (G), H0 (G) and H− (G). Remark 2.4. (i) Let Q be a trace class operator on L2(G). Let e L2(G) H1(G) { j}∞j=1∈ ∩ 0 ∩ L4(G) be a CONS in L2(G) such that there exists a nondecreasing sequence of positive numbers λ j ∞= , lim j λ j = and e j = λ je j for all j. Let Qek = qkek with k∞=1 qk < { } j 1 →∞ ∞ −△ , and bk(t) be a sequence of iid Brownian motions in R. Then the L2(G)-valued Q- ∞ { } k Wiener process is taken as W(t)= k∞=1 √qkb (t)ek. (ii) Let Q be a trace class operator on L2(G). Similarly, we can define a complete orthonormal system e , a nondecreasing sequence of positive numbers ρ such { j}∞j=1 { j}∞j=1 that e = ρ e , and positive numbers q′ such that Qe = q′ e and ∞ q′ < . Let −△ j j j j j j j j=1 j ∞ = j 2 W(t) ∞j=1 q ′jb (t)e j. Then W(t) is an L (G)-valued Q-Wiener process. Thus according to Definition 2.2 and 2.3, LQ, the space of linear operators E such that 1 2 2 EQ 2 is a Hilbert-Schmidt operator from L (G) to L (G), is well-defined, and so is LQ. In this paper we consider a filtered complete probability space (Ω, , P; t t 0), where F {F } ≥ t is the natural filtration of W(t) and W(t) , augmented by all the P-null sets of . Introducing{F } randomness to system{ (2.2),} { and suppose} the terminal value of the tideF is given, one can construct the following backward stochastic tidal dynamics equations: ∂u(t) = Au(t) γ u(t) + w0(t) (u(t) + w0(t)) g ξ(t) + f(t) + Z(t) dW(t) ; ∂t − − | | − ∇ dt ∂ξ(t) + (Ru(t)) = Z(t) dW(t) ; (2.3)  ∂t ∇ dt u(T) = φ and ξ(T) = ψ,   2 1 2 2 2 2 where f L (0, T; H− ), φ L (Ω; L (G)) and ψ L (Ω; L (G)).  T T  ∈ ∈ F ∈ F Definition 2.5. A quaternion of t-Adapted processes (u, Z, ξ, Z) is called a solution of backward tidal dynamics equationF (2.3) if it satisfies the integral form of the system T u(t) = φ + Au(s) + γ u(s) + w0(s) (u(s) + w0(s)) + g ξ(s) f(s) ds t | | ∇ − T W  t Z(s)d (s);  − T T ξ(t) = ψ + (Ru(s))ds Z(s)dW(s);  t ∇ − t  P-a.s., and the following holds:  2 2 2 2 1 (a) u L (Ω; L∞(0, T; L (G))) L (Ω; L (0, T; H0(G))); ∈ F2 2 ∩ F (b) Z L (Ω; L (0, T; LQ)); ∈ F 750 HONG YIN

2 2 2 2 1 (c) ξ L (Ω; L∞(0, T; L (G))) L (Ω; L (0, T; H0(G))); ∈ F2 2 ∩ F (d) Z L (Ω; L (0, T; LQ)). ∈ F We list below some commonly used results and omit some of the proofs. Readers may refer to Adams[1], Kesavan[11], Ladyzhenskaya[12], Manna, Menaldi and Sritharan[23], and Temam[32] for more details.

Lemma 2.6. For any real-valued, compact supported smooth functions x and y in R2, the following holds: 2 xy x∂ x 1 y∂ y 1 , L2 ≤ 1 L 2 L x 4 2 x 2 x 2 . L4 ≤ L2 ∇ L2 Lemma 2.7. Let X be a normed linear space. Let O be an open subset of X, and K be a convex subset of O. Let J : O R be twice diﬀerentiable in O. Then J is convex if and only if, for all u and v K, → ∈ d2 J′′(v; u, u) = J(v + θu + αu) θ,α= 0. dθdα | 0 ≥ Lemma 2.8. Denote B(u) , γ u + w0 (u + w0). Then B( ) is a continuous operator from L4(G) into L2(G), and for all u|and v in| L4(G),

B(u) B(v), u v 0. − − ≥ Lemma 2.9. (a) For any u and v H1(G), and u has a smooth second derivative, ∈ 0 = 2 (Au, u) κh u H1 0 and

(Au, v) C2 u H1 v H1 ≤ 0 0

for some constant C2 = κh + 2ρ cos θ. (b) For any u and w0 L4(G), ∈

B(u) 2 C u 4 , L ≤ 3 L

where C3 = supx G γ(x). (c) For any u, v and∈w0 L4(G), ∈

B(u) B(v) 2 C u 4 + v 4 u v 4 , − L ≤ 3 L L − L and

2 2 B(u) B(v), u v C u + v u v 2 . | − − | ≤ 4 L4 L4 − L (d) For any u, v H1(G) and w0 L4(G), ∈ 0 ∈ 3 1 2 2 B(u) B(v), u v C u 4 + v 4 u v u v . 5 L L L2 H1 | − − | ≤ − − 0 BACKWARD STOCHASTIC TIDAL DYNAMICS EQUATION 751

3. A Priori Estimates In this section we are going to show some a priori estimates for a projected system. These projections are useful for the Galerkin approximation scheme employed in Section 5 and Section 6. For any N N, let ∈ L2 (G) , span e , e , , e N { 1 2 N} 2 1 1 be the N-dimensional subspace of L (G). Likewise, we can define H0N (G), H− N(G), L2 (G), H1 (G) and H 1(G). Note that since e L2(G) H1(G) L4(G), we have N 0N N− { j}∞j=1∈ ∩ 0 ∩ 2 1 1 L N (G) = H0N(G) = H− N (G). Similarly, we have 2 1 1 LN (G) = H0N (G) = HN− (G). 2 2 Let PN be the orthogonal projection from L (G) to L N (G). Let N N W (t) , PN W(t) and W (t) , PNW(t). N N i N N i Note that by Remark 2.4, W (t) = i=1 √q ib (t)ei and W (t) = i=1 qi′b (t)ei. Let N N N t be the natural filtration of W (t) and W (t) , and we introduce the following {Fprojections:} { } { } fN (t) , P f(t), φN , E(P φ N ) and ψN , E(P ψ N ). N N |FT N |FT The projected backward tide dynamics system is given by ∂uN (t) N N N N N N N ∂t = Au (t) B (u (t)) g ξ (t) + f (t) + Z (t)dW (t); N − − − ∇ ∂ξ (t) + (RN uN (t)) = ZN (t)dWN (t); (3.1)  ∂t ∇ uN (T) = φN and ξN (T) = ψN ,  N  N 0N 0N 4 where B (u) , γ u + w (u + w ) for all u L (G).  | | ∈ 2 Proposition 3.1. Suppose that the terminal conditions satisfy φ L∞ (Ω; L (G)), ψ T 2 2 1 ∈ F ∈ L∞ (Ω; L (G)), and the external force f L (0, T; H− (G)). Then for any solution of T ∈ systemF (3.1), the following is true: N N 2 2 2 1 2 2 (u , Z ) L∞([0, T] Ω; L (G)) L (Ω; L (0, T; H0(G))) L (Ω; L (0, T; LQ)), ∈ F × ∩ F × F N N 2 2 2 1 2 2 (ξ , Z ) L∞([0, T] Ω; L (G)) L (Ω; L (0, T; H0(G))) L (Ω; L (0, T; LQ)). ∈ F × ∩ F × F Proof. An application of the Itôformula to uN (t) 2 yields L2 T uN (t) 2 + ZN (s) 2 ds L2 LQ t T N 2 N N N N N N = φ L2 + 2 Au (s) + B (u (s)) + g ξ (s) f (s), u (s) ds t ∇ − T 2 ZN (s)dWN(s), uN (s) . (3.2) − t By Lemma 2.6 and 2.9, we have N N = N N = N 2 2 Au (s), u (s) 2(Au (s), u (s)) 2κh u (s) H1 , (3.3) 0 752 HONG YIN

2 BN (uN(s)), uN(s) N N 2C u (s) 4 u (s) 2 ≤ 3 L L 3 1 4C uN (s) 2 uN (s) 2 3 L2 H1 ≤ 0 N 2 + N 2 3C3 u (s) L2 C3 u (s) H1 , (3.4) ≤ 0 2 g ξN (s), uN(s) ∇ = 2g ξN (s), uN (s) − ∇ N N 2g ξ (s) 2 u (s) 2 ≤ L ∇ L N 2 + N 2 g ξ (s) L2 g u (s) H1 (3.5) ≤ 0 and N N N 2 + N 2 2 f (s), u (s) f (s) H 1 u (s) H1 . (3.6) − ≤ − 0 Thus (3.2) becomes T uN (t) 2 + ZN (s) 2 ds L2 LQ t T N 2 + + + + N 2 + N 2 + N 2 φ L2 (2κh C3 g 1) u (s) H1 3C3 u (s) L2 g ξ (s) L2 ≤ t 0 T N 2 N N N + f (s) H 1 ds 2 Z (s)dW (s), u (s) . (3.7) − − t Applying the Itˆoformula to ξN (s) 2 to get L2 T ξN (t) 2 + ZN (s) 2 ds L2 LQ t T T = ψN 2 + 2 (RNuN (s)), ξN (s) ds 2 ZN (s)dWN(s), ξN (s) . (3.8) L2 ∇ − t t The term 2 (RNuN (s)), ξN (s) ∇ =2 RN uN (s), ξN (s) + 2 uN (s) RN, ξN (s) ∇ ∇ N N N N N N 2 R L u (s) H1 ξ (s) L2 + 2 u (s) L2 R L2 ξ (s) L2 ≤ ∞ 0 ∇ N 2 + N 2 + N 2 C1 u (s) L2 u (s) H1 2 ξ (s) L2 . (3.9) ≤ 0 Thus substituting (3.9) into (3.8), and adding up (3.7) and (3.8), one gets T T N 2 N 2 N 2 N 2 EFr u (t) + EFr ξ (t) + EFr Z (s) ds + EFr Z (s) ds L2 L2 LQ LQ t t T r N 2 + r N 2 + r + + + + N 2 EF φ L2 EF ψ L2 EF (2κh C3 g 1 C1) u (s) H1 ≤ t 0 N 2 N 2 N 2 + (3C3 + C1) u (s) L2 + (g + 2) ξ (s) 2 + f (s) H 1 ds (3.10) L − BACKWARD STOCHASTIC TIDAL DYNAMICS EQUATION 753 for 0 r t, P-a.s. Since ≤ ≤ N 2 u (s) H1 0 = uN (s), uN(s) −△ N N = λiei, u (s) = i 1 λ uN (s) 2 , ≤ N L2 where λi, as stated in Section 2, is the eigenvalue of with respect to ei. Thus equation (3.10) becomes −△ T r N 2 + r N 2 + r N 2 + N 2 EF u (t) L2 EF ξ (t) L2 EF u (s) H1 ξ (s) H1 ds 0 0 t T T + E r ZN (s) 2 ds + E r ZN (s) 2 ds (3.11) F LQ F LQ t t T r N 2 r N 2 r N 2 N 2 N 2 EF φ L2 + EF ψ L2 + EF K(N) u (s) L2 + K(N) ξ (s) L2 + f (s) H 1 ds, ≤ t − P-a.s., where K(N) is a constant dependingon N only. By means of the Gronwall inequality and letting r = t, one obtains T N 2 + N 2 + N 2 + N 2 sup u (t) L2 ξ (t) L2 E u (s) H1 ξ (s) H1 ds t [0,T] 0 0 0 ∈ T T + E ZN (s) 2 ds + E ZN (s) 2 ds LQ LQ 0 0 T t N 2 t N 2 N 2 K(N) sup EF φ L2 + sup EF ψ L2 + f (s) H 1 ds , (3.12) ≤ t [0,T] t [0,T] 0 − ∈ ∈ P-a.s., which completes the proof. Proposition 3.2. Suppose that the terminal conditions satisfy φ Ln (Ω; L2(G)), ψ T n 2 2 1 ∈ F ∈ L (Ω; L (G)), and the external force f L (0, T; H− (G)), for all n N and n 2. The T ∈ ∈ ≥ followingF is true for any solution of system (3.1): N N n 2 n n 1 2 2 (u , Z ) L∞(0, T; L (Ω; L (G))) L (Ω; L (0, T; H0(G))) L (Ω; L (0, T; LQ)), ∈ F ∩ F × F N N n 2 n n 1 2 2 (ξ , Z ) L∞(0, T; L (Ω; L (G))) L (Ω; L (0, T; H0(G))) L (Ω; L (0, T; LQ)). ∈ F ∩ F × F Proof. First of all, the case when n = 2 can be proved by applying the Gronwall inequality to equation (3.11) in Proposition 3.1, taking the expectation, and then taking supremum over the time interval [0, T]. Secondly,suppose the proposition holds for all 2 m n 1. Let us show that the proposition is still true for m = n. An application of the≤ Itˆoformula≤ − yields n2 n T uN (t) n + uN (s) n 2 ZN (s) 2 ds L2 − L−2 LQ 2 t T N n N n 2 N N N N N N = φ + n u (s) − Au (s) + B (u (s)) + g ξ (s) f (s), u (s) ds L2 L2 ∇ − t 754 HONG YIN

T N n 2 N N N n u (s) − Z (s)dW (s), u (s) . − L2 t Taking the expectation on both sides, one obtains T n2 n T E uN (t) n + E uN (s) n ds + E uN (s) n 2 ZN (s) 2 ds L2 H1 − L−2 LQ t 0 2 t T N n N n 2 N N N N =E φ L2 + nE u (s) L−2 Au (s) + B (u (s)) + g ξ (s) ds t ∇ T T N n 2 N N N n nE u (s) −2 u (s) 1 f (s) H 1 ds + E u (s) 1 ds L H0 − H − t t 0 T T N n N n N n E φ L2 + K(n, N)E u (s) L2 ds + K(n, N)E ξ (s) L2 ds ≤ t t T N N n 1 + n λN f (s) 1 E u (s) −2 ds H− L t T T N n N n N n E φ L2 + K(n, N)E u (s) L2 ds + K(n, N)E ξ (s) L2 ds ≤ t t T N n 1 N + n λ sup E u (s) f (s) 1 ds. (3.13) N L−2 H− s [0,T] t ∈ where K(n, N) is a constant depending on n and N only. Similar, one can show that T 2 T N n N n n n N n 2 N 2 E ξ (t) 2 + E ξ (s) 1 ds + − E ξ (s) −2 Z (s) ds L H L LQ t 0 2 t T E ψN n + K(n, N)E uN (s) n + ξN (s) n ds. (3.14) ≤ L2 L2 L2 t Adding up (3.13) and (3.14) to get T T N n + N n + N n + N n E u (t) L2 E ξ (t) L2 E u (s) H1 ds E ξ (s) H1 ds t 0 t 0 T E φN n + E ψN n + K(n, N)E uN (s) n + ξN (s) n ds ≤ L2 L2 L2 L2 t T N n + K(n, N) f (s) 1 ds, H− t which completes the proof after an application of the Gronwall inequality.

4. Well-posedness of the Projected System In this section, we are going to show the well-posedness of the projected system (3.1). In order to do so, we need to truncate the system. For every M N, let L to be the ∈ M Lipschitz C∞ function given as follows:

1 if u H1 < M 0 LM( u H1 ) = 0 if u H1 > M + 1 0  0  0 LM( u H1 ) 1 otherwise  ≤ 0 ≤   BACKWARD STOCHASTIC TIDAL DYNAMICS EQUATION 755

N N Proposition 4.1. LM( x H1 )B (x) LM( y H1 )B (y) L2 C(N, M) x y H1 for any 0 − 0 ≤ − 0 x, y L2 (G) and M N, where C(N, M) is a constant depending on N, M andG only. ∈ N ∈ 2 Proof. Let x and y be any two elements in L N (G). Without lose of generality, we assume that x H1 y H1 , and let us discuss it in the following 3 cases: 0 ≤ 0 Case I. x 1 > M + 1. H0 N N 2 2 By the definition of L , L ( x 1 )B (x) L ( y 1 )B (y) = 0 x y . Thus M M H M H L2 H1 0 − 0 ≤ − 0 N we see that LMB is Lipschitz. Case II. y H1 M + 1. 0 ≤ It is clear that N N 2 LM( x H1 )B (x) LM( y H1 )B (y) 2 0 − 0 L N N N 2 = LM( x 1 )B (x) LM( y 1 )B (y), ei H0 H0 = | − | i 1 N N N N N 2 = LM( y 1 )B (x) LM( y 1 )B (y) + LM( x 1 )B (x) LM( y 1 )B (x), ei H0 H0 H0 H0 = | − − | i 1 N N N N 2 N 2 2 2 LM( y 1 ) B (x) B (y), ei + 2 B (x), ei LM( x 1 ) LM( y 1 ) H0 H0 H0 ≤ = | − | = | | | − | i 1 i 1 N N 2 N N 2 2 N 2 2 2 LM( y H1 ) B (x) B (y), ei + 2CM B (x), ei x y 1 , (4.1) 0 H0 ≤ = | − | = | | − i 1 i 1 where CM is Lipschitz coefficient of LM. By Lemma 2.6, Lemma 2.9, and the Poincaré inequality, one has N N N 2 B (x) B (y), ei = | − | i 1 N N N 2 = B (x) B (y) L2 = − i 1 2C2 x 2 + y 2 x y 2 ≤ 3 L4 L4 − L4 2 2 + 2 2 4C3CG x H1 y H1 x y H1 , ≤ 0 0 − 0 where CG is a constant depending on G only. Also N N 2 2 2 2 2 B (x), ei C2 x L4 C2CG x 1 . H0 = | | ≤ ≤ i 1 Thus (4.1) becomes N N 2 LM ( x H1 )B (x) LM( y H1 )B (y) 2 0 − 0 L 2 2 2 2 2 2 2 2 1 + + 8C3CG LM( y H ) x H1 y H1 2CMC2CG x H1 x y H1 . ≤ 0 0 0 0 − 0 N N 2 Since x H1 and y H1 are all bounded by M +1, LM( x H1 )B (x) LM( y H1 )B (y) 2 0 0 0 − 0 L ≤ C(N, M) x y 2 , where C(N, M) is only related to N, M, and G. H1 − 0 756 HONG YIN

Case III. y H1 > M + 1 and x H1 M + 1. 0 0 ≤ Then by the definition of LM, LM( y H1 ) = 0. Thus 0 N N 2 2 2 2 2 LM( x H1 )B (x) LM( y H1 )B (y) L2 2CMC2CG x H1 x y H1 . 0 − 0 ≤ 0 − 0 Thus we have shown that N N LM( x H1 )B (x) LM( y H1 )B (y) L2 C(N, M) x y H1 , 0 − 0 ≤ − 0 where C(N, M) is a constant which is only related to N, M and G. Let us state without proof an useful result from Yong and Zhou [34]. k k m k k m Proposition 4.2. For any (y, z) R R × , assume thath(t, y, z) : [0, T] R R × Ω k ∈ × 2 2 k × × × → R is t t 0-adapted with h( , 0, 0) L (Ω; L (0, T; R )). Moreover, there exists an L > 0, such that{F } ≥ ∈ F h(t, y, z) h(t, y¯, z¯) L y y¯ + z z¯ k | k −m | ≤ {| − | | −2 |} k t [0, T], y, y¯ R and z, z¯ R × P-a.s. For any given ξ L (Ω; R ), the BSDE ∀ ∈ ∈ ∈ ∈ FT dY(t) = h(t, Y(t), Z(t))dt + Z(t)dW(t), t [0, T), a.s. ∈ (4.2) Y(T) = ξ,  admits a unique adapted solution (Y( ), Z( )) [0, T], where  ∈M [0, T] = L2 (Ω; C([0, T]; R)) L2 (Ω; L2(0, T; R)) M F × F and it is equipped with the norm T 2 2 1 Y( ), Z( ) [0,T] = E( sup Y(t) ) + E Z(t) dt 2 . M { 0 t T | | 0 | | } ≤ ≤ Now we are able to prove the main result of this section. Theorem 4.3. System (3.1) admits a unique adapted solution (uN, ZN , ξN , ZN ) in 2 2 2 1 2 2 L∞([0, T] Ω; L (G)) L (Ω; L (0, T; H0(G))) L (Ω; L (0, T; LQ)) F × ∩ F × F 2 2 2 1 2 2 L∞([0, T] Ω; L (G)) L (Ω; L (0, T; H0(G))) L (Ω; L (0, T; LQ)), × F × ∩ F × F 2 2 provided that the terminal conditions satisfy φ L∞ (Ω; L (G)), ψ L∞ (Ω; L (G)), and ∈ T ∈ T the external force f L2(0, T; H 1(G)). F F ∈ − Proof. First of all, for any M R, let us define a truncated system as follows: ∈ ∂uNM (t) NM NM N NM NM N = Au (t) LM( u (t) 1 )B (u (t)) g ξ (t) + f (t) ∂t H0 − NM − N − ∇  +Z (t)dW (t);  ∂ξNM (t) (4.3)  + (RN uNM(t)) = ZNM(t)dWN (t);  ∂t ∇ uNM (T) = φN and ξNM (T) = ψN .  From Lemma 4.1, it is clear that all coefficients of the above system are Lipschitz continu-   Ω 2 2 Ω 2 1 ous for fixed N and M. Let us fix ζ(t) L∞([0, T] ; LN(G)) L ( ; L (0, T; H0N (G))). Consider ∈ F × ∩ F ∂uNM (t) NM NM N NM N NM N = Au (t) LM( u (t) 1 )B (u (t)) g ζ(t) + f (t) + Z (t)dW (t); ∂t H0 NM − N − − ∇ u (T) = φ .  (4.4)  BACKWARD STOCHASTIC TIDAL DYNAMICS EQUATION 757

Let us map system (4.4) to RN . It is obviously that the image of system is equivalent to system (4.4). Since the coeﬃcients in the image system are Lipschitz, Proposition 4.2 guarantees the existence of a unique adapted solution of the image system. By the equivalence between two systems, we claim that system (4.4) admits a unique adapted solution (uNM, ZNM ). Clearly, for this uNM, the following system

NM ∂ξ (t) + (RNuNM (t)) = ZNM (t)dWN (t); ∂t (4.5) NM ∇ N ξ (T) = ψ  admits a unique adapted solution (ξNM , ZNM). Hence we can deﬁne an operator Φ, such NM  that Φ(ζ) , ξ . We would like to show that Φ is a contraction mapping. For any ζ1 and Ω 2 2 Ω 2 1 Φ = NM Φ = NM ζ2 L∞([0, T] ; LN(G)) L ( ; L (0, T; H0N (G))), let (ζ1) ξ1 and (ζ2) ξ2 . Denote∈ F × ∩ F uˆ , uNM uNM , ξˆ , ξNM ξNM , ζˆ , ζ ζ , 1 − 2 1 − 2 1 − 2 Zˆ , ZNM ZNM , Zˆ , ZNM ZNM . 1 − 2 1 − 2 Similar to the proof of Proposition 4.1, one can verify that NM N NM NM N NM LM( u (t) H1 )B (u (t)) LM( u (t) H1 )B (u (t)), uˆ(t) | 1 0 1 − 2 0 2 | 2 C(M, CG, C3) uˆ(t) H1 , ≤ 0 where C(M, CG, C3) is a constant depending on M, CG, C3 only. Let η be a positive number such that ρ 2κ λ + C(M, C , C )λ + λ η > max( N , h N G 3 N N ). 2 2 Applying the Itˆoformula to ξˆ(t) 2 e2ηt to get L2 T ξˆ(t) 2 e2ηt + Zˆ(s) 2 e2ηsds L2 LQ t T T = 2η ξˆ(s) 2 + 2 (RNuˆ(s)), ξˆ(s) e2ηsds 2e2ηs Zˆ(s)dWN (s), ξˆ(s) − L2 ∇ − t t T ˆ 2 ˆ 2ηs 2η ξ(s) 2 + 2C1(1 + CG) uˆ(s) H1 ξ(s) L2 e ds ≤ − L 0 t T 2e2ηs Zˆ(s)dWN (s), ξˆ(s) , − t where the estimates are obtained similar to (3.9). Thus for any 0 r t, ≤ ≤ T T 2 2ηt 2 2ηs 2 2ηs EFr ξˆ(t) e + EFr ξˆ(s) e ds + EFr Zˆ(s) e ds L2 H1 LQ t 0 t T r 2 2 2ηs ˆ + ˆ + + 1 ˆ 2 EF 2η ξ(s) L2 ξ(s) H1 2C1(1 CG) uˆ(s) H ξ(s) L e ds ≤ − 0 0 t T r ˆ 2 ˆ 2ηs EF ( 2η + ρN ) ξ(s) 2 + 2C1(1 + CG) uˆ(s) H1 ξ(s) L2 e ds ≤ − L 0 t T 2 2 C (1 + CG) r 2ηs 1 2 EF e uˆ(s) H1 ds, (4.6) ≤ 2η ρ 0 t − N 758 HONG YIN

2 2ηt P-a.s., where ρN is defined in Remark 2.4. An application of the Itôformula to uˆ(t) L2 e yields T uˆ(t) 2 e2ηt + Zˆ (s) 2 e2ηsds L2 LQ t T 2 NM N NM = 2η uˆ(s) 2 + 2 Auˆ(s) + LM( u (s) H1 )B (u (s)) − L 1 0 1 t NM N NM 2ηs LM( u (s) H1 )B (u (s)) + g ζˆ(s), uˆ(s) e ds − 2 0 2 ∇ T 2 Zˆ (s)dWN (s), uˆ(s) e2ηs − t T 2 2ηs 2η + 2κ λ + C(M, C , C )λ uˆ(s) + 2g λ ζˆ(s) 2 uˆ(s) 2 e ds ≤ − h N G 3 N L2 N L L t T 2 Zˆ (s)dWN (s), uˆ(s) e2ηs. − t Thus for any 0 r t, ≤ ≤ T T E r uˆ(t) 2 e2ηt + E r uˆ(s) 2 e2ηsds + E r Zˆ (s) 2 e2ηsds F L2 F H1 F LQ t 0 t T 2 EFr 2η + 2κ λ + C(M, C , C )λ + λ uˆ(s) ≤ − h N G 3 N N L2 t 2ηs + 2g λ ζˆ(s) 2 uˆ(s) 2 e ds N L L T 2 2ηs g λN 2 EFr e ζˆ(s) ds, (4.7) ≤ 2η 2κ λ C(M, C , C )λ λ L2 t − h N − G 3 N − N P-a.s. Equations (4.6) and (4.7) imply T T 2 2ηt 2 2ηs 2 2ηs EFr ξˆ(t) e + EFr ξˆ(s) e ds + EFr Zˆ(s) e ds L2 H1 LQ 0 t t 2 + 2 2 T C1(1 CG) g λN 2ηs 2 EFr e ζˆ(s) ds, ≤ 2η ρ 2η 2κ λ C(M, C , C )λ λ L2 − N − h N − G 3 N − N t P-a.s. Hence we take η to be large enough such that T 1 T r ˆ 2 2ηs r 2ηs ˆ 2 EF ξ(s) H1 e ds EF e ζ(s) H1 ds, 0 ≤ 2 0 t t P-a.s. Taking the expectation and letting t = 0, we see that Φ is a contraction mapping 2 Ω 2 1 2 Ω 2 1 from L ( ; L (0, T; H0N (G))) to L ( ; L (0, T; H0N (G))). By the contraction mapping theorem,F a unique adapted solutionF (uNM, ZNM , ξNM , ZNM) of (4.3) is guaranteed. As N 2 shown in Proposition 3.1, supt [0,T] u (t) L2 K(N), where K(N) is a constant associated ∈ ≤ with N only. Since for finite-dimensional spaces, the norms L2 and 1 are equivalent, H0 N we know that u (s) H1 is also uniformly bounded for every N. By the definition of the 0 truncation LM, it is clear that (3.1) and (4.3) are equivalent when M is large enough. Thus letting M approach infinity, the limit of the solution (uNM , ZNM, ξNM , ZNM) is the unique adapted solution of the projected system (3.1). The regularity of the solution can be obtained by Proposition 3.1. BACKWARD STOCHASTIC TIDAL DYNAMICS EQUATION 759

Continuity of the solution to the projected system (3.1) can also be obtained along similar lines of the proof of Theorem 6.1. We shall skip the proof and postpone it to Section 6. Thus the well-posedness of the projected system has been fully investigated.

5. Existence In this section, we are going to show the existence of an adapted solution of system (2.3). The Galerkin approximation scheme and Minty-Browder technique will be employed. In order to assure an uniform bound on a priori estimates, we make the following assumptions. Such an approach is commonly taken in the study of stochastic Euler equations by several authors so that a dissipative effect arises. Also they are standard hypotheses in the theory of stochastic PDEs in infinite dimensional spaces (see Chow [5], Kallianpur and Xiong [10], Prévôt and Röckner [29]). 1 1 (A.1) (Continuity): f: H0 H− is a continuous operator; (A.2) (Coercivity): There→ exist positive constants α and β, such that 2 2 Au f(u), u α u L2 β u H1 ; − ≤ − 0 2 2 Au f(u), Au α u H1 β Au H1 ; − ≤ 0 − 0 1 (A.3) (Monotonicity): There exist ν > 1 and α > 0, such that for any u and v in H0, and M N, ∈ νA(u v) (f(u) f(v)), RM(u v) α u v 2 , − − − − ≤ − L2 M 2 where R is the projection of R into L M(G); (A.4) (Linear growth): For any u H1 and some positive constant α, ∈ 0 f(u), u α u 2. | | ≤ The system (2.3) can now be written as ∂u(t) = Au(t) γ u(t) + w0(t) (u(t) + w0(t)) g ξ(t) + f(u(t)) + Z(t) dW(t) ; ∂t − − | | − ∇ dt ∂ξ(t) + (Ru(t)) = Z(t) dW(t) ; (5.1)  ∂t ∇ dt u(T) = φ and ξ(T) = ψ,  and the corresponding projected system is  ∂uN (t) N N N N N N N N ∂t = Au (t) B (u (t)) g ξ (t) + f (u (t)) + Z (t)dW (t); N − − − ∇ ∂ξ (t) + N N = N N (5.2)  ∂t (R u (t)) Z (t)dW (t);  ∇ uN (T) = φN and ξN (T) = ψN .  Under these assumptions, we are able to prove a very important monotonicity result,  which is the essence of proof of the existence theorem.

4 4 0 1 Lemma 5.1. For any u, v L 3 ([0, T]; L (G)) L (0, T; H (G)), and M N, deﬁne ∈ ∩ 0 ∈ T 4 4 1 4 3 C1C5 3 3 , 4 4 r(t) 2α + 5 u(s) L + v(s) L ds. t 2 3 ()ν 1 κhC0 − Then 1 A(u v) + B(u) B(v) (f(u) f(v)) + r˙(t)(u v), RM(u v) 0. − − − − 2 − − ≤ 760 HONG YIN

Proof. From the monotonicity assumption and Lemma 2.9, one has A(u v) + B(u) B(v) (f(u) f(v)), RM(u v) − − − − − = νA(u v) (f(u) f(v)), RM(u v) + B(u) B(v), RM(u v) − − − − − − + (1 ν)A(u v), RM(u v) − − − 3 1 2 2 2 2 α u v + C C u 4 + v 4 u v u v + (1 ν)κ C u v L2 1 5 L L L2 H1 h 0 H1 ≤ − − − 0 − − 0 C4C4 1 4 2 3 1 5 3 3 2 + 4 + 4 α u v L2 8 u L v L u v L2 ≤ − 2 3 ()ν 1 κhC0 − − 1 = r˙(t) u v 2 , − 2 − L2 which completes the proof. The coercivity assumption assures a uniform a priori estimate. The following a priori estimate is very useful for the Galerkin approximationwhich will be used in Theorem 5.3. 2 Proposition 5.2. (i) Suppose that the terminal conditions satisfy φ L∞ (Ω; L (G)) and T 2 ∈ F ψ L∞ (Ω; L (G)). Then for any solution of system (5.2), the following is true: ∈ FT N N 2 2 2 1 2 2 (u , Z ) L∞([0, T] Ω; L (G)) L (Ω; L (0, T; H0(G))) L (Ω; L (0, T; LQ)), ∈ F × ∩ F × F N N 2 2 2 (ξ , Z ) L∞([0, T] Ω; L (G)) L (Ω; L (0, T; LQ)). ∈ F × × F Moreover, T N 2 + N 2 + N 2 sup u (t) L2 E u (s) H1 ds sup ξ (t) L2 t [0,T] 0 0 t [0,T] ∈ ∈ T T + E ZN (s) 2 ds + E ZN (s) 2 ds K, (5.3) LQ LQ ≤ 0 0 P-a.s. for some constant K, independent of N. (ii) Suppose the terminal conditions satisfy φ Ln (Ω; L2(G)) and ψ Ln (Ω; L2(G)) ∈ T ∈ T for all n N and n 2. The following is true for anyF solution of system (5.2)F: ∈ ≥ N N n 2 2 2 1 2 2 (u , Z ) L∞(0, T; L (Ω; L (G))) L (Ω; L (0, T; H0(G))) L (Ω; L (0, T; LQ)), ∈ F ∩ F × F N N n 2 2 2 (ξ , Z ) L∞(0, T; L (Ω; L (G))) L (Ω; L (0, T; LQ)). ∈ F × F Moreover, T N n + N n 2 N 2 + N n sup E u (t) L2 E u (s) L−2 u (s) H1 ds sup E ξ (t) L2 t [0,T] 0 0 t [0,T] ∈ ∈ T T + E ZN (s) n ds + E ZN (s) n ds K, (5.4) LQ LQ ≤ 0 0 for some constant K, independent of N. 1 1 (iii) Let the terminal conditions satisfy φ L∞ (Ω; H (G)) and ψ L∞ (Ω; H (G)). ∈ T 0 ∈ T 0 Then for any solution of system (5.2), the followingF is true: F N N 1 2 2 (u , Z ) L∞([0, T] Ω; H0(G)) L (Ω; L (0, T; LQ)), ∈ F × × F N N 1 2 2 (ξ , Z ) L∞([0, T] Ω; H0 (G)) L (Ω; L (0, T; LQ)). ∈ F × × F BACKWARD STOCHASTIC TIDAL DYNAMICS EQUATION 761

Moreover, N 2 + N 2 sup u (t) H1 sup ξ (t) H1 t [0,T] 0 t [0,T] 0 ∈ ∈ T T + E ZN (s) 2 ds + E ZN (s) 2 ds K, (5.5) LQ LQ ≤ 0 0 P-a.s. for some constant K, independent of N. Proof. The proof is very similar to Proposition 3.1, with few modiﬁcations of some estimates. It is clear that

4 3 27C β N N N 3 N 2 + N 2 2 B (u (s)), u (s) u (s) L2 u (s) H1 , ≤ 2β 4 0 and 4g2 β N N N 2 + N 2 2 g ξ (s), u (s) ξ (s) L2 u (s) H1 . ∇ ≤ β 4 0 Thus under part one of Assumption (A.2), we have T β T uN (t) 2 + ZN (s) 2 ds + uN (s) 2 ds L2 LQ H1 2 0 t t T 4 2 N 2 3 27C3 N 2 4g N 2 φ L2 + α + u (s) L2 + ξ (s) L2 ds ≤ t 2β β T 2 ZN (s)dWN(s), uN (s) . − t Since 2 (RNuN (s)), ξN (s) ∇ β 4C2 N 2 + N 2 + 1 + N 2 C1 u (s) L2 u (s) H1 C1 ξ (s) L2 , ≤ 4 0 β we have β T r N 2 + r N 2 + N 2 EF u (t) L2 EF ξ (t) L2 u (s) H1 ds 4 t 0 T T + E r ZN (s) 2 ds + E r ZN (s) 2 ds F LQ F LQ t t N2 N 2 EFr φ + EFr ψ ≤ L2 L2 T 4 2 2 3 27C 4g 4C r 3 N 2 1 N 2 + EF α + + C1 u (s) L2 + + + C1 ξ (s) L2 ds, t 2β β β for 0 r t, P-a.s. An application of the Gronwall inequality and letting r = t completes the proof.≤ ≤ We skip the proof of part (ii) and (iii) since they are very similar to part (i) and the proof of Proposition 3.2. Note that the proof of (iii) uses the second half of the coercivity assumption. 762 HONG YIN

Under our assumptions, the well-posedness of system (5.2) can be obtained similarly to Theorem 4.3. We shall skip the proof. Now we are ready to present the main result of this paper. 1 Theorem 5.3. Suppose that the terminal conditions satisfy φ L∞ (Ω; H (G)) and ψ T 0 1 ∈ F ∈ L∞ (Ω; H (G)). Then there exists an adapted solution (u, Z, ξ, Z) of system (5.1), such T 0 thatF 1 2 2 (u, Z) L∞([0, T] Ω; H0(G)) L (Ω; L (0, T; LQ)), ∈ F × × F 1 2 2 (ξ, Z) L∞([0, T] Ω; H0 (G)) L (Ω; L (0, T; LQ)). ∈ F × × F Proof. For technical reasons, let us introduce a new system. For any M1 N, M1 N, let M1 2 ∈ ≤ R be the projection of R to L M1 (G). Then clearly previous results on projected system (5.2) hold for

NM1 ∂u (t) NM1 N NM1 NM1 N NM1 NM1 N ∂t = Au (t) B (u (t)) g ξ (t) + f (u (t)) + Z (t)dW (t); ∂ξNM1 (t) − − − ∇ + M1 NM1 = NM1 N  ∂t (R u (t)) Z (t)dW (t);  ∇ uNM1 (T) = φN and ξNM1 (T) = ψN .   NM1 NM1 NM1 NM1 Let the unique adapted solution be (u , Z , ξ , Z ). First of all, let us establish several limits of convergent sequences. They are necessary when we perform the Galerkin approximation scheme. By Proposition 5.2, uNM1 , ξNM1 , ZNM1 { }∞N=1 { }∞N=1 { }∞N=1 and ZNM1 are all uniformly bounded in respective spaces. Thus there exist u, ξ, Z, { }∞N=1 and Z, and a subsequence Nk, such that w Nk M1 2 2 1 u u in L (Ω; L (0, T; H0(G))), −→ F N M w 2 ξ k 1 ξ in L∞([0, T] Ω; L (G)), −→ F × w Nk M1 2 2 Z Z in L (Ω; L (0, T; LQ)), −→ F w Nk M1 2 2 Z Z in L (Ω; L (0, T; LQ)). −→ F 1 1 Since A is a continuous mapping from H0(G) to H− (G), we know that

Au H 1 C u H1 , − ≤ 0 1 for all u H0(G) and some constant C. Thus combined with the assumptions on f, one gets ∈ w Nk M1 Nk Nk M1 2 2 1 Au f (u ) F1 in L (Ω; L (0, T; H− (G))), − −→ F for some function F1 and some subsequence Nk. By Lemma 2.9,

1 3 N NM1 N NM1 NM1 2 NM1 B (u (t)) H 1 CG B (u (t)) L2 CGC3 u (t) L4 2 4 C C3 u (t) H1 . − ≤ ≤ ≤ G 0 Thus w Nk Nk M1 2 2 1 B (u ) F2 in L (Ω; L (0, T; H− (G))), −→ F for some function F2 and some subsequence Nk. For every t, let us deﬁne 2 2 2 2 1 t : L (Ω; L (0, T; LQ)) L (Ω; L (0, T; H− (G))) L F → F T J J(s)dW(s). → t BACKWARD STOCHASTIC TIDAL DYNAMICS EQUATION 763

Clearly t is a bounded linear operator. Hence it maps weakly convergent sequences to weakly convergentL sequences, and T T N M N w 2 2 1 Z k 1 (s)dW k (s) Z(s)dW(s) in L (Ω; L (0, T; H− (G))). −→ t t F Similarly, one can prove that T T w AuNk M1 (s) fNk (uNk M1 (s)) + BNk (uNk M1 (s)) ds F (s) + F (s) ds, − −→ 1 2 t t 2 2 1 in L (Ω; L (0, T; H− (G))) and F T T N M N w 2 2 1 Z k 1 (s)dW k (s) Z(s)dW(s) in L (Ω; L (0, T; H− (G))). −→ t t F One can also show that 2 2 2 1 ξ : L∞([0, T] Ω; L (G)) L (Ω; L (0, T; H− (G))) L F × → F T ξ ξ(s)ds → ∇ t N M 2 is a bounded linear operator. Since ξ k 1 L∞([0, T] Ω; L (G)), we have ∈ F × T T N M w 2 2 1 ξ k 1 (s)ds ξ(s)ds in L (Ω; L (0, T; H− (G))). ∇ −→ ∇ t t F Likewise, we have T T M N M w M 2 2 1 (R 1 u k 1 (s))ds (R 1 u(s))ds in L (Ω; L (0, T; H− (G))). ∇ −→ ∇ t t F Thus we have shown that T T u(t) =φ + F (s) + F (s) + g ξ(s) ds Z(t)dW(s), (5.6) 1 2 ∇ − t t and T T ξ(t) = ψ + (RM1 u(s))ds Z(s)dW(s) (5.7) ∇ − t t hold P-a.s. For notational convenience, let us denote Nk by N again. For any M2 N, let Ω 1 ≤ v L∞([0, T] ; H0M (G)). Deﬁne ∈ F × 2 T 4 4 1 3 C1C5 3 4 , 3 r(t) 2α + 5 K ds, t 2 3 ()ν 1 κhC0 − where

NM1 K = sup sup u 4 sup u 4 ∞ + sup v 4 . L L = L (t,ω) [0,T] Ω ∪ (t,ω) [0,T] Ω N 1 (t,ω) [0,T] Ω ∈ × ∈ × ∈ × By Lemma 5.1, it is easy to see that 1 AuNM1 (t) + BN (uNM1 (t)) fN(uNM1 (t)) + r˙(t)uNM1 (t) − 2 1 Av(t) BN (v(t)) + fN(v(t)) r˙(t)v(t), RM1 uNM1 (t) RM1 v(t) 0. − − − 2 − ≤ 764 HONG YIN

Integrating both sides and taking the expectation, one gets T r(s) NM N NM N NM 1 NM E e− Au 1 (s) + B (u 1 (s)) f (u 1 (s)) + r˙(s)u 1 (s), − 2 0 RM1 uNM1 (s) RM1 v(s) ds − T r(s) N N 1 M NM M E e− Av(s) + B (v(s)) f (v(s)) + r˙(s)v(s), R 1 u 1 (s) R 1 v(s) ds. ≤ 0 − 2 − (5.8)

An application of the Itˆoformula to e r(s) √RM1 uNM1 (s) 2 yields − L2 T √ M1 N 2 r(0) √ M1 NM1 2 r(s) NM1 M1 NM1 E R φ L2 Ee− R u (0) L2 + 2E e− g ξ (s), R u (s) ds − 0 ∇ T r(s) √ M1 NM1 2 E e− R Z (s) LQ ds − 0 T r(s) √ M1 NM1 2 = E e− r˙(s) R u (s) L2 ds − 0 T r(s) NM N NM N NM M NM 2E e− Au 1 (s) + B (u 1 (s)) f (u 1 (s)), R 1 u 1 (s) ds − 0 − T r(s) NM N NM N NM 1 NM = 2E e− Au 1 (s) + B (u 1 (s)) f (u 1 (s)) + r˙(s)u 1 (s), − − 2 0 RM1 uNM1 (s) ds. (5.9) Applying the Itˆoformula to ξNM1 (s) 2 to get L2 T E 2 g (RM1 uNM1 (s)), ξNM1 (s) ds − 0 ∇ T =E 2 g ξNM1 (s), RM1 uNM1 (s) ds 0 ∇ T =gE ψN 2 gE ξNM1 (0) 2 gE ZNM1 (s) 2 ds. (5.10) L2 − L2 − LQ 0 Substituting (5.10) into (5.9), one gets

M N 2 r(0) M NM1 2 N 2 r(0) NM1 2 E √R 1 φ Ee− √R 1 u (0) + gE ψ gEe− ξ (0) L2 − L2 L2 − L2 T T r(s) NM 2 r(s) NM 2 gE e Z 1 (s) ds E e √RM1 Z 1 (s) ds − LQ − LQ − 0 − 0 T r(s) NM N NM N NM 1 NM = 2E e− Au 1 (s) + B (u 1 (s)) f (u 1 (s)) + r˙(s)u 1 (s), − − 2 0 RM1 uNM1 (s) ds. By the lower semi-continuity of the norms, we have T r(s) NM N NM N NM 1 NM 2 lim inf E e− Au 1 (s) + B (u 1 (s)) f (u 1 (s)) + r˙(s)u 1 (s), N − 2 →∞ 0 RM1 uNM1 (s) ds. BACKWARD STOCHASTIC TIDAL DYNAMICS EQUATION 765

M1 2 r(0) M1 NM1 2 = E √R φ 2 + lim inf Ee− √R u (0) 2 L N L − →∞ 2 r(0) NM1 2 gE ψ 2 + g lim inf Ee− ξ (0) 2 L N L − →∞ T T r(s) NM1 2 r(s) √ M1 NM1 2 + g lim inf E e− Z (s) L ds + lim inf E e− R Z (s) L ds N Q N Q →∞ 0 →∞ 0 M 2 r(0) M 2 2 r(0) 2 E √R 1 φ + Ee− √R 1 u(0) gE ψ + gEe− ξ(0) ≥ − L2 L2 − L2 L2 T T r(s) 2 r(s) M 2 + gE e− Z(s) + E e− √R 1 Z(s) ds. (5.11) LQ LQ 0 0 r(s) √ M1 2 2 Again applying the Itˆoformula to e− R u(s) L2 and ξ(s) L2 in (5.6) and (5.7) to get

M 2 r(0) M 2 2 r(0) 2 E √R 1 φ Ee− √R 1 u(0) + gE ψ gEe− ξ(0) L2 − L2 L2 − L2 T T r(s) 2 r(s) 2 gE e Z(s) ds E e √RM1 Z(s) ds − LQ − LQ − 0 − 0 T r(s) 1 M = 2E e− F (s) + F (s) + r˙(s)u(s), R 1 u(s) ds. (5.12) − 1 2 2 0 Hence (5.11) and (5.12) imply T r(s) NM N NM N NM 1 NM 2 lim inf E e− Au 1 (s) + B (u 1 (s)) f (u 1 (s)) + r˙(s)u 1 (s), N − 2 →∞ 0 RM1 uNM1 (s) ds. T r(s) 1 M 2E e− F (s) + F (s) + r˙(s)u(s), R 1 u(s) ds. (5.13) ≥ 1 2 2 0 Together with (5.8), one gets T 1 r(s) M1 M1 E e− F1(s) + F2(s) + r˙(s)u(s), R u(s) R v(s) ds 0 2 − T r(s) N N 1 M NM M lim inf E e− Av(s) + B (v(s)) f (v(s)) + r˙(s)v(s), R 1 u 1 (s) R 1 v(s) ds N 2 ≤ →∞ 0 − − T r(s) 1 = lim inf E e− Av(s) + B(v(s)) f(v(s)) + r˙(s)v(s), N − 2 →∞ 0 P RM1 uNM1 (s) RM1 v(s) ds. (5.14) N − w NM1 2 2 1 Since u u in L (Ω; L (0, T; H0(G))), it is easy to show that −→ F w P RM1 uNM1 RM1 u N −→ 2 2 1 in L (Ω; L (0, T; H0(G))) as well. Thus (5.14) becomes F T 1 r(s) M1 M1 E e− F1(s) + F2(s) + r˙(s)u(s), R u(s) R v(s) ds 0 2 − T r(s) 1 M M E e− Av(s) + B(v(s)) f(v(s)) + r˙(s)v(s), R 1 u(s) R 1 v(s) ds. ≤ − 2 − 0 766 HONG YIN

1 Since the above inequality holds for all v L∞([0, T] Ω; H (G)) and all M2 N, we 0M2 ∈ F 1 × ∈ know that it holds true for all v L∞([0, T] Ω; H0(G)). Let us choose v=u + λw where 1 ∈ F × w L∞([0, T] Ω; H0(G)) and λ > 0, one gets ∈ F × T r(s) M1 E e− F1(s) + F2(s), R w ds 0 T r(s) 1 M E e− Av(s) + B(v(s)) f(v(s)) + λ r˙(s)w(s), R 1 w(s) ds. ≥ − 2 0 Letting λ vanish to 0, and by the arbitrariness of w and the continuity of the coeﬃcients, we know that F (s) + F (s) = Au(s) + B(u(s)) f(u(s)) P-a.s. 1 2 − for all M N. The regularity of the solution is guaranteed by Proposition 5.2. The proof 1 ∈ can then be completed by letting M1 go to inﬁnity.

6. Uniqueness and Continuity In this section we deal with the uniqueness and continuity of the solution. Again we 1 assume the uniform bound of the terminal conditions under H0-norm. Such circumstances arise in certain other nonlinear stochastic partial diﬀerential equations such as stochastic Euler equations.

1 Theorem 6.1. Suppose that the terminal conditions satisfy φ L∞ (Ω; H (G)) and ψ T 0 1 ∈ F ∈ L∞ (Ω; H0(G)). Then system (5.1) admits a unique adapted solution (u, Z, ξ, Z) in FT 1 2 2 L∞([0, T] Ω; H0(G)) L (Ω; L (0, T; LQ)) F × × F 1 2 2 L∞([0, T] Ω; H0(G)) L (Ω; L (0, T; LQ)). × F × × F Moreover, the solution is continuous with respect to the terminal conditions in 2 2 2 2 L∞([0, T]; L (Ω; L (G))) L (Ω; L (0, T; LQ)) F × F 2 2 2 2 L∞([0, T]; L (Ω; L (G))) L (Ω; L (0, T; LQ)). × F × F Proof. Suppose that (u1, Z1, ξ1, Z1) and (u2, Z2, ξ2, Z2) are solutions of system (5.1) according to terminal conditions (φ1, ψ1) and (φ2, ψ2), respectively. Denote uˆ = u u , Zˆ = Z Z , Zˆ = Z Z , 1 − 2 1 − 2 1 − 2 ξˆ = ξ ξ , φˆ = φ φ , ψˆ = ψ ψ . 1 − 2 1 − 2 1 − 2 Then the diﬀerences satisfy ∂uˆ(t) ˆ ˆ dW(t) ∂t = Auˆ(t) B(u1(t)) + B(u2(t)) g ξ(t) + f(u1(t)) f(u2(t)) + Z(t) dt ; ˆ − − − ∇ − ∂ξ(t) + (Ruˆ(t)) = Zˆ(t) dW(t) ;  ∂t ∇ dt uˆ(T) = φˆ and ξˆ(T) = ψ.ˆ  (6.1)  Deﬁne

T 4 4 1 3 C1C5 3 4 , 3 r(t) 2α + 5 K ds, t 2 3 ()ν 1 κhC0 − BACKWARD STOCHASTIC TIDAL DYNAMICS EQUATION 767 where

K = sup u1 L4 + sup u2 L4 . (t,ω) [0,T] Ω (t,ω) [0,T] Ω ∈ × ∈ × Then an application of the Itˆoformula to e r(s) √R uˆ(s) 2 and ξˆ(s) 2 yields − L2 L2 T T Ee r(t) √Ruˆ(t) 2 + gE ξˆ(t) 2 + gE Zˆ(s) 2 ds + E e r(s) √RZˆ (s) 2 ds − L2 L2 LQ − LQ t t T 2 2 r(s) =E √REφˆ + g ψˆ + 2E e− Auˆ(s) + B(u (s)) B(u (s)) f(u (s)) + f(u (s)) L2 L2 1 − 2 − 1 2 t 1 + r˙(s)uˆ(s), Ruˆ(s) ds 2 E √REφˆ 2 + g ψˆ 2 . ≤ L2 L2 Thus we have shown the uniqueness and continuity of solutions.

Now we have established the well-posedness of the backward stochastic tidal dynamics equation. Such well-posedness holds when the terminal conditions are uniformly bounded 1 under H0-norm. One may want to relax the conditions on terminal values to a weaker sense, such as uniformly boundednessin L2 sense. However, such problems are still open. The difficulty lies in the nonadaptiveness nature of the backward stochastic differential equations. For instance, the function r defined in Lemma 5.1 is not adaptive to the forward filtration. So in the proof of Theorem 5.3 and Theorem 6.1, we redefined r so that it is adaptive to the system. For this approach, we have to improve the regularity of the solution u appeared in the definition of r. Such obstacles do not arise in the forward stochastic systems.

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Hong Yin:Department of Mathematics,State University of New York,Brockport,Brockport, NY 14420 E-mail address: [email protected]