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Well posedness of a stochastic hyperviscosity-regularized 3D Navier-Stokes equation

We analyse the well posedness of a stochastic hyperviscosity-regularized 3D Navier-Stokes equation; this is the Navier-Stokes equation in which the Laplace operator is replaced by its a-power for a>1. We prove existence and uniqueness for this equation assuming a>= 5/4.


Introduction
We call stochastic Navier-Stokes problem the following: Here u = u(t, x) is the 3-dimensional velocity vector field defined for t ≥ 0 and x ∈ D ⊆ R 3 , p = p(t, x) is the scalar pressure field, ν > 0 is the coefficient of kinematic viscosity, u 0 = u 0 (x) is the initial velocity, f = f (t, x) and n = n(t, x) are, respectively, the deterministic and stochastic forcing terms. If the spatial domain has a boundary, we assume that u vanishes on ∂D.
When there is no noise term n, this reduces to the deterministic Navier-Stokes problem which models the motion of viscous fluids. For the 3-dimensional setting, both in the deterministic and in the stochastic case we know the existence of a weak solution but uniqueness is proved in a smaller class, where existence is not known. The question of proving existence and uniqueness on any finite time interval and with any initial data for the deterministic Navier-Stokes equation is one of the Millennium Prize problems (see, e.g., [3]). However, for the 3-dimensional problem there are results for small initial data or locally in time, whereas the 2-dimensional problem is well posed (see, e.g., [12,13] for the deterministic problem and [7,4] for the 2-dimensional stochastic problem with additive noise).
There have been many attempts to modify the 3-dimensional Navier-Stokes equation in order to prove an existence and uniqueness result. The first models go back to Lions [8]. For more recent results, we focus on two particular cases: [9] and [11]. In [9] the Laplacian operator −∆ is replaced with (−∆) α (for α > 1) in the deterministic Navier-Stokes equation; the stochastic problem with a similar modification (−ν∆ replaced with −ν 0 ∆ + ν 1 (−1) α ∆ α ) is studied in [11]. Setting α > 1 we obtain a model for hyperviscous fluids (see [11] and references therein).
Our aim is to analyse the well posedness of the stochastic version of the modified Navier-Stokes equation considered by Mattingly-Sinai in [9], that is We shall consider the model of additive noise, i.e. n is independent of u. This is the simplest case, which reduces the technicalities. However, the case of multiplicative noise can be treated in a similar way.
In Section 3, we shall prove an existence and uniqueness result for α ≥ 5 4 , as conjectured in [5]. The bound α > 5 4 appeared first in [9] for the deterministic problem. Regularity results will be given in Section 4.
Finally, we point out that our technique is different from that of [9] or [11]; indeed, we use tools from [12] and [7].

Notations and preliminaries
Let the spatial domain be a torus, i.e. the spatial variable x belongs to T = [0, L] 3 and periodic boundary conditions are assumed.
We introduce the classical spaces for the Navier-Stokes equation (see, e.g., [13,12] for all the results in this section). D ∞ is defined as the space of infinitely differentiable divergence free periodic fields u : T → R 3 , with zero mean ( T u(x)dx = 0). Let H 0 be the closure of D ∞ in the [L 2 (T )] 3 -topology; it is the subspace of [L 2 (T )] 3 of all fields u such that div u = 0, the normal component of u on the boundary is periodic, T u (x) dx = 0. We endow H 0 with the inner product u, v = T u(x) · v(x) dx and the associated norm |·|. Similarly, for m ∈ N let H m be the closure of D ∞ in the [H m (T )] 3 -topology.
Let A : D(A) ⊂ H 0 → H 0 be the operator Au = −∆u (componentwise) with D(A) = H 2 . This is called Stokes operator and it is a strictly positive unbounded self-adjoint operator in H 0 , whose eigenvectors h j form a complete orthonormal basis of the space H 0 ; the eigenvalues λ j are strictly positive and 0 < λ 1 ≤ λ 2 ≤ . . . with λ j ∼ j 2/3 for j → ∞. Since the spatial domain is the torus, we know the expressions of the eigenvectors with their eigenvalues (see, e.g., [6]).
The power operators A α are defined for any α ∈ R. If u = j u j h j , then We set |u| 2α = |A α u| and the space H s can be defined (for any s ∈ R) as the closure of D ∞ in the | · | s -metric. For s < 0 the space H s is the dual space of H −s with respect to the H 0 -topology. The space H s+r is dense and compactly embedded in H s for any r > 0.
Notice that |u| m is equivalent to the usual [H m (T )] 3 -norm.
In particular We have H 0 = span{h k } and we set H n = span{h k : |k| ≤ n}; moreover, we denote by π n the projection operator from H 0 onto H n . The operators A and π n commute. By Π we denote the projector operator from [L 2 (T )] 3 onto H 0 . The operator −A generates in H 0 (and in any H s ) an analytic semigroup of for every u, v, w ∈ H 1 . By the incompressibility condition, we have We shall use the following estimates (see Lemma 2.1 in [13]): and similarly Here and in the following, we denote by c a positive constant, which may vary from place to place.

Main theorem
We apply the projection operator Π to the first equation in (2). We get an Itô equation in an infinite dimensional Hilbert space: assuming the noise is of white type in time and with spatial covariance independent of u. This means that w is a cylindrical Wiener process in H 0 defined on a complete probability space with filtration (Ω, F , {F t } t≥0 , P) (i.e. given a sequence {β j } j of i.i.d. standard Wiener processes, we represent the Wiener process in series as w(t) = j β j (t)h j ). For simplicity we consider the operator in front of w to be a power of the Stokes operator; this is the model studied in [5]. For α = 1, this corresponds to the stochastic Navier-Stokes equation as analysed for instance in [7] for the 2-dimensional setting. The technique to study equation (8) comes from [1,14,2,7]. First we consider the linear equation, that is the modified stochastic Stokes equation Then, we define v := u−z; this unknown solves the following equation, obtained subtracting equation (9) to equation (8) and bearing in mind the bilinearity of the operator B: Let [0, T ] be any finite time interval. We now state our main result.
4 then there exists a unique process u which is a strong solution of (8) such that u is progressively measurable in these topologies and is a Markov process in H 1 .

Existence
We study pathwise the problems for the unknowns z and v. This will imply an existence result for u.
For the linear problem we have (see, e.g., Proposition 4.1 in [5], based on then equation (9) has a unique strong solution z such that Now, we work pathwise for the equation satisfied by v and therefore also for u. v is progressively measurable in these topologies.
Proof. From Lemma 3.2 we have that z ∈ C([0, T ]; H α ) P-a.s., since γ > 3 4 . We take the scalar product of equation (10) with v and use (4)-(5): Then d dt |v| 2 ≤ c|z| 2 α |v| 2 + c|z| 4 α and from Gronwall lemma: sup 0≤t≤T |v(t)| 2 < ∞. Moreover, integrating in time the first inequality above, we have T 0 |v(t)| 2 α dt < ∞. Now we take the scalar product of equation (10) with Av: We use (6) and Young inequality to obtain 1 2 As usual, from d dt Gronwall inequality, with the result v ∈ L 2 (0, T ; H α ) proved before, implies sup 0≤t≤T |v(t)| 2 1 < ∞ and integrating in time (13) we get T 0 |v(t)| 2 1+α dt < ∞. The technique to prove existence is classical (see [12]). We consider first the finite dimensional problem in the unknown v n , obtained projecting equation (10) onto H n . This is the Galerkin approximation, for any n = 1, 2, . . . . The above estimates hold uniformly also for the Galerkin sequence: for any n  for constants c 1 and c 2 independent of n. Any finite dimensional (Galerkin) problem has a solution. By passing to the limit as n → ∞ we get an existence result for (10). We also need that dvn dt is uniformly bounded in L 2 (0, T ; H 1−α ). We verify easily that dvn dt ∈ L 2 (0, T ; H 1−α ) and the norm is bounded uniformly for all n, since A α v ∈ L 2 (0, T ; H 1−α ) and according to (6) all the bilinear terms in (10) are in L 2 (0, T ; H 0 ). Therefore, we have a compact embedding (see Theorem 2.1, Ch. III in [12]) so the Galerkin sequence stays in a compact subset of L 2 (0, T ; H 1 ). Therefore there exists a subsequence converging to v as follows: The strong convergence allows one to pass to the limit in the bilinear term (see Lemma 3.2, Ch. III in [12]). Finally, v ∈ C([0, T ]; H 1 ) (see Lemma 1.2, Ch. III in [12]). The limit v fulfils all the estimates found above: We conclude noting that by interpolation L ∞ (0, T ;

Pathwise uniqueness
We consider two solutions u 1 and u 2 of (8) obtained in the previous section; we have that, for α ≥ 5 4 , Since the operator B is bilinear, this becomes Taking the scalar product of (15) with AU in H 0 , we get with U (0) = 0. We estimate the r.h.s. according to (6)- (7) 1 2 Thus, by Young inequality: and, by Gronwall inequality Then U (t) = 0 for all t ≥ 0, because U (0) = 0. This means that pathwise we have u 1 (t) = u 2 (t) for all t ≥ 0.

Remark 3.5 i) The Markov property of u comes from the same properties for
the Galerkin approximations and from the uniqueness result (see, e.g., [7]).
ii) The pathwise estimate (16) implies also the Feller property in H 1 , that is given a sequence of solutions u j with initial data u j 0 , if lim j u j 0 = u 0 in H 1 then lim j Eφ(u j (t)) = Eφ(u(t)) for any t ∈ [0, T ] and any continuous bounded function φ : H 1 → R.

Regularity results
Considering as phase space other Hilbert spaces H s , we get different bounds on α to obtain that the dynamics of the stochastic Navier-Stokes equation is well posed in such H s . As it has been pointed out in [9], the smaller is s (with u 0 ∈ H s ) the bigger is α. Theorem 3.1 deals with s = 1. In this section, we show that problem (8) is well-posed in the space H 0 if α > 3 2 , and in the spaces H 0 -regularity We have the following result u is progressively measurable in these topologies and a Markov process in H 0 .
Proof. Existence is proved by means of a priori estimates as in the proof of Proposition 3.3; to be precise, for α ≥ 5 4 we get that there exists a solution u ∈ C([0, T ]; H 0 ) ∩ L 2 (0, T ; H α ) P-a.s. if z has paths in C([0, T ]; H α ), i.e. if γ > 3 4 . Pathwise uniqueness is obtained according to the result by Prodi [10], requiring u ∈ L s (0, T ; [L q (T )] 3 ) P-a.s. for 2 s + 3 q ≤ 1. However, using an interpolation result and Sobolev embedding we have Then sup 0≤t≤T |v(t)| < ∞, T 0 |v(t)| 2 1 dt < ∞. Moreover, from Lemma 2.1 in [13] we have that B : . This gives a compact embedding and therefore there exists a subsequence of the Galerkin sequence that converges to v as follows: Lemma 1.2, Ch. III in [12]).

Remark 4.3
To compare our result with [11], we have that [11], for its model, deals with the phase space H 0 assuming α ≥ 2.
H s -regularity with s ≥ 2 We need the following estimates: Proof. First, consider (17) for m = 1. We have Then use the continuous embeddings H 1 (T ) ⊂ L 4 (T ) and H 2 (T ) ⊂ L ∞ (T ). For m = 2, 3, . . . we use that H m is a multiplicative algebra if m > 3 2 ; then |B(u,ũ)| m ≤ c|u| m |ũ| m+1 for m = 2, 3, . . . which is even stronger than (17). 2 We have the following result For simplicity, we provide the proof for s = 2. In this way we show the difference with respect to the case s = 1 considered in the previous section. However, the proof would go along the same lines for s > 2 using (17).
Since we already know from Proposition 3.3 that v ∈ L 2 (0, T ; H 1+α ) ⊂ L 2 (0, T ; H 2 ), it follows as usual by Gronwall lemma that v ∈ L ∞ (0, T ; H 2 ) ∩ L 2 (0, T ; H 2+α ). From now on, the proof goes as in Proposition 3.3. Pathwise uniqueness: the estimates hold for any α ≥ 1 but the regularity required on u i holds for α ≥ 5 4 . This shows that in H 2 it is "easier" to prove uniqueness than existence. Set U = u 1 − u 2 as in Section 3.2; now u 1 , u 2 ∈ C([0, T ]; H 2 ). Taking the scalar product of (15) with A 2 U in H 0 , we get with U (0) = 0. As before, we estimate the r.h.s. by means of (17), and get we conclude that |U (t)| 2 = 0 for any t ∈ [0, T ].