Characterization of the law for 3D stochastic hyperviscous fluids
aa r X i v : . [ m a t h . P R ] M a r Characterization of the law for 3D stochastichyperviscous fluids
Benedetta Ferrario ∗ September 5, 2018
Abstract
We consider the 3D hyperviscous Navier-Stokes equations in vor-ticity form, where the dissipative term − ∆ ~ξ of the Navier-Stokes equa-tions is substituted by ( − ∆) c ~ξ . We investigate how big the correc-tion term c has to be in order to prove, by means of Girsanov trans-form, that the vorticity equations are equivalent (in law) to easierreference equations obtained by neglecting the stretching term. Thisholds as soon as c > , improving previous results obtained with c > in a different setting in [5, 14]. MSC2010 : 76M35, 60H15, 35Q30.
Keywords : Hyperviscous fluids, well-posedness, Girsanov formula.
The stochastic Navier-Stokes equations, governing the motion of a homoge-neous and incompressible viscous fluid, are ∂~v∂t − ν ∆ ~v + ( ~v · ∇ ) ~v + ∇ p = ~f + ~n ∇ · ~v = 0 (1)where the unknown are the velocity ~v and the pressure p ; the data are theviscosity ν >
0, the deterministic forcing term ~f and the random one ~n .Working in a bounded three dimensional spatial domain with suitableboundary conditions, it is known that for initial velocity of finite energy and ∗ Universit`a di Pavia, Dipartimento di Matematica ”F. Casorati”, via Ferrata 5, 27100Pavia, Italy. Email: [email protected] ~n = ~
0) and to [9] for the stochastic ones (the case ~n = ~ ∂~v∂t + ν ( − ∆) c ~v + ( ~v · ∇ ) ~v + ∇ p = ~f + ~n ∇ · ~v = 0 (2)We consider c >
0, whereas it reduces to the Navier-Stokes system for c = 0.This model is widely used in computer simulations (see e.g. [11], [18] andreferences therein). It turns out that large enough values of the parameter c provide better mathematical properties of system (2).As far as the well posedness of (2) is concerned, the condition c ≥ allows to prove that there exists a unique global solution for the hyperviscousNavier-Stokes equations (2). This is based on the fact that the operator( − ∆) c has a more regularizing effect than the Laplacian itself and c ≥ provides a sufficient regularity to prove uniqueness of the global weaksolution. The result has been proved first for integer values of c ≥
1, both inthe stochastic (see [19]) and deterministic case (see [15]). Then, these resultshave been improved allowing c to be non integer (see [6] for the stochasticcase and [16] for the deterministic one).A further question concerns the characterization of the law of the pro-cess solving (2) with a stochastic force. When ~f = ~ ~n is a Gaussianrandom field, white in time and coloured in space, Gallavotti (see [12], Ch6.1) suggested to use Girsanov transform to relate the law of the stochasticNavier-Stokes equations with that of the stochastic Stokes equations, whichare linear equations obtained from the Navier-Stokes ones by neglecting thenon linear term ( ~v · ∇ ) ~v . The formula given in [12] when c = 0 is formal, butthis idea can be used also for the hyperviscous fluids. Actually, a rigorousresult has been proved in [14], [5]: for c > the law of the process ~v solving ∂~v∂t + ν ( − ∆) c ~v + ( ~v · ∇ ) ~v + ∇ p = ~n ∇ · ~v = 0 (3)is equivalent to the law of the process ~z solving the stochastic hyperviscous2tokes system ∂~z∂t + ν ( − ∆) c ~z + ∇ p = ~n ∇ · ~z = 0 (4)This holds in the 2D and in the 3D setting and implies that all what holdsa.s. for the hyperviscous Stokes problem (4) holds a.s. for the hyperviscousNavier-Stokes problem (3) as well. In other words: the advection term ( ~v ·∇ ) ~v takes second place to the dissipative term ( − ∆) c ~v for c large enough. Thismeans that hyperviscosity with c > changes drastically the nature of theequations of motion of the fluid. This remark already appeared in [11], wherethe authors discuss artifacts arising in numerical simulation of hyperviscousfluids. The mathematical representation of the law of ~v by means of Girsanovtransform, which reduces the analysis of the law of ~v to the analysis of thelaw of the linear problem for ~z , gives evidence in support of the fact thathyperviscous fluid models with c > are far away from the real turbulentfluids.But, what happens for smaller values of the correction term, i.e. for c ≤ ? To answer this question, we change the auxiliary process. First of allwe write the Navier-Stokes system in vorticity form ∂~ξ∂t + ν ( − ∆) c ~ξ + ( ~v · ∇ ) ~ξ − ( ~ξ · ∇ ) ~v = ∇ × ~n ∇ · ~v = 0 ~ξ = ∇ × ~v (5)Notice that the first equation can be rewritten as ∂~ξ∂t + ν ( − ∆) c ~ξ + P [( ~v · ∇ ) ~ξ ] − P [( ~ξ · ∇ ) ~v ] = ∇ × ~n where P is the projection operator onto the space of divergence free vectorfields (see details in Section 2).The idea is to simplify the vorticity equation by neglecting only the vor-ticity stretching term, getting ∂~η∂t + ν ( − ∆) c ~η + P [( ~v · ∇ ) ~η ] = ∇ × ~n ∇ · ~v = 0 ~η = ∇ × ~v (6)This system has the same structure as the 2D vorticity system, but we con-sider it in the 3D setting. Indeed, in the 2D setting the vorticity is a vector3rthogonal to the plane where the fluid moves and therefore the term ( ~ξ · ∇ ) ~v vanishes. Therefore, systems (5) and (6) are different only in the 3D setting.Let us compare them.From the mathematical point of view we shall prove that system (6) iswell posed for any c ≥
0, whereas the well posedness of the full system (5)has been proved by assuming c ≥ .On the other hand, the vorticity stretching term ( ~ξ · ∇ ) ~v is essentialin 3D fluids (see e.g. [10] Ch 9); it is responsible of the peculiar featuresof 3D turbulence, which is very different from and more involved than 2Dturbulence. Thus one expects the dynamics of ∂~ξ∂t − ν ∆ ~ξ + P [( ~v · ∇ ) ~ξ − ( ~ξ · ∇ ) ~v ] = ∇ × ~n ∇ · ~v = 0 ~ξ = ∇ × ~v to be very different from that of ∂~η∂t − ν ∆ ~η + P [( ~v · ∇ ) ~η ] = ∇ × ~n ∇ · ~v = 0 ~η = ∇ × ~v Now, the question is: what happens if we introduce hyperviscosity ( − ∆) c ?Our main theorem states the equivalence of laws of the solution processes ofsystems (5) and (6) under the assumption c > . Again our result givesevidence that the hyperviscous models with c > do not represent well thereal 3D turbulence, since the effect of the vorticity stretching term are notrelevant when c > .Finally, we present this paper. In the next section we define the functionalspaces and the noise term. Section 3 presents various technical results. Thenwe start to analyze the main equations: the linear problem in Section 4,the auxiliary problem (6) in Section 5 and the full vorticity problem (5) inSection 6. The main result on the equivalence of the laws is proved in Section7. We denote a 3D vector as ~k = ( k (1) , k (2) , k (3) ); we define Z = Z \ { ~ } and Z = { k (1) > } ∪ { k (1) = 0 , k (2) > } ∪ { k (1) = 0 , k (2) = 0 , k (3) > } . Then4or any ~k ∈ Z , there exist two unit vectors ~b ~k, and ~b ~k, , orthogonal to eachother and belonging to the plane orthogonal to ~k ; we choose these vectorsin such a way that ( ~b ~k, ,~b ~k, , ~k | ~k | ) is a right-handed orthonormal frame and ~b ~k,j = − ~b − ~k,j .We work on the 3D torus, that is we deal with functions defined on R and [ − π, π ] -periodic. We set D = [ − π, π ] . As usual, in the periodic casewe assume that the mean value of the vectors we are dealing with is zero.This gives a simplification in the mathematical treatment, but it does notprevent to consider non zero mean value vectors. Actually, if we can analysethe problem for zero mean vectors then the problem without this assumptioncan be dealt with in a similar way (see [21]).The velocity vector ~v is divergence free by assumption and the vorticityvector ~ξ is divergence free by construction. We can write any zero mean,periodic, divergence free vector ~u in Fourier series as ~u ( ~x ) = X ~k ∈ Z [ u ~k, ~b ~k, + u ~k, ~b ~k, ] e i~k · ~x , ~x ∈ R where u ~k, , u ~k, ∈ C , with the condition u ~k,j = − u − ~k,j in order to have a realvector ~u ( ~x ).When needed, we use the notation ~v and ~ξ to make precise that we dealwith the velocity or vorticity vector. For instance, we have ~ξ = ∇ × ~v , butwe can also express the velocity in terms of the vorticity, solving − ∆ ~v = ∇ × ~ξ ∇ · ~v = 0 ~v periodic (7)More explicitly ~ξ ( ~x ) = X ~k ∈ Z ( ξ ~k, ~b ~k, + ξ ~k, ~b ~k, ) e i~k · ~x = ⇒ ~v ( ~x ) = i X ~k ∈ Z | ~k | ( ξ ~k, ~b ~k, − ξ ~k, ~b ~k, ) e i~k · ~x (8)We now define the functional spaces. Let L denote the subspace of[ L ( D )] consisting of zero mean, periodic, divergence free vectors (this con-dition has to be understood in the distributional sense): L = n ~u ( ~x ) = X ~k ∈ Z [ u ~k, ~b ~k, + u ~k, ~b ~k, ] e i~k · ~x : X ~k ∈ Z ( | u ~k, | + | u ~k, | ) < ∞ o h ~u, ~v i = (2 π ) X ~k ∈ Z ( u ~k, v ~k, + u ~k, v ~k, )The space L is a closed subspace of [ L ( D )] ; we decide to put the subindexin L in order to distinguish them.Moreover, for any integer n we define the projection operator Π n as alinear bounded operator in L such thatΠ n X ~k ∈ Z [ u ~k, ~b ~k, + u ~k, ~b ~k, ] e i~k · ~x = X < | ~k |≤ n [ u ~k, ~b ~k, + u ~k, ~b ~k, ] e i~k · ~x and we set H n = Π n L .For p > L p = L ∩ [ L p ( D )] These are Banach spaces with norms inherited from [ L p ( D )] .We denote by P the projection operator from [ L p ( D )] onto L p . We havethat P [( ~v · ∇ ) ~ξ − ( ~ξ · ∇ ) ~v ] = 0. Indeed, the vorticity transport term ( ~v · ∇ ) ~ξ and the vorticity stretching term ( ~ξ · ∇ ) ~v are not divergence free vector fields;so P [( ~v · ∇ ) ~ξ ] = ( ~v · ∇ ) ~ξ and P [( ~ξ · ∇ ) ~v ] = ( ~ξ · ∇ ) ~v . However, their differenceis divergence free, being given by the curl form ∇ × [( ~v · ∇ ) ~v ]. Moreover, if ~φ is a divergence free vector field (i.e. P ~φ = ~φ ), then h P [( ~ξ · ∇ ) ~v ] , ~φ i = h ( ~ξ · ∇ ) ~v, ~φ i For any a ∈ R we define the fractional powers of the Laplace operator;formally, if ~u ( ~x ) = X ~k ∈ Z [ u ~k, ~b ~k, + u ~k, ~b ~k, ] e i~k · ~x then ( − ∆) a ~u ( ~x ) = X ~k ∈ Z | ~k | a [ u ~k, ~b ~k, + u ~k, ~b ~k, ] e i~k · ~x Thus, for b ∈ R we define the Hilbert spaces H b = { ~u ( ~x ) = X ~k ∈ Z [ u ~k, ~b ~k, + u ~k, ~b ~k, ] e i~k · ~x : X ~k ∈ Z | ~k | b ( | u ~k, | + | u ~k, | ) < ∞} h ~u, ~v i b = (2 π ) X ~k ∈ Z | ~k | b ( u ~k, v ~k, + u ~k, v ~k, ) ≡ h ( − ∆) b ~u, ( − ∆) b ~v i The duality between H b and H − b (or between [ H b ( D )] and [ H − b ( D )] ) isagain denoted by h· , ·i .For b > p >
2, we define the generalized Sobolev spaces H bp H bp = { ~u ∈ L p : ( − ∆) b ~u ∈ L p } which are Banach spaces with norms k ~u k H bp = k ( − ∆) b ~u k L p When b ∈ N , H bp are the Sobolev spaces. We recall the Sobolev embeddingtheorem (see [17] Ch 1 § • if 1 < p < q < ∞ with q = p − a − b , then the following inclusion holds H ap ⊂ H bq and there exists a constant C (depending on a − b, p, q ) such that k ~v k H bq ≤ C k ~v k H ap • if 1 < p < ∞ with 3 < ap , then the following inclusion holds H ap ⊂ L ∞ and there exists a constant C (depending on a, p ) such that k ~v k L ∞ ≤ C k ~v k H ap The Poincar´e inequality holds, because of the zero mean value assump-tion, and therefore k ~u k H bp is equivalent to ( k ~u k pL p + k ~u k pH bp ) /p , which appearsusually in the definition of the generalized Sobolev spaces.Moreover for ~ξ = ∇ × ~v , the norms k ~v k H bp and k ~ξ k H b − p are equivalent (see(8)).For any t > b >
0, the linear operator e − t ( − ∆) b , formally defined as e − t ( − ∆) b X ~k ∈ Z [ u ~k, ~b ~k, + u ~k, ~b ~k, ] e i~k · ~x = X ~k ∈ Z e − t | ~k | b [ u ~k, ~b ~k, + u ~k, ~b ~k, ] e i~k · ~x
7s a contraction operator in L p for any p ≥ d~n of theform d ( − ∆) − b ~w , where ~w is a cylindrical Wiener process in L (see, e.g., [4]).We can represent it as follows. Suppose we are given a Brownian stochasticbasis, i.e. a probability space (Ω , F , P ) and a filtration ( F t ) t ≥ ; we denoteby E the mathematical expectation with respect to P . Let { β ~k, , β ~k, } ~k ∈ Z be a double sequence of complex valued independent Brownian motions on (cid:0) Ω , F , ( F t ) t ≥ , P (cid:1) ; namely, the sequence {ℜ β ~k,j , ℑ β ~k,j } ~k ∈ Z ; j =1 , consists ofreal valued processes that are independent, adapted to ( F t ) t ≥ , continuousfor t ≥ t = 0, with increments on any time interval [ s, t ] thatare N (0 , t − s )-distributed and independent of F s .Moreover, for − ~k ∈ Z let β ~k,j = − β − ~k,j . Then ~w ( t, ~x ) = X ~k ∈ Z [ ~b ~k, β ~k, ( t ) + ~b ~k, β ~k, ( t )] e i~k · ~x (9)is a cylindrical Wiener process in L . Its paths do not live in the space C ( R + ; L ); they are less regular in space. Indeed E k ( − ∆) a ~w ( t ) k L = 2 t X ~k ∈ Z | ~k | a which is finite if and only if a < − .Within this setting, we write system (5) for the vorticity as d~ξ + (cid:16) ( − ∆) c ~ξ + P [( ~v · ∇ ) ~ξ ] − P [( ~ξ · ∇ ) ~v ] (cid:17) dt = ( − ∆) − b d ~w ∇ · ~v = 0 ~ξ = ∇ × ~v (10)We have put ν = 1 for simplicity and consider b, c ≥ Definition 1.
Given (Ω , F , ( F t ) t ≥ , P ) and an L -cylindrical Wiener process ~w , we say that a process ~ξ is a basic solution to system (10) on the finitetime interval [0 , T ] with initial condition ~ξ (0) = ~ξ ∈ L if ~ξ ∈ C ([0 , T ]; L ) ∩ L (0 , T ; L ) P − a.s. (11)8 nd it satisfies the first equation of (10) in the following sense:for any t ∈ [0 , T ] , for any ~φ ∈ H c ∩ H − b h ~ξ ( t ) , ~φ i + Z t h ~ξ ( s ) , ( − ∆) c ~φ i ds − Z t h ( ~v ( s ) · ∇ ) ~φ, ~ξ ( s ) i ds + Z t h ( ~ξ ( s ) · ∇ ) ~φ, ~v ( s ) i ds = h ~ξ , ~φ i + h ( − ∆) − ~w ( t ) , ( − ∆) − b ~φ i (12) P -a.s. The latter relationship is obtained by multiplying the first equation of(10) by ~φ , integrating in space and time and finally by integration by partin the trilinear terms. Indeed, −h ( ~v ( s ) · ∇ ) ~φ, ~ξ ( s ) i = h ( ~v ( s ) · ∇ ) ~ξ ( s ) , ~φ i = h P [( ~v ( s ) ·∇ ) ~ξ ( s )] , ~φ i and h ( ~ξ ( s ) ·∇ ) ~φ, ~v ( s ) i = −h ( ~ξ ( s ) ·∇ ) ~v ( s ) , ~φ i = −h P [( ~ξ ( s ) ·∇ ) ~v ( s )] , ~φ i , since ~φ is a divergence free vector. Remark 1.
We remark that all the terms in (11) are meaningful. We showthe basic estimates for the trilinear terms, by means of H¨older and Sobolevinequalities: (cid:12)(cid:12)(cid:12)(cid:12)Z t h ( ~v ( s ) · ∇ ) ~φ, ~ξ ( s ) i ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ k ~φ k H Z t k ~v ( s ) k L k ~ξ ( s ) k L ds ≤ C k ~φ k H Z t k ~v ( s ) k H k ~ξ ( s ) k L ds ≤ C k ~φ k H Z t k ~ξ ( s ) k L k ~ξ ( s ) k L ds ≤ C k ~φ k H k ~ξ k L ∞ (0 ,T ; L ) k ~ξ k L (0 ,T ; L ) and similarly (cid:12)(cid:12)(cid:12)(cid:12)Z t h ( ~ξ ( s ) · ∇ ) ~φ, ~v ( s ) i ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k ~φ k H Z t k ~ξ ( s ) k L k ~v ( s ) k L ds ≤ C k ~φ k H k ~ξ k L (0 ,T ; L ) k ~ξ k L ∞ (0 ,T ; L ) Here and in the following, we denote by C a generic constant, which mayvary from line to line. However a subscript denotes that the constant dependson the specified parameters. Remark 2.
To prove the well posedness of system (10) , we shall exploit thepathwise technique used the first time in [2] and later on in a more usefulway in [8]. We shall transform the stochastic equation of Itˆo type (10) into a andom equation which behaves like a deterministic equation when studied for P -a.e. ω ∈ Ω , that is we find estimates for the paths of the solution process.The solution process will enjoy more properties as a stochastic process;as in the 2D setting, we shall prove pathwise uniqueness and continuousdependence on the initial data in L . Thus our solution will be a strongsolution from the point of view of stochastic differential equations (see e.g.[13]), and a Feller and Markov process in L . For these details, see [9] andreferences therein. This is a technical section, where we present the estimates to be used inproving the well posedness of system (10) and (6).First, we present a classical result.
Lemma 3.
Let ~u, ~v, ~w : R → R be smooth D -periodic and divergence freevector fields. Then h P [( ~u · ∇ ) ~v ] , ~w i = −h P [( ~u · ∇ ) ~w ] , ~v i (13) In particular h P [( ~u · ∇ ) ~v ] , ~v i = 0 (14) Proof.
First h P [( ~u · ∇ ) ~v ] , ~w i = h ( ~u · ∇ ) ~v, ~w i = X i,j =1 Z D u ( i ) ( ~x ) ∂ i v ( j ) ( ~x ) w ( j ) ( ~x ) d~x Then by integration by parts we get (13). The relationship (14) is obtainedfrom (13) by taking ~w = ~v .By density, the above results hold for all vectors giving meaning to theabove expressions. One can find estimates on the trilinear term in [21].Here we present particular estimates, not included in [21], and useful in thesequel. Their proofs are based on Sobolev embeddings theorems and H¨olderinequalities. Lemma 4.
Let c ≥ . Then there exists a positive constant C (dependingon c ) such that for any ǫ > we have |h ( ~u · ∇ ) ~u , ~u i| ≤ ǫ k ~u k H c + Cǫ k ~u k H k ~u k L (15)10 h ( ~u · ∇ ) ~u , ~u i| ≤ ǫ k ~u k H c + Cǫ k ~u k L k ~u k H c (16) |h ( ~u · ∇ ) ~u , ~u i| ≤ ǫ (cid:13)(cid:13) ~u (cid:13)(cid:13) H c + Cǫ k ~u k H k ~u k L (17) for all vectors making finite each r.h.s.Proof. We begin with the first inequality: |h ( ~u · ∇ ) ~u , ~u i| ≤ k ~u k L k∇ ~u k L k ~u k L by H¨older inequality ≤ C k ~u k H k ~u k H k ~u k L by Sobolev embedding H ⊂ L ≤ C c k ~u k H k ~u k H c k ~u k L ≤ ǫ k ~u k H c + C c ǫ k ~u k H k ~u k L by Cauchy inequalityFor the second inequality, we proceed in a similar way: |h ( ~u · ∇ ) ~u , ~u i| ≤ k ~u k L k∇ ~u k L k ~u k L ≤ C k ~u k L k ~u k H k ~u k H ≤ C c k ~u k L k ~u k H c k ~u k H c Then we apply Cauchy inequality to get the desired result.For the third inequality, we have h ( ~u · ∇ ) ~u , ~u i = −h ( ~u · ∇ ) ~u , ~u i from (13). Then we get (17) from (15). Lemma 5.
Let c ≥ . Then there exists a positive constant C (dependingon c ) such that for any ǫ > we have |h ( ~u · ∇ ) ~u , ~u i| ≤ ǫ (cid:13)(cid:13) ~u (cid:13)(cid:13) H c + Cǫ (cid:13)(cid:13) ~u (cid:13)(cid:13) H c (cid:13)(cid:13) ~u (cid:13)(cid:13) L for all vectors making finite the r.h.s..Proof. First we consider the range of values ≤ c < . We have − c + − c + ≤ H c ⊂ L − c , H c ⊂ L − c . Thus, H¨older and Sobolevinequalities give |h ( ~u · ∇ ) ~u , ~u i| ≤ k ~u k L − c k∇ ~u k L − c k ~u k L ≤ C c k ~u k H c k∇ ~u k H c k ~u k L ≤ C c k ~u k H c k ~u k H c k ~u k L c ≥ , we use the Sobolev embeddings H ⊂ L and H ⊂ L . Therefore, again we estimate |h ( ~u · ∇ ) ~u , ~u i| ≤ k ~u k L k∇ ~u k L k ~u k L ≤ C k ~u k H k∇ ~u k H k ~u k L ≤ C c k ~u k H c k ~u k H c k ~u k L Applying Cauchy inequality we conclude the proof.
When we neglect the non linearites in system (10) for the vorticity, we get ( d~ζ + ( − ∆) c ~ζ dt = ( − ∆) − b d ~w ∇ · ~ζ = 0 (18)Here the second equation keeps track of the fact that the vorticity vector isdivergence free. So ~ζ is the usual Ornstein-Uhlenbeck process, well studiedin the literature. Here we assume ~ζ (0) = ~
0. Therefore the mild solution of(18) is ~ζ ( t ) = Z t e − ( − ∆) c ( t − s ) ( − ∆) − b d ~w ( s ) (19)(see e.g. [4]). We have Proposition 6.
Let b + c > a + 12 (20) Then, for any m ∈ N ~ζ ∈ C ( R + ; H a m ) P − a.s Proof.
The proof is basically the same as that in [3] proving that ~ζ has P -a.e.path in C ( R + ; H a ). Working on the torus, we can improve that result getting ~ζ ∈ C ( R + ; H a m ).The factorization method uses that ~ζ ( t ) = sin( πα ) π Z t t − s ) − α e − ( − ∆) c ( t − s ) ~Y α ( s ) ds (21)for 0 < α <
1, with ~Y α ( s ) = Z s s − r ) α e − ( − ∆) c ( s − r ) ( − ∆) − b d ~w ( r )12ow we prove that under assumption (20) there exists α ∈ (0 , ) such that E k ~Y α k mL m (0 ,T ; H a m ) < ∞ (22)for any m ∈ N .For fixed ~x and t , [( − ∆) a/ ~Y α ]( t, ~x ) is a Gaussian random variable givenby the sum of independent Gaussian random variables( − ∆) a/ ~Y α ( t, ~x ) = X ~k ∈ Z | ~k | a X j =1 Z t t − s ) α e −| ~k | c ) ( t − s ) | ~k | − b ~b ~k,j dβ ~k,j ( s ) e i~k · ~x Therefore the variance of ( − ∆) a/ ~Y α ( t, ~x ) is the sum of the variance of eachaddend: E | ( − ∆) a/ ~Y α ( t, ~x ) | = X ~k ∈ Z | ~k | a − b Z t t − s ) α e − | ~k | c ) ( t − s ) ds = X ~k ∈ Z | ~k | a − b Z t r α e − | ~k | c ) r dr = X ~k ∈ Z | ~k | a − b | ~k | c )(2 α − Z t | ~k | c ) u α e − u du ≤ X ~k ∈ Z | ~k | a − b | ~k | c )(2 α − Z ∞ u α e − u du = C α X ~k ∈ Z | ~k | a − b +2(1+ c )(2 α − where the constant C α is finite for any α < .Since ( − ∆) a/ ~Y α ( t, ~x ) is a centered Gaussian random variable, for anyinteger m we have E | ( − ∆) a/ ~Y α ( t, ~x ) | m = C m (cid:16) E | ( − ∆) a/ ~Y α ( t, ~x ) | (cid:17) m ≤ C m,α X ~k ∈ Z | ~k | a − b +2(1+ c )(2 α − m Integrating with respect to the variables t ∈ [0 , T ] and ~x ∈ D we get E k ~Y α k mL m (0 ,T ; H a m ) ≤ C m,α T (2 π ) X ~k ∈ Z | ~k | a − b +2(1+ c )(2 α − m a − b + 2(1 + c )(2 α − < − b + c > a + 12 + 2 α (1 + c ) (23)If (20) holds then there exists α > α we have proved (22).Now, given (22), with a trivial modification of the proof of Lemma 2.7 in[3], from (21) we get E sup ≤ t ≤ T k ~ζ ( t ) k mH a m ≤ C m,T k ~Y α k mL m (0 ,T ; L m ) and the continuity result. As explained before, we consider the system obtained from (10) by neglectingthe term P [( ~ξ · ∇ ) ~v ] in the first equation. This is d~η + ( − ∆) c ~η dt + P [( ~v · ∇ ) ~η ] dt = ( − ∆) − b d ~w ∇ · ~v = 0 ~η = ∇ × ~v (24)We call it the vorticity transport system, since its first equation is a reducedform of the vorticity equation in (10): in (24) vorticity is only transported,not stretched.Let us point out a feature of the equation of ~η . The nonlinearity ( ~v · ∇ ) ~η has a peculiar form similar to that appearing in the regularized form of Leray- α models for fluids (see e.g. [1]), that is the first entry of the bilinear term P [( ~v · ∇ ) ~η ] is not the unknown ~η itself but indeed ~v , which has one ordermore of regularity with respect to ~η (recall that if ~η ∈ H bp then ~v ∈ H b +1 p ).Therefore, even if ~η satisfies a nonlinear equation, the quadratic term ( ~v · ∇ ) ~η in (24) (with ~η = ∇ × ~v ) behaves better than ( ~v · ∇ ) ~v in (1) and this makesthe difference in the analysis of systems (24) and (1).As far as the technique is concerned, we point out that in order to getexistence and uniqueness results, we could look for mean estimates. However,for our purpose it is enough to get pathwise estimates (see Theorem 12).Moreover, the advantage of the pathwise approach is twofold: the existence14esult is obtained asking weaker assumption on the covariance of the noiseand the regularity results are easily obtained. To see the first advantage,thanks to (13), with the usual techniques (see e.g. [2], [9]) we can get E (cid:20) k ~η ( t ) k L + 2 Z t k ~η ( s ) k H c ds (cid:21) ≤ k ~η (0) k L + T r (cid:0) ( − ∆) − b (cid:1) t This requires
T r (cid:0) ( − ∆) − b (cid:1) < ∞ , i.e. X ~k ∈ Z | ~k | − b < ∞ which holds when b > . But Theorem 8 allows to get existence of a basicsolution ~η for b > − c . Since our task in Theorem 12 will be to estimate k ( − ∆) b P [( ~η · ∇ ) ~v ] k L it is clear than the smaller is b the easier is our task.For this aim, we set ~β = ~η − ~ζ and exploit that the noise is independentof the unknowns; then ∂ ~β∂t + ( − ∆) c ~β + P [( ~v · ∇ )( ~β + ~ζ )] = ~ ∇ · ~v = 0 ∇ × ~v = ~β + ~ζ (25)System (25) is studied pathwise. We have the following result Proposition 7. i) Assume that ( c ≥ b + c > Then, for any ~β (0) ∈ L there exists a solution to (25) such that ~β ∈ C ([0 , T ]; L ) ∩ L (0 , T ; H c ) P − a.s. ii) Assume that ( c ≥ b + c > Then, for any ~β (0) ∈ H the solution given in i) enjoys also ~β ∈ C ([0 , T ]; H ) ∩ L (0 , T ; H c ) P − a.s. ii) Assume that ( c ≥ b + c > Then, for any ~β (0) ∈ H the solution given in i) enjoys also ~β ∈ C ([0 , T ]; H ) ∩ L (0 , T ; H c ) P − a.s. Proof.
We proceed pathwise. The technique to prove existence is to considerfirst the finite dimensional problem, obtained by applying the projectionoperator Π n to (25). The goal is to find suitable a priori estimates, uniformlyin n . Thus, when any finite dimensional (Galerkin) problem has a solutionwe pass to the limit as n → ∞ to get an existence result for (25). Thistechnique, based on finite dimensional approximation, is well known (see e.g.[20, 21]). Therefore we look for a priori estimates for the full system (25);they hold for any Galerkin approximation as well, but we skip the details forthe limit as n → ∞ .i) We multiply the l.h.s. of the first equation of (25) by ~β ( t ) and integrateover D . Using (13)-(14) and then H¨older and Sobolev inequalities, we get12 ddt k ~β ( t ) k L + k ~β ( t ) k H c = −h P [( ~v ( t ) · ∇ ) ~ζ ( t )] , ~β ( t ) i = h ( ~v ( t ) · ∇ ) ~β ( t ) , ~ζ ( t ) i≤ C k ~v ( t ) k L k ~β ( t ) k H k ~ζ ( t ) k L ≤ C c k ~v ( t ) k H k ~β ( t ) k H c k ~ζ ( t ) k L ≤ C k ~β ( t ) + ~ζ ( t ) k L k ~β ( t ) k H c k ~ζ ( t ) k L Cauchy inequality gives12 ddt k ~β ( t ) k L + k ~β ( t ) k H c ≤ k ~β ( t ) k H c + C k ~ζ ( t ) k L k ~β ( t ) k L + C k ~ζ ( t ) k L (26)Therefore, Gronwall inequality applied to ddt k ~β ( t ) k L ≤ C k ~ζ ( t ) k L k ~β ( t ) k L + C k ~ζ ( t ) k L gives sup ≤ t ≤ T k ~β ( t ) k L ≤ C ( b, c, T, k ~β (0) k L , k ~ζ k L ∞ (0 ,T ; L ) )Integrating in time (26) we get Z T k ~β ( t ) k H c dt ≤ ˜ C ( b, c, T, k ~β (0) k L , k ~ζ k L ∞ (0 ,T ; L ) )16e remind that ~ζ ∈ C ([0 , T ]; L ) if 2 b + c > , according to Proposition6. Then these a priori estimates give ~β ∈ L ∞ (0 , T ; L ) ∩ L (0 , T ; H c ).Moreover, ∂ ~β∂t = − ( − ∆) c ~β − P [( ~v · ∇ ) ~β ] − P [( ~v · ∇ ) ~ζ ]Given the regularity of ~β we have that the r.h.s. belongs to L (0 , T ; H − − c );indeed ( − ∆) c ~β ∈ L (0 , T ; H − − c ) and the two latter terms belong to L (0 , T ; H − ). Let us see this; we proceed as before |h ( ~v · ∇ ) ~β, ~u i| = |h ( ~v · ∇ ) ~u, ~β i| ≤ k ~v k L k∇ ~u k L k ~β k L This gives k ( ~v · ∇ ) ~β k H − = sup k ~u k H > |h ( ~v · ∇ ) ~β, ~u i|k ~u k H ≤ k ~v k L k ~β k L ≤ C k ~v k H k ~β k H ≤ C ( k ~β k L + k ~ζ k L ) k ~β k H Similarly we deal with ( ~v · ∇ ) ~ζ : k ( ~v · ∇ ) ~ζ k H − ≤ k ~v k L k ~ζ k L ≤ C k ~ζ k L + k ~ζ k L k ~β k L We recall that the space { ~β ∈ L (0 , T ; H c ) : ∂ ~β∂t ∈ L (0 , T ; H − − c ) } iscompactly embedded in L (0 , T ; L ).These are the basic results to implement the Galerkin approximation.As far as the continuity is concerned, the fact that ~β ∈ L (0 , T ; H c )and ∂ ~β∂t ∈ L (0 , T ; H − − c ) implies ~β ∈ C ([0 , T ]; L ) (see Ch III Lemma 1.2 of[20]).ii) We need a priori estimates and we proceed as in the previous step. Wemultiply the l.h.s. of the first equation of (25) by − ∆ ~β ( t ) and integrate on D . We get12 ddt k ~β ( t ) k H + k ~β ( t ) k H c = h ( ~v ( t ) · ∇ )( ~β ( t ) + ~ζ ( t )) , ∆ ~β ( t ) i We estimate the r.h.s. as follows h ( ~v · ∇ )( ~β + ~ζ ) , ∆ ~β i ≤ k ( ~v · ∇ )( ~β + ~ζ ) k L k ∆ ~β k L ≤ k ~v k L ∞ k ~β + ~ζ k H k ~β k H ≤ C k ~v k H k ~β + ~ζ k H k ~β k H since H ⊂ L ∞ ≤ C c k ~β + ~ζ k H k ~β k H c ≤ k ~β k H c + C k ~β k H + k ~ζ k H ddt k ~β ( t ) k H + k ~β ( t ) k H c ≤ C k ~β k H + k ~ζ k H and we conclude as before using Gronwall Lemma and the fact that ~β ∈ L (0 , T ; H ) from i) and ~ζ ∈ C ([0 , T ]; H ) from Proposition 6, gettingsup ≤ t ≤ T k ~β ( t ) k H ≤ C ( b, c, T, k ~β (0) k H , k ~ζ k L ∞ (0 ,T ; H ) ) Z T k ~β ( t ) k H c dt ≤ ˜ C ( b, c, T, k ~β (0) k H , k ~ζ k L ∞ (0 ,T ; H ) )Continuity in time is obtained as before.iii) We multiply the l.h.s. of the first equation of (25) by ( − ∆) ~β ( t ) andintegrate on D . We get12 ddt k ~β ( t ) k H + k ~β ( t ) k H c = −h ( ~v ( t ) · ∇ )( ~β ( t ) + ~ζ ( t )) , ( − ∆) ~β ( t ) i We estimate the r.h.s. as follows. First, we use the estimate for the product;by means of the Sobolev embedding H ⊂ L ∞ we get k f g k H ≤ k g ∇ f k L + k f ∇ g k L ≤ k∇ f k L ∞ k g k L + k f k L ∞ k∇ g k L ≤ C k f k H k g k L + C k f k H k g k H Hence, for the trilinear term we get h ( ~v · ∇ )( ~β + ~ζ ) , ( − ∆) ~β i = h ( − ∆) [( ~v · ∇ )( ~β + ~ζ )] , ( − ∆) ~β i≤ k ( ~v · ∇ )( ~β + ~ζ ) k H k ~β k H ≤ C (cid:16) k ~v k H k ~β + ~ζ k H + k ~v k H k ~β + ~ζ k H (cid:17) k ~β k H ≤ C k ~β + ~ζ k H k ~β + ~ζ k H k ~β k H ≤ C c k ~β + ~ζ k H k ~β + ~ζ k H k ~β k H c ≤ k ~β k H c + C k ~β + ~ζ k H k ~β k H + C k ~β + ~ζ k H k ~ζ k H This gives ddt k ~β ( t ) k H + k ~β ( t ) k H c ≤ C k ~β ( t )+ ~ζ ( t ) k H k ~β ( t ) k H + C k ~ζ ( t ) k H k ~β ( t ) k H + C k ~ζ ( t ) k H Since ~β ∈ C ([0 , T ]; H ) from step ii) and ~ζ ∈ C ([0 , T ]; H ) from Proposition6, we get firstsup ≤ t ≤ T k ~β ( t ) k H ≤ C ( b, c, T, k ~β (0) k H , k ~ζ k L ∞ (0 ,T ; H ) )18nd then Z T k ~β ( t ) k H c dt ≤ ˜ C ( b, c, T, k ~β (0) k H , k ~ζ k L ∞ (0 ,T ; H ) )Continuity in time is obtained as before. This concludes the proof.Now we come back to the unknown ~η = ~β + ~ζ . The definition of basicsolution is the same as that for ~ξ given at the end of Section 2, with theobvious modification of the equation by neglecting P [( ~ξ · ∇ ) ~v ]. Theorem 8. i) Assume that ( c ≥ b + c > Then, for any ~η (0) ∈ L there exists a unique process ~η which is a basicsolution to (24) such that ~η ∈ C ([0 , T ]; L ) ∩ L (0 , T ; L ) P -a.s.Moreover there is continuous dependence on the initial data: given two initialdata ~η (0) , ~η ⋆ (0) ∈ L we have k ~η (0) − ~η ⋆ (0) k L → ⇒ k ~η − ~η ⋆ k C ([0 ,T ]; L ) → ii) Assume that ( c ≥ b + c > Then, for any ~η (0) ∈ H the solution given in i) enjoys also ~η ∈ C ([0 , T ]; H ) P − a.s. iii) Assume that ( c ≥ b + c > Then, for any ~η (0) ∈ H the solution given in i) enjoys also ~η ∈ C ([0 , T ]; H ) P − a.s. roof. The existence comes from the existence results on ~β , ~ζ . Moreover ~ζ ∈ C ([0 , T ]; L q ) ∀ q < ∞ and by Sobolev embedding ~β ∈ L (0 , T ; H c ) ⊂ L (0 , T ; H ) ⊂ L (0 , T ; L )Merging toghether the regularity of these processes we get our results for ~η .As far as continuous dependence on the initial data is concerned, let ustake two basic solutions ~η and ~η with ~η (0) = ~η (0) ∈ L ; at least we have ~η , ~η ∈ C ([0 , T ]; L ) ∩ L (0 , T ; L )We define ~y = ~η − ~η ; then the system fulfilled by ~y can be written as ∂~y∂t + ( − ∆) c ~y + P [( ~v · ∇ ) ~y ] + P [(( ~v − ~v ) · ∇ ) ~η ] = ~ ∇ · ~v = ∇ · ~v = 0 ~y = ∇ × ( ~v − ~v )We estimate the following term, as usual: |h [( ~v − ~v ) · ∇ ] ~η , ~y i| = |h [( ~v − ~v ) · ∇ ] ~y, ~η i|≤ k ~y k H c + C k ~η k L k ~v − ~v k H from (15) ≤ k ~y k H c + C k ~η k L k ~y k L Then taking the scalar product of the the first equation for ~y with ~y , inte-grating on the spatial domain and using (13), we get ddt k ~y ( t ) k L + k ~y ( t ) k H c ≤ C k ~η ( t ) k L k ~y ( t ) k L Recall that ~η ∈ L (0 , T ; L ). Applying Gronwall lemma to ddt k ~y ( t ) k L ≤ C k ~η ( t ) k L k ~y ( t ) k L we get sup ≤ t ≤ T k ~y ( t ) k L ≤ k ~y (0) k L e C R T k ~η ( t ) k L dt This gives the continuous dependence on the initial data; uniqueness is ob-tained when ~y (0) = ~
0. 20
The vorticity equation
Now we consider the full nonlinear system (10). If the initial velocity is moreregular, say ~v (0) ∈ H (i.e. ~ξ (0) ∈ L ), one can prove a local existence anduniqueness result for c = 0; global existence holds only for c ≥ (see [6]). Inthis paper we improve the results for c ≥ considering initial data ~ξ (0) ∈ H and H .We need a preliminary result for the velocity, fulfilling (3) with the noiseobtained from a Wiener process ~w vel such that ∇ × ~w vel = ( − ∆) − b ~w , that is ~w vel ( t, ~x ) = X ~k ∈ Z | ~k | − b − [ − ~b ~k, β ~k, ( t ) + ~b ~k, β ~k, ( t )] e i~k · ~x Therefore (3) becomes ( d~v + ( − ∆) c ~v dt + ( ~v · ∇ ) ~v dt + ∇ p dt = d ~w vel ∇ · ~v = 0 (27) Proposition 9.
Assume that ( c ≥ b > Then for any ~v (0) ∈ L there exists a process ~v with P -a.e. path in L ∞ (0 , T ; L ) ∩ L (0 , T ; H c ) , solving (27) .Proof. We know the result for c = 0 (see [9]); the case c > d k ~v ( t ) k L ;the details can be found in [9]. We have E k ~v ( t ) k L + 2 Z t E k ~v ( s ) k H c ds ≤ k ~v (0) k L + t X ~k ∈ Z | ~k | − b +1) The series in the r.h.s. converges if and only if 2(2 b + 1) >
3, i.e. b > .These estimates improves the regularity: ~v ∈ L (0 , T ; H c ), P -a.s.Now we consider the unknown ~ξ . Let ~δ := ~ξ − ~ζ ; bearing in mind theequations for ~ξ and ~ζ we have that this new unknown satisfies ∂~δ∂t + ( − ∆) c ~δ + P [( ~v · ∇ ) ~δ − ( ~δ · ∇ ) ~v + ( ~v · ∇ ) ~ζ − ( ~ζ · ∇ ) ~v ] = ~ ~v and ~δ are linked through ~δ = − ~ζ + ∇ × ~v .Our aim is to find existence and regularity results for ~δ in order to obtainthe same results for ~ξ . This requires c ≥ .As in the previous section we look for pathwise results. Proposition 10. i) Assume that ( c ≥ b > Then, for any ~δ (0) ∈ L there exists a solution to (28) such that ~δ ∈ C ([0 , T ]; L ) ∩ L (0 , T ; H c ) P − a.s. ii) Assume that c ≥ b > b + c > Then, for any ~δ (0) ∈ H the solution given in i) enjoys also ~δ ∈ C ([0 , T ]; H ) ∩ L (0 , T ; H c ) P − a.s. iii) Assume that c ≥ b > b + c > Then, for any ~δ (0) ∈ H the solution given in i) enjoys also ~δ ∈ C ([0 , T ]; H ) ∩ L (0 , T ; H c ) P − a.s. Proof. i) First, notice that if c ≥ and b > then 2 b + c > > . ThereforeProposition 6 provides that for any finite p we have ~ζ ∈ C ([0 , T ]; L p ) a.s..We deal with (28) as we did with (25). So12 ddt k ~δ ( t ) k L + k ~δ ( t ) k H c = h ( ~δ · ∇ ) ~v − ( ~v · ∇ ) ~ζ + ( ~ζ · ∇ ) ~v, ~δ i From Lemma 5 h ( ~δ · ∇ ) ~v, ~δ i ≤ k ~δ k H c + C k ~v k H c k ~δ k L |h ( ~v · ∇ ) ~ζ , ~δ i| ≤ k ~δ k H c + C k ~v k H k ~ζ k L From (16) of Lemma 4 |h ( ~ζ · ∇ ) ~v, ~δ i| ≤ k ~δ k H c + C k ~v k H k ~ζ k L Summing up, we get ddt k ~δ ( t ) k L + k ~δ ( t ) k H c ≤ C k ~v ( t ) k H c k ~δ ( t ) k L + C k ~ζ ( t ) k L k ~v ( t ) k H c From Proposition 9, we know that ~v ∈ L (0 , T ; H c ); moreover ourassumption with Proposition 6 give ~ζ ∈ C ([0 , ] T ; H ). Then by Gronwalllemma we get sup ≤ t ≤ T k ~δ ( t ) k L < ∞ and integrating in time Z T k ~δ ( t ) k H c dt < ∞ The continuity in time is obtained as in Proposition 7.ii) We need a priori estimates and we proceed as in the previous step. Wemultiply the l.h.s. of the first equation of (28) by − ∆ ~δ ( t ) and integrate on D . We get12 ddt k ~δ ( t ) k H + k ~δ ( t ) k H c = h ( ~v ( t ) · ∇ )( ~δ ( t ) + ~ζ ( t )) , ∆ ~δ ( t ) i − h (( ~δ ( t ) + ~ζ ( t )) · ∇ ) ~v ( t ) , ∆ ~δ ( t ) i We estimate the latter term in the r.h.s. as usual: |h (( ~δ + ~ζ ) · ∇ ) ~v, ∆ ~δ i| ≤ k ~δ + ~ζ k L k∇ ~v k L k ∆ ~δ k L ≤ C k ~δ + ~ζ k H k∇ ~v k H k ~δ k H ≤ C c k ~δ + ~ζ k H k ~δ k H c ≤ k ~δ k H c + C k ~δ k H + C k ~ζ k H With this estimate and dealing with the other trilinear term as in theproof of Proposition 7 ii), we obtain ddt k ~δ ( t ) k H + k ~δ ( t ) k H c ≤ C k ~δ ( t ) k H + k ~ζ ( t ) k H ~δ ∈ L (0 , T ; H ) from the previous step and ~ζ ∈ C ([0 , T ]; H ) fromProposition 6, we conclude as in the proof of Proposition 7 ii).iii) We multiply the l.h.s. of the first equation of (28) by ( − ∆) ~δ ( t ) andintegrate on D . We get12 ddt k ~δ ( t ) k H + k ~δ ( t ) k H c = −h ( ~v ( t ) · ∇ )( ~δ ( t ) + ~ζ ( t )) , ( − ∆) ~δ ( t ) i + h (( ~δ ( t ) + ~ζ ( t )) · ∇ ) ~v ( t ) , ( − ∆) ~δ ( t ) i We are left to estimate the latter trilinear term. First, we use the estimatefor the product; by means of the Sobolev embeddings H ⊂ L ∞ and H ⊂ L we get k f g k H ≤ k g ∇ f k L + k f ∇ g k L ≤ k∇ f k L k g k L ∞ + k f k L k∇ g k L ≤ C k f k H k g k H + C k f k H k g k H Hence, for the trilinear term we get h (( ~δ + ~ζ ) · ∇ ) ~v, ( − ∆) ~δ i ≤ k (( ~δ + ~ζ ) · ∇ ) ~v k H k ~δ k H ≤ C k ~δ + ~ζ k H k∇ ~v k H k ~δ k H ≤ C c k ~δ + ~ζ k H k ~δ + ~ζ k H k ~δ k H c ≤ k ~δ k H c + C k ~δ + ~ζ k H k ~δ k H + C k ~δ + ~ζ k H k ~ζ k H Therefore, keeping in mind the proof of Proposition 7 iii) to estimate theother trilinear term, we obtain ddt k ~δ ( t ) k H + k ~δ ( t ) k H c ≤ C k ~δ ( t )+ ~ζ ( t ) k H k ~δ ( t ) k H + C k ~δ ( t )+ ~ζ ( t ) k H k ~ζ ( t ) k H Since ~δ ∈ L (0 , T ; H ) from the previous step and ~ζ ∈ C ([0 , T ]; H ) fromProposition 6, we conclude as in the proof of Proposition 7 iii).Now we have the result for ~ξ = ~δ + ~ζ . Theorem 11. i) Assume that ( c ≥ b > Then, for any ~ξ (0) ∈ L there exists a unique process ~ξ which is a basicsolution to (10) such that ~ξ ∈ C ([0 , T ]; L ) ∩ L (0 , T ; L )24 -a.s.Moreover there is continuous dependence on the initial data: given twoinitial data ~ξ (0) , ~ξ ⋆ (0) ∈ L we have k ~ξ (0) − ~ξ ⋆ (0) k L → ⇒ k ~ξ − ~ξ ⋆ k C ([0 ,T ]; L ) → ii) Assume that c ≥ b > b + c > Then, for any ~ξ (0) ∈ H the solution given in i) enjoys also ~ξ ∈ C ([0 , T ]; H ) P − a.s. iii) Assume that c ≥ b > b + c > Then, for any ~ξ (0) ∈ H the solution given in i) enjoys also ~ξ ∈ C ([0 , T ]; H ) P − a.s. Proof. i) If c ≥ and b > then 2 b + c > . Therefore Proposition 6 providesthat for any finite p we have ~ζ ∈ C ([0 , T ]; L p ) a.s.. We merge the results ofProposition 10 for ~δ with those of Proposition 6 for ~ζ to get existence of ~ξ and its regularity. This is the same as in Theorem 8.As far as continuous dependence on the initial data is concerned, weproceed as in the proof of Theorem 8. The additional term does not give anyproblem; we estimate it as follows. Set ~y = ~ξ − ~ξ ; then the system fulfilledby ~y can be written as ∂~y∂t + ( − ∆) c ~y + P [( ~v · ∇ ) ~y + (( ~v − ~v ) · ∇ ) ~ξ − ( ~ξ · ∇ )( ~v − ~v ) − ( ~y · ∇ ) ~v ] = ~ ∇ · ~v = ∇ · ~v = 0 ~y = ∇ × ( ~v − ~v )Therefore, in the equation fulfilled by k ~y ( t ) k L , in addition to the termsappearing in the proof of Theorem 8 we also have h ( ~ξ · ∇ )( ~v − ~v ) , ~y i + h ( ~y · ∇ ) ~v , ~y i
25e have |h ( ~ξ · ∇ )( ~v − ~v ) , ~y i| ≤ k ~ξ k L k∇ ( ~v − ~v ) k L k ~y k L ≤ C k ~ξ k L k ~y k L k ~y k H ≤ C c k ~ξ k L k ~y k L k ~y k H c ≤ k ~y k H c + C k ~ξ k L k ~y k L and |h ( ~y · ∇ ) ~v , ~y i| ≤ k ~y k L k∇ ~v k L k ~y k L ≤ C k ~y k L k ~ξ k L k ~y k H ≤ C c k ~y k L k ~ξ k L k ~y k H c ≤ k ~y k H c + C k ~ξ k L k ~y k L Therefore ddt k ~y ( t ) k L ≤ C (cid:0) k ~ξ ( t ) k L + k ~ξ ( t ) k L (cid:1) k ~y ( t ) k L By Gronwall lemma, we get continuous dependence on the initial data.Uniqueness is obtained when ~y (0) = ~ Let T : ~ξ ~v be the mapping giving the solution to (7).We write system (10) as ( d~ξ + ( − ∆) c ~ξ dt + P [( T ~ξ · ∇ ) ~ξ ] dt − P [( ~ξ · ∇ ) T ~ξ ] dt = ( − ∆) − b d ~w ∇ · ~ξ = 0 (29)and system (24) as ( d~η + ( − ∆) c ~η dt + P [( T ~η · ∇ ) ~η ] dt = ( − ∆) − b d ~w ∇ · ~η = 0 (30)Denote by L ~ξ and L ~η the laws of the processes ~ξ and ~η respectively,when defined on a finite time interval [0 , T ]. Let σ T ( ~η ) denote the σ -algebragenerated by { ~η ( t ) } ≤ t ≤ T .We recall the main result of [5], [7], in a form adapted to our context;indeed in those papers it was sufficient to assume weak existence (withoutuniqueness) for system (29). 26 heorem 12. Assume (30) and (29) have a unique basic solution with thesame initial data in H . If P { R T k ( − ∆) b P [( ~η ( t ) · ∇ ) T ~η ( t )] k L dt < ∞} = 1 , (31) P { R T k ( − ∆) b P [( ~ξ ( t ) · ∇ ) T ~ξ ( t )] k L dt < ∞} = 1 , (32) then the laws L ~ξ and L ~η , defined as measures on the Borel subsets of C ([0 , T ]; H ) ,are equivalent.In particular for the Radon-Nykodim derivative we have d L ~ξ d L ~η ( ~η ) = E h e R T h ( − ∆) b P [( ~η ( t ) ·∇ ) T ~η ( t )] ,d ~w ( s ) i− R T k ( − ∆) b P [( ~η ( t ) ·∇ ) T ~η ( t )] k L ds (cid:12)(cid:12) σ T ( ~η ) i (33) P -a.s.Finally, L ~ξ is unique. From this we get our main result.
Theorem 13.
Let ( c > b = 1 If ~η (0) = ~ξ (0) ∈ H , then the laws L ~ξ and L ~η are equivalent and (33) holds.Proof. We use Theorems 11, iii); notice that the conditions on b and c arefulfilled if b = 1 and c > . We have only to check estimates (31) - (32)with b = 1. This follows easily, since H is a multiplicative algebra and kT ~ξ k H ≤ C k ~ξ k H ; therefore k P [( ~ξ · ∇ ) T ~ξ ] k H ≤ C k ~ξ k H k∇T ~ξ k H ≤ C k ~ξ k H kT ~ξ k H ≤ C k ~ξ k H and finally we use that the paths are in C ([0 , T ]; H ).We point out that the restriction c > cannot be weakened with thistechnique using k ( − ∆) b P [( ~ζ · ∇ ) T ~ζ ] k L ≤ k ( ~ζ · ∇ ) T ~ζ k H b ≤ C k ~ζ k H b for b large enough. Indeed, Proposition 6 provides ζ ∈ C ([0 , T ]; H b ) a.s. if c > . And the paths of ~ξ, ~η cannot have better behavior than those of ~ζ . Acknowledgments.
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