In this article we develop an existence and uniqueness theory of variational solutions for ..... We can easily infer from the above hypotheses that each term in (9) is ...... [49] Stein, E., Singular Integrals and Differentiability Properties of Functions.
Variational Solutions for Partial Differential Equations Driven by a Fractional Noise David Nualart and Pierre-A. Vuillermot Facultat de Matem`atiques, Universitat de Barcelona UMR-CNRS 7502, Universit´e Henri-Poincar´e, Nancy Abstract In this article we develop an existence and uniqueness theory of variational solutions for a class of non autonomous stochastic partial differential equations of parabolic type defined on a bounded open subset D ⊂ Rd and driven by an infinite-dimensional multiplicative fractional noise. We introduce two notions of such solutions for them and prove their existence and their indistinguishability by assuming that the noise is derived from “ an L2 (D)-valued fractional Wiener process W H with Hurst ” parameter H∈
1 ,1 γ+1
, whose covariance operator satisfies appropriate
integrability conditions, and where γ ∈ (0, 1] denotes the H¨ older exponent of the derivative of the nonlinearity in the stochastic term of the equations. We also prove the uniqueness of solutions when the stochastic term is an affine function of the unknown random field. Our existence and uniqueness proofs rest upon the construction and the convergence of a suitable sequence of Faedo-Galerkin approximations, while our proof of indistinguishability is based on certain density arguments as well as on new continuity properties of the stochastic integral we define with respect to W H .
1
Introduction and Outline
It is well-known that the self-similarity and the long-range correlation properties of fractional Brownian motion make this stochastic process quite relevant to the mathematical modelling of a host of applications in engineering, the natural sciences and mathematical finance, to name only a few (see, for instance, [10], [12], [13], [36], [37], [46] and their numerous references). Many of these applications call for the integration of ordinary or partial differential equations driven by a finite or an infinite-dimensional fractional noise, which, in turn, requires the development of a suitable calculus. Accordingly, various forms of stochastic analysis with respect to fractional Brownian motion have recently been put forth by many authors, which has led to the introduction of fundamental tools such as continuity criteria for stochastic integrals, versions of Itˆo’s
1
formulae, maximal inequalities, Wong-Zaka¨ı approximations and related properties (see, for instance, [2], [3], [4], [6], [11], [15], [16], [17], [21], [26], [32], [44], [45], [52] and [53]), as well as to the proof of existence and uniqueness theorems for stochastic differential or integral equations as in [23], [28], [29], [31] or [40]. In addition, several works have recently been devoted to the investigation of the existence, the uniqueness, the regularity properties and the long-time behavior of solutions to semilinear parabolic partial differential equations driven by an infinite-dimensional fractional noise; thus, in [22], the authors have proved an existence and uniqueness theorem for a class of stochastic evolution equations driven by an additive noise; in [25], the author has analyzed a heat equation driven by a multiplicative, multiparameter fractional noise and estimated the Lyapunov exponents of the corresponding solutions; finally, the authors of [18] and [38] have proved the existence and the uniqueness of mild solutions to various classes of stochastic evolution equations driven by a fractional noise in a Hilbert space, by using techniques from fractional calculus and semi-group theory. In this article we develop an existence and uniqueness theory of variational solutions for a class of non autonomous, semilinear, stochastic partial differential equations driven by an infinite-dimensional multiplicative noise, which we can define in the following way: for d ∈ N+ let D ⊂ Rd be open, bounded, with a smooth boundary ∂D and satisfying the cone property (see, for instance, [1]); we write Ls (D) for the usual Lebesgue spaces of real-valued functions on D and k.ks for the corresponding norms, along with (., .)2 for the standard inner product in L2 (D); in this space we consider a linear, self-adjoint, positive trace-class operator C, which implies that C is an integral transform whose generating kernel we denote by κ; in the sequel we write (ei )i∈N+ for an orthonormal basis of L2 (D) consisting of eigenfunctions of the operator C and (λi )i∈N+ for the sequence of the corresponding eigenvalues. Let (BiH (t))t∈R+ i∈N+ be a sequence of one-dimensional, independent, identically distributed fractional Brownian motions with Hurst parameter H∈ (0, 1), defined on the complete probability space (Ω, F, P) and starting at the origin. We also introduce the L2 (D)-valued fractional Wiener process (W H (., t))t∈R+ by setting W H (., t) :=
+∞ X
1
λi2 ei (.)BiH (t)
(1)
i=1
where the series converges a.s. in the strong topology of L2 (D) by virtue of the basic properties of the BiH (t)’s and the fact that C is trace-class. From these basic properties, we can also conclude that (W H (., t))t∈R+ is a centered Gaussian process whose covariance is given by E (W H (., s), v)2 (W H (., t), vˆ)2 1 2H 2H = s + t2H − |s − t| (Cv, vˆ)2 2 Z 1 2H 2H = s + t2H − |s − t| dxdyκ(x, y)v(x)ˆ v (y) (2) 2 D×D 2
1 for all s, t ∈ R+ and all v, vˆ ∈ L2 (D). Let T ∈ R+ ∗ , H∈ 2 , 1 and let us consider the class of real, parabolic, initial-boundary value problems formally defined by du(x, t) = (div (k(x, t)∇u(x, t)) + g(u(x, t))) dt + h(u(x, t))W H (x, dt), (x, t) ∈ D × (0, T ] , u(x, 0) = ϕ(x), x ∈ D, ∂u(x, t) = 0, ∂n(k)
(x, t) ∈ ∂D × (0, T ] ,
(3)
where the last relation stands for the conormal derivative of u relative to the matrix-valued vector field k. We assume that the following hypotheses hold: 2 (K) The function k : D × (0, T ] 7→ Rd is Lebesgue-measurable and we have ki,j (.) = kj,i (.) for every i, j ∈ {1, ..., d}. Moreover, there exist constants k, k ∈ R+ ∗ such that the inequalities 2
2
k |q| ≤ (k(x, t)q, q)Rd ≤ k |q|
(4)
hold uniformly in (x, t) ∈ D ×(0, T ] for all q ∈ Rd , where we have written (., .)Rd for the standard Euclidean inner product in Rd . (L) The functions g, h : R 7→ R are Lipschitz continuous. As we shall see below, it will be convenient to separate Hypothesis (L) from the following, stronger hypothesis concerning h and H: (Hγ ) The derivative h0 : R 7→ R of h exists, is H¨older continuous with expo nent γ ∈ (0, 1] and bounded; moreover we have H∈
1 γ+1 , 1
.
2
(I) The initial condition ϕ is an L (D)-valued random variable. It is worth remembering here that for H= 12 , the sequence ((BiH (t))t∈R+ )i∈N+ reduces to a sequence of one-dimensional, independent, standard Brownian motions ((Bi (t))t∈R+ )i∈N+ ; accordingly, the series (1) reduces to the representation W (., t) :=
+∞ X
1
λi2 ei (.)Bi (t)
i=1
of a standard L2 (D)-valued Wiener process with nuclear covariance (see, for instance, [14]), and in this case we can trace an existence theory of variational solutions for (3) to the classical works [30] and [43]. We also note that such solutions have played an important rˆole in a number of situations, particularly in those dealing with the analysis of long-time behavior phenomena in population dynamics and population genetics (see, for instance, [5], [7], [8], [9], [42] and the references therein), aside from being indistinguishable from other important notions of solution as was recently proved in [48]. In contrast, for H6= 21 we are neither aware of any works concerning the existence of variational solutions for models as general as (3), nor are we aware of any connections between such solutions and other notions such as the mild solutions constructed in [38]. Accordingly, we shall organize the remaining part of this article in the following way: in Section 2 we state our main result concerning the existence, the indistinguishability and the uniqueness of two notions of variational solution for (3), 3
namely, variational solutions of type I and of type II. We devote Section 3 to the proof of the result stated in Section 2; in particular, we prove our existence and uniqueness result by investigating the convergence of a suitable Faedo-Galerkin approximation scheme; our arguments there rely upon the availability of good a priori estimates and upon a new compact embedding theorem, whose proofs involve mathematical tools such as the L2 (R)-boundedness properties of certain maximal functions in the sense of [49], as well as the Hardy-Littlewood-P´olya theorem regarding the L2 (R)-boundedness properties of integral transforms with homogeneous kernels, respectively (see, for instance, [24] or [49]). We also prove the indistinguishability of the two types of solution-random fields by invoking certain density arguments along with new continuity properties of the stochastic integral we define with respect to (W H (., t))t∈R+ . Finally, we refer the reader to [41] for a short announcement of our results.
2
Statement and Discussion of the Main Result
In the remaining part of this article we give a meaning to (3) by defining all the stochastic integrals through pathwise generalized Stieltjes integrals as is the case in [38], [40] and [52], to which we refer the reader for the basic definitions and properties (see also [51]); the techniques involved are those from fractional calculus (see, for instance, [47]). We write H 1 (D) for the usual Sobolev space of real-valued functions on D, (., .)1,2 for the corresponding inner product and 1 ,1 k.k1,2 for the induced norm; throughout the article we assume that H∈ γ+1 even in preparatory statements where the other requirements of Hypothesis γ ; we (Hγ ) are not all needed and fix once and for all an α ∈ 1 − H, γ+1 then introduce the Banach space B α,2 ([0, T ] ; L2 (D)) consisting of all Lebesguemeasurable mappings u : [0, T ] 7→ L2 (D) endowed with the norm k.kα,2 defined by !2 2 kukα,2
:=
esssup ku(., t)k2
Z +
T
Z dt
t∈[0,T ]
0
t
dτ
ku(., t) − u(., τ )k2
0
α+1
(t − τ )
!2 < +∞.
(5) The first notion of variational solution we introduce is one where we require the relevant test functions to depend only on the space variable. In addition to (K), (L), (Hγ ), (I) above, this calls for the following hypothesis regarding the spectral properties of the covariance operator C: (C) We have ei ∈ L∞ (D) for each i and +∞ X
1
λi2 kei k∞ < +∞.
(6)
i=1
Two remarks are in order regarding this hypothesis; on the one hand we can easily obtain the essential boundedness of the ei ’s from an integrability condition
4
of the form
Z
2
dy |κ(x, y)| ∈ L∞ (D),
x 7→ D
since we can rewrite the eigenvalue equation Cei = λi ei as Z −1 ei (x) = λi dyκ(x, y)ei (y) D
for almost every x ∈ D when λi 6= 0; this follows immediately from Schwarz inequality and the fact that kei k2 = 1 for each i; on the other hand, owing to the existence of the continuous embedding L∞ (D) 7→ L2 (D), we note that P+∞ 21 1 2 < +∞. Therefore, Hypothesis (C) defines a (6) implies i=1 λi := Tr C restricted set of trace-class covariance operators among those whose square root is itself trace-class. We shall see below that such a restriction is intimately related to the nature of the stochastic calculus we use to give a meaning to (3). Definition 1. We say the L2 (D)-valued random field (uI (., t))t∈[0,T ] defined on (Ω, F, P) is a variational solution of type I to Problem (3) if the following two conditions hold: (1) We have uI ∈ L2 (0, T ; H 1 (D)) ∩ Bα,2 ([0, T ] ; L2 (D)) a.s., which means that the relations Z T 2 dt kuI (., t)k1,2 0
Z = 0
T
2 2 dt kuI (., t)k2 + k∇uI (., t)k2 < +∞
(7)
and kuI kα,2 < +∞
(8)
hold a.s.. (2) The integral relation Z Z dxv(x)uI (x, t) = dxv(x)ϕ(x) D D Z t Z − dτ dx (∇v(x), k(x, τ )∇uI (x, τ ))Rd 0 D Z t Z + dτ dxv(x)g(uI (x, τ )) 0 D Z tZ + dxv(x)h(uI (x, τ ))W H (x, dτ ) 0
(9)
D
holds a.s. for every v ∈ H 1 (D) and every t ∈ [0, T ], where we have defined the stochastic integral as Z tZ Z t +∞ X 1 dxv(x)h(uI (x, τ ))W H (x, dτ ) := λi2 (v, h(uI (., τ ))ei )2 BiH (dτ ). 0
D
i=1
0
(10) 5
We can easily infer from the above hypotheses that each term in (9) is well defined and finite a.s.; in particular, we shall show in Section 3 that our definition of the stochastic integral with respect to (W H (., t))t∈R+ as an infinite sum of one-dimensional, pathwise, generalized Stieltjes integrals represents a real-valued random variable by virtue of the fact that h is Lipschitz continuous and Hypothesis (C). The second notion of variational solution we introduce involves test functions that depend on both the space and the time variable, a natural requirement since Problem (3) is parabolic. In order to define these for every t ∈ (0, T ], we introduce the Sobolev space H 1 (D × (0, t)) of all real-valued functions v ∈ L2 (D × (0, t)) that possess distributional derivatives vxi , vτ ∈ L2 (D × (0, t)), whose norm we denote by 2 kvk1,2,t
Z
2
dxdτ |v(x, τ )| +
= D×(0,t)
i=1
Z
2
dxdτ |vτ (x, τ )| .
+
d Z X
2
dxdτ |vxi (x, τ )|
D×(0,t)
(11)
D×(0,t)
We have the following. Definition 2. We say the L2 (D)-valued random field (uII (., t))t∈[0,T ] defined on (Ω, F, P) is a variational solution of type II to Problem (3) if the first condition of Definition 1 holds, and if the integral relation Z Z Z t Z dxv(x, t)uII (x, t) = dxv(x, 0)ϕ(x) + dτ dxvτ (x, τ )uII (x, τ ) D D 0 D Z t Z − dτ dx (∇v(x, τ ), k(x, τ )∇uII (x, τ ))Rd 0 D Z t Z + dτ dxv(x, τ )g (uII (x, τ )) 0 D Z tZ + dxv(x, τ )h (uII (x, τ )) W H (x, dτ ) (12) 0
D
1
holds a.s. for every v ∈ H (D × (0, t)) and every t ∈ [0, T ], where x 7→ v(x, 0) ∈ L2 (D) and x 7→ v(x, t) ∈ L2 (D) denote the Sobolev traces of v on D and D × {τ ∈ R : τ = t}, respectively, and where we have defined the stochastic integral as in Definition 1. From the above hypotheses again we can prove that every term in (12) is well defined and finite a.s.. We also note that the structure of (12) is similar to that of (9) up to the appearance of the term that involves the partial derivative vτ . In Section 3 we will in fact carry out a detailed analysis of each term in (12) that will lead to a proof of the indistinguishability of the two notions. The main result of this article is indeed the following.
6
Theorem. Assume that Hypotheses (K), (L), (H γ ) (I) and (C) hold. Then, Problem (3) possesses a variational solution uI,ϕ of type I and a variational solution uII,ϕ of type II such that uI,ϕ (., t) = uII,ϕ (., t) a.s. in L2 (D) for every t ∈ [0, T ]. Moreover, if h is an affine function, uI,ϕ is the only variational solution of type I to (3) while uII,ϕ is its only variational solution of type II. We do not know whether the uniqueness statement still holds for an arbitrary h satisfying Hypothesis (Hγ ), and we refer the reader to the next section for a short discussion of this point. We shall devote the remaining part of our article to proving the above theorem.
3
Proof of the Main Result
In what follows we write c for all the irrelevant constants that occur in the various estimates unless we specify these constants otherwise; we begin by proving that the stochastic integral we defined in (10) is finite a.s., and that it enjoys an important continuity property which we shall invoke below. Proposition 1. Let u ∈ Bα,2 ([0, T ] ; L2 (D)); assume that Hypothesis (L) holds for h and that Hypothesis (C) holds; then, the integral t
Z
(v, h(u(., τ ))ei )2 BiH (dτ )
0
exists as a generalized Stieltjes integral in the sense of [52] for every i ∈ N+ and we have Z t +∞ X 1 H 2 (v, h(u(., τ ))ei )2 Bi (dτ ) < +∞ (13) λi 0
i=1
a.s. for every v ∈ L2 (D) and every t ∈ [0, T ]; moreover, for any sequence (vN )N ∈N+ such that vN → v strongly in L2 (D) as N → +∞, we have +∞ X
lim
N →+∞
=
+∞ X
1
0
i=1 1
Z
λi2
i=1
Z
λi2
0
t
(vN , h(u(., τ ))ei )2 BiH (dτ )
t
(v, h(u(., τ ))ei )2 BiH (dτ )
(14)
a.s. for every t ∈ [0, T ]. Proof. Since kukα,2 < +∞, it is sufficient to prove that there exists an a.s. finite, positive random variable rαH such that the estimate +∞ X i=1
Z t H λi (v, h(u(., τ ))ei )2 Bi (dτ ) ≤ crαH kvk2 1 + kukα,2 1 2
0
7
(15)
holds a.s. for every v ∈ L2 (D) and every t ∈ [0, T ]. For each i ∈ N+ , we first define the functions fi : [0, T ] 7→ R by Z fi (t) = (v, h(u(., t))ei )2 = dxv(x)h(u(x, t))ei (x). (16) D
From (16) and by using successively the fact that h is Lipschitz continuous, the essential boundedness of the ei ’s along with Schwarz inequality, we obtain |fi (t)| ≤ c kei k∞ kvk2 (1 + ku(., t)k2 ) and |fi (t) − fi (τ )| ≤ c kei k∞ kvk2 ku(., t) − u(., τ )k2 for every t, τ ∈ [0, T ]; consequently, from Proposition 4.1 of [40] and (16) we infer that there exists a finite, positive random variable Λα (BiH ), depending only on α, BiH and having moments of all orders, such that the estimate +∞ X
Z t H (v, h(u(., τ ))ei )2 Bi (dτ ) λi 1 2
i=1 +∞ X
≤
0
1 2
λi Λα (BiH )
t
dτ 0
i=1
≤c
Z
+∞ X
1 2
|fi (τ )| +α τα !
λi kei k∞ Λα (BiH )
kvk2
τ
Z
dσ 0
Z
1+ 0
i=1
t
|fi (τ ) − fi (σ)|
!
α+1
(τ − σ)
ku(., τ )k2 dτ + τα
Z
t
Z dτ
0
τ
dσ 0
ku(., τ ) − u(., σ)k2 α+1
(τ − σ) (17)
P+∞ 1 holds a.s. since τ 7→ τ −α is integrable on (0, t). But i=1 λi2 kei k∞ E(Λα (BiH ) ) ≤ P+∞ 1 c i=1 λi2 kei k∞ < +∞ by virtue of the fact that the BiH ’s are identically disP+∞ 1 tributed and that Hypothesis (C) holds, which implies rαH := i=1 λi2 kei k∞ Λα (BiH ) < +∞ a.s.; moreover, the function τ 7→ τ −2α is also integrable on (0, t) since α < 12 , so that by using Schwarz inequality relative to the measure dτ on (0, t) in the last two integrals of (17) along with (5) we obtain +∞ X
Z t 1 λi2 (v, h(u(., τ ))ei )2 BiH (dτ ) 0 i=1 ≤ crαH kvk2 1 + kukα,2 < +∞
a.s.. Now let (wn )n∈N+ be an orthonormal basis of L2 (D) such that (cn wn )n∈N+ be an orthonormal basis of H 1 (D) for some suitably chosen coefficients cn (the existence of such a basis follows from standard elliptic theory, see, for instance, [20] or [39]). An immediate consequence of the above proposition is the following result, whose proof follows from standard arguments and is therefore omitted. 8
!
Corollary 1. Let uI ∈ L2 (0, T ; H 1 (D)) ∩ B α,2 ([0, T ] ; L2 (D)) and assume that Hypotheses (K), (L), (I) and (C) hold; then uI is a variational solution of type I to (3) if, and only if, the relation d Z X
(wn , uI (., t))2 = (wn , ϕ)2 −
dτ wn,xi , ki,j (., τ )uI,xj (., τ ) 2
0
i,j=1 t
Z
dτ (wn , g(uI (., τ )))2
+ +
t
0 +∞ X
1
t
Z
(wn , h(uI (., τ ))ei )2 BiH (dτ )
λi2 0
i=1
(18)
holds a.s. for every n ∈ N+ and every t ∈ [0, T ]. The preceding result motivates the following definition of the Faedo-Galerkin approximation scheme we introduce to prove our main theorem. For each N ∈ N+ , let VN be the finite-dimensional subspace of L2 (D) generated by {w1 , ..., wN } and write ϕN for the orthogonal projection of the initial condition ϕ onto VN . We also introduce the Banach space B α,∞ ([0, T ] ; VN ) of all Lebesgue-measurable mappings u : [0, T ] 7→ VN endowed with the norm k.kα,β defined by ! Z t ku(., t) − u(., τ )k2 kukα,β = sup exp [−βt] ku(., t)k2 + dτ (19) α+1 (t − τ ) t∈[0,T ] 0 for every β ∈ R+ . Definition 3. We say the VN -valued random field (uN (., t))t∈[0,T ] defined on (Ω, F, P) is a Faedo-Galerkin approximation to Problem (3) if the following two conditions hold: (1) We have uN ∈ B α,∞ ([0, T ] ; VN ) a.s.. (2) The relation (wn , uN (., t))2 = (wn , ϕN )2 −
d Z X i,j=1
Z
dτ wn,xi , ki,j (., τ )uN,xj (., τ ) 2
t
+ +
0
t
dτ (wn , g(uN (., τ )))2 0 +∞ X i=1
1 2
Z
λi
0
t
(wn , h(uN (., τ ))ei )2 BiH (dτ )
holds a.s. for every n ∈ {1, ..., N } and every t ∈ [0, T ]. We have the following result.
9
(20)
Proposition 2. Assume that Hypotheses (K), (L), (H γ ), (I) and (C) hold. Then there exists a unique sequence ((uN (., t)t∈[0,T ] )N ∈N+ of Faedo-Galerkin approximations to (3). The proof of the preceding proposition rests on several preparatory results regarding the construction of a suitable fixed point for the mapping ΦN defined by N X ΦN (u)(., t) := (wn , ΦN (u)(., t))2 wn (21) n=1
for u ∈ B
α,∞
([0, T ] ; VN ), where
(wn , ΦN (u)(., t))2 := (wn , ϕN )2 −
d Z X i,j=1
Z
0
dτ wn,xi , ki,j (., τ )uxj (., τ ) 2
t
+ +
t
dτ (wn , g(u(., τ )))2 0 +∞ X
1 2
t
Z
(wn , h(u(., τ ))ei )2 BiH (dτ )
λi
0
i=1
(22)
a.s. for every n ∈ {1, ..., N } and every t ∈ [0, T ]. We begin by introducing the scalar counterpart of B α,∞ ([0, T ] ; VN ), namely, the Banach space B α,∞ ([0, T ] ; R) of all Lebesgue-measurable functions f : [0, T ] 7→ R endowed with the norm |.|α,β defined by ! Z t |f (t) − f (τ )| |f |α,β := sup exp [−βt] |f (t)| + dτ . (23) α+1 (t − τ ) t∈[0,T ] 0 Our goal is then to estimate kΦN (u)kα,β as a function of kukα,β by first estimating each time-dependent term on the right-hand side of (22) with respect to the norm |.|α,β as a function of kukα,β . For this purpose and for every n ∈ {1, ..., N }, we define the three functions Fn (u), Gn (u), Hn (u) : [0, T ] 7→ R by d Z X
Fn (u)(t) =
i,j=1
Z Gn (u)(t) =
t
0
dτ wn,xi , ki,j (., τ )uxj (., τ ) 2 ,
t
dτ (wn , g(u(., τ )))2
(24)
0
and Hn (u)(t) =
+∞ X i=1
1
Z
λi2 0
t
(wn , h(u(., τ ))ei )2 BiH (dτ )
respectively.
10
(25)
Lemma 1. Let u ∈ Bα,∞ ([0, T ] ; VN ), assume that Hypotheses (K), (L) and (C) hold and let rαH be the random variable introduced in the proof of Proposition 1; then, for every β ∈ [1, +∞) we have Fn (u), Gn (u), Hn (u) ∈ B α,∞ ([0, T ] ; R) along with the estimates |Fn (u)|α,β ≤ cβ 2α−1 kukα,β , |Gn (u)|α,β ≤ cβ 2α−1 1 + kukα,β , |Hn (u)|α,β ≤ crαH β 2α−1 1 + kukα,β
(26)
for every n ∈ {1, ..., N }, where the last inequality holds a.s. and where the constant c is independent of β but depends on α and N . Proof. For each n we define the functions fn (u) : [0, T ] 7→ R by d X
fn (u)(t) =
wn,xi , ki,j (., t)uxj (., t) 2 .
i,j=1 ∞
We first note that ki,j ∈ L (D × (0, T )) for every i, j ∈ {1, ..., d} as a consequence of the upper bound in (4) of Hypothesis (K), so that by using the equivalence of the norms k.k1,2 and k.k2 on VN we obtain |fn (u)(t)| ≤ c
d X
kwn,xi k2 uxj (., t) 2 ≤ c ku(., t)k2
(27)
i,j=1
since kwn k2 = 1; furthermore, we have successively Z t ku(., τ )k2 sup exp [−βt] dτ α (t − τ ) t∈[0,T ] 0 Z t exp [−β(t − τ )] ≤ sup dτ × sup exp [−βt] ku(., t)k2 α (t − τ ) t∈[0,T ] t∈[0,T ] 0 Z +∞ exp [−βτ ] ≤ dτ × sup exp [−βt] ku(., t)k2 τα t∈[0,T ] 0 ≤ β 2α−1 Γ(1 − α) × sup exp [−βt] ku(., t)k2
(28)
t∈[0,T ]
since β ∈ [1, +∞), where Γ is Euler’s Gamma function. Therefore, from (23), Proposition 4.3. of [40] along with (27), (28) and (19) we get ! Z t |Fn (u)(t) − Fn (u)(τ )| |Fn (u)|α,β = sup exp [−βt] |Fn (u)(t)| + dτ α+1 (t − τ ) t∈[0,T ] 0 Z t ku(., τ )k2 ≤ c sup exp [−βt] dτ α (t − τ ) t∈[0,T ] 0 ≤ cβ 2α−1 sup exp [−βt] ku(., t)k2 ≤ cβ 2α−1 kukα,β , t∈[0,T ]
11
which is the first estimate in (26). We can prove the second estimate in (26) in an essentially similar way, by invoking the Lipschitz continuity of g. As for the stochastic term, we first define the functions fn,i (u) : [0, T ] 7→ R by fn,i (u)(t) = (wn , h(u(., t))ei )2 . On the one hand, since h is Lipschitz continuous we have Z t |fn,i (u)(t) − fn,i (u)(τ )| |fn,i (u)(t)| + dτ α+1 (t − τ ) 0 Z t ku(., t) − u(., τ )k2 ≤ c kei k∞ 1 + ku(., t)k2 + dτ (t − τ )α+1 0
(29)
(30)
for every t ∈ [0, T ]; on the other hand, by elementary changes of variables we may write and estimate the following integral as Z t 1 1 dτ exp [−β(t − τ )] + (t − τ )2α τα 0 Z βt Z βt exp [−τ ] exp [−τ ] = β 2α−1 dτ + β α−1 dτ 2α τ (βt − τ )α 0 0 Z βt exp [−τ ] ≤ β 2α−1 Γ(1 − 2α) + β 2α−1 dτ ≤ cβ 2α−1 (31) (βt − τ )α 0 uniformly in t ∈ [0, T ] since β ∈ [1, +∞). Consequently, from Proposition 4.1 of [40], (30), (31) and the fact that rαH < +∞ a.s., we infer that the inequalities ! Z t |Hn (u)(t) − Hn (u)(τ )| |Hn (u)|α,β = sup exp [−βt] |Hn (u)(t)| + dτ α+1 (t − τ ) t∈[0,T ] 0 Z t 1 1 H ≤ crα sup exp [−βt] dτ + α (t − τ )2α τ t∈[0,T ] 0 Z τ ku(., τ ) − u(., σ)k2 × 1 + ku(., τ )k2 + dσ (τ − σ)α+1 0 ! Z t 1 1 ≤ crαH sup dτ exp [−β(t − τ )] + 1 + kuk α,β (t − τ )2α τα t∈[0,T ] 0 ≤ crαH β 2α−1 1 + kukα,β hold a.s. for every n ∈ {1, ..., N }. An immediate consequence of the preceding lemma is the following invariance result. Lemma 2. Let u ∈ B α,∞ ([0, T ] ; VN ) and assume that Hypotheses (K), (L), (I) and (C) hold; then we have ΦN (u) ∈ Bα,∞ ([0, T ] ; VN ) a.s. and there exists 12
a sufficiently large random variable β0 ∈ [1, +∞) depending on N such that the closed ball n o BN := u ∈ B α,∞ ([0, T ] ; VN ) : kukα,β0 ≤ 2 (1 + N kϕN k2 ) (32) remains invariant under ΦN , that is, ΦN (BN ) ⊆ BN a.s.. Proof. The function ΦN (u) is clearly measurable since u is and we have PN kΦN (u)(., t)k2 ≤ n=1 |(wn , ΦN (u)(., t))2 | and kΦN (u)(., t) − ΦN (u)(., τ )k2 ≤ PN n=1 |(wn , ΦN (u)(., t) − ΦN (u)(., τ ))2 | a.s. for all t, τ ∈ [0, T ] as a consequence of (21); from this and the definition of the norms (23) and (19), along with (22) and (26), we infer that the inequalities kΦN (u)kα,β ≤
N X
|(wn , ΦN (u)(., .))2 |α,β
n=1
≤ N kϕN k2 + c 1 + rαH β 2α−1 1 + kukα,β
(33)
hold a.s. for every β ∈ [1, +∞). Consequently we have ΦN (u) ∈ B α,∞ ([0, T ] ; VN ) a.s. and u ∈ BN implies ΦN (u) ∈ BN a.s. from (33) when c 1 + rαH β 2α−1 ≤ 12 ; h (1−2α)−1 it is then sufficient to choose β0 ∈ 1 ∨ 2c 1 + rαH , +∞ . The mapping ΦN (u) : [0, T ] 7→ VN also enjoys an important H¨older continuity property, which we describe as follows. Lemma 3. Let u ∈ B α,∞ ([0, T ] ; VN ) and assume that Hypotheses (K), (L), (I) and (C) hold; then we have 1−α kΦN (u)(t) − ΦN (u)(t∗ )k2 ≤ c(1 + rαH ) exp [β0 T ] 1 + kukα,β0 |t − t∗ | (34) a.s. for all t, t∗ ∈ [0, T ]. Proof. According to (21), it is sufficient to prove that the three functions Fn (u), Gn (u), Hn (u) : [0, T ] 7→ R are H¨older continuous; from (27) and (19) we first get Z t |Fn (u)(t) − Fn (u)(t∗ )| ≤ c dτ ku(., τ )k2 ≤ c exp [β0 T ] kukα,β0 |t − t∗ | t∗
1−α
≤ c exp [β0 T ] kukα,β0 |t − t∗ |
for all t, t∗ ∈ [0, T ] such that t ≥ t∗ . In a similar way we obtain 1−α |Gn (u)(t) − Gn (u)(t∗ )| ≤ c exp [β0 T ] 1 + kukα,β0 |t − t∗ |
13
since g is Lipschitz continuous. As for the function Hn (u), we begin by observing that the definition of the stochastic integral in [40] along with (29) and (30) imply the estimates Z t Z τ Z t |fn,i (u)(τ )| |fn,i (u)(τ ) − fn,i (u)(σ)| H H + α dσ f (u)(τ )B (dτ ) ≤ Λ (B ) dτ α i i ∗ n,i (τ − t∗ )α (τ − σ)α+1 t∗ t t∗ Z t 1 ≤ Λα (BiH ) exp [β0 T ] |fn,i (u)|α,β0 dτ + α (τ − t∗ )α t∗ 1−α
≤ cΛα (BiH ) exp [β0 T ] |fn,i (u)|α,β0 |t − t∗ | 1−α ≤ cΛα (BiH ) kei k∞ exp [β0 T ] 1 + kukα,β0 |t − t∗ | a.s.. Consequently, we have +∞ X
Z t H |Hn (u)(t) − Hn (u)(t )| ≤ λi fn,i (u)(τ )Bi (dτ ) t∗ i=1 1−α ≤ crαH exp [β0 T ] 1 + kukα,β0 |t − t∗ | ∗
1 2
a.s. for all t, t∗ ∈ [0, T ]. The preceding estimates immediately lead to (34). Regarding the nonlinearities g and h, we have thus far only used the fact that they are Lipschitz continuous; we now elucidate the forthcoming rˆole of Hypothesis (Hγ ) by means of the following result. Lemma 4. Let u, u∗ ∈ Bα,∞ ([0, T ] ; VN ) and assume that Hypothesis (H γ ) holds; then we have the estimate kh(u(., t)) − h(u∗ (., t)) − h(u(., τ )) + h(u∗ (., τ ))k2 ≤ c ku(., t) − u∗ (., t) − u(., τ ) + u∗ (., τ )k2 γ γ + c ku(., t) − u∗ (., t)k2 (ku(., t) − u(., τ )k2 + ku∗ (., t) − u∗ (., τ )k2 )
(35)
for all t, τ ∈ [0, T ]. Proof. From the mean-value theorem we have h(u(x, t)) − h(u∗ (x, t)) − h(u(x, τ )) + h(u∗ (x, τ )) Z 1 = dθ (u(x, t) − u∗ (x, t) − u(x, τ ) + u∗ (x, τ )) h0 (θu(x, τ ) + (1 − θ)u∗ (x, τ )) Z
0 1
+
dθ(u(x, t) − u∗ (x, t))
0
× (h0 (θu(x, t) + (1 − θ)u∗ (x, t)) − h0 (θu(x, τ ) + (1 − θ)u∗ (x, τ ))) for almost every x ∈ D, which implies that |h(u(x, t)) − h(u∗ (x, t)) − h(u(x, τ )) + h(u∗ (x, τ ))| ≤ c |u(x, t) − u∗ (x, t) − u(x, τ ) + u∗ (x, τ )| γ γ + c |u(x, t) − u∗ (x, t)| (|u(x, t) − u(x, τ )| + |u∗ (x, t) − u∗ (x, τ )| ) 14
by virtue of Hypothesis (Hγ ). We also have t 7→ h(u(., t)) ∈ Bα,∞ ([0, T ] ; VN ) since (Hγ ) implies that h is Lipschitz continuous; therefore, we may invoke the equivalence of the norms k.k2 and k.k∞ on VN to obtain kh(u(., t)) − h(u∗ (., t)) − h(u(., τ )) + h(u∗ (., τ ))k2 ≤ c ku(., t) − u∗ (., t) − u(., τ ) + u∗ (., τ )k2 γ γ + c ku(., t) − u∗ (., t)k2 (ku(., t) − u(., τ )k∞ + ku∗ (., t) − u∗ (., τ )k∞ ) ≤ c ku(., t) − u∗ (., t) − u(., τ ) + u∗ (., τ )k2 γ γ + c ku(., t) − u∗ (., t)k2 (ku(., t) − u(., τ )k2 + ku∗ (., t) − u∗ (., τ )k2 ) for all t, τ ∈ [0, T ]. Our last preparatory result is the following. Lemma 5. Assume that Hypotheses (K), (L), (H γ ) and (C) hold; then, the nonlinear functional γ : B α,∞ ([0, T ] , VN ) 7→ [0, +∞] defined by Z t γ ku(., t) − u(., τ )k2 γ (u) = sup dτ (36) α+1 (t − τ ) t∈[0,T ] 0 is lower semicontinuous. Moreover, its restriction to ΦN (BN ) is real-valued and bounded. Proof. Let (um )m∈N+ ⊆ Bα,∞ ([0, T ] , VN ) be a strongly convergent sequence to u, which implies in particular that sup kum (., t) − u(., t)k2 → 0
t∈[0,T ]
as m → +∞; Fatou’s lemma then allows us to conclude that the estimates Z t γ ku(., t) − u(., τ )k2 dτ α+1 (t − τ ) 0 Z t γ kum (., t) − um (., τ )k2 ≤ lim inf dτ ≤ lim inf γ (um ) α+1 m→+∞ 0 m→+∞ (t − τ ) hold uniformly in t ∈ [0, T ], which proves the lower semicontinuity of γ . As for the second statement of the lemma, if u ∈ ΦN (BN ) there exists u ∈ BN such that u = ΦN (u); therefore, by using successively (34) and (32) we obtain Z t γ kΦN (u)(., t) − ΦN (u)(., τ )k2 γ (u) = sup dτ α+1 (t − τ ) t∈[0,T ] 0 Z t (1−α)γ γ |t − τ | ≤ c(1 + rαH )γ exp [β0 γT ] 1 + kukα,β0 sup dτ α+1 (t − τ ) t∈[0,T ] 0 γ ≤ c(1 + rαH )γ exp [β0 γT ] 1 + kukα,β0 γ
≤ c(1 + rαH )γ exp [β0 γT ] (1 + N kϕN k2 ) < +∞ 15
(37)
a.s. since α
0 a.s.; relation (72) then implies that ! Z t 2 supσ∈[0,τ ] kuN (., σ)k2 2 2 dτ sup kuN (., σ)k2 ≤ kϕk2 + (r1 () + r2 ) 1 + τα σ∈[0,t] 0 a.s. for every t ∈ [0, T ], so that by a Gronwall’s inequality argument we get sup
2
sup kuN (., t)k2 < +∞
N ∈N+ t∈[0,T ]
29
(73)
!
2
a.s. since τ 7→ τ −α supσ∈[0,τ ] kuN (., σ)k2 is integrable on (0, T ); we then go back to (72) without the square of the L2 (D)-norm on the left-hand side and use (73) to infer the uniform bound Z t 2 sup dτ k∇uN (., τ )k2 < +∞ N ∈N+
0
a.s. for every t ∈ [0, T ], which, together with (71) and (73) once again, leads to !2 Z t Z t Z τ kuN (., τ ) − uN (., σ)k2 2 sup < +∞ dτ k∇uN (., τ )k2 + dτ dσ α+1 + (τ − σ) N ∈N 0 0 0 (74) a.s. for every t ∈ [0, T ]; but (73) and (74) are equivalent to (46) and (47). Our goal now is to prove that there exists a subsequence of (uN )N ∈N+ converging to a variational solution uI,ϕ of type I for (3); to this end we note that (45) implies the boundedness of (uN )N ∈N+ in L2 (0, T ; H 1 (D)); therefore, there exists uI,ϕ ∈ L2 (0, T ; H 1 (D)) with the assumption that uN * uI,ϕ weakly in L2 (0, T ; H 1 (D)) as N → +∞ a.s., by passing to a suitable subsequence if necessary. In a similar way, we may assume that uN *∗ uI,ϕ in the weak-* topology of L∞ (0, T ; L2 (D)) a.s., since (45) along with the continuity of t 7→ kuN (., t)k2 imply the boundedness of (uN )N ∈N+ in L∞ (0, T ; L2 (D)) as well; consequently we have uI,ϕ ∈ L2 (0, T ; H 1 (D)) ∩ L∞ (0, T ; L2 (D)) (75) a.s.. In fact we have much more, as stated in the following result. Lemma 13. The hypotheses are the same as in Proposition 3. Then we have uI,ϕ ∈ L2 (0, T ; H 1 (D)) ∩ B α,2 ([0, T ] ; L2 (D)) a.s.. Proof. According to (44) and (75), it remains to prove that T
Z
t
Z dt
0
dτ
kuI,ϕ (., t) − uI,ϕ (., τ )k2
!2
α+1
(t − τ )
0
< +∞
(76)
a.s.. To this end we define the functional Ψ : L2 (0, T ; H 1 (D)) 7→ R+ 0 by T
Z Ψ(u) :=
Z dt
0
t
dτ
ku(., t) − u(., τ )k2
0
α+1
(t − τ )
!2 ,
and notice that it is sufficient to prove the inequality Ψ(uI,ϕ ) ≤ lim inf Ψ(uN ) N →+∞
30
(77)
a.s. by virtue of (47). We first remark that Ψ is convex and proper, in that it does not take the value −∞ and takes finite values on L2 (0, T ; H 1 (D)) ∩ B α,2 ([0, T ] ; L2 (D)); therefore, for the proof of (77) we may assume that uN → uI,ϕ strongly in L2 (0, T ; H 1 (D)) a.s., or else replace (uN )N ∈N+ by a sequence of suitable convex combinations of uN ’s converging strongly to uI,ϕ a.s. and apply standard arguments (see, for instance, [19]). This means we may further assume that uN (., t) → uI,ϕ (., t) strongly in H 1 (D) a.s., and a fortiori strongly in L2 (D) a.s., for almost every t ∈ (0, T ). In this way, by two successive applications of Fatou’s lemma we get !2 Z T Z t kuN (., t) − uN (., τ )k2 Ψ(uI,ϕ ) ≤ dt lim inf dτ α+1 N →+∞ 0 (t − τ ) 0 !2 Z T Z t kuN (., t) − uN (., τ )k2 ≤ dt lim inf dτ α+1 N →+∞ (t − τ ) 0 0 ≤ lim inf Ψ(uN ) ≤ sup Ψ(uN ) < +∞ N →+∞
N ∈N+
a.s.. The preceding considerations make it natural to think that uI,ϕ may eventually provide a solution to (3); we now show that this is indeed the case. Proposition 4. Assume that Hypotheses (K), (L), (H γ ), (I) and (C) hold. Then, the random field (uI,ϕ (., t))t∈[0,T ] of Lemma 13 is a variational solution of type I to (3). According to Corollary 1, we can reduce the proof of this proposition to showing that (18) holds. We will accomplish this in several stages by investigating the limit N → +∞ in (20), in which context the following result will turn out to be essential to control the nonlinear terms. Lemma 14. The embedding L2 (0, T ; H 1 (D)) ∩ B α,2 ([0, T ] ; L2 (D)) ,→ L2 (0, T ; L2 (D))
(78)
is compact. While (78) is evidently continuous by virtue of (44), the proof of Lemma 14 requires some additional considerations. Let I0α+ be the linear operator of fractional integration on L2 (0, T ; L2 (D)) defined by Z t 1 u(., τ ) α I0+ u(., t) := dτ (79) Γ(α) 0 (t − τ )1−α for every u ∈ L2 (0, T ; L2 (D)); it is easily verified that the right-hand side of (79) exists as an L2 (D)-valued Bochner integral for almost every t ∈ (0, T ), or 31
even as an H 1 (D)-valued Bochner integral when u ∈ L2 (0, T ; H 1 (D)). Moreover, I0α+ is bounded in L2 (0, T ; L2 (D)), as well as in L2 (0, T ; H 1 (D)). Now let D0 (0, T ; L2 (D)) be Schwartz’s space of L2 (D)-valued distributions defined on (0, T ) and write Z 1 d t u(., τ ) d α dτ = I01−α u(., t) (80) D0+ u(., t) := Γ(1 − α) dt 0 (t − τ )α dt + for the fractional derivative of any u ∈ L2 (0, T ; L2 (D)), where we mean the time derivative on the right-hand side of (80) in the sense of D0 (0, T ; L2 (D)) (see, for instance, [33] for a definition of this space); from (79) and (80) we have α Dα 0+ I0+ u = u
(81)
for every u ∈ L2 (0, T ; L2 (D)). Let us now write Ran I0α+ for the range of I0α+ in L2 (0, T ; L2 (D)) and consider the vector space W α,2 (0, T ; H 1 (D), L2 (D)) := L2 (0, T ; H 1 (D)) ∩ Ran I0α+ , endowed with the inner product Z T Z (u, u e)W α,2 := dt (u(., t), u e(., t))1,2 + 0
0
T α dt(Dα e(., t))2 0+ u(., t), D0+ u
(82)
2 2 whose related norm we denote by k.kW α,2 . The fact that Dα 0+ u ∈ L (0, T ; L (D)) α,2 in this case is an immediate consequence of (81). We also have W (0, T ; H 1 (D), L2 (D)) ⊆ L2 (0, T ; L2 (D)) and kukL2 (0,T ;L2 (D)) ≤ kukW α,2 for every u ∈ W α,2 (0, T ; H 1 (D), L2 (D)); in order to prove Lemma 14, however, we need the following result whose proof rests upon the use of the Hardy-Littlewood-P´olya theorem regarding the L2 boundedness properties of integral transforms with homogeneous kernels (see, for instance, [24] and [49]).
Lemma 15. The vector space W α,2 (0, T ; H 1 (D), L2 (D)) is a Hilbert space with respect to (82), and the embedding W α,2 (0, T ; H 1 (D), L2 (D)) ,→ L2 (0, T ; L2 (D))
(83)
is compact. Proof. Aside from its trivial linear and metric structure the space we consider is complete; for if (um )m∈N+ is a Cauchy sequence therein, then um → u α 2 2 strongly in L2 (0, T ; H 1 (D)), and that Dα 0+ um → u0+ strongly in L (0, T ; L (D)) 2 1 α 2 2 when m → +∞, for some u ∈ L (0, T ; H (D)) and some u0+ ∈ L (0, T ; L (D)), respectively. But um = I0α+ u em for some u em ∈ L2 (0, T ; L2 (D)), so that from the α α α boundedness of I0+ and (81) we have u = I0α+ uα 0+ and D0+ u = u0+ , which implies the strong convergence um → u in W α,2 (0, T ; H 1 (D), L2 (D)). We now prove the compactness of (83) by reduction to a classic result of [33]; let 32
Hα,2 (R; H 1 (D), L2 (D)) be the Hilbert space consisting of all u∗ ∈ L2 (R; H 1 (D)) whose Fourier transform Z Fu∗ (k) := dt exp [−2πikt] u∗ (t) R α
∗
2
satisfies k7→ |k| Fu (., k) ∈ L (R; L2 (D)), endowed with the norm Z Z 2 2 2α 2 ku∗ kHα,2 := dt ku∗ (., t)k1,2 + dk |k| kFu∗ (., k)k2 . R
R
We write Rα 0+ for the restriction map α,2 (R; H 1 (D), L2 (D)) 7→ L2 (0, T ; H 1 (D)) Rα 0+ : H ∗ ∗ α defined by Rα 0+ u (., t) = u (., t) for almost every t ∈ (0, T ), and RanR0+ for its α range; it is then plain that R0+ is continuous, and that we can endow RanRα 0+ with a Hilbert space structure whose related norm is
kukRan Rα = inf ku∗ kHα,2 0+
(84)
where the infimum is taken over all u∗ ∈ Hα,2 (R; H 1 (D), L2 (D)) such that ∗ Rα 0+ u = u (see, for instance, [33] or [34] for typical quotient space constructions of this kind). Now the compactness of H 1 (D) ,→ L2 (D) (see, for instance, [1]) along with Theorem 5.2. of [33] allow us to conclude that the embedding 2 2 Ran Rα 0+ ,→ L (0, T ; L (D))
is compact. Therefore, in order to prove the compactness of (83) we need only show that there exists the continuous embedding W α,2 (0, T ; H 1 (D), L2 (D)) 7→ Ran Rα 0+ .
(85)
Fix an arbitrary u ∈ W α,2 (0, T ; H 1 (D), L2 (D)) and let u∗ be the zero extension of u outside (0, T ), that is, u(., t) if t ∈ (0, T ) u∗ (., t) = (86) 0 if t ∈ R \ (0, T ). We first claim that u∗ ∈ Hα,2 (R; H 1 (D), L2 (D)); obviously we have u∗ ∈ L2 (R; H 1 (D)), so that the more difficult part of the claim consists in provα ing that k7→ |k| Fu∗ (., k) ∈ L2 (R; L2 (D)); in order to see this we introduce the ∗ auxiliary function Dα + u defined by Z 1 d t u∗ (., τ ) ∗ Dα u (., t) := dτ (87) + Γ(1 − α) dt −∞ (t − τ )α for almost every t ∈ R, where we mean the time-derivative on the right-hand side in the sense of D0 (R; L2 (D)); we can then write (87) as 0 if t ∈ (−∞, 0) α ∗ D u(., t) if t ∈ (0, T ) (88) Dα u (., t) = 0 + R T+ u(.,τ ) − α dτ if t ∈ (T, +∞) Γ(1−α) 0 (t−τ )α+1 33
by virtue of (86). We proceed by showing that
α ∗
D+ u 2 ≤ c Dα 0+ u L (R;L2 (D))
(89)
L2 (0,T ;L2 (D))
for some c ∈ R+ . According to (88), (89) will be a consequence of the estimate Z
+∞
T
Z
T u(., τ )
dt dτ
0 (t − τ )α+1
2
2
.
≤ c Dα 0+ u 2
L (0,T ;L2 (D))
(90)
2
But since u ∈ Ran I0α+ we may write for some u e ∈ L2 (0, T ; L2 (D)) the equalities Z
T
u(., τ ) (t − τ )α+1 0 ! Z T Z T 1 1 = dσ dτ u e(., σ) Γ(α) 0 (t − τ )α+1 (τ − σ)1−α σ α Z T σ u e(., T − σ) 1 dσ = αΓ(α) 0 t−T t−T +σ dτ
(91)
for almost every t ∈ (T, +∞), according to (79) and following an explicit evaluation of the inner integral. Consequently, from (91) we obtain
Z
2
T u(., τ )
dt dτ α+1
(t − τ ) T 0 2
Z +∞ Z T u(., τ )
= dt dτ
(t + T − τ )α+1 0 0
Z
+∞
Z ≤c
+∞
Z dt
0
0
T
2
2
σ α ke u(., T − σ)k2 dσ t t+σ
!2 ,
(92)
α −1 where the kernel (σ, t) 7→ σt (t + σ) is homogeneous of degree minus one. Therefore, since α < 12 we infer from (92) and a straightforward adaptation of the Hardy-Littlewood-P´ olya theorem we referred to above the estimate
2 Z +∞ Z T u(., τ )
2 dt dτ ≤ c ke ukL2 (0,T ;L2 (D)) , α+1
(t − τ ) T 0 2
∗ which is (90) by virtue of (81). Hence we have (89) and in particular Dα +u ∈ 2 2 α ∗ L (R; L (D)), which entails the existence of the Fourier transform FD+ u in this very space by Plancherel’s theorem; but from (87), a direct calculation of this Fourier transform gives ∗ α ∗ FDα + u (., k) = (2πik) Fu (., k)
34
α α ∗ where (ik)α := |k| exp iαπ 2 sgn k ; consequently we have k 7→ (2πik) Fu (., k) ∈ 2 2 ∗ α,2 1 2 L (R; L (D)) and thus u ∈ H (R; H (D), L (D)) as claimed, which indeed proves the set-theoretical inclusion in (85). It remains to prove the continuity of this inclusion; but from what we just proved along with (84), (86), the fact that the Fourier transform is unitary in L2 (R; L2 (D)) together with (89) we have kukRan Rα ≤ ku∗ kHα,2 ≤ c kukW α,2 0+
for every u ∈ W α,2 (0, T ; H 1 (D), L2 (D)). We are now ready for the following. Proof of Lemma 14. By virtue of Lemma 15, it is sufficient to prove that there exists the continuous embedding L2 (0, T ; H 1 (D)) ∩ B α,2 ([0, T ] ; L2 (D)) 7→ W α,2 (0, T ; H 1 (D), L2 (D)). For every u ∈ L2 (0, T ; H 1 (D)) ∩ B α,2 ([0, T ] ; L2 (D)), we note that the integral Z t u(., t) − u(., τ ) dτ (t − τ )α+1 0 exists as an L2 (D)-valued Bochner’s integral for almost every t ∈ (0, T ) by virtue of the convergence of the integral in (5). This along with an adaptation of some arguments in [47] allows us to show that the inclusion B α,2 ([0, T ] ; L2 (D)) ⊆ Ran I0α+ holds, and that the expression Dα 0+ u(., t) =
1 Γ(1 − α)
u(., t) +α tα
Z
t
dτ 0
u(., t) − u(., τ ) (t − τ )α+1
(93)
for the fractional derivative (80) is valid for every u ∈ Ran I0α+ and almost every t. Consequently, from (93) and (5) we infer the estimates Z T
2
dt Dα 0+ u(., t) 2 0 !2 Z T Z T Z t 2 ku(., t)k2 ku(., t) − u(., τ )k2 ≤ c dt + dt dτ α+1 t2α (t − τ ) 0 0 0 2
≤ c kukα,2 < +∞
(94)
since α < 12 . The desired result then follows from (94) and (44). We can now state two important consequences of Lemma 14. Lemma 16. The hypotheses are the same as in Proposition 3. Then, we may assume that g(uN ) → g(uI,ϕ ) (95) 35
strongly in L2 (0, T ; L2 (D)) a.s., and that α Dα 0+ h(uN )ei * D0+ h(uI,ϕ )ei
(96)
weakly in L2 (0, T ; L2 (D)) a.s. for every i ∈ N+ when N → +∞. Proof. From Proposition 3 and Lemma 14 we may assume that uN → uI,ϕ strongly in L2 (0, T ; L2 (D)) a.s., and hence that (95) holds since g is Lipschitz continuous. In a similar way we have h(uN ) → h(uI,ϕ ) strongly in L2 (0, T ; L2 (D)) a.s., and a fortiori h(uN ) * h(uI,ϕ )
(97)
weakly in L2 (0, T ; L2 (D)) a.s.. Furthermore, since h is Lipschitz continuous and has a bounded derivative h0 , we infer from (5), (44) and (45) that the uniform estimate sup kh(uN )k1,α,2 ≤ c 1 + sup kuN k1,α,2
N ∈N+
< +∞
(98)
N ∈N+
and the inequality kh(uI,ϕ )k1,α,2 ≤ c 1 + kuI,ϕ k1,α,2
(99)
hold a.s., so that h(uN ), h(uI,ϕ ) ∈ L2 (0, T ; H 1 (D)) ∩ Bα,2 ([0, T ] ; L2 (D)), the sequence (h(uN ))N ∈N+ being bounded in this space. Therefore, from (98), (99), (3), the continuity of the embedding (83) along with (97), we may assume that h(uN ) * h(uI,ϕ )
(100)
weakly in W α,2 (0, T ; H 1 (D), L2 (D)) a.s., passing if necessary to a suitable subsequence. But from the very definition of this fractional Sobolev space we see that the operator defined by (80) is linear and bounded from W α,2 (0, T ; H 1 (D), L2 (D)) into L2 (0, T ; L2 (D)); this and (100) then imply that α Dα 0+ h(uN ) * D0+ h(uI,ϕ )
weakly in L2 (0, T ; L2 (D)) a.s., which leads in particular to (96) since ei ∈ L∞ (D) for each i ∈ N+ . In light of the preceding results, our strategy to prove Proposition 4 first amounts to proving the weak convergence in L2 (0, T ; R) of each term in (20) considered as a function of t to the corresponding term in (18). For this we need to introduce the scalar counterpart of I0α+ , namely, the linear operator 2 iα 0+ of fractional integration on L (0, T ; R) defined by iα 0+ f (t) :=
1 Γ(α)
Z
t
dτ 0
36
f (τ ) (t − τ )1−α
(101)
for every f ∈ L2 (0, T ; R); as is the case for I0α+ , it is easily verified that iα 0+ is a bounded operator on L2 (0, T ; R), and that the corresponding fractional derivative operator dα 0+ can be defined by Z d t f (τ ) d 1 dα f (t) := dτ = i1−α f (t) (102) 0+ Γ(1 − α) dt 0 (t − τ )α dt 0+ for every f ∈ L2 (0, T ; R) in the sense of D0 (0, T ; R). As above we have α dα 0+ i0+ f = f
for every f ∈ L2 (0, T ; R), as well as Z t 1 f (t) f (t) − f (τ ) dα f (t) = + α dτ 0+ Γ(1 − α) tα (t − τ )α+1 0
(103)
(104)
α 2 for almost every t and every f ∈ Ran iα 0+ , the range of i0+ in L (0, T ; R).
Proof of Proposition 4. From now on we fix n ∈ N+ and choose N > n in (20). It is plain that all three linear contributions in (20) converge weakly in L2 (0, T ; R) to the corresponding contribution in (18) a.s.. Furthermore, since g is Lipschitz continuous we have Gn (uN ) → Gn (uI,ϕ ) strongly in L2 (0, T ; R) a.s. as a consequence of (95) where Gn (u) is given by (24), so that a fortiori Gn (uN ) * Gn (uI,ϕ ) weakly in L2 (0, T ; R) a.s.. It remains to prove the weak convergence Hn (uN ) * Hn (uI,ϕ )
(105)
2
in L (0, T ; R) a.s. when N → +∞, where Hn (u) is given by (25). For each M ∈ N+ , let us define the corresponding partial sums by Z t M X 1 HnM (uN )(t) := λi2 fn,i (uN )(τ )BiH (dτ ) 0
i=1
and HnM (uI,ϕ )(t) :=
M X
1
t
Z
fn,i (uI,ϕ )(τ )BiH (dτ )
λi2 0
i=1
respectively, where fn,i (u) is given by (29). We begin by proving that HnM (uN ) * Hn (uN ) weakly in L2 (0, T ; R) a.s. when M → +∞, uniformly in N ; we have Z T dtχ(t) Hn (uN )(t) − HnM (uN )(t) 0
=
+∞ X i=M +1
1
Z
λi2
T
Z dtχ(t)
0
0
37
t
fn,i (uN )(τ )BiH (dτ )
(106)
for an arbitrary χ ∈ L2 (0, T ; R), where we can estimate the stochastic integral as Z t H sup fn,i (uN )(τ )Bi (dτ ) + N ∈N 0 H ≤ c kei k∞ Λα (Bi ) 1 + sup kuN kα,2 < +∞ N ∈N+
by repeating the arguments of the proof of Proposition 1 and by invoking the a priori estimates (45). Therefore, by combining the last two relations we get Z T sup dtχ(t)(Hn (uN )(t) − HnM (uN )(t)) N ∈N+ 0 ≤ c kχkL2 (0,T ;R)
+∞ X
1
λi2 kei k∞ Λα (BiH ) ≤
i=M +1
a.s. for every χ ∈ L2 (0, T ; R) and every M ≥ M (), where the existence of P+∞ 1 M () ∈ N+ follows from the convergence of i=1 λi2 kei k∞ Λα (BiH ) a.s.. This proves (106) with the required uniformity in N . An entirely similar argument allows us to prove that HnM (uI,ϕ ) * Hn (uI,ϕ ) (107) weakly in L2 (0, T ; R) a.s. when M → +∞, where we may choose the corresponding M () as before. We then proceed by showing that HnM () (uN ) * HnM () (uI,ϕ )
(108)
weakly in L2 (0, T ; R) a.s. when N → +∞; for this we begin by proving the weak convergence α dα (109) 0+ fn,i (uN ) * d0+ fn,i (uI,ϕ ) in L2 (0, T ; R) a.s. for each i; in fact, for an arbitrary χ ∈ L2 (0, T ; R) we have χ ⊗ wn ∈ L2 (0, T ; L2 (D)) so that (96) implies the relation Z lim
N →+∞
0
T
dtχ(t)(wn , Dα 0+ h(uN )(t)ei )2
Z = 0
T
dtχ(t)(wn , Dα 0+ h(uI,ϕ )(t)ei )2
(110) a.s.. Furthermore, since h(uN ), h(uI,ϕ ) ∈ L2 (0, T ; H 1 (D)) ∩ Bα,2 ([0, T ] ; L2 (D)) we have h(uN ), h(uI,ϕ ) ∈ Ran I0α+ according to the remark in the proof of Lemma 15, which immediately gives fn,i (uN ), fn,i (uI,ϕ ) ∈ Ran iα 0+ ; therefore, from (104), the definition of fn,i (u) and (93) we obtain α dα 0+ fn,i (uN )(t) = (wn , D0+ h(uN )(t)ei )2
(111)
α dα 0+ fn,i (uI,ϕ )(t) = (wn , D0+ h(uI,ϕ )(t)ei )2
(112)
and
38
a.s. for every i and almost every t, so that the combination of (110), (111) and (112) indeed leads to (109). In order to see how (108) follows from (109), we now write M () X 1Z t 1−α H HnM () (uN )(t) = (−1)α λi2 dτ dα (113) 0+ fn,i (uN )(τ )dt− Bi,t− (τ ) 0
i=1
and M ()
HnM () (uI,ϕ )(t)
α
= (−1)
X
1 2
Z
λi
i=1
0
t 1−α H dτ dα 0+ fn,i (uI,ϕ )(τ )dt− Bi,t− (τ )
(114)
as is possible by the definition of the stochastic integral in this case, where H (−1)α = exp [iπα], Bi,t (τ ) = BiH (τ ) − BiH (t) and − ! Z t H H H Bi,t (τ ) Bi,t (τ ) − Bi,t (σ) (−1)1−α − − − 1−α H + (1 − α) dσ . dt− Bi,t− (τ ) = Γ(α) (t − τ )1−α (σ − τ )2−α τ Moreover, we have sup 0
