Well-Posedness of Evolutionary Navier-Stokes Equations with Forces of Low Regularity on Two-Dimensional Domains

Existence and uniqueness of solutions to the Navier-Stokes equation in dimension two with forces in the space $L^q( (0,T); \mathbf{W}^{-1,p}(\Omega))$ for $p$ and $q$ in appropriate parameter ranges are proven. The case of spatially measured-valued inhomogeneities is included. For the associated Stokes equation the well-posedness results are verified in arbitrary dimensions with $1


Introduction
In this paper we investigate the following Navier-Stokes system    ∂y ∂t − ν∆y + (y · ∇)y + ∇p = f in Q = Ω × I, div y = 0 in Q, y = 0 on Σ = Γ × I, y(0) = y 0 in Ω, (1.1) with focus on low regularity assumptions on the inhomogeneity f . Here, I = (0, T ) with 0 < T < ∞, and Ω ⊂ R d denotes a connected bounded domain with a C 3 boundary Γ. Our interest in this problem is two-fold. First, it has received very little attention in the literature so far. Indeed the only result which we are aware of is given in [24], where f is chosen in W 1,∞ (I; W −1,p (Ω)), with W −1,p (Ω) = d i=1 W −1,p (Ω), d ∈ {2, 3}, and p ∈ ( d 2 , 2]. It is mentioned there, that likely the result is not optimal, and the natural question arises whether, and how, it can be improved. Secondly we are interested in control problems with sparsity constraints, subject to (1.1) as constraint. In this case it is natural to demand that for almost every t ∈ I the forcing function is a vector valued Borel measure, i.e. f (t, ·) ∈ M(Ω) = d i=1 M (Ω), where M (Ω) is the space of real and regular Borel measures in Ω, see e.g. [7] and the references on sparse control given there. To treat (1.1) with spatially measure valued controls it is natural to consider the space W −1,p (Ω) with p ∈ [1, d d−1 ), since in this case M(Ω) ⊂ W −1,p (Ω). In dimension 3 this requires to consider the space W −1,p (Ω) with p ∈ [1, 3 2 ). However, even in the stationary case the existence of a solution is an open issue for d = 3 and this range of values for p, see [18] or [24]. For this reason we restrict our attention to the case d = 2 throughout the paper, unless specifically mentioned otherwise. For the two-dimensional case the result in [24] guarantees the existence of a solution to (1.1) for f ∈ W 1,∞ (I; M(Ω)). But this regularity requirement with respect to time is not practical for control theory purposes.
Thus the focus of our work is the investigation of (1.1) for f ∈ L q (I; W −1,p (Ω)) in the case Ω ⊂ R 2 , and p < 2. For this purpose we also require results on the Stokes equation associated to (1.1). Surprisingly, even this case has not yet been analysed for f ∈ L q (I; W −1,p (Ω)). We carry out such an analysis which will be independent of the spatial dimension d.
Before we start, let us summarize, very selectively, some relevant literature. In the stationary case the investigation of the Navier-Stokes system with data less regular than L 2 (Q) dates back to [20], who considers the case f ∈ W −1,2 (Ω). In [24] the range of admissible forcing functions is increased to f ∈ W −1,p (Ω), with p ∈ ( d 2 , 2) For more recent results we refer to [10], [12], [18], and the references there. For the evolutionary system well-posedness for forcing functions in the Hilbert spaces L 2 (I; L 2 (Ω)), and L 2 (I; W −1,2 (Ω)), with d ∈ {2, 3}, is well understood, see e.g. [5], or [28]. The analysis of the Stokes problem associated to (1.1) with forcing functions in the Bochner spaces L q (I; L p (Ω)), with 1 < p, q < ∞, has attracted much attention. We refer to [17] for an informative summary, including the development of the maximal regularity techniques for this scenario. Well-posedness of the Navier-Stokes system with f ∈ L q (I; L p (Ω)) has been investigated in [16] or [30]. As mentioned above, the only work that we are aware of where (1.1) with forcing functions in Sobolev spaces with negative exponents has been investigated is [24].
The plan of the paper is the following. In Section 2 the well-posedness results for the Stokes and the Navier-Stokes equations with f ∈ L q (I; W −1,p (Ω)) under proper conditions on p and q are presented. Some selected proofs are postponed to Sections 3 and 4. Section 5 presents a sensitivity analysis with respect to the right hand side, and Section 6 an asymptotic stability analysis. The proof of a technical result on the nonlinearity appearing in (1.1) is given in the Appendix.

Well-Posedness Results
The aim of this section is to prove the well-posedness of the following Navier-Stokes equations in dimension 2    ∂y ∂t − ν∆y + (y · ∇)y + ∇p = f in Q, div y = 0 in Q, y = 0 on Σ, y(0) = y 0 in Ω, where ν > 0 is the kinematic viscosity coefficient, f ∈ L q (I; W −1,p (Ω)), and y 0 ∈ Y 0 = H+B p,q (Ω). The parameters p and q are fixed throughout this manuscript, and it is assumed that hold; with the exception of Corollary 2.6. Observe that these assumptions imply that q > 4. This condition (2.2) is essential for the well-posedness of the bilinear form introduced in Lemma 2.1 as well as in the proofs of Theorem 2.4 and Proposition 2.7. The low regularity of the force is due to the assumption p < 2. For p ≥ 2 the solvability of (2.1) is well known. The space Y 0 is endowed with the norm which makes it a Banach space. The assumption Ω ⊂ R 2 is imposed throughout this paper, except in Theorem 2.5, which addresses the Stokes equation. Now we introduce the following spaces: They are Banach spaces with the norms The solution of (2.1) will be found in Y. Before proving the existence of such a solution, let us present the following technical lemma. Its proof is given in the Appendix.
As usual, we can remove the pressure from the equation (2.1) by using divergence free test functions.
A distribution p in Q is called an associated pressure if the equation is satisfied in the distribution sense. Then, (y, p) is called a solution of (2.1).
Given y satisfying (2.3), the pressure p is obtained by using De Rham's theorem; see [27,. The details are obtained in a similar way as in the proof for the case of the Stokes equation, see step (iv) of the proof of Theorem 2.5 in Section 3. Remark 2.3. Given s ∈ (1, ∞) and g ∈ W −1,s (Ω), we have that g : W 1,s ′ 0 (Ω) −→ R is a linear and continuous mapping. We know that W s ′ (Ω) is a closed subspace of W 1,s ′ 0 (Ω). Therefore, for every element g ∈ W −1,s (Ω) we can consider its restriction to W s ′ (Ω), and g W s ′ (Ω) ′ ≤ g W −1,s (Ω) holds. Moreover, from Hahn-Banach Theorem we know that every element of W s ′ (Ω) ′ is the restriction of an element of W −1,s (Ω). It is important to observe that the restriction of an element g ∈ W −1,s (Ω) to W s ′ (Ω) can be zero even though g = 0. Actually, given an element g ∈ W s ′ (Ω) ′ , there are infinitely many elements in W −1,s (Ω) whose restriction to W s ′ (Ω) coincide with g. As a consequence, the variational solution y of (2.1), as defined by (2.3), only depends on the restriction of f to W p ′ (Ω). Thus, different elements f can lead to the same solution y, but the pressure p changes. The pressure depends on the action of f on the whole domain W 1,p ′ 0 (Ω). As pointed out in Section 1, the embeddings W(0, T ) ⊂ C([0, T ]; H) and W q,p (0, T ) ⊂ C([0, T ]; B p,q (Ω)) hold. Hence, Y ⊂ C([0, T ]; Y 0 ) and, consequently, the initial condition y(0) = y 0 with y 0 ∈ Y 0 makes sense.
The next theorem is the main result of this section.
Theorem 2.4. Suppose that (2.2) and Ω ⊂ R 2 hold. Then, system (2.1) has a unique solution (y, p) ∈ Y ×W −1,q (I; L p (Ω)/R). Furthermore, there exists a nondecreasing function For the proof of this result we will use the next two theorems. The first one concerns the associated Stokes equation and holds in arbitrary dimension d. It is given by div y S = 0 in Q, y S = 0 on Σ, y S (0) = y S0 in Ω. (2.5) Given g ∈ L r (I; W −1,s (Ω)) and y S0 ∈ B s,r (Ω) with 1 < r, s < ∞, analogously to Definition 2.2, we say that y S ∈ W r,s (0, T ) is a variational solution of (2.5) if for every ψ ∈ W s ′ (Ω) (2.6) A distribution p S in Q is called an associated pressure if the equation is satisfied in the distribution sense.
Given g ∈ L r (I; W −1,s (Ω)) and y S0 ∈ B s,r (Ω) with 1 < r, s < ∞, there exists a unique solution (y S , p S ) ∈ W r,s (0, T ) × W −1,r (I; L s (Ω)/R) of (2.5). Moreover, there exists a constant C r,s such that y S Wr,s(0,T ) ≤ C r,s g L r (I;W s ′ (Ω) ′ ) + y S0 Bs,r(Ω) . (2.7) As mentioned at the end of section 1, the embedding W r,s (0, T ) ⊂ C([0, T ]; B s,r (Ω)) holds. Moreover, the trace mapping y ∈ W r,s (0, T ) → y(0) ∈ B s,r (Ω) is continuous and surjective. This motivates our choice for the initial condition y S0 ∈ B s,r (Ω). Though Theorem 2.5 is expected to hold by experts, it seems that there is no proof available in the literature. For this reason it is given in the next section. There, in Remark 3.1, we shall also assert that Theorem 2.5 holds for domains which are only Lipschitz, provided that conditions on the ranges of r and s are met.
As a consequence of Theorem 2.5 we get the following corollary.
Proof of Theorem 2.4. We are going to prove the existence of a solution y = y N + y S with y N ∈ W(0, T ) and y S ∈ W q,p (0, T ). To this end we write y 0 = y N 0 + y S0 with y N 0 ∈ H and y S0 ∈ B p,q (Ω). Using Theorem 2.5 with r = q and s = p, we define the function y S ∈ W q,p (0, T ) as the unique solution of the system (2.12) Now, we take y N as the solution of 13) The existence and uniqueness of the solution y N ∈ W(0, T ) of the above system follows from Proposition 2.7 by taking ν 0 = 1, e 1 = e 2 = y S ∈ Y, and g(t) = −B(y S (t), y S (t)). As a consequence of Lemma 2.1 we have that g ∈ L 2 (I; H −1 (Ω)). Now, setting y = y N + y S and p = p N + p S , and adding equations (2.12) and (2.13) we obtain that y ∈ Y, p ∈ W −1,q (I; L p (Ω)/R)), and (y, p) is a solution of (2.1). Moreover, (2.4) follows from (7.6) to estimate g, (2.7) and (2.10).

Subtracting the equations satisfied for both solutions we have
(2.14) The right hand side of the above equation can be written in the form g = −B(y 1 , y) − B(y, y 2 ), which belongs to L 2 (I; H −1 (Ω)) by Lemma 2.1. Let us prove that y ∈ W(0, T ). First, we observe that due to the properties of p and q, in particular p < 2 and q > 2, we have (Ω) and, consequently, g ∈ L 2 (I; W −1,p (Ω)). We also have that p ∈ W −1,2 (I; L p (Ω)/R). Now, from Theorem 2.5 we infer that (y, p) is the unique solution of (2.14) in On the other hand, (2.14) can be considered as a Stokes system with the right hand side belonging to L 2 (I; H −1 (Ω)). Hence, it is well known that there exists a unique element (ŷ,p) ∈ W(0, T ) × W −1,∞ (I; L 2 (Ω)/R) solution of (2.14). Using again that we deduce from Theorem 2.5 that (ŷ,p) = (y, p). Thus, we have y ∈ W(0, T ). Therefore, we can multiply the equation (2.14) by y and after integration by parts it yields From this inequality we deduce that Since y(0) = 0, we infer from Gronwall's inequality that y = 0, and with (2.14) p is the zero element of W −1,∞ (I; L p (Ω)/R).
The solution of (1.1) enjoys a better regularity than the one established in the previous remark for q ≥ 8 and under additional assumption on y 0 . Theorem 2.9. Let us assume that q ≥ 8 and y 0 = y N 0 + y S0 ∈ B 2,4 (Ω) + B p,q (Ω). Then the variational solution y of (1.1) belongs to L q (I; L 4 (Ω)) and depends continuously in this topology on f and y 0 . Moreover, the estimate holds for an increasing monotone function η q : [0, ∞) −→ [0, ∞) independent of f and y 0 , with η q (0) = 0.
(Ω)) follows. This embedding, (2.19) and Hölder's inequality imply for 0 <t ≤ T Let us take R > 0 and z any element of the closed ballB R (0) of L 8 (0,t; L 4 (Ω)). Then for some 0 <t ≤ T small enough we obtain from the above inequality Hence, we have a compact mapping z ∈B R (0) → y z ∈B R (0). From Schauder's fixed point theorem we infer the existence of a fixed point. Since the solution of (1.1) is unique, this fixed point must be y N and, consequently, y N belongs to L 8 (0,t; L 4 (Ω)). From (2.16) with g z replaced by g y we infer that y ∈ W 4,2 (0,t) and satisfies (2.19). Therefore, there exists a maximal time T * ≤ T and a solution in the space W 4,2 . We know there are two possibilities: either T * = T and the theorem is proved, or T * < T and lim t→T * y N W4,2(0,t) = ∞ and y N W4,2(0,t) < ∞ ∀t < T * . Let us prove that the second option can not occur. Given ε > 0, we know from Remark 2.8 that there exists t ε > 0 close enough to T * so that From (2.19) and (2.18) with z replaced by y, and continuous embedding W 4,2 (t ε ,t) ⊂ C([t ε ,t]; L 4 (Ω)) we get for everyt ∈ (t ε , T * ) and any ε > 0 which proves that the explosion is not possible.
Estimate (2.15) follows from (2.7), the above inequality, and the estimate obtained in Remark 2.8 for y N L 4 (0,T ;L 4 (Ω)) . Finally, the continuous dependence of y with respect to f and y 0 can be proved using Theorem 2.5 and (2.15).
For the above proof the L ∞ (I; L 4 (Ω)) regularity of the solution of (2.16) is crucial. This is obtained if q ≥ 8.

Proof of Theorem 2.5
We separate the proof in several steps.
(i) Notation and preliminaries. Let L s σ (Ω) denote the closure in L s (Ω) of the space {φ ∈ C ∞ 0 (Ω) : div φ = 0}. Given f ∈ L s (Ω) we define the Helmholtz projection P s : where ∆H = div f in Ω, and n · (∇H − f ) = 0 with n equal to the unit outer normal vector to Γ. We have that range(P s ) = L s σ (Ω), P 2 s = P s , and P ′ s = P s ′ , for the dual operator. The Stokes operator in L s σ (Ω) is defined by It is a closed bijective operator when considered on the dense domain D(A s ) ⊂ L s σ (Ω). This operator enjoys maximal parabolic regularity, [14], [16], [17, page 147]. More precisely, for has a unique solutionỹ ∈ L r (I; W 2,s (Ω)∩W s (Ω))∩W 1,r (I; L s σ (Ω)). Moreover, the inequality ∂ỹ ∂t L r (I;L s (Ω)) + ỹ L r (I;W 2,s (Ω)) ≤ C g L r (I;L s (Ω)) + ỹ 0 (3.3) holds for some C independent of (g,ỹ 0 ). Above the norm ofỹ 0 is taken in the interpolation space (L s σ (Ω), W s (Ω) ∩ W 2,s (Ω)) 1− 1 r ,r . The fractional power A s ). Moreover, the norms A s y L s (Ω) and y W 2,s (Ω) are equivalent on D(A s ), see e.g. [10]. We shall use in an essential manner that and −∆ is understood with homogenous boundary conditions. This was verified in [15], for domains with a 'smooth' boundary. Using the classical result in [11] on the characterisation of D ((−∆   1 2 )), we obtain that D(A 2 denotes complex interpolation. The second equality follows from the fact that the Stokes operator on L s σ (Ω) admits an H ∞ calculus [23], see also [17, pg. 149]. Next we note that the Helmhotz projection satisfies P s = P s P s and that the range of P s is given by L s σ (Ω). Hence L s σ (Ω) is a complemented subspace of L s (Ω), see [31, pg. 22]. Using standard regularity results for the Stokes equation [3] it follows from the definition of P s , that it is a bounded linear operator from L s (Ω) to L s σ (Ω) and from W 2,s (Ω) to W s (Ω) ∩ W 2,s (Ω). As a consequence we obtain that [L s σ (Ω), W s (Ω) ∩ W 2,s (Ω)] 1 2 = W s (Ω), by a general result on interpolation couples involving subspaces, see [31, pg.118].
(ii) Extending the operator A s . In the following we use arguments inspired by [9, §6] and [4, §11] where the case of second order elliptic operators is considered. First, note that we can utilize the above arguments for s replaced by its conjugate s ′ . Hence A 1 2 where equality holds in W s ′ (Ω) ′ . Indeed, for all y ∈ W s (Ω) and z ∈ W s ′ (Ω) we get where the identity y, A Let us study some properties of this operator. First, we observe that from the definitions of A s and A s it follows that A s y = A s y for all y ∈ D(A s ). Hence, A s is an extension of A s . Now, given y ∈ D(A 2 ) ∩ D(A and consequently (A s ′ ) ′ ) −1 y S0 , henceg ∈ L r (I; L s σ (Ω)) and sỹ (t) ∈ W s (Ω) for almost all t ∈ I. Hence, we have that y S ∈ L r (I; W s (Ω)). Additionally, from ∂ỹ ∂t ∈ L r (I; L s σ (Ω)) we deduce that ∂yS ∂t ∈ L r (I; W s ′ (Ω) ′ ). Consequently, we have that y s ∈ W r,s (0, T ).

Remark 3.1.
Recently interesting work has been carried out on the treatment of the semigroup associated to the Stokes equation on L s σ (Ω) under the assumption that the boundary Γ is only Lipschitz continuous. This requires a restriction on the range of the parameter s. Let us summarize how these results can be utilized for our treatment of the Stokes equation in W −1,s (Ω) if the assumption on the regularity of Γ is relaxed to that of Lipschitz continuity.
Following [25] we define the Stokes operator in L s σ (Ω) is defined by A s y = −ν∆y + ∇p with domain D(A s ) = {y ∈ W 1,s 0 (Ω) : div y = 0, −ν∆y + ∇p ∈ L s σ (Ω) for some p ∈ L s (Ω)}. We have decided to use the same notation as for the definition of A s given in (3.1). But this should not be problematic since it is only used within this remark. It was proved in [25] that there exists an ǫ > 0 such that for all s satisfying 12) and all d ≥ 3, the operator A s is sectorial with angle 0 and with 0 in the resolvent set, and that −A s generates a bounded analytic semigroup on L s σ (Ω). As a consequence it is densely defined and closed. Moreover W s (Ω) ∩ W 2,s (Ω) ⊂ D(A s ) and A s = −νP s ∆ on W s (Ω) ∩ W 2,s (Ω). In [19] it was further verified that ǫ > 0 could further be chosen such that A s has maximal L s σ (Ω) regularity and that it admits a bounded H ∞ − calculus for all s satisfying (3.12). This implies that the fractional powers of A s are well-defined and that s ) = W s (Ω) was established in [29] in the case of Lipschitz domains, see also [30].
Summarizing, there exists ǫ > 0 such that for all s satisfying (3.12), and all d ≥ 3, the structural properties of Step (i) of the proof of Theorem 2.5 hold. The assertion of Theorem 2.5 can therefore be obtained as before. It appears to be the case that restriction d ≥ 3 dated back to the work in [25], where the proof utilizes the fundamental system of the Stokes system.

Proof of Proposition 2.7
Proof of Proposition 2.7. Let us consider the classical operator associated with the Stokes system A : is orthonormal for the Hilbert product in H: (ψ i , ψ j ) L 2 (Ω) = δ ij . Let us denote by V k the subspace generated by {ψ 1 , . . . , ψ k }. Now we define the orthogonal L 2 (Ω)-projection operator P k : H −→ V k given by Following the classical Faedo-Galerkin approach, we discretize (2.11) It is clear that (4.1) has a maximal solution y k defined in an interval I k . We shall derive a point-wise a-priori bound for y k on I for each k from which I k = I follows. We will also prove a-priori estimates which allow to pass to the limit in (4.1). I -Estimates in L ∞ (I; L 2 (Ω)). Multiplying the equation (4.1) by g k,j (t) and adding from j = 1 to k we infer 1 2 d dt y k (t) 2 L 2 (Ω) + a(y k (t), y k (t)) + b(y k (t), e 2 (t), y k (t)) = g(t), y k (t) V ′ ,V , where we have used the identities b(y k (t), y k (t), y k (t)) = b(e 1 (t), y k (t), y k (t)) = 0. Let us estimate the term b(y k (t), e 2 (t), y k (t)). To this end we write e 2 = e 2H + e 2W with e 2H ∈ L 2 (I; V) ∩ L ∞ (I; L 2 (Ω)) and e 2W ∈ L q (I; W p (Ω)). Then, we have b(y k (t), e 2 (t), y k (t)) = b(y k (t), e 2H (t), y k (t)) + b(y k (t), e 2W (t), y k (t)).
Now we observe that the estimates I and II imply that {y k } ∞ k=1 is bounded in L 2 (I; V) ∩ L ∞ (I; H), hence bounded in Y. Then, using Lemma 2.1 we deduce form the above identity and the inequality P k ψ H 1 0 (Ω) ≤ ψ H 1 This inequality along with the estimates (4.6) and (4.7) proves the boundedness of {y ′ k } ∞ k=1 in L 2 (I; V ′ ). Hence, {y k } ∞ k=1 is bounded in W(0, T ) and the second inequality of (2.10) holds.
Finally, using the above estimates, it is standard to pass to the limit in (4.1), taking a subsequence if necessary, and to deduce that {y k } ∞ k=1 converges weakly in W(0, T ) and weakly * in L ∞ (I; H) to a solution y N of the system (2.11); see, for instance, [21,. Moreover, since every y k satisfies (2.10), then y N does it as well. Further, since y N ∈ L 2 (I; V) and y ′ N ∈ L 2 (I; V ′ ) we deduce that y N ∈ W(0, T ). Moreover, applying De Rham theorem we infer the existence of p N ∈ W −1,∞ (I; L 2 (Ω)/R) such that (y N , p N ) is solution of (2.9); see [5,. Now, using Lemma 2.1 and Gronwall's inequality the uniqueness of a solution follows in a standard way; cf. [5, Chaper V-1.3.5].

Sensitivity analysis of the state equation
In this section we analyze the differentiability of the mapping G : L q (I; W −1,p (Ω)) −→ Y associating to each element f ∈ L q (I; W −1,p (Ω)) the solution y f ∈ Y of (2.3).
Proof. Let us define the space F = L 2 (I; V ′ ) + L q (I; W p ′ (Ω) ′ ) endowed with the norm Thus, F is a Banach space. We also consider the operators Associated with these two continuous operators we define where y = y 1 + y 2 with y 1 ∈ W(0, T ) and y 2 ∈ W q,p (0, T ). It is immediate to check that Ay is independent of the chosen representation y = y 1 + y 2 , and it is continuous. Now, we introduce the mapping where y 0 is the initial condition in (2.1). Recall that Y ⊂ C([0; T ]; Y 0 ) holds. Hence, y ∈ Y → y(0) ∈ Y 0 is a linear and continuous mapping. Moreover, Lemma 2.1 implies that y ∈ Y → B(y, y) ∈ L 2 (I; H −1 (Ω)) ⊂ L 2 (I; V ′ ) ⊂ F is bilinear and continuous. By definition of W(0, T ) and W q,p (0, T ) we also have that ∂ ∂t : Y → F is a linear and continuous operator. All together this implies that G is a C ∞ mapping.
Given f ∈ L q (I; W p ′ (Ω) ′ ), we denote by y f ∈ Y the solution of (2.3). Then, we have that is a linear and continuous mapping. Actually, it is an isomorphism. Let us prove this. Given an arbitrary element (g, z 0 ) ∈ F × Y 0 , we set g = g N + g S and z 0 = z N 0 + z S0 with g N ∈ L 2 (I; V ′ ), g S ∈ L q (I; W p ′ (Ω) ′ ), z N 0 ∈ H, and z S0 ∈ B p,q (Ω). Now, we show the existence and uniqueness of a solution z ∈ Y of the equation We decompose the system in two parts and The existence and uniqueness of a solution z S ∈ W q,p (0, T ) of (5.5) follows from Theorem 2.5. In equation (5.6), we have that z N 0 ∈ H, y f ∈ Y, and from Lemma 2.1 we get that the right hand side of the partial differential equation belongs to L 2 (I; H −1 (Ω)). Hence, from Proposition 2.7 we infer the existence and uniqueness of a solution y N ∈ W(0, T ) of (5.6). Now, setting y = y N + y S ∈ Y, we deduce that y is a solution of (5.4). The uniqueness follows from Gronwall's inequality, arguing as in the proof of Theorem 2.4. Now, it is enough to apply the implicit function theorem to deduce the existence of a func-tionG : Hence,G(f ) = y f is the solution of (2.3). Moreover, by differentiation with respect to f of the identity G(G(f ), f ) = 0, setting z g = DG(f )g for g ∈ L q (I; W −1,p (Ω)), and using (5.3) and Rham's theorem equation (5.1) follows. The equation (5.2) follows easily from the identity Finally, the theorem follows by observing that G : L q (I; W −1,p (Ω)) −→ Y is given by is the restriction operator, that is linear and continuous.