On The Two-Dimensional Stokes Problem in Exterior Domains: The Maximum Modulus Theorem

The two-dimensional Stokes IBVP on (0,T)×Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(0,T)\times \Omega $$\end{document} is investigated under the assumptions that Ω⊂R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset {\mathbb {R}}^2$$\end{document} is a smooth exterior domain, the initial datum v0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_0$$\end{document} belongs to L∞(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty (\Omega )$$\end{document} and (v0,∇ϕ)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(v_0,\nabla \phi )=0$$\end{document} for all ϕ∈Lℓoc1(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi \in L^1_{\ell oc}(\Omega )$$\end{document} with ∇ϕ∈L1(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla \phi \in L^1(\Omega )$$\end{document}. The well-posedeness in L∞((0,T)×Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty ((0,T)\times \Omega )$$\end{document} and the maximum modulus theorem are achieved, in particular one deduces that the Stokes semigroup on L∞(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty (\Omega )$$\end{document} is a bounded analytic semigroup.

We are interested to prove Theorem 1. (Maximum Modulus Theorem) For all v 0 ∈ L ∞ (Ω) enjoying (2), there exists a unique solution (v, π) to problem (1) such that for all q > 2,  We do not know an estimate of the pressure field π v like the one given in (4). In our approach to the uniqueness, this kind of estimate is crucial. In this regard, we deem it appropriate to make a digression in relation to the problem of the uniqueness of the solutions to the problem (54).
In the L ∞ -setting the difficulties to obtain a sharp result are connected essentially with the lack of the Helmholtz decomposition of L ∞ , decomposition that in L q -setting, for all q ∈ (1, ∞), holds. The lack of the Helmholtz decomposition does not allow to state a posteriori the existence of the pressure field as in the case of the L q -theory or to define the Stokes operator like in the L q -theory.
This critical result is produced considering the two dimensional boundary value problem for steady Stokes system in exterior domains, which admits the same pathologic solutions. That is, a solution admits a value at infinity that is not a datum of the problem.
The result given in [18] is not in contradiction with the ones contained in [3,4] (or with the statement of Theorem 1 of the present paper). Actually, the pressure field π does not verify the condition d(x, ∂Ω)|∇π(t, x)| ≤ c with c independent of x, property exhibits in [3,4] for the solutions (or estimates (4) of the present paper). Since the solution with limit u ∞ (t) obtained in [18] is unique, 1 as consequence we get that no solution established in [3,4] admits u ∞ (t) as limit at infinity (as well as the one of Theorem 1).
The present paper is devoted to the memory of Professor Carlo Miranda, he was an Eminent Mathematician in Napoli, this year is the 40th anniversary of his death.

Outline of The Proof
Before outlining the proof of Theorem 1, we consider useful to recall what approaches the present author employed in previous papers studying the question in nD, n ≥ 3, exterior domains and the result of non-uniqueness in 2D.
In n-dimensional case, n ≥ 3, the results are proved by means of a suitable coupling of the results proved in [3] and in [16], subsequently the same approach is reconsidered in [6]. As already recalled, the first paper is concerned with local in time estimates and the second paper is concerned with the extension of the estimates to large time.
In the two-dimensional case the result of the first paper still works, while the result of the second paper does not work. The result in [16] is based on a technique of duality which does not work in twodimension, roughly speaking, because the solution ϕ(t, x) of the (local) adjoint problem has the behavior ||ϕ(t)|| ∞ ≤ c||ϕ|| 1 t −1 where the exponent −1 is sharp. Actually, in [16] one translates the original question into the study of the problem ω t − Δω +∇π ω = −F (1) t + G , ∇·ω = 0 , in (0, T )×Ω , ω = 0 on (0, T )×∂Ω , ω = 0 on {0}×Ω , where F (1) t and G are suitable functions. In this way the difficulty becomes the fact that G, with compact support in x ∈ Ω and belonging to L ∞ ((0, T ) × Ω), has no behavior for large t. By a duality approach, 83 Page 4 of 29 P. Maremonti JMFM in order to obtain an estimate for ||ω(t)|| ∞ , one has to tackle the estimate related to for which, due to the sharpness behavior of ||ϕ(s)|| ∞ ≤ c||ϕ 0 || 1 s −1 , one at most is able to deduce O(||ϕ 0 || 1 log(t + e)). Hence, via this approach, no uniform bound in t holds for the L ∞ -norm of the solution ω(t, x). In the paper [18], roughly speaking, the previous problem of ω becomes (2) plays the same role of F (1) but with different properties. The function F (2) in [18], for all t > 0, is the For all t > 0, the extension F (2) is just the solution to the boundary value problem in Ω of the steady Stokes system. Thanks to this construction, for all v 0 ∈ L ∞ which enjoys (2), we are able to prove the existence of a solution to problem (1) and estimates (3)-(4) 1,2 of Theorem 1 for v. But we are not able to furnish estimate (4) 3 for the pressure field, which is substituted by π := U (t) · x + π, with ∇π ∈ L q (Ω).
with v 0R equal to v 0 in a neighbourhood of ∂Ω and with compact support in Ω. The disadvantage is that for all R > 0 we can construct a different solution ( U depends on R).
In this note, the chief aim is to avoid the difficulties arises by the sharpness of the estimates for the solutions of the two-dimensional adjoint problem.
As it will be clear by the arguments that we develop in the sequel, we realize the task blending the ideas contained in papers [16] and [18]. It is appropriate to say: in medio stat virtus.
We consider an initial datum v 0 ∈ L ∞ (Ω) that enjoys (2). We make the decomposition where g is a smooth cutoff function with g = 1 in neighborhood of ∂Ω and g = 0 for large x, and b 0 is a Bogovskiȋ solution to the problem ∇ · b 0 = ∇g · v 0 . The peculiarity of the decomposition is in v 0c with compact support in Ω and v c 0 with support in Ω but far from the boundary ∂Ω. Of course, this last property plays an important role in the construction of the solution (see the comments after the the following items). We consider v = v c + v c and π := π v c + π vc . The pairs (v c , π v c ) and (v c , π vc ) are solutions to problem (1) with initial datum v c 0 and v 0c , respectively. The solution (v c , π vc ) is already known from [3,4]. In fact, the compact nature of the support of the initial datum v 0c allows us to employ the result of the L q -setting (cf. [8,9]), thus the estimate The solution (v c , π v c ) is instead constructed as it follows. We look for v c := U − h U + F + W + ω and π := π ω .
In the previous formula: • U is the solution to the Cauchy problem with initial datum v c 0 extended to zero on R 2 , • h is a cutoff function with support depending on t, and, for all t > 0, h = 1 in a ball including ∂Ω, • the trace on ∂Ω of −U + U has a suitable extension F from ∂Ω into Ω with compact support, • W is a solution to the Bogovskiȋ problem ∇ · W = ∇h · U with compact support in Ω, • finally, ω is the solution to the Stokes problem with homogeneous boundary and initial datum, but with a right-hand side given by The chief properties of U are the behaviors in t for the derivatives of U in L ∞ (Ω L ), where Ω L ⊃ ∂Ω is bounded, which are not singular in t = 0, and U with its derivatives evaluated in L ∞ (Ω L )-norm go to 0, letting t → 0. All this is a consequence of the special initial datum v c 0 , cf. sect. 2.1. The extension F is obtained by the same technique employed in [16] for F (1) . But the new fact is that the boundary value of 2 ) holds, that let us to obtain a decay for ΔF = ||v 0 || ∞ O(t − 1 2 ) (ΔF has the same meaning of G in (5)) (for the construction of F see sect. 2.2). This property from one side allows us to find the right estimate to discuss (6), from another side leads to discuss the additional term −h U . The role of this term is to realize the homogeneous boundary value of the solution on (0, T ) × ∂Ω. We recall that at t = 0 we have U = 0 pointwise on {0} × Ω, so no correction is due in order to obtain the initial value on {0} × Ω.
Hence, h has compact support in the ball B 7 4 R , and ∇h has compact support in the shell {x ∈ Ω : q −2 , that are decaying in t for q > 2 and q > 1, respectively. We take advantage this behavior in t in order to discuss the term Δh U . This is a new fact with respect to the behavior of the term G of the n-dimensional case, that arose the difficulty of the estimate (6) in the two-dimensional case. Instead, in the estimates the time derivative of h U , as a matter of course, go on without difficulties. However, the term h U is not divergence free. Hence, in order to preserve the divergence free of the solution v c , we introduce the function W . The function W is a solution of the problem ∇ · W = ∇h · U in the shell Ω(R) : with homogenous boundary value. The shell Ω(R) is variable in t, but, for all t > 0, there is the homothety with the shell S := {1 < |x| < 2}. Considering a solution to ∇ · W S (t) = ∇h S · U (t) in the shell S, with homogeneous boundary value on ∂S, then a solution W in Ω(R) is calculated in the following way: . We find the suitable estimates for W and its derivatives considering the ones related to W S and employing the homothety property of the domain Ω(R). It is important to stress that W t exists, but W t does not solve the time derivative of the Bogowskiȋ problem. Since no interest there is for this last fact, and since W t is a "linear" combination of the spatial derivatives of W and of the time derivative of the solution on the fixed shell S, using the homothety property of the domain, we can deduce all the estimates related to W t (cf. sect. 2.4).
The plan of the paper follows the items detected for the construction of the auxiliary function U, F, h, W . They are discussed and proved in Sect. 2. In Sect. 3 we give the statement of the results due to K. Abe and Y. Giga, that furnish Theorem 1 for initial data with compact support. In sect. 4 we solve the Stokes problem related to ω. Finally, in Sect. 5 we give the proof of Theorem 1.
The symbol B ρ (x 0 ) denotes a ball in R 2 with center x 0 and radius ρ, in the case of x 0 = 0, we simply write B ρ .
In the following we consider R > 3 diam (R 2 − Ω). We set Ω R := Ω ∩ B R . For a Lebesgue's measurable set D, the symbol |D| denotes the measure. By the symbol L(q, σ)(Ω) we mean the G.G. Lorentz spaces and with || · || (q,σ) its norm. In particular, we consider L(q, ∞)(Ω) ≡ L q w (Ω), q ∈ (1, ∞), endowed with the Lorentz norm For a function g(t, x) and t ≥ 0, we denote by supp x) the support in the variable x.
In the following the symbol c denotes a numerical constant whose value is inessential for our aims.

Heat Solution
We denote by H(t, x) the heat fundamental solution and we indicate by where R+δ and M c q as given in (9). Then, for all k, h ∈ N ∪ {0}, there exists a constant c(δ, L) such that Proof. By the definition of M c q , we recall that the left hand side of (10) 1 has to be considered for (t, x) ∈ R + × B R (O). For k, h ∈ N ∪ {0} and μ > 0, we have the well known estimate If L ≥ R + δ and x ∈ B R (0), we get that |x − z| ∈ [R + δ, 2L] implies |z| ∈ [δ, 3L], as well as |x − z| > 2L implies |z| > L. Hence, applying Hölder's inequality, by virtue of the definition of the support of v 0 , we deduce Proof. We set where χ 2R0 denotes the characteristic function of the ball B 2R0 . Of course, we have u = u 2R0 + u 2R0 too. Hence we get Hence, letting t → 0, we achieve (12).

The Extension F
We recall some results concerning the boundary value problem in a smooth bounded domain D of the steady Stokes system: with M independent of a. In particular, we deduce Proof. The proof of lemma can be found in [10] Ch.IV Lemma 6.1.
The following is an a priori estimate Then there exists a field π u such that where c is a constant independent of u.
Proof. This result is contained in [10,22]. Actually, in our hypotheses, for u we can consider the Helmholtz decomposition of Δu, hence, formally u is a solution to the boundary value problem Then the estimates and regularity follow from the result in [10,22] for solution to the Stokes problem in exterior domains.
We recall some results concerning the Bogovskiȋ problem. Let E be a smooth bounded domain and with the compatibility condition E gdx = 0.
It is known that one solves problem (16) by considering the domain E as a union of domains C k , k = 1, . . . , N, star-shaped with respect to the balls B(k) of a fixed radius; moreover, using a smooth partition of unity, say N k=1 ψ k (x) = 1, with suppψ k ⊂ C k . Then, a vector field satisfying (16) can be written in the where We also recall that, for each k = 1, . . . , N, B k is an operator with weakly singular kernel. Actually, B k j [ · ] is the integral operator with the kernel and ∂ ∂xj B k is an operator with singular kernel of Calderon-Zigmund kind.
with c(R) independent of A. In particular, we get F (x) ∈ C 1 (Ω).
Proof. Let us consider the boundary value problem (13) for D ≡ Ω R , with boundary data a = A on ∂Ω and a = 0 on |x| = R. By virtue of Lemma 3 there exists a unique solution Moreover, we consider a Bogovskiȋ's solution V to the equation (16) assuming . By virtue of the estimate of Lemma 5, we get Setting F = V h R + V we have proved estimate (20). The regularity in Ω is a consequence of the ones doable for V and V (see [10]).
with c independent of t.
Proof. For all t ≥ 0, we consider the extension F = V h R + V given in Lemma 6. Hence, recalling the definition of V and V , there exists Hence, via estimate (21) for D k t V and via representation formula (18) for V t , in our hypotheses estimate (23) follows by the same arguments developed for the estimates (20).
We set where U is the solution to the heat equation furnished in Sect. 2.1 and corresponding to v 0 with supp v 0 ⊂ Ω − B R+δ and enjoying (2).

Lemma 7. Let
with c independent of v 0 .
Proof. By virtue of Corollary 1, estimate (25) easily follows achieving the estimate the task is to justify the exponent − 1 2 on the right hand side of estimate (25). The assumption Ω smooth exterior domain leads to assert ∂Ω ≡ p ∪ m=1 ∂Ω m . For any continuous function g, the mean value 2 is In order to estimate we initially remark that, by virtue of (26), for i = 1, 2 and for all k ∈ N ∪ {0} there exist ξ h , h = 1, . . . , p, such that where D k t U i (t, ξ h ) is mean value of the integral on ∂Ω h . Hence, by virtue of Lagrange's theorem and assumptions on U , from (27) we get where in the last step we take (10) into account. This justify the estimate for the L q (∂Ω) norm of D k t A. Instead, for the seminorm we have Hence, considering again estimate (10), a fortiori there is for the exponent the increment − 1 2 . 2 We recall that in our notations we denoted by |D| the measure of any Lebesgue's measurable set.

A Special Bogovskiȋ Problem
For all t > 0, we consider the Bogovskiȋ problem where we set R := R + √ t and Ω(R) : Of course, for all R > 0 the domain Ω(R) is homothetic to the shell S := 1 < |z| < 2. For problem (33), since the compatibility condition holds, Lemma 5 holds too. However, here we are interested to state the result employing the following approach, that is more suitable for the special domain Ω(R).

Lemma 8.
There exist a constant c and a smooth solution W (t, x) to problem (33) with compact support in Ω(R) and such that, for all t > 0, and with Proof. For all z ∈ S, we set h S (z) := h(|z|). We consider the following problem Taking into account that h S (z) = 1 on |z| = 1 and h S (z) = 0 on |z| = 2, since U is independent of z, for the Bogovskiȋ problem (36) the compatibility condition holds, and so, by virtue of Lemma 5, we establish the existence of a solution W S (t, z) with compact support in the shell S. Easily one verifies that W (t, is a solution to problem (33) with compact support in Ω(R). Being Ω(R) homothetic with the shell S, via estimate (17) for m = 1 and via the following trivial chain, we deduce (34) 1 : Analogously, via (17) for m = 2, we get Deriving W with respect to t, we get we point out that the last term has to be considered as the "Eulerian derivative" which arises via formula (18) written for solution W S (t, z) where, thanks to the static position, we transport the time derivative on U (t). Now, let us consider ∇W t . From (38) it follows that Since W and ∇W have compact support in Ω(R) and where, taking the homothety between the sets Ω(R) and S into account, we argued as made in estimate (37). Hence, we arrive at (35) 1 for ∇W , and thanks to the Poincaré inequality we complete the proof of where again we stress that the last term is meant as the "Eulerian derivative". We set and K(t, ξ) := W S tt (t, ξ) . Hence, recalling that |ξ| = | x R+ √ t | ≤ 2, employing the homothetic transformation for the coordinates, via (17), we get then, first via (17), and subsequently applying the homothetic change of variables, we arrive at Since W tt has compact support and the sets Ω(R) and S are homothetic, we get where again we consider the "Eulerian derivative", that is, via formula (18) written for solution W S (t, z), we transported the time derivative on U (t), and then, for the estimate of the Bogovskiȋ solution, we toke into account that the kernel in (18) is weakly singular with exponent α = 1. This completes the proof of (35).

Lemma 9. Let W be a solution to the Bogovskiȋ problem (33) stated in Lemma 8 and U ≡ H[v 0 ] as in Lemma 1. Then we get
and for all t > 0, where we set and where c is a constant independent of v 0 and t. Proof. We have where all the estimates are consequence of (10). Moreover, as a consequence of (10) 2 , for all μ > 0, we have For q ≥ 1, we get Hence, as a matter of course, the right hand side of (34) 1 is bounded by c(R + √ t) 2 q −1 ||v 0 || ∞ and the one of (34) 2 is bounded by c(R + √ t) 2 q −2 ||v 0 || ∞ . Hence we get (39) for ∇W and ∇∇W . Employing Gagliardo-Nirenberg inequality, for q > 2, we get Since estimate (39) is achieved for ||∇W (t)|| q , via the Poincaré inequality, we obtain Moreover, by means of the estimates (43) for ∇h and ∇∇h, via (41), the right hand side of (35 q −1 ||v 0 || ∞ , which furnishes (40) 2 , and then, again via (35) 1 we arrive at (40) 1 .
Estimate given in (43) Hence, for the terms involving the L 2 -norm on the right hand side of (35) 2 we get For the estimate of the term involving the L 2 w -norm on the right hand side of (35) 2 , being
Of course, since (44)-(45) are stated in a neighborhood of t = 0 and since μ > 0 can be chosen as we want, estimates (44) and (45) are not given in sharpness way, but they are given functional to our aims.

Some Integral Estimates
The symbols F, h, and W have the same meaning given in previous section. We recall that we assumed R > 3diam(R 2 − Ω) and L > R, as well in Lemma 1 we assumed suppv 0 ⊂ B c R+δ , δ > 0 . In this section, for all η > 0, the function ϕ ∈ C(η, T ; J 1,q (Ω), q ∈ (1,2], is such that for all t > 0, for r = ∞ we just consider L ∞ , and c is a constant independent of ϕ. We set Lemma 11. For all q ∈ [1, 2), there exists a constant C := C(R) such that Proof. We separately look for the estimate of each integral term, called I i (t), i = 1, 2, 3, of the sum. Applying Hölder's inequality, we get where increasing we employed (25) for F , R < L and (47) 2 for ϕ. Applying Hölder's inequality and employing (31), we get Since |supp Applying Holder's inequality, we get By virtue of estimate (40) 2 for W t and estimate (47) 2 for ϕ, we obtain The above estimates furnish (48).

Lemma 12.
For all q ∈ [1, 2) there exists a constant c such that where constant c is independent of v 0 and ϕ 0 .
Proof. We look for separately the estimate of each integral, called I i (t), i = 1, 2, 3, of the sum. Via Hölder's inequality, we get where increasing we employed (25) for F and (47) 1 for ϕ. Via Hölder's inequality, and employing (43) for Δh and (47) 1 for ϕ, we obtain After applying Hölder's inequality, we get where in the last step of the estimate we taken into account (47) 2 for ϕ and (39) for ΔW . Collecting the estimates for I i (t), we arrive at the wanted one for I 2 (t).
We set

Lemma 14.
For all q ∈ [1,2], there exists a constant C := C(R) such that Proof. We initially point out that μ > 0 can be chosen in Lemma 10 in such a way that the integral is well posed. Moreover, we recall that L > R. In order to deduce (51) we look for separately the estimate of each integral, called J i (t), i = 1, 2, 3, of the sum. Integrating by parts we get where in t = 0 we used the bound (44) for μ > 3. Applying Hölder's inequality, we get where for F we toke estimate (25) into account, as well estimates (47) 2 for ϕ. Since R < L we arrive at (51) for J 1 . After integrating by parts, applying Hölder's inequality, we obtain where in t = 0, we toke the bound (44) 2 for μ > 3 into account. Recalling that ≤ 1, employing (31) for h, (10) 1 for U and U t , and (47) 2 for ϕ, since |supp Recalling that |supp Now, applying (10) 1 for the term || U t (t)|| ∞ and for the term || U || ∞ , being R < L, we realize For the last term we have D 2 estimates (10) for U and developing (32) 2 for h ττ , we get where increasing we used the semigroup property of U . Collecting the estimates related to J i 2 , we arrive at Integrating by parts, applying Hölder's inequality, we get where in t = 0 we toke the bound (44). Employing (40) 1,3 for W , we get where we considered (47) 2 for ϕ.
In problem (54) the initial condition is given in the weak form (ϑ(0), ϕ) = (w 0 , ϕ), ϕ ∈ C 0 (Ω), in order to state the initial boundary value problem with an initial data w 0 belonging to the weaker Lebesgue space L p (Ω), p ≥ 1. With the weak formulation in L p , p ∈ (1, ∞), the continuity of the equation of divergence (also in weak form) at t = 0 is lost, as well as the zero value of the normal component of the solution at the initial instant t = 0. Of course, if the datum is an element of J p (Ω) ⊂ L p (Ω), p > 1, then, the problem is just the classical one. For our aims the case w 0 ∈ C 1 0 (Ω) ⊂ L 1 (Ω) has a special interest. Actually, we look for an estimate in L ∞ (Ω) by means of the variational formulation ||a|| ∞ = sup (ω(t), θ) ||θ|| 1 of the auxiliary solution ω(t, x) to problem (67).
In Theorem 2 the initial boundary value problem (54) can be considered for Ω bounded or exterior, indifferently.
Proof. For the proof of the above theorem see [17] Theorem 2.1. Actually, the quoted reference is the two-dimensional version of Theorem 3.2 given in [14] .

Remark 1.
We stress that the property ψ ∈ C([0, T ); J q (Ω)) is meant in the sense that lim t→0 ||ψ(t) − P q (w 0 )|| q = 0. In the case of w 0 ∈ C 1 0 (Ω) in t = 0 at most the weak limit property holds, the one stated in the theorem. Of course, if we assume w 0 ∈ J q (Ω), q ∈ (1, ∞), the result becomes the classical one, in particular the continuity in norm holds.
By virtue of trace theorems, for any element of a ∈ W 1− 1 q ,q (∂Ω) admits an extension from ∂Ω into Ω which is an element of W 1,q (Ω ). We denote by the same symbol a the element of space trace W 1− 1 q ,q (∂Ω) and its extension as element of W 1,q (Ω ).

A Nonlinear Generalization of The Gronwall Inequality
Lemma 20. Let y(t) be a nonnegative function that satisfies the integral inequality (a(s)y(s) + b(s)y σ (s)ds , A 0 ≥ 0, σ ∈ [0, 1) , where a(t) and b(t) are continuous nonnegative functions for t ≥ t 0 . Then, the following inequality holds