Evolutionary, symmetric p-Laplacian. Interior regularity of time derivatives and its consequences

We consider the evolutionary symmetric $p$-Laplacian with safety $1$. By symmetric we mean that the full gradient of $p$-Laplacian is replaced by its symmetric part, which causes breakdown of the Uhlenbeck structure. We derive the interior regularity of time derivatives of its local weak solution. To circumvent the space-time growth mismatch, we devise a new local regularity technique of iterations in Nikolskii-Bochner spaces. It is interesting by itself, as it may be modified to provide new regularity results for the full-gradient $p$-Laplacian case with lower-order dependencies. Finally, having the regularity result for time derivatives, we obtain respective regularity of the main part. The Appendix on Nikolskii-Bochner spaces, that includes theorems on their embeddings and interpolations, may be of independent interest.


Introduction
Let Du denote the symmetric part of the gradient, i.e. Du = ∇u+∇ T u 2 . We consider the following symmetric p-Laplace system u, t − div A(Du) = 0, that generalizes the generic examples with p ≥ 2. The precise assumptions on A are presented in Section 4.2.
In this paper we obtain local regularity results connected with the second-order energy estimates in time (i.e. testing (1) with the localized u, tt , roughly speaking). Contrary to the result of [23], where we derive local regularity related to the second-order energy estimates in space, now we face serious difficulties, even on the level of formal estimates. These difficulties arise from the space-time mismatch in localization terms (for more on this, see subsection 6.1). We cope with these difficulties by means of a new interior iteration technique in Nikolskii-Bochner spaces, inspired by Bulíček, Ettwein, Kaplický & Pražak [20]. We believe that our approach is interesting in itself and may be fruitful for obtaining new regularity results for other p-nonlinear systems of PDEs, for instance for the full p-Laplacian with lower-order terms (A(x, ∇u) or A(x, u, Du)). In addition to obtaining the quantitative result, we also provide (qualitative) inequalities.

Motivation and known results
From the perspective of non-Newtonian hydrodynamics and nonlinear elasticity, it would be essential to repeat for the symmetric p-Laplacian the C 1,α loc -regularity result, available for the full gradient p-Laplacian since the works of Uhlenbeck [53], Tolksdorff [50] (stationary case) and DiBenedetto & Friedman [17] (evolutionary case). Unfortunately, the pointwise structure of the symmetric p-Laplacian seems to be resistant to the methods used in the full p-Laplacian case 1 to get boundedness of gradients.
Nevertheless, one can obtain certain regularity results for the symmetric p-Laplacian. For the stationary case, Beirão da Veiga Beirão da Veiga & Crispo [9] and [10] provide smoothness of a periodic-boundary value problem, provided p is close to 2. Results for generic boundary-value problems, developed for the full p-Navier-Stokes system, are of course available for the symmetric p-Laplacian. In particular, one has smoothness of solutions to basic initial-boundary value problems in 2d case, see Kaplický, Málek & Stará [39] and Kaplický [37], [38] as well as existence of strong solutions for 3d case, compare [40] by Málek, Nečas & Růžička and also [7], [8] by Beirão da Veiga and [11] by Beirão da Veiga, Kaplický & Růžička. Let us mention also here a recent local regularity study for the p-Laplacian, symmetric p-Laplacian and p-Stokes type problems by Frehse & Schwarzacher [29]. Since their results corresponds strongly to ours, we compare them in a more detailed manner in Subsection 5.3. For small data regularity results, one may refer to Crispo & Grisanti [24].
As remarked, the regularity results cited above concern certain basic boundary-value problems. Local (interior) regularity results are much more scarce. In [22] the partial C 1,α loc regularity theory has been developed. One should mention also [30] by Fuchs and Seregin. On the other hand, the regularity results for the full-gradient case are abundant. For the evolutionary p-case, let us restrict ourselves to referring to the classical monograph by DiBenedetto [25] and a simple proof of C 1,α loc regularity by Gianazza, Surnachev & Vespri [31].

Plan of the paper
In Section 4 we provide needed preliminaries, including notation, assumptions on growth of stress tensor A, definition of local weak solution to (1) and elements of Nikolskii-Bochner spaces. For traceability, details concerning Nikolskii-Bochner, including the equivalence between their definitions, their embeddings and interpolations, are gathered in Section 7 -Appendix. Another reason for creating Appendix is the fact that its results may be of independent interest. Section 5 presents our main results concerning (1) and Section 6 is devoted to their proofs.

Notation
Constants denoted by C, K may change from line to line of estimates and are larger than 1. If a more careful control over a constant is needed, we denote its dependence on certain parameters writing C(parameter) and generally suppress marking its dependence on irrelevant parameters. Such constant may also vary.
A space-time point z = (x, t) is taken from Ω × I =: Ω I , where Ω is a spatial domain (an open, bounded connected set) in R d and I = (I L , I R ) is an open, bounded time interval in R. As we develop the interior regularity theory, further assumptions on Ω I are immaterial. B r (x) denotes the ball with the radius r centered at a point x, I ρ (t) -the interval centered at t and with radius ρ and Q r Iρ (z) denotes the space-time cylinder B r (x) × I ρ (t). We use, in particular, cylinders with parabolic scaling, i.e. cylinders of the type Q r I r 2 (z), which we denote briefly by Q r (z) and refer to as parabolic cylinders. Where possible, we drop dependence on x, z, t.
Symbol '⋐' denotes embedding of the closure, i.e. A ⋐ B iff A ⊂ B. For a set S in an Euclidean space we denote its ε-interior by 'S ε ':= {s ∈ S| dist(s, ∂S) ≥ ε}.
We will write ∆ h u := u • T h − u, where T h f (x, t) := f (x, t + h). We use the standard notation for function spaces, often dropping the underlying domain, when there is no danger of confusion. For example, we write L q (W k,p ), W r,s (W 1,q ) for Sobolev-Bochner spaces and N k,p (I; X) for Nikolskii-Bochner space, see Section 4.4.
As we are interested in systems of PDEs, the underlying spaces are vector valued. For instance L q (W k,p ) = L q I; (W k,p (Ω)) N . In fact we will have the equality of the dimension of the spatial domain Ω and of the target space, i.e. N = d.
We will introduce additional notation where necessary.
The prototype tensors A 1 , A 2 satisfy Assumption 1. They are given through The formulation of Assumption 1 is clearly inspired by the Orlicz-structure-type assumptions. In this context let us mention that in this paper we could have used Boyd indices q 1 , q 2 of ϕ instead of the pure p-growth. This would result in less tangible results and further technical complications, hence we do not follow this direction.
In order to gain a quadratic structure in our estimates, we introduce the following square root of an N -function ϕ and the associated tensor V Definition 1 (φ and V). Let ϕ be an N -function. We definē ϕ ′ := tϕ ′ (t), V := ∂ Qφ (|Q|).
We have (A(P ) − A(Q)) : see for instance Proposition 5 of [23]. Let us also make a technical assumption Assumption 2. We fix now an übercylinderQ and a nonempty untercylinderQ such thatQ ⋐Q and Q ⋐ Ω I ,Q ⊂ Q 1 . All the subsequent analysis will happen in-between them.
Assumption 2 serves only the purpose of not writing explicitly all the dependences of constants on the size of a domain. Indeed, their scaling is now controlled by the choice ofQ,Q and hence not written explicitly. As we derive in this paper local (interior) estimates, the important part of Assumption 2 lies in the untercylinderQ. We need this lower bound in the Poincaré-type estimates, for instance in Proposition 2.

Weak solutions
We will use the following standard definition of the weak solution Take an open interval I, a domain Ω ⊂ R d and a stress tensor A compatible with Assumption 1. A function u ∈ C(I; L 2 (Ω)) ∩ L p (I; W 1,p (Ω)) is the local weak solution of the system (1) on Ω I iff for any subinterval (t 1 , for an arbitrary test function w ∈ W 1,1 0 (I; L 2 (Ω)) ∩ L p (I; W 1,p 0 (Ω)). The above definition implies that u has the generalized time derivative in (L p (I; W 1,p 0 (Ω))) * , compare Theorem 2.8 in [21]. Let us mention here, that instead of assuming C(I; L 2 (Ω)) in Definition 2 we can start with L ∞ (I; L 2 (Ω)) there and regain C(I; L 2 (Ω)) via duality and just-mentioned regularity of the time derivative. For a time difference Lemma 1. Take a local weak solution u of (1) on Ω I . Fix small δ > 0 such that I δ , Ω δ are nonempty. For almost any real τ, |τ | ≤ δ 2 the generalized time derivative of the difference ∆ τ u belongs to For every (t 1 , t 2 ) ⊂ I δ and for an arbitrary test function w ∈ L p (I δ ; For almost every t ∈ I δ and for an arbitrary test function v ∈ W 1,p 0 (Ω δ ) we have for almost every τ .
We will need this result to derive rigorous estimates in Section 5. It is shown as Lemma 2.9 in [21].

Nikolskii-Bochner spaces
As already mentioned, due to technical reasons, explained in Subsection 6.1 we resort in proofs to Nikolskii-Bochner spaces. We provide here only their definition, as some of our results use them. The broader presentation of Nikolskii-Bochner spaces is shifted to Section 7 -Appendix for the sake of clarity of our exposition.
Observe that in the case α ∈ N we have r 0 = α + 1, i.e. we use higher-order differences than these giving Sobolev-Bochner space.

Results
We will use Observe that Let us also define We denote by 2 * either the Sobolev embedding exponent 2d d−2 for d ≥ 3 or any finite number q for d = 2.

A lightweight version of the main results
For the sake of clarity we start with qualitative results, i.e. we restrict ourselves to indicating what regularity class solutions belong to, without presenting the inequalities.
First we cover the case γ 0 ≤ 1, where we get low-regularity in time.
For d = 2 Lemma 3 implies that locally ∇u ∈ L ∞ (BM O). In fact it can be raised to Hölder continuity of ∇u. It is the subject of current research, based on quadratic approximations of A. ) for d = 3.
Proof. N 2α p ,p (W 1,p ) with α < γ 0 of Lemma 2 gives Hölder continuity of u, provided p > d, α > 1 /2, which is equivalent to p ∈ (d, 3+2d+ ). Focusing on the physically relevant d = 2, 3 we see that the case p ∈ (2, 3) has already been covered by Lemma 3 and that we can easily complete the case p = 3. Namely for p = 3 we use another information of Lemma 2, i.e. L p (W 1, 2 * p 2 ) to raise infinitesimally in space, by interpolation, the already used N 2α p ,p (W 1,p ).

Full main results
Theorem 1. Take an untercylinder and an übercylinderQ ⋐Q ⋐ Ω I fixed by Assumption 2 and a local weak solution u to (1) on Ω I .

7
Theorem 1 can be extended to systems dependent on lower order terms z, u. More importantly, it can be a starting point to proving temporal interior regularity results for evolutionary p-Laplace, p-Stokes and p-Navier-Stokes systems, that are apparently strongly needed (see for instance Acerbi, Mingione & Seregin [1], Bae & Jin [6] and Duzaar, Mingione & Steffen [27]). Of course there, the main additional difficulty would be pressure. Now, let us present the eponymous consequences of regularity of time derivatives. Having results of Theorem 1 (ii) and (iii), let us consider system (1) as an inhomogenous stationary one on time levels with its r.h.s. in and derive the regularity of its l.h.s. from the information on u, t (t). It is a common approach in evolutionary PDEs that proves fruitful also in the nonlinear case -recall the paper of Nečas and Šverák [43], where they study the regularity of solutions to general nonlinear, quadratic, strongly elliptic systems. In our case, the stationary estimate reads Theorem 2 (Regularity via the stationary estimates). Take a local weak solution u to (1) on Ω I . If u, t ∈ L ∞ (I; L 2 (Ω)) and ϕ(|∇u|) ∈ L ∞ (I; L 1 (Ω)), then for any concentric balls Remark 1. The statement of the main theorems, especially Theorem 1, may seem discouraging. Let us clarify that they are simply a qualitative version of Lemmas from Subsection 5.  [19]. For some more details of it, see Subsection 6.1. This approach has certain limitations: a lot of smoothness is needed a priori and, more importantly, one can deal only with p ≤ 2+ 4 d (possibly +δ thanks to a Gehring-type argument). Observe that the range of p's, where we obtain full time derivatives, is larger for d > 4 than the range given by parabolic embedding approach (compare Lemma 3). Moreover, unlike in the parabolic embedding approach, we can obtain some gain in regularity for any p, see Lemma 2. It implies, in particular, Hölder continuity of u for p ∈ [2, 3+2d+ ) for d = 3, see Corollary 1. This second range is larger than [2, 10 3 ] of parabolic embedding. Finally, one can improve our iterative scheme and allow for larger range of p's in Lemma 3, whereas bound p ≤ 2 + 4 d (+δ) in the other technique seems to be unavoidable. This improvement of our iterative scheme involves using higher space regularity, provided by [23], at each iteration step.

Comparison with Frehse & Schwarzacher [29]
The main difference between our results and these of [29] concerns localization. We derive space-time local estimates and actually the biggest difficulty that we face, compare Subsection 6.1, follows from spacetime mismatch in lower-order terms that stem from using a cutoff function. On the contrary, [29] concerns global in space estimates, hence no need of a cutoff function there (localization in time there follows simply from integration over a portion of time interval).
In fact there is no intersection of our result with [29]. While one of our main goals is to overcome the troubles with localization and to prove that u t ∈ L ∞ (L 2 ), Frehse & Schwarzacher consider this regularity as starting point, see [29,Proposition 4.1].
In this context it is interesting that from our results in Lemmas 2 and 3 it follows that u ∈ N α,2 (L 2 ) locally on Ω I for any α Which is similar to u ∈ N 3/2,2 (L 2 ) obtained in [29]. Since our method is completely different to the one in [29], it would be interesting to find if their method could improve the result presented here.
Finally, let us observe that in [29] the authors do not need to set any restriction on the growth p, do not need to rely on structure with safety-1 and they provide some results also for real fluid-dynamics systems.

Proofs
Let us first indicate the main trouble, by trying to derive formal estimates. Compare subsection 5.1 of our paper on regularity of space derivatives [23].

A priori estimates
In order to detect the difficulties that are of a structural origin, we begin with a priori estimates. Let us test formally (1) with (u, t ψ), t , where ψ is a cutoff function for Ω ′ I ′ ⋐ Ω I .We arrive at for any τ ∈ I ′ . Observe, that the term ϕ ′′ (|Du|)|u, t | 2 at the r.h.s. of (10) is more troublesome than the respective ϕ ′′ (|Du|)|∇u| 2 of (21) in [23]. This difficulty arises especially in the case of an unbounded ϕ ′′ , i.e. in the superquadratic regime, which is of interest for us. Using the assumed here p growth of A we have ϕ ′′ (t) ≤ C(1 + t p−2 ). Hence one can choose r > 2 and split ϕ ′′ (|Du|)|u, t | 2 into After integration over Ω I , the second summand (11) may be externally controlled with the use of the spatial estimate of in [23] and the parabolic embedding where V = L ∞ (I 1 ; L 2 (Ω 1 )) ∩ L p (I 1 ; W 1,p (Ω 1 )). The entire information from the l.h.s. of (12) controls through the parabolic embedding u, t in L 2+ 4 d . Hence we can close the estimate for certain p's, dealing with the larger support of the r.h.s. of (12) by means of a Giaquinta-Modica-type Lemma, where we use the smallness of δ. This is essentially the parabolic embedding method, mentioned in Subsection 5.3. Such method has been used in [1] by Acerbi, Mingione and Seregin to derive the spatial regularity of a full-gradient p(x)-Laplacian. Unfortunately, this approach requires very smooth approximate solutions to the considered system, in order to be made rigorous. By superseding the physically justified symmetric gradient with the full one, the authors gain such a natural smooth approximate solution, i.e. the safety-1 p-Laplacian with a smooth dependence on lower-order terms. One can also provide enough smoothing for the symmetricgradient case, for instance by adding the extra smoothening term −ε∆ m u to the system (1), with a large m. It is a rather technical approach and it is still limited to p ≤ 2 + 4 d (+δ) due to the parabolic embedding. We will evade the need of the above mentioned use of smooth approximations to (1) and another technicalities with a method inspired by [20] of Bulíček, Ettwein, Kaplický and Pražák.
The key observation is that we already possess a certain time differentiability (of a fractional degree) directly from a weak solution. Observe that in order to obtain the information on the time continuity of weak L ∞ (I; L 2 (Ω)) ∩ L p (I; W 1,p 0 (Ω)) solutions one interpolates, roughly speaking, u t ∈ (L p (I; W 1,p 0 (Ω))) * and u ∈ L p (I; W 1,p 0 (Ω)). However, one can interpolate between the same spaces in the scale of Besov spaces and obtain certain information on fractional time derivatives of u. Next we will try to improve this information iteratively.

Energy estimates in Nikolskii-Bochner spaces
In the next three sections we derive regularity results in Nikolskii-Bochner spaces. We strongly rely on the assumption of the p-structure for the system (1), in order to describe precisely the Nikolskii-Bochner space resulting from a given estimate. As already mentioned, it is also possible to obtain analogous results for a growth function ϕ satisfying certain growth restrictions expressed via Boyd indices, but then results are less tangible, compare Remark 2.
Recall that Q ̺ is the parabolic cylinder B ̺ × I ̺ 2 , Q δ := {z ∈ Q | z + s ∈ Q for any |s| ≤ δ} and for an interval I we denote I rh = {t ∈ I| t + rh ∈ I}. Recall also that Definition 2 of a local weak solution to (1) already includes the fact that A satisfies growth Assumption 1.

High time-regularity estimates in Nikolskii-Bochner spaces
The result of this subsection will be used in the following Section 6.3 as a high-time and low-space regularity ingredient of interpolation Lemma 10, i.e. a N α2,p2 (I; W −1,q2 0 (Q)) part of the r.h.s. of (97).

Lemma 4.
We take an untercylinder and an übercylinderQ ⋐Q ⋐ Ω I fixed by Assumption 2. Fix any δ > 0 so small thatQ ⊂ Ω δ I and any smooth time-independent η ∈ C ∞ (Ω). Take a local weak solution u to (1) on Ω I . Then for any Q ρ such thatQ ⊂ Q ρ ⊂Q and any α ∈ [0, 1) provided the above r.h.s.'s are meaningful. For the case p = 2 we apply the convention »∞ 0 = 1« (In the sense that the term (1 + |Du| L p (ΩI ) ) p−2 2 in the r.h.s.'s above is not present for p = 2.).
Further we estimate the first term of the r.h.s. above by (3). This, plugged into (16), gives . (17) So far, we have not used the p-structure in our estimates. We do it now, estimating the last term of (17 Next we split the square root in (19) by the Hölder inequality with powers p p−2 , p 2 to obtain where we have also increased the domain of integration in the terms containing Du. Use (20) in (19) to get for any w ∈ L p (( Both cases (i) and (ii), i.e. estimates (18), (21) imply that for any h where we have used for the case p = 2 the convention ∞ 0 = 1. Hence, applying sup h∈(0, δ /2] to both sides of the above estimate we have Observe that where the inequality holds by a generous use of Sobolev embedding and the Hölder inequality. Now, in the case α ∈ [0, 1), we can use the reduction inequality (92) of Proposition 1 with choices r := 1, β := 1, δ := δ 2 and X := W −1,p ′ (B ρ ) for the l.h.s. of (22). It gives We want to have a full norm on the l.h.s. of (24), therefore we add (23) to (24) and obtain (13).
In the case α = 1 we drop out of the Nikolskii-Bochner class into the Sobolev-Bochner class. In order to deal with first-order (lower) seminorm here, we add (24) to that holds for any weak solution. Consequently where we can write the Sobolev norm thanks to Corollary 2. It says also that the l.h.s. above dominates We use this in (26), add to both sides of the resulting estimate (23) and obtain (14).

High space-regularity estimate in Nikolskii-Bochner spaces
We have Lemma 5. We take an untercylinder and an übercylinderQ ⋐Q ⋐ Ω I fixed by Assumption 2. Fix any δ > 0 so small thatQ ⊂ Ω δ I . Take a local weak solution u to (1) on Ω I . For any Q ρ1 , Q ρ2 such thaẗ for any α ∈ [0, 1) and provided the r.h.s.'s above are meaningful. For the case p = 2 we apply the convention »∞ 0 = 1«.
Now we choose in (29) ησ so that it equals 1 on After multiplication by h −2α and use of Assumption 1, (30) becomes Now we follow the proof of Lemma 4. We split the last term with the Hölder inequality with powers p p−2 , p 2 and obtain We apply sup h∈(0, δ /2] to the both sides of the above estimate and end up with In order to gain control over the full norms, we add to (33) the low-order estimate Hence we arrive at (27) (the case α ∈ [0, 1)). Finally we also get (28) via Corollary 2 (the case α = 1).

A keystone inequality
Recall that our goal in this section is to develop estimates that shall be used in Lemma 10. For the time being, Lemma 4 provides its high-time and low-space regularity leg. To have its low-time and high-space regularity ingredient we would like to control for certain α 1 , p 1 , q 1 . However, Lemma 5 only controls We bridge this gap in the following lemma.
Proof. Let us consider ϕ(|∆ h D(uη)|). It holds Hence, using |η| ≤ 1 we get and similarly, via |∇η| ∞ ≥ 1, Using the above two inequalities in (37), we obtain Since Assumption 1 and (3) hold, from (39) we arrive at We use (40) to write Next, we use (38) in the l.h.s. of (41) to get We can add to both sides harmlessly The commutativity of ∆ h and D and the Korn inequality allow us to obtain from (43) Adding to both sides of (44) we form the Nikolskii-Bochner norms in the l.h.s. of (44). It gives in the case α ∈ [0, 1) the wanted estimate (34) and in the case α = 1, via Corollary 2, estimates (35), (36).
Remark 2. One can restate Lemma 6 without Assumption 1 using instead of the p-growth -the Boyd indices q 1 , q 2 in an Orlicz-growth setting.

Iteration in Nikolskii-Bochner spaces
Endowed with the estimates of Section 6.2, we are now ready to use the interpolation Lemma 10 recursively, raising the time-regularity of a local weak solution to (1). For clarity of exposition, let us begin the presentation of our iteration procedure without the quantitative details.

A qualitative iteration
First we present how one raises the regularity in a single step of our iteration.
An iteration step. We take the inductive assumption that for a certain α ∈ [0, 1) and we want to raise the fractional time differentiability of u up to (α i+1 ) with the aid of energy estimates of Section 6.2. We do the following (ii) The estimate (45) via Lemma 4 gives the high-time and low-space information Moreover, (45) via Lemma 6 and the assumption (α i ) gives also the low-time and high-space information (iii) We intend to interpolate the high-time and low-space (46) with the low-time and high-space (47) pieces of information by means of Lemma 10. In order to use it we need zero space-trace functions. Therefore we choose an appropriate spatial cutoff function η, which equals 1 on B ρi+1 slightly smaller than Bρ i . Now we take in Lemma 10 the following parameters The choice (48) is admissible in Lemma 10. As a result we obtain where (iii.i) For the case p = 2 we simply choose θ = 1 in Lemma 10 and get Hence we can proceed immediately to the next step (α i+1 ) with α i+1 = α i + 1 /2.
(iii.ii) In the case p > 2 we have the following parameters in Lemma 10 any finite number otherwise.
(iv.i) The first one is to choose θ that gives p 0 = p. Knowing q 0 ≥ p 0 , we then get Unfortunately, in order to have p 0 = p we need θ = 0, see (51). Consequently α ′ < α i , so there is no increase of fractional time differentiability. (iv. ii) The second one is choose θ that gives q 0 = p. This condition used in (51) implies and consequently We have thus obtained that N α ′ ,p0 (L q0 ) of (49) is N α ′ ,p0 (L p ). Recall that p 0 ≤ q 0 (= p) now. Therefore to end up with (α i+1 ) we need to decrease α ′ to an α i+1 such that Lemma 11 provides for (54) the following condition Hence we have uη ∈ N αi+1,p (Iρ2 i ; L p (Bρ i )). Particularly, as η ≡ 1 on B ρi+1 we have reached the next step where the obtained information holds on a slightly smaller cylinder.
The upper bound for the possible regularity gain. We can carry on with our iterative scheme unless α i ≥ 1, because the energy estimates for the Nikolskii-Bochner spaces, derived in Section 6.2, hold within the range α ∈ [0, 1). Nevertheless, if the iteration provides us with strictly increasing fractional differentiabilities α i < 1, i = 0, 1, . . . , i 0 − 1, α i0 ≤ 1, α i0+1 > 1, we can cross the full differentiability and reach any α < α i0+1 by interpolation. More precisely, the energy estimates hold only for α < 1, but there is no such bound in the interpolation Lemma 10. Hence an allowed iteration step is a i0 − δ(ǫ) → a i0+1 − ǫ, where for any small ǫ > 0 we have δ(ǫ) > 0, due to the strict monotonicity and continuity of the iteration step with respect to the parameter of the fractional differentiability.
In other words, if the iteration process formally exceeds 1 and the first obtained there fractional differentiability is 1 + ε, we can obtain for any ε − < ε that The iterative regularity gain. The L p (W 1,p ) regularity of a local weak solution to (1) allows us to start the iteration process (56) with α 0 = 0.
In the case p = 2 the fractional differentiability grows along the iteration (56) arithmetically compare (50) of the substep (iii.i) in Subsection 6.3.1. Here we perform two steps of iteration that begins with α 0 = 0 to get N 1,p (L p ) and decrease a little this regularity, as explained before, to cross the full differentiability. Hence we get u ∈ N 3 2 −δ, p (L p ) for an arbitrarily small δ. In the case p > 2 the fractional differentiability grows along the iteration (56) slowly geometrically compare substep (55) in Subsection 6.3.1. We have A ∈ (0, 1), B > 0 hence For α 0 = 0 we get Therefore, in the case B 1−A ≤ 1, for any α < B 1−A , after a finite number of iterations we get Otherwise, i.e. when B 1−A > 1, we can cross the full differentiability in a finite number of iterations, as outlined at the end of the previous section, to get u ∈ N 1+ε − ,p (L p ).
As we have the explicit iteration formula (59), we realize that Observe that it also agrees with the result of the case p = 2, as there A + B = 3 2 . Remark 3. In both cases we terminate the iterations after a finite number of steps. This will be important in the following subsection on the qualitative iterations ( i.e. iterations, where we also keep track of inequalities), because it will allow to deal easily with the constants of estimates.
A remark on a trouble. Observe that for p > 2 we are in a worse situation than in [20] in two aspects.
(i) Firstly, because the localization produces in estimates a lower order term in N α,p (L p ) that we need to control, contrary to [20], where the bad term belongs to N α,2 L 2 .
(ii) Secondly, because our low-time and high-space information has 2α p < α order of fractional timedifferentiability.

Consequently
(i) In order to control space regularity of the lower order term in N α,p (L p ) we need to interpolate known N 2α p ,p (W 1,p ) and N 1+α,p ′ (W −1,p ′ ) not with θ = 1 2 , but closer to the high-space regularity leg. This diminishes the gained fractional-time differentiability 2 .
(ii) Moreover, the gained fractional-time differentiability is smaller because the high-space regularity leg enjoys lower fractional-time differentiability 2α p < α.

Quantitative iteration
For brevity, we write Let us also recall that which is B 1−A from (60). Observe that Having presented the qualitative iteration, we are ready to provide a quantitative regularity result based on our iterative scheme. It is divided into two lemmas. The first one considers the situation when we cannot reach the full differentiability by our iteration (case γ 0 ≤ 1). The second one covers the full differentiability case (case γ 0 > 1).
Recall that at every iteration step (56) we decrease a little the underlying cylinder. It needs to be taken care of, because we want to obtain our regularity result for Q r starting from Q R , where r, R may be arbitrarily close to each other. Therefore we define the intermediate cylinders Q ρ i in-between Q r , Q R as follows. The cylinder Q ρ i is concentric with Q r and ρ i := r + (R − r)2 −i . These intermediate cylinders decrease from Q ρ 0 = Q R to Q ρ ∞ = Q r . We will use also a cylinder Q ρ (i,i+1) which is in the middle between Q ρ i and Q ρ i+1 , 2 One can choose N α,2 (W 1,2 ) instead of N 2α p ,p (W 1,p ) as the high-space-regularity endpoint for interpolation. Then a formal computation indicates that a possible gain of fractional time differentiability, i.e. α i+1 ≥ α i occurs only for p ≤ 2 + 4 d+1 . This is always majorized by 2 + 4 d of parabolic embedding and worse than our high-regularity bound 2 + 2 √ d+1 for d ≥ 3, so of no interest to us.
Proof. Along the proof of Lemma 7, for any α i0 < 1 we needs a finite number of steps of the iteration (56) to reach It has been already explained in Subsection 6.3.1, that performing one iteration step beyond the full differentiability is allowed. Let us recall briefly the argument. At the level of any α i0 < 1 the fractional energy estimates of Section 6.2 are valid. Therefore we take in (56) the iterative assumption (α i0 ) and proceed with the step i 0 → i 0 + 1, because both interpolation and embeddings work without the upper bound α < 1 for the fractional differentiability. For the quantitative result, we additionally take into account the possible increase of the constant, caused by this last step i 0 → i 0 + 1. There is also no problem with second-order differences, as Lemma 9 allows to increase the order of differences in a Nikolskii-Bochner space. Hence one obtains for any γ < γ 1 , compare (62) with 1 + ε = γ 1 , The r.h.s. of (78) controls |u| δ thanks to (99) of the embedding Lemma 11. It gives thesis via (28), (14), (35), (36). We use (103), (89) to pass from the norms containing term δ 2 to the standard norms, where applicable 3 , and get thesis.

Proof of Theorem 1
It is a combination of Lemma 8 and Lemma 7.

Proof of Theorem 2
Proof. We follow closely the proof of Theorem 1 of [23]. The main difference lies in the fact that now we do not obtain any regularity from the evolutionary part, but we assume it. This in turn allows us to use an arbitrary σ ∈ D(I ε ) in place of the cutoff function σ from the proof of Theorem 1 of [23]. Now we proceed with some more details. However, since the approach is very close to [23], we refer to [23] or to Chapter 5 of [21] for full explanations. Let us define the space difference is the space shift.
Step 1. (Estimate for differences with growing spatial support.) Let us fix ε > 0 so small that the übercylinder Q R+2ε ⋐ Ω I . We take any concentric Q ρ1 ⋐ Q ρ2 , such that Q ρ2 ⊂ Q R . A smooth η cuts off between B ρ1 and B ρ2 with |∇η| ∞ ≤ 2 ρ2−ρ1 and σ is an arbitrary function from D(I ε ). Similarily as in Lemma 1, a local weak solution u of (1) on Ω I satisfies for real |l| ≤ ε 2 , where e i denotes i-th canonical vector in R d Let us focus on the term I at the r.h.s. of (79). Commutativity of time derivatives with space differences yields 3 Observe that neither (103) nor (89) covers the case |u| W 2,p ′ W −1,p ′ (Qr ) , but we do not need to get rid of δ 2 -term there, because (14) concerns already the standard δ-free norm of |u| W 2,p ′ W −1,p ′ .
Our choice of the cutoff function gives supp η ⊂ B R . Hence our choice of Q R+2ε gives supp η ⊂ Ω 2ε . This along with the assumed |s| ≤ δ /2, after the discrete integration by parts, yields Next we estimate the difference ∆ (−lei,0) with the derivative and obtain In the above estimate we have also used that the spatial support of the integrand of the last term is contained in B R+ε ⊂ Ω ε . It follows from supp η ⊂ B ρ2 ⊂ B R . This last information and the Tonelli Theorem allow us to write For the second inequality in (80) we have used characteristics of η and we have increased l to h such that ε 2 ≥ h ≥ l. This h will be used later as the proper denominator of difference quotients. The Korn inequality gives us subsequently For the second inequality in (81) we have estimated the difference |∆ (lei ,0) u| 2 with the space gradient.
for any r ≤ R. After dropping the homogenization terms in f h,σ (r) we arrive at for any h such that |h| ∈ (0, ε 2 ], any σ ∈ D(I ε ). For a fixed, positive h, the estimate (86) can be written as holding for any smooth nonnegative ψ ∈ D(I ε ). The term B h belongs to L ∞ (I ε ) thanks to our assumptions. Similarly A h ∈ L ∞ (I ε ), where we use Assumption 1 and p ≥ 2. The arbitrariness of a nonnegative ψ ∈ D(I ε ) implies therefore that for a.e. t ∈ I Hence for a.e. t ∈ I and h such that |h| ∈ (0, ε 2 ] we get which gives our thesis via the difference quotient argument.

Appendix
Here we provide needed results on Nikolskii-Bochner spaces and some comments In order to put the Nikolskii-Bochner spaces in a broader analytic context, let us briefly mention Besov-Bochner spaces. We introduce them by the real interpolation, along [3] of Amann. By (·, ·) θ,q we understand the real interpolation functor. For k > 0 we fix natural k 1 , k 2 such that k 1 < k < k 2 and put θ = k − k1 /k2 − k1. Any intermediate space (W k1,p (I; X); W k2,p (I; X)) θ,q is the same one; we call it the Besov-Bochner space B k,p q (I; X) (see [3], especially Section 4 there). The Nikolskii-Bochner space with a non-integer α can now be identified with a Besov-Bochner space with q = ∞. Indeed, formula (6.1) of [3], Section 2 gives Observe that in fact Nikolskii-Bochner spaces are defined in [3] in a slightly different way then in our case. Instead of higher-order differences of (8), Amann uses in his definition differences of derivatives. As Besov spaces admit numerous equivalent descriptions, these definitions are equivalent 4 .

References
The literature on standard Besov spaces (as opposed to the Besov-Bochner case) is broad. In this case one can trace back the theory to the original works of Nikolskii [44] and Besov [15] 5 , with important earlier contributions by Marchaud [41] and Zygmund [55]. The classical references for function spaces are the books by Triebel, for instance [51]. For anisotropic Besov spaces, useful for the evolutionary problems, one refers to Besov, Illin, Nikolskii [16].
This versatility carries over the Besov-Bochner spaces (which are commonly referred to in literature as the vector-valued Besov spaces), but the results are less clearly stated and scattered, compare [5], [42], [47]. This forces us to prove the needed results by ourselves.

Results on Nikolskii-Bochner spaces
Here we gather needed results. In order to allow this section to serve as a quick reference, the proofs of these results are gathered in subsection 7.3.

Equivalences between differences and derivatives
Now let us compare our Definition 3 of a Nikolskii-Bochner space (the definition by differences) with a definition that involves a mixture of weak derivatives and differences. This is a result that corresponds to one of the equivalences mentioned in the subsection 7.1. Recall that we work all the time within Remark 2, so the length of I, present implicitly in the estimates, is bounded from below and from above. We denote the β-th order weak time derivative by f, The fact that derivatives control differences reads Proposition 1 (Reduction). For r, β, j ∈ N, α ∈ R such that r > α ≥ 0 provided the r.h.s. is meaningful. Hence in particular We call Proposition 1 the reduction proposition, because it says that derivatives can be reduced to differences.
In order to provide the result converse to the reduction 6 Propositon 1, we resort to an extension of a function from N α,p (I; X) to an element of N α,p (R; X) and the interchangeability of derivatives with differences in general Besov-type spaces. After having invoked these general results we can also immediately reprove and generalize Propositon 1. We keep it however for the shortness of its proof. The reverse of Proposition 1 reads Proposition 2 (Accession). Take r, β ∈ N, α ∈ R. For any r > α > β ≥ 0 one can access differences to derivatives as follows f, provided the r.h.s. is meaningful. Hence, in particular f ∈ N 1+α,p (I; X) =⇒ f, t ∈ N α,p (I; X).

Interpolation
Besov spaces are stable under the real interpolation. In case of Nikolskii spaces, their interpolation actually reduces to the Hölder inequality. More precisely, we have Proposition 3. Assume that norms of Banach spaces Z, X, Y satify with b ∈ [0, 1]. Then we have for any α 1 , α 2 ≥ 0 and any r > α 1 ∨ α 2 , δ > 0 where Let us now derive from Proposition 3 an interpolation result, that is tuned to our further needs (and not the most general possible). From now on we use the notation where, as before, r 0 = [α] + 1 is the »natural« Nikolskii-Bochner difference step.

Then for
The formulation of Lemma 10 may appear slightly awkward. For instance, one would expect interpolation parameter from [0, 1] and we have it effectively in [0, 1 2 ] as well as some presence of p 2 in (98b). This, together with an unnatural (from the viewpoint of interpolation theory) requirement q 1 ≥ q ′ 2 , p 1 ≥ p ′ 2 , is caused by the fact that we deal with the negative differentiability W −1,q2 by a duality formula and next we use a standard interpolation. None of these suboptimalities affect our further results. To prove them we use the case q 1 = q ′ 2 , p 1 = p ′ 2 , where p 0 , q 0 coincide with the expected interpolation values. Of course our energy estimates depend on peculiarities of Lemma 10, but regularity classes remain intact.

Embeddings
As before, in the following embedding result we do not write explicitly the dependence of constants on size of the underlying domain I. In view of Assumption 2 this dependence is irrelevant. We have Lemma 11. Fix a non-empty unterintervalÏ and an überintervalÏ. Take any interval I such thatÏ ⊂ I ⊂Ï, a Banach space X and p ∈ [1, ∞], α ∈ R + .

Sobolev-Bochner space W 1,p (I; X)
The space N 1,p (I; X) is defined with the second order differences. As we use only the first order estimates in the energy estimates of Section 6.2, let us mention here the standard Sobolev-Bochner space W 1,p (I; X). We have We have for any δ 1 ≤ δ 2 from (0, 1] |f | δ1,W 1,p (I;X) ≤ |f | δ2,W 1,p (I;X) ≤ 3 δ 1 |f | δ1,W 1,p (I;X) .
In the case p = ∞ any element of W 1,p (I; X) has a C 0,1 (I; X) (Lipschitz) representative and there is equivalence of norms.

Proofs
Proof of Lemma 9.
Step 1. We show that N α,p (I; X) is a Banach space by using the fact that L p (I; X) is a Banach space, compare Proposition 23.2 in [54]. More precisely, we take a Cauchy sequence {f n } n∈N in N α,p (I; X). The norm of N α,p (I; X) contains the L p (I; X)-norm, so we have f such that lim n→∞ f n = f in L p (I; X). As {f n } n∈N is the Cauchy sequence in N α,p (I; X), we have h −α |∆ r h f n | L p (I rh ;X) ≤ C for any n ∈ N. For any h ∈ (0, 1) one finds n h such that so f is indeed in N α,p (I; X).
Step 2. Now we show (89). The important part of the semi norm [f ] r,δ,N α,p (I;X) is the one for small δ's. More precisely, for δ 1 ≤ δ 2 in view of the formula (8). For the last summand of the r.h.s. of (104) we have |∆ r h f | L p (I rh ;X) ≤ 2 r |f | L p (I;X) .
Step 3. Finally, we want to show (90). Decreasing the number of differences is easy because .
In order to add a number of differences, we derive a Marchaud-type inequality. A computation gives The above formula can be found in the monograph [13] by Bennett and Sharpley, p. 333. It implies Observe that the used interval I 2hr is the largest admissible, if we want to stay in (105) within I, i.e. the domain of f . Formula (105) gives Let us take any natural j. One has Using for the r.h.s. of (107) the inequality (106) with h := 2 m h we get We want to increase in the l.h.s. of (108) the domain of integration to I hr . For the missing part we have where I −a := (I L + a, I R ), provided r2 j |h| ≤ |I| 2 (110) Next, we consider (108) with h and with −h, add and via (109) arrive at After translations, the "−h" terms give a copy of the "+h" terms, so we have Consequently, for any h > 0 We split and estimate the first term of the r.h.s. of (112) as follows where to obtain the factor 1 2 we need j ≥ 1 2(r−α) ; we choose In (114) we can see clearly why one needs for the definition of Nikolskii spaces the difference step r to be sharply larger than the differentiability parameter α. The estimate (113) gives via (112) This estimate yields [f ] r,δ,N α,p (I;X) ≤ 2r 2 r − α [f ] r+1,δ,N α,p (I;X) + δ −α |f | L p (I;X) , where we use (114) and r ≥ 1 to write rj ≤ 2r 2 /r − α. Hence we have the missing right inequality in (90) in the case r − r 0 = 1. The general case, i.e.
Proof of Proposition 1. The fact that u has a weak time derivative u, t is equivalent to the fact that u has a representative u ∈ AC(I; B) for which holds for t 0 , t ∈ I, where the equality is in B.
First consider the case β = 1. The assumed finiteness of the r.h.s. of (92) and the definition of a Nikolskii-Bochner allow us to use the representation formula (117). This and the Tonelli Theorem give for the case of I being the real line or a half-line, hence we need to execute the formula (7.2) on the extended functionf . It reads |f, Next, we use the defining formula (118) for the first integral on the r.h.s. above. The remaining term there is integrable for α > β. For the second integral of the r.h.s. above we use the triangle inequality. We get The last inequality follows from (123). Putting together (125) and (126) we arrive at (93).
Proof of Proposition 3. We write where we have used the Hölder inequality.
Proof of Lemma 10. We get our result in two steps: a duality and a standard interpolation. More precisely, the idea of the proof is as follows. First we obtain an interpolation of the type |f | δ,N α(θ),p(θ) (I;W k(θ),q(θ) (Q)) ≤ C|f | 1−θ δ,N α 1 ,p 1 (I;W 1,q 1 0 (Q)) |f |θ δ,N α 2 ,p 2 (I;W −1,q 2 (Q)) with k(θ) positive. Next we check what is the optimal q * (θ) of embedding W k(θ),q(θ) (Q) ֒→ L q * (θ) . Unfortunately, on domains, the interpolation theory of Sobolev spaces with a real differentiability parameter seems incomplete. There is a problem with the local description for spaces with negative differentiability, see the Triebel's classic [51], p. 209. This has not changed much in modern times, compare Section 4.1.4. of [52]. Therefore, we carry out our interpolation into two steps. First we use simply a duality formula which deals with the negative Sobolev space W −1,q2 . Next we use the complex interpolation.
Step 1. (Use of duality.) Recall that we have restricted ourselves to α 1 ≤ α 2 < 2, so it is sufficient to use the difference step r = 2 in Nikolskii seminorms. In view of the assumed q 1 ≥ q ′ 2 , p 1 ≥ p ′ 2 and the Hölder inequality we have For the domain-independence of the constant above, recall Remark 2 (this applies to the further embedding constants as well). The duality between W 1,q ′ 2 0 (Q) and W −1,q2 (Q) gives for any g ∈ W It allows us to write via Proposition 3 [f ] 1 2 2,δ,N α 2 ,p 2 (I;W −1,q 2 (Q)) .
As intended, now we don't have any dependence on negative differentiability in the l.h.s. of (130).
Step 2. (The complex interpolation.) Now we interpolate between the l.h.s. of (130) and N α1,p1 (I; W 1,q1 0 (Q)). We want to use again Proposition 3, this time using the complex interpolation for the spaces on Q. Hence first we need to identify the space where [·, ·] θ denotes the complex interpolation functor. Here we see the benefit of splitting our proof into two steps. Namely, for a natural, nonnegative k and l ∈ (1, ∞) we have W k,l = F k,l  [51]. Therefore In view of the theorem in Section 3.3.1 of [51], for positive p 0 , p 1 and real s 0 > s 1 we have the continuous embedding F s0,p0 This gives in our case for any finiteq 0 , provided Otherwise the largest allowedq 0 (finite and > 1) is .
Embedding (132) in these cases can be written together as with q 0 =q 0 ifq 0 is positive and any finite q 0 otherwise. This is precisely our condition (98c). Quantitatively, (133) reads |g| L q 0 (Q) ≤ C |g| θ L 2 (Q) |g| 1−θ W 1,q 1 (Q) . We use it in Proposition 3 to get where Estimating the first term of the r.h.s. of (134) with (130), we arrive at Analogously to (136) we obtain the estimate for the low-order part of the Nikolskii-Bochner norm, i.e.
Where applicable, we use (90) of Lemma 9 to decrease in (138) the differentiability step from 2 to the »natural one«, which is either 1 or 2. It forces us to assume (91), which is reflected here by (96), as we have restricted ourselves to r ≤ 2. Hence we get from (138) The fact that f has zero space-trace gives via (139) the desired estimate (97). Recall that p 0 , α come from (135).
Proof of Lemma 11. Ad (i We sum (140) over β varying from 0 to [α] to get (99). Ad (ii). Let us first observe that in the case α = α ′ the embedding follows from the Hölder inequality. Moreover, it suffices to show (100) in the case p ≤ q, because having this case we show the complementary one p > q with the Hölder inequality. For the case p ≤ q and α > α ′ , we proceed in a few steps.
Next, we extend (143) over α ′ = 0 and α = 1 and modify it, so that its constant does not blow up as α ′ → 0 (with all the other parameters fixed). This extension will be performed at the cost of assuming the sharp inequality in the first condition of (141). Namely, we substitute (141) with Substep 1.1 Let us consider (144) in the case p = q. First, for α < 1, in both norms of (100) we have the difference step r = 1, compare Definition 3. Therefore the definition of a Nikolskii-Bochner seminorm, where we use the restriction δ ≤ 1, gives |f | 1,δ,N α ′ ,p (I;X) ≤ |f | 1,δ,N α,p (I;X) .
δ ≤ |I| Now we intend to use the previous substep K + 1 times. To this end we define α 0 := α ′ , α i := [α ′ ] + i, i = 1, . . . K, α K+1 := α The second inequality in (167) is valid in view of which we justify analogously to (165). Conclusion of Step 2. We put together (159) and the condition (161) with (164) augmented with the condition (167). Hence, after a generous estimate of constants, we obtain part (ii) of the thesis in the case α > α ′ and p ≤ q. The rest follows from the Hölder inequality (compare two first sentences of this proof of part (ii) of the thesis). Specifically, for the case p > q we use first the α > α ′ and p = q estimate, which has β = α − α ′ and next the Hölder inequality. This is reflected in presence of the term β ∧ (α − α ′ ) in the definition of the constant C (100) .
One can also provide a higher-order version of this estimate, analogous to the one from the part (ii), but we do not need it here.
Proof of Corollary 2. The equivalence of norms is standard. Therefore let us only observe that one direction is a verbatim of our reduction result. The other follows from the definition of a weak derivative, the formula for the discrete integration by parts and a limit passage. For (103), we repeat proof of (89) of Lemma 9. The Rademacher part of this corollary for the one-dimensional case follows from from the absolute continuity of W 1,1 (I; X) functions and (117).