Almost everywhere Hölder continuity of gradients to non-diagonal parabolic systems

We present a local almost everywhere C1,α-regularity result for a general class of p-nonlinear non-diagonal parabolic systems. The main part of the considered systems depends on space-time variable, solution and symmetric part of the gradient of solution. To obtain our result, we adapt for the symmetric-gradient case techniques developed for the full-gradient case by Duzaar, Mingione and coauthors.


Introduction
The problem of local Hölder continuity of gradients for the evolutionary p-Laplace system has been resolved in a series of papers by DiBenedetto and coauthors, summed up in a monograph [7], with crucial earlier (stationary) contributions of Uhlenbeck [23], Tolksdorf [22] and the Russian school. From the perspective of mathematical physics, it is interesting to replace ∇u by its symmetric part Du = (∇u + ∇ T u)/2; then such a symmetric p-Laplace system is a simplification of the hydrodynamic model of a non-Newtonian flow (referred to as p-Navier-Stokes in the following). In fact, for p > 11/5, the generalization from a p-Stokes system to the respective p-Navier-Stokes is not essential from the perspective of regularity theory (compare [16]).
It turns out that the amendment from ∇u to Du in a system of the p-Laplace type, a supposedly harmless one, diminishes dramatically our understanding of C 1,α -regularity of such system. The main difficulty lies in the fact that most of the methods, which are successful in the full gradient case, depend essentially on a pointwise structure, peculiar only to full-gradient p-Laplace system. In this article we show, however, that the p-caloric approximation approach can still be used to obtain almost everywhere regularity. We consider parabolic systems of the following type u ,t − div A(z, u, Du) = 0 (1.1) the prototype of which is the following symmetric p-Laplace system with safety-1 Let us provide the reader with a short account of relevant known results. In [17] an extensive short-time maximal regularity theory in Sobolev-Slobodeckii spaces for p-Navier-Stokes is presented by Prüss and Bothe. However, not much is known on the global-in-time C 1,α -regularity of such systems in arbitrary dimension d (or at least for physically plausible d ≥ 3), even for the prototype case. It is worth mentioning that for p ∈ (12/5; 10/3) Ladyzhenskaya and Seregin have shown in [19] an almost-everywhere regularity result for the complete three-dimensional hydrodynamic system. One can also easily see from the theory developed in [13] by Kaplický, Málek, Stará for the two-dimensional p-Navier-Stokes that system (1.2) and some of its generalizations enjoy C 1,α -regularity in the case of twodimensions. The current research status is a little clearer in the case of stationary simplifications. There is a well developed C 1,α -theory for the stationary p-Laplace and p-Navier-Stokes systems with p < 2 by Beirão da Veiga and collaborators (see [2] and references therein). In the case of p ≥ 2, one can refer to [1], where Apushkinskaya, Bildhauer and Fuchs obtain partial C 1,α -regularity for three-dimensional p-Stokes and full C 1,α -regularity in the two-dimensional case.
In this paper we follow the theory based on p-caloric approximations, which has been developed for the full-gradient case and very general main parts in [9], [10] by Duzaar, Mingione and coauthors. We apply their ideas for the symmetricgradient case. At some points we could have merely quoted the respective results from [10]; instead, for reader's convenience, most of the proofs are presented with concern for the clarity of exposition.
However the p-caloric approach seems to be very well-suited also for our symmetric-gradient case, let us emphasize that our result seems to be new not only for general system (1.1), but even for its prototype (1.2). As a byproduct, which may be interesting by itself, we obtain also a Campanato-type theory for linear parabolic systems satisfying Legendre-Hadamard conditions, for which we couldn't find a satisfactory reference.

Notation and statement of the result
The expression A =: B means that A defines B. Denote a space-time point z = (x, t) ∈ Ω × (−T, 0) =: Q, where Ω ⊂ R d . As we develop a local interior regularity theory, any further assumptions on domain Q are unnecessary. B r (x), Q r (z) denote, respectively, the ball with the radius r centered at a point x and the parabolic cylinder B r (x) × (t − r 2 , t). ∂ ΓQ denotes parabolic boundary of cylinderQ.
For a tensor ξ ∈ R d×d denote its symmetric part by ξ s := (ξ + ξ T )/2. For any matrix M ∈ R d 2 ×d 2 denote its coefficients by M i j kl ; its action on tensor ξ with coefficients ξ kl is M i j kl ξ kl (here and further on we use the summation convention). Sym d×d denotes set of d × d symmetric tensors.
We use standard notation for function spaces; L p (τ, t; W 1, p (B (z))) will be sometimes abbreviated to L p (W 1, p ), when there is no danger of confusion regarding underlying cylinder.
Let us emphasize that constants denoted by C may change from line to line of estimates and are bigger than 1. If a more careful control over a constant is needed, we denote their dependence on certain parameters writing C( parameter) and generally suppress marking their dependence on irrelevant parameters; such constants may also vary. For clarity we also use some fixed constants, which we denote by C subscri pt . Now let us present a list of assumptions for the studied generalization (1.1) to (1.2). For any tensors ξ, η ∈ R d×d • main part A satisfies properties of and the local modulus of continuity ω satisfies: ω(·, s), ω(t, ·) are nondecreasing, ω(t, 0) = 0 and ω(t, ·) is continuous at zero, ω p (t, ·) is concave. The main result reads.
Let us repeat that, to our best knowledge, even for the prototype system (1.2) this result is new.

Outline of the paper
The rest of the article is devoted to the proof of Theorem 2.1. For traceability, let us first present the outline of the paper.
-In Sect. 4 auxiliary lemmas are gathered. This includes a Campanato-type regularity theory for linear parabolic systems satisfying Legendre-Hadamard conditions, see Lemma 4.4, and the symmetric caloric approximation lemma-Lemma 4.5. The latter states that every function which is close in a certain weak sense to a solution of a linear parabolic system, with its main part dependent on symmetric gradient, is indeed close to a solution of a linear parabolic system in a strong sense. -Section 5 is devoted to showing, by means of linearization and Caccioppoli inequality, that an appropriately rescaled weak solution to (1.1) satisfies locally certain inequalities that resemble assumptions of the caloric approximation lemma. This is done via Lemmas 5.2, 5.3 and summed up in Corollary 5.4. -Section 6 combines results of the previous sections and gives the proof of Theorem 2.1. Namely, thanks to Corollary 5.4, around points which satisfy certain regularity assumptions, one can use caloric approximation for (rescaled) solution of (1.1), which thanks to the regularity of linear systems gives proper shrinking of excess energies (Lemma 6.1). This yields, by iteration, the Hölder continuity of gradients (Lemma 6.3).
Only the crucial results are proved directly after their statements; for the sake of clarity, the remaining proofs are transferred to the Sect. 7-Appendix.

Useful auxiliary results
This section begins with a Simon-type compactness result for parabolic spaces, which can be found as Theorem 2.5 in [10].
Lemma 4.1. Take p ∈ (1, ∞), three Banach spaces X ⊂⊂ Y ⊂ Z and a sequence g k , which is uniformly bounded in L p (−T, 0; X ) and satisfies then g k contains a subsequence convergent in the space L p (−T, 0; Y ).
The next result collects properties needed to perform analysis of excess energies (see Definition 5.1). For proof see [10] Lemma 2.1; the last inequality can be found in proof of Lemma 4.8 there. Compare also [15].
the linear part ∇l (2) z 0 , of which is close to (∇u) z 0 , and shrinks as follows For the minimizer in the case of general s ≥ 2 holds Subsequently let us state the Korn's inequality. For hints for proof, see the Appendix.

Lemma 4.3. (Korn's inequality)
For u ∈ W 1, p (B r (x)) following inequalities hold with C K or p independent on radius of B r (x) Next lemma, which may be of independent interest, collects needed results on linear parabolic systems with main part depending on symmetric gradient. Recall that AM denotes constant coefficient matrix A with elements a where for constant coefficient matrix A holds: then u is locally smooth and satisfies for any p, q ∈ [1, ∞], ≤ r/2, arbitrarỹ where u (m) denotes either ∇ (2m) u or ∂ (m) t u and |a| s = N n=1 |a n | s for a ∈ R N . As outlined in the introduction, we end this section by stating a local result which says that a function, which is approximately solving a certain linear system in a weak sense (such function is called approximatively weakly symmetrical caloric in the following), is indeed close to some solution to this system in an appropriate strong L 2 − L p sense. The idea can be traced back to L. Simon, see [20]. The proof, up to few technicalities connected with symmetric gradient, is identical with its counterpart in [10] and can be found in the Appendix. We work now with fixed p ≥ 2 and cylinder Q (z 0 ) (therefore they does not appear as parameters). Let us introduce some definitions.
Observe that A is weakly symmetrizing, as a i j kl = a ji lk implies Aξ s η = Aξ s η s . In the following two definitions δ > 0, γ ≥ 0 are number parameters.

Local estimates
Let us emphasize that in this section the dependence of constants C on irrelevant parameters is suppressed. All cylinders Q z 0 ( ) ⊂ Q used below have ≤ 1. First let us define local excess energies For briefness, using the energies defined above we often drop certain parameters, writing for example φ p ( ), ψ p ( ). First we state an auxiliary algebraic lemma needed for the estimates of this section.

Lemma 5.1. (Algebraic inequalities) Assume that for a matrix A condition (2.1d)
is valid. Then, for any: z ∈ Q z 0 ( ), u ∈ R d , P ∈ Sym d×d , affine function l(x) the following inequalities hold If, additionally, A satisfies (2.1c), (2.2c), then it also holds Proof of this lemma has been shifted to appendix.
Proof. Use time-independence of l to get from weak formulation of (1.1) that for which by adding and subtracting certain terms yields We need to estimate the right-hand-side of (5.7). First, estimate second integral on the r.h.s. of (5.7) with respect to the splitting of Q (z 0 ) into without loss of generality we may assume that neither Q s nor Q b is empty.
On Q s holds |l(z 0 )| + |Dl| + |Du − Dl| ≤ M + 1 in view of assumptions on l, so by (2.2d) Merging (5.11) and (5.12) one has where the last two inequalities hold by concavity of ω p (t, ·) and p ≥ 2. Therefore we can estimate (5.9) as follows Consider now (nonempty) Q b . One has for any s > 1 From |Dl| + |l(z 0 )| ≤ M and (2.1c), (2.2c) we estimate (5.10) where the last inequality holds in view of (5.15) with s = p and s = p . Combine estimates (5.14) and (5.17) to get It remains to estimate the first term in (5.7); use (5.2a) with P ≡ Du to get (2.1d) Inequality (5.19) used to estimate the first term of the right-hand-side of (5.6) gives Inequalities (5.18), (5.20) used in (5.7) give thesis.
be a weak solution to (1.1) with structure conditions (2.1c-2.2a) and p ≥ 2. Then the following inequalities hold for any Q (z 0 ) and constants being nondecreasing functions of their parameters where l is an affine function depending only on x and satisfying |l(z 0 )| + |∇l| ≤ M and β ∈ (0, 1) is given by (2.1d).
Next, we restate the linearization lemma (Lemma 5.2) using local inequalities of Lemma 5.3 in a way useful for further computations. To proceed, introduce the following useful quantities where Q (z 0 ) ⊂ Q is an arbitrary local cylinder with ≤ 1.

Partial regularity
If and ω M + 1,Ẽ z 0 ,l (2) z 0 , , Proof. We need certain care to avoid a logical loop. Therefore let us first explicitly define constants:  We have already fixed in the statement of lemma M > 0, α ∈ (0, 1). Now let us fix certain parameters: Observe that by assumptions (2.2a), (2.2b) holds i.e. the constant coefficients matrix, resulting from linearization around z 0 , belongs to the set of elliptic bilinear, symmetrizing forms as defined in Definition 4.1. The imbedding results from (2.2c) with (6.1); λ is given by (2.2a) and Λ-by (6.5).
Consequently, let us fix via Lemma 4.5 Take γ ≡ C 5.4 (M)γ l (2) ,δ ε ( ). (6.8) Observe that assumptions of Corollary 5.4 are fulfilled; this and assumption (6.2) give for v defined as in (5.46), inequalities By (6.8) and (6.2) one has also 0 ≤ γ = C 5.4 (M) E z 0 ,l (2) Obervation (6.6) with inequalities (6.9), (6.10), (6.11) imply that v belongs to the set H ( /2; δ, Λ, γ ) of approximatively weakly symmetrical caloric functions (compare Definition 4.2). Consequently in view of the symmetric caloric approximation lemma, i.e. Lemma 4.5, there exists a caloric function h that locally approximates v; more precisely Having such approximation of v by h, we are ready to show (6.3); to this end, estimate ψ s,z 0 ,l (2) σ (σ ), (which for s being 2, p constitute by definitionẼ z 0 ,l (2) σ (σ )) as follows where the second inequality holds in view of (4.5) of Lemma 4.2, the third one by minimization property of l (s) and the equality is given by definition (5.46) of v. To proceed further denote the mean integral over space (emphasizing its time dependence) by and estimate the second integral in the r.h.s. of (6.13) as follows Observe that we cannot take (h) x 0 , θ 4 (t) instead of (h) z 0 , θ 4 directly in the second inequality of (6.13), as only time-independent affine functions are admissible there.
Consider the right-hand-side of (6.14). One has for every t, so Poincaré inequality in space followed by integration over time gives ; the last inequality is valid as h is a symmetrical caloric function. Simultaneously we have thanks to a smoothness of h and the mean-value property Using the inequality (4.9) of Lemma 4.4 with m = 1, q = ∞, p = s to estimate the r.h.s. of (6.16) one arrives at where the second inequality results from h being a symmetrical caloric function. Combine (6.15) and (6.17) to estimate the right-hand-side of (6.14) where (robustly) C (6.20) ≡ 2 5 p−3 max s∈{2; p} C (4.5) (n, s) C (4.9) (λ, Λ, s) +C (4.11) (λ, Λ, s) (6.21) recall that we have taken σ = θ/4; this with the definition of C (6.4) gives from (6.20) where the second inequality is given by choice of ε, see (6.5); in the same inequality we have chosen σ so that C (6.4) 2 7 σ 2 ≤ σ 2α , which gives thesis.
Let us now state inequalities used for the singular set description in the following iteration of Lemma 6.1 performed in Lemma 6.3.

Lemma 6.2.
Under the assumptions of Lemma 6.1, the following inequalities hold Proof. Recall that η (x) denotes a standard space-mollifier, supported in B (x 0 ). Define where for the second inequality we use Poincaré in space and computation analogous to that of (6.29) and for the third one: estimate (5.22) and again Poincaré (both for standard and generalized integral means). (6.31) is (6.24); used in (6.30), it gives which is (6.23). Finally to obtain (6.23) from (6.23), estimate from below l.h.s. of (6.27), using first (4.5), then the minimization property of l then inQ(z 0 ), denoting a certain vicinity of z 0 , holds where β is given by (2.1d).
Proof. In view of Lemma 4.2 (u) z 0 , = l (2) (x 0 ); this, assumptions (6.34), (6.35) and pointwise estimate |Dg| ≤ |∇g| imply that we can find sequence n → 0 for which the following hold for a certain M < ∞ where Hölder inequality and p ≥ 2 is used to control ψ 2 inẼ with ψ p being l.h.s. of (6.23). Moreover (6.37), (6.38) with inequality (6.23) give again via Hölder inequality In order to replace |(∇u) z 0 , | in (6.38) with ∇l (2) z 0 , n , perform estimate using inequality (4.3) of Lemma 4.2 Fix r ∈ (0, 0 /2) and choose j such that Then by the minimizing property of a mean value (the first inequality), Caccioppoli inequality (5.24) (the middle inequality), (6.46) (the third one) we get where the last but one inequality is obtained as (6.50). The estimate (6.51) states that lim j→∞ (∇u) σ j 2 =Γ exists and that |(∇u)˜z ,σ j ( /2) −Γ | ≤C(M, β)σ β j (6.52) This combined with (6.50) results in: where the last inequality holds in view of (6.49). As (6.53) is valid for anyz, being an arbitrary point fromQ(z 0 ), imbedding of Campanato into Hölder spaces gives We are done with the partial regularity result for the gradient. Let us now focus on an analogous property for the solution itself, stated in the following result. Proof. For any x, y ∈ B˜ (x) ⊂Q(z 0 ) the following pointwise estimate for a weak solution u to (1.1), u ∈ C(−T, 0; L 2 (Ω)) ∩ L p (−T, 0; W 1, p (Ω)) holds where M( f ) is a maximal function. The first inequality is given by Bojarski-Hajłasz inequality (see [3], Theorem 3) and the second is a consequence of boundedness of gradients given by Lemma 6.3. Adding (5.22), which holds for every time level, and twice (6.55): the first one withτ = τ, y = y 1 and the second one withτ = t, y = y 2 we obtain thanks to boundedness of ∇u

Appendix
Here we present results which have been removed from the main part of this article for the sake of traceability.
where R is the set of rigid motions, i.e. affine functions with antisymmetric linear part. (7.1) with h := u − (Du)(x − x 0 ) yields: we have also pointwisely |∇h| 2 ≥ |Dh| 2 , so for h : The independence of K on radius r comes from scaling.
Let us remark that the Korn's inequality can be traced back to the papers of Friederichs [12] and Korn himself [14]. Next we show the needed result on linear systems.

Proof (of Lemma 4.4).
Smoothness is a standard result for systems with coefficients depending on full gradient and satisfying Legendre-Hadamard conditions. See [11], [18]. To prove inequalities we modify slightly the technique of Campanato [5]. Scaling v(y, s) = u(r y, r 2 s) justifies that u solves locally (4.7) in Q r iff v solves (4.7) locally in Q 1 . Therefore we consider first v satisfying Take a smooth cutoff functions from C ∞ 0 (Q 1 ) that satisfies and as a result for arbitrary q ≥ 1 and ρ/r ≤ 1/2 both ⎡ which together with Giusti's technique, allowing in local inequalities to decrease the power of integrability on right-hand-sides below 2 by interpolation, implies thesis.
Let us now turn to the proof of symmetric caloric approximation lemma.
then we would have in view of (7.22) which with (7.21) is almost a contradiction to (7.17) with an exception, that we require in (7.17) h ∈ H (1/2; A k , γ k ) and instead have f ∈ H (1; A, γ ). We compensate this difference by proceeding as follows. Consider the following linear boundary-value problem ω k ,t − div A k Dω k = 0 in Q 3/4 ω k = f on ∂ Γ Q 3/4 (7.24) (7.24) and (7.20) give which by ellipticity of A k and Korn's inequality gives Observe that in view of inequality (4.9) of Lemma 4.4, |ω k | L 2 (Q 3/4 ) controls norms on Q 1/2 of ω k of arbitrary high order. This and (7.26) yield We show now that ω k contradicts (7.17) for large k. One has where the convergence stems from (7.27) and (7.22). Similarly Finally we obtain estimate for I I I in view of (7.34) I I I ≤ C(M)|u − l| β (7.45) and inequalities for I, I I, I I I give desired (5.3). Inequality (5.4) follows similarly and more straightforwardly, when instead of (7.34) we observe that assumption (2.1d) implies |A(z, u, q) − A(z,ũ, q)| ≤ C 1 + |q| p−1 . (7.46)