Almost everywhere H\"older continuity of gradients to non-diagonal parabolic systems

We present a local almost everywhere regularity result for a general nonlinear non-diagonal parabolic system, which main part depends on symmetric part of the gradient.


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 [2], with crucial earlier (stationary) contributions of Uhlenbeck [21], Tolksdorf [20] 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 hydrodynamic one is not essential from the perspective of regularity theory (compare [14]). It turns out that the amendment from ∇u to Du in the p-Laplace system, a supposedly harmless one, diminishes dramatically our understanding of C 1,α -regularity of such system. The reason is that most of the relevant methods successful in the full gradient case turn out to be useless, because they rely essentially on pointwise structure. In this article we show, however, that the caloric approximation approach can still be used to obtain almost everywhere regularity. We consider parabolic systems of the following type u ,t − divA(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 [15] 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) Seregin has shown in [17] an almost-everywhere regularity result for the complete three-dimensional hydrodynamic system. One can also easily see from the theory developed in [12] 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 two-dimensions. 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 Bairão da Veiga and collaborators (see [3] 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 symmetric-gradient 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, 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 is defined as 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). ∂ Γ Q denotes parabolic boundary of cylinder Q. 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 it coefficients by M ij kl ; its action on tensor ξ with coefficients ξ kl is M ij 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 generally 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 subscript . 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 A(z, u, ξ s )ξ s ≥ λ|ξ s | p (being strongly elliptic), (2.1) A(z, u, ξ s )η ≥ A(z, u, ξ s )η s (being weakly symmetrizing), where β ∈ (0, 1) and K : [0, ∞) → [1, ∞) is a non-decreasing real function; and local modulus of continuity ω satisfying: 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) the results is new.

Outline of the paper
The rest of the article is devoted to the proof of the result stated above. For traceability, let us first present the outline of the paper. In Section 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, in the context of symmetric gradients, that every function which is close to a solution of a linear parabolic system in a certain weak sense is indeed close to a solution of a linear parabolic system in a strong sense. Next sections are devoted to the proof of Theorem 2.1, the main steps of which are as follows.
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. For proof see [10] Lemma 2.1; the last inequality can be found in proof of Lemma 4.8 there. Compare also [13].
. There is the unique minimizer l (s) (x) to Q (z0) |u − l| s among affine, time-independent functions l; moreover z0, of which is close to (∇u) z0, 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. (Korn's inequality) For u ∈ W 1,p (B r (x)) following inequalities hold with K 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 ij kl acting on tensor M with elements m kl , i.e. AM = a ij kl m kl . Again we refer to the Appendix for the proof. Lemma 4.4. (Campanato-type regularity theory for linear parabolic systems satisfying Legendre-Hadamard conditions) Let u ∈ L 2 (−T, 0; W 1,2 (Ω)) be a local solution to u ,t − divA Du = 0, i.e let it satisfy where for constant coefficient matrix A holds: then u is locally smooth and satisfies for any where u (m) denotes either ∇ (2m) u or ∂ (m) t u and |a| s = N n=1 |a i | 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 δ-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 [18]. 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 sweakly ymmetrizing, as a ij 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. 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). Fix M . Assume that for matrix A condition (2.4) is valid. Then, for any z ∈ Q z0 ( ) ⊂ Q with ≤ 1, any u ∈ R d , P ∈ Sym d×d and any affine function l(x) the following inequalities hold If, additionally, A satisfies (2.3), (2.7), then it also holds Proof of this lemma has been shifted to appendix. For Proof. Use time-independence of l to get from weak formulation of (1.1) that for any ϕ ∈ C ∞ 0 (Q (z 0 )) holds which by adding and subtracting certain terms yields To obtain our thesis we need to estimate the right-hand-side of (5.8). First, estimate second integral on the r.h.s. of (5.8) with respect to the splitting of Q (z 0 ) into without loss of generality 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 (2.8) where the last two inequalities hold by concavity of ω p (t, ·) and p ≥ 2 ( this is in fact the only place here where we use assumption for p). Therefore we can estimate (5.10) as follows Consider now (nonempty) Q b . One has for any s > 1 where the last inequality holds in view of (5.16) with s = p and s = p . Combine estimates (5.15) and (5.17) to get It remains to estimate the first term in (5.8); use (5.2) with P ≡ Du to get (2.4) Inequality (5.19) used to estimate the first term of the right-hand-side of (5.7) gives Inequalities (5.18), (5.20) used in (5.8) give thesis.
) be a weak solution to (1.1) with structure conditions (2.3 -2.5). Then the following inequalities hold for any Q (z 0 ) ⊂ Q with ≤ 1 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 (2.4).
by sending ε → 0 (this holds pointwisely in time, because u ∈ C(L 2 )). Estimate the r.h.s. of (5.26) using that |∇η ( where the last inequality comes from adding estimates ( the last inequality holds, because |σ | ≤ 2 −2 on t 0 − 2 , t 0 − 2 /4 in view of (5.25). As Q (z0) A(z 0 , l(z 0 ), Dl)Dϕ = 0, for the main part holds hence, using the assumption (2.5) one obtains Inequalities (5.29), (5.30), (5.32) show that testing (1.1) with ϕ = θ 2 σ(u − l) yields the following estimate Let us estimate I by (5.4) with P ≡ Du getting for ≤ 1 with which we estimate II Estimates for I and II give together In view of β < 1, ≤ 1 one has Consequently, (5.37) takes the form First, use inequality (5.39) for τ = t 0 , neglecting the first term of the left-hand-side. This estimate is uniform in ε, so we obtain Next, drop second part of left-hand-side of (5.39) and consider any τ in interval of admissibility t 0 − ρ 2 4 , t 0 ; this via Steklov averages argument gives rise to Next, we restate the linearization lemma (Lemma 5.2) using local inequalities of Lemma 5.3 in a way useful for further computations.
Proof. We suppress parameters of the excess energies writing E,Ẽ for E z0,l ,Ẽ z0,l and similarily for moments ψ, φ. Take The last three inequalities come, respectively, from: definition (5.46) of v and choice (5.49) of constant C 5.4 (M ); the Definition 5.3 of γ l,δ ( ) and the fact that δ ≤ 1; the Caccioppoli inequality (5.23) and the Definition 5.2 of perturbed excess energyẼ. As C lin , C Cacc are bigger than 1, (5.51) implies (5.48).
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 Let us perform estimates, using first (4.5), then the approximative minimization property of standard mean value with respect to The first integral in the right-hand-side of (6.28) is majorized in view of Poincaré inequality in space (for generalized integral means) by whereas for the second one, in view of the inequality (5.21) of Lemma 5.3 holds Summing up, (6.28) takes the form where for the second inequality we use Poincaré in space and computation analogous to that of (6.30) and for the third one: estimate (5.22) and again Poincaré (both for standard and generalized integral means). This inequality is (6.26); it used in (6.31) gives which gives (6.25). Finally to obtain (6.24) from (6.25), estimate from below l.h.s. of (6.28), 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 (2.4).
Finally we see that Proof of Theorem 2.1. results from Lemmas 6.3 and 6.4.

Conclusions
The natural next step is to perform an analysis of the Hausdorff dimension of the singular set, at least for less general systems, for example for which the dependence of the main part A on u is waived. This, together with the non-linear Calderon-Zygmund L q estimates will be the joint content of the forthcoming paper, as there is a natural connection between the singular set estimates and the restriction on q.
It would be interesting, using new results on parabolic approximation, to perform similar analysis for p-Stokes system. Finally let us mention, that it seems that for a certain range of p's, close to 2, full C 1,α regularity for symmetric p-Laplace holds; this is also currently work in progress. However, the ultimate goal in this field, namely the full interior C 1,α -regularity for symmetric p-Laplace system, without restrictions on p and the space dimensions, seems to be essentially open.

Appendix
Here we present results which have been removed from the main part of this article for the sake of traceability.
Proof of Korn's inequality (4.7) in Lemma 4.3. Use inequality from [7] K B1(x) where R is the set of rigid motions, i.e. affine functions with antisymmetric linear part. (8.1) 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.
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], [16]. To prove inequalities we modify slightly the technique of Campanato [6]. Scaling v(y, s) = u(y/r, s/r 2 ) justifies that u solves locally (4.8) in Q r iff v solves (4.8) locally in Q 1 . Therefore we consider first v satisfying we test (4.12) with θ 2 σ 2 v, which yields: (i) for the main part: (ii) for the evolutionary part: with constant C depending on λ, |A|, K p . By linearity of (8. 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.