Simon Bortz

AND Moritz Egert

Olli Saari

NOTE ON TIME-REGULARITY FOR WEAK SOLUTIONS TO PARABOLIC SYSTEMS OF p-LAPLACE TYPE

Keywords: . 2010 Mathematics Subject Classification. 35K55, 42B15 non-linear parabolic systems, weak solutions, time regularity

We show that local weak solutions to parabolic systems of p-Laplace type are Hölder continuous in time with values in a spatial Lebesgue space and Hölder continuous on almost every time line. We provide an elementary and self-contained proof building on the local higher integrability result of Kinnunen and Lewis.

Introduction

Let d ě 2 and 2d{pd `2q ă p ă 8 and N ě 1. Consider the following parabolic system of p-Laplace type: [START_REF] Arendt | Vector-valued Laplace transforms and Cauchy problems[END_REF] Bu i Bt " div A i pt, x, ∇uq `Bi pt, x, ∇uq, i " 1, . . . , N, in I ˆQ,

where I Ă R is an interval, Q Ă R d a cube, and A i and B i satisfy certain structural conditions. These are the same as in [START_REF] Kinnunen | Higher integrability for parabolic systems of p-Laplacian type[END_REF] and do not require any smoothness of A i and B i , see Section 2.1. In a celebrated paper, Kinnunen and Lewis have obtained the higher integrability of the gradient of weak solutions.

Theorem 1 (Theorem 2.8 in [START_REF] Kinnunen | Higher integrability for parabolic systems of p-Laplacian type[END_REF]). There exists δ ą 0 depending on p, d and the structural constants c 1 , c 2 and c 3 such that if u P L 2 pI ˆQq X L p pI; W 1,p pQqq is a weak solution to (1), I 1 Ť I and Q 1 Ť Q, then u P L p`δ pI 1 ; W 1,p`δ pQ 1 qq.

The norm of u in L p`δ pI 1 ; W 1,p`δ pQ 1 qq depends on the same constants, on N, I, I 1 , Q, Q 1 , the structural constant c 4 and the norms }u} L 2 pIˆQq and }u} L p pI;W 1,p pQqq .

The case p " 2 is due to earlier work of Giaquinta and Struwe [START_REF] Giaquinta | On the partial regularity of weak solutions of nonlinear parabolic systems[END_REF]. The significance of these results is highlighted by the otherwise lacking regularity for solutions to parabolic systems, which can be essentially discontinuous.

In this short note we prove the following in-time Hölder continuity as an addendum to the Kinnunen-Lewis result.

Theorem 2. Let α :" 1 2 p 1 p ´1 p`δ q and q :" 2 1´2α , where δ ą 0 is from Theorem 1. If u P L 2 pI ˆQq X L p pI; W 1,p pQqq is a weak solution to (1), I 1 Ť I and Q 1 Ť Q, then there is a representative u P C α pI 1 ; L q pQ 1 qq.

The norm of u in C α pI 1 ; L q pQ 1 qq has the same dependencies as in Theorem 1. Moreover, for a.e. x P Q 1 the restriction u| I 1 ˆtxu is α-Hölder continuous.

Experts in interpolation theory of vector-valued Triebel-Lizorkin type spaces (see [START_REF] Denk | General parabolic mixed order systems in L p and applications[END_REF]) will realize that Theorem 2 can be obtained from midway complex interpolation of the smoothness properties

u P L p`δ pI 1 ; W 1,p`δ pQ 1 qq & u P W 1,p 1 pI 1 ; W ´1,p 1 pQ 1 qq,
where the second one follows from the equation [START_REF] Arendt | Vector-valued Laplace transforms and Cauchy problems[END_REF]. Since 2{q " 1{pp `δq `1{p 1 , we find u P H 1{2,q pI 1 ; L q pQ 1 qq. As time is a one-dimensional variable, this breaks the threshold 1 ´q{2 ă 0 in embeddings of such vector-valued Bessel potential spaces and leads to Hölder continuity. Still, we believe that an elementary and self-contained proof to deduce Theorem 2 from Theorem 1 will be of interest for a broader audience and the purpose of our note is to provide such an argument.

The abstract strategy sketched above is our guide in doing so. First, we smooth and localize the weak solution u and use the equation to write the t-derivative of the approximant as a global negative order Bessel potential (Lemma 3). Second, we use the scalar-valued Mihlin Fourier multiplier theorem and Hadamard's three lines theorem from complex analysis to bound a fractional order potential (Proposition 6). Third, a Fourier analytic characterization of Hölder continuity can be used to obtain the desired regularity of the approximant and further that of the local solution itself (Section 5).

We close this introduction with a brief comparison to our previous work with P. Auscher in [START_REF] Auscher | On regularity of weak solutions to linear parabolic systems with measurable coefficients[END_REF], where we obtained regularity as in Theorem 2 for linear operators and p " 2 by a more involved approach. See also [START_REF] Zatoń | Tent space well-posedness for parabolic Cauchy problems with rough coefficents[END_REF] for a generalization to higher order systems. The flexibility in the definition of the structure functions A and B, allows us to use Theorem 2 for inhomogenous linear systems of the form Bu i Bt ´div A i pt, x, ∇uq ´Bi pt, x, ∇uq " div F i `fi , i " 1, . . . , N, in I ˆQ,

where F, f P L 2`η for some η ą 0. The condition on f is more restrictive than in [START_REF] Auscher | On regularity of weak solutions to linear parabolic systems with measurable coefficients[END_REF]. This is needed here -as in the classical Lions theory [6] -because we use u P W 1,p 1 pI 1 ; W ´1,p 1 pQ 1 qq as a priori information.

Acknowledgement. This research was supported by the CNRS through a PEPS JCJC project and by DFG through DFG SFB 1060 and DFG EXC 2047.

Preliminaries

2.1. Structural assumptions. We summarize the assumptions of [START_REF] Kinnunen | Higher integrability for parabolic systems of p-Laplacian type[END_REF]. The matrixvalued function A : I ˆQ ˆpR d q N Ñ R dˆN has columns given by A i " A i pt, x, Vq, i " 1, . . . , N and the vector-valued function B : I ˆQ ˆpR d q N Ñ R N has scalar entries

B i " B i pt, x, Vq, i " 1, . . . , N.
Both are (Lebesgue) pd `1q-measurable functions on I ˆQ, whenever V " Vpt, xq is pd `1q-measurable on I ˆQ. For example, A and B could be of Carathéodory type. We also assume there are positive constants c j , j " 1, 2, 3, such that for almost every pt, xq P I ˆQ and every

V P pR d q N , |A i pt, x, Vq| ď c 1 |V| p´1 `h1 pt, xq, |B i pt, x, Vq| ď c 2 |V| p´1 `h2 pt, xq, for i " 1, . . . N and N ÿ i"1 xA i pt, x, Vq, V i y ě c 3 |V| p ´h3 pt, xq.
Here x¨, ¨y is the standard inner product on R d and h j , j " 1, 2, 3, are measurable functions on I ˆQ satisfying

}p|h 1 | `|h 2 |q p{pp´1q `|h 3 |} L qpIˆQq " c 4 ă 8,
for some q ą 1.

2.2. Weak solutions. The space L p pI; W 1,p pQqq consists of all functions f P L p pIQ q so that for almost every t P I the function f pt, ¨q is in the usual Sobolev space W 1,p pQq and

} f } L p pI;W 1,p pQqq :" } f } L p pIˆQq `}|∇ f |} L p pIˆQq ă 8.
We use the same notation for R N valued functions with the obvious interpretation.

We then say that u is a weak solution to (1) if u P L 2 pI ˆQq X L p pI; W 1,p pQqq and if ż

I ż Q N ÿ i"1 ˆ´u i Bφ i Bt `xA i pt, x, ∇uq, ∇φ i y ´Bi pt, x, ∇uqφ i ˙dx dt " 0
holds for all φ " pφ 1 , . . . , φ N q P C 8 c pI ˆQq.

Potential spaces.

We define the Fourier transform on the Schwartz space SpR d`1 ; Cq as usual by

F f pτ, ξq " ij e ´iτt´ixξ,xy f pt, xq dx dt
and extend it to the tempered distributions by duality. The partial Fourier transforms with respect to only space or time variables are denoted by the subscripts x and t. We define the Bessel potentials of order s P C through

J s f " F ´1pp1 `| ¨|2 q ´s{2 F f q.
Again, a subscript x or t tells with respect to which variable the potential is taken.

If s ą 0 and f P L p pR d q for some p P p1, 8q, then J s x f is given as a convolution with an integrable function

J s x f " G s x ˚x f, G s x pxq " 1 p4πq s 2 Γp s 2 q ż 8 0 δ s´d 2 e ´π|x| 2 δ ´δ 4π dδ δ . (2)
See Section V.3.1 in [START_REF] Stein | Singular Integrals and Differentiability Properties of Functions[END_REF] for this classical formula. The Bessel potential space H 1,p pR d q " tJ 1

x f : f P L p pR d qu with norm g Þ Ñ }J ´1 x g} L p pR d q coincides with W 1,p pR d q up to equivalence of norms. See Section V.3.3 in [START_REF] Stein | Singular Integrals and Differentiability Properties of Functions[END_REF].

2.4.

Mollification. The definition of weak solutions does not imply any a priori regularity in time direction. This causes technical problems, which in the context of this paper can be overcome through a mollification argument. Let ϕ P C 8 c pR d`1 q be an even function with integral one that we fix from this point on. For ǫ ą 0 we denote the mollification of a function g with ϕ by g ǫ pt, xq :"

ij 1 ǫ n`1 ϕ ˆt ´s ǫ , x
´y ǫ ˙gps, yq dy ds.

A priori potential estimate

We rephrase integrability and differentiability of localized weak solutions to ( Let u be a weak solution to (1) in I ˆQ and define v :" χu. Then there is an admissible constant C such that for any ǫ ą 0,

}J ´1 x pv ǫ q} L p`δ pR d`1 q ď C, }J ´1 t J 1 x pv ǫ q} L p 1 pR d`1 q ď C.
Proof. We use the symbol À for inequalities that hold up to a multiplicative admissible constant. We obtain from Young's inequality, the choice of χ and Theorem 1 that

}v ǫ } L p`δ pR;W 1,p`δ pR d qq À }u} L p`δ pI;W 1,p`δ pQqq ď C.
By coincidence of Sobolev and potential spaces, the left-hand side is comparable to }J ´1 x pv ǫ q} L p`δ pR d`1 q . Hence, we have the first estimate. To prepare the second estimate, we fix φ P SpR d`1 ; R N q normalized such that }φ} L p pR;W 1,p pR d qq " 1. By Hölder's inequality we have that ˇˇˇij xv ǫ , φy dx dt ˇˇˇ" ˇˇˇij xv, φ ǫ y dx dt ˇˇˇÀ }u} L 8 pI;L 2 pQqq }φ ǫ } L p pI;L 2 pQqq .

The Caccioppoli inequality (Lemma 3.2 in [START_REF] Kinnunen | Higher integrability for parabolic systems of p-Laplacian type[END_REF] with a " 0) yields

}u} L 8 pI;L 2 pQqq À }u} L 2 pIˆQq `}u} L p pI;W
Next, we get from the equation for u, using the summation convention for i " 1, . . . , N and omitting the variable of integration dx dt for the sake of readability, ´ij xB t pv ǫ q, φy " ij ˆxu, B t pχφ ǫ qy ´xu, pB t χqφ ǫ y " ij ˆxA i pt, x, ∇uq, ∇pχφ ǫ q i y ´Bi pt, x, ∇uqpχφ ǫ q i ´xu, pB t χqφ ǫ y "

ij xpχA i pt, x,
|II| `|III| À c 4 `pc 1 `c2 q}∇u} p´1 L p pIˆQq .
For IV, we can argue as in (3) with φ ǫ replaced by pB t χqφ ǫ , in order to give

|IV| À }u} L 2 pIˆQq `}u} L p pI;W 1,p pQqq .
Summarizing these estimates, we get ˇˇˇij xv ǫ , φy dx dt ˇˇˇ`ˇˇˇij xB t pv ǫ q, φy dx dt ˇˇˇď C This is true for any φ P SpR d`1 q normalized in L p pR; W 1,p pR d qq. In view of the equivalence of Sobolev and potential spaces on R d , this is the same as taking φ " J 1 x ψ, where ψ P SpR d`1 q is normalized in L p pR d`1 q. Hence, we get ˇˇˇij xJ 1 x v ǫ , ψy dx dt ˇˇˇ`ˇˇˇij xB t pJ 1 x v ǫ q, ψy dx dt ˇˇˇď C Since SpR d`1 q is dense in L p 1 pR d`1 q, we obtain

}J 1 x v ǫ } L p pR d`1 q `}B t pJ 1
x v ǫ q} L p pR d`1 q ď C Now, we invoke the equivalence of Sobolev and potential spaces in t and apply Fubini's theorem to conclude the bound for }J ´1 t J 1 x pv ǫ q} L p pR d`1 q .

Interpolation estimate of a mixed potentials

We begin by recalling (a simple version of) the Mihlin multiplier theorem.

Proposition 4 (Theorem 8.2 in [START_REF] Muscalu | Classical and multilinear harmonic analysis[END_REF]). Let n ě 1, let m : R n Ñ C satisfy, for all multi-index of order |α| ď n `2 and all ξ ‰ 0, |B α ξ mpξq| ď M|ξ| ´|α| . Then, for any q P p1, 8q there is a constant C " Cpn, qq, such that for all φ P SpR n q and for F the Fourier transform on SpR n q, }F ´1pmF φq} L q pR n q ď C M}φ} L q pR n q .

The multiplier theorem entails quantitative bounds for Bessel potentials. Lemma 5. Let a P r0, 8q and b P R. For all q P p1, 8q there is a constant C " Cpd, qq such that for all φ P SpR d q, }J 2a`2ib

x φ} L q pR d q ď Cp1 `a `|b|q d`2 }φ} L q pR d q .

The same holds for J 2a`2ib t upon replacing d by 1.

Proof. We put hpσq " p1 `σq ´a´ib . According to the Mihlin multiplier theorem, we have }J 2a`2ib

x φ} L q pR d q ď C M}φ} L q pR d q , where M " max

|α|ďd`2 sup ξ‰0 |ξ| |α| |B α ξ php|ξ| 2 qq|.
Let α be any multi-index. By induction on the length of α, we find numerical constants c β pαq, one for each multi-index β with 2β i ď α i , i " 1, . . . , d, such that

B α ξ php|ξ| 2 qq " ÿ β c β pαqh p|α|´|β|q p|ξ| 2 qξ α´2β .
Since the higher order derivatives of h satisfy

|h pkq pσq| ď cpkqp1 `a `|b|q k p1 `σq ´a´k ,
we can take M " Cpdqp1 `a `|b|q d`2 . The one dimensional case is clearly included in the computation.

We deduce the following interpolation inequality. In the proof we shall use the notion of holomorphic functions f : Ω Ă C Ñ L 2 pR d`1 ; C N q. Holomorphy is defined via convergence of difference quotients. If f is locally bounded, then it is equivalent to holomorphy of z Þ Ñ ť f pzqφ dx dt for all φ P SpR d`1 ; C N q. The reader can refer to Appendix A of [START_REF] Arendt | Vector-valued Laplace transforms and Cauchy problems[END_REF] for further background. Proposition 6. Let f P SpR d`1 ; C N q, let θ P p0, 1q and let q 0 , q θ , q 1 P p1, 8q be such that 1{q θ " p1 ´θq{q 0 `θ{q 1 . Then there is a constant C " Cpd, N, q 0 , q 1 q such that

}J 2θ´1 x J ´θ t f } L q θ pR d`1 q ď C}J ´1 x f } 1´θ L q 0 pR d`1 q }J 1 x J ´1 t f } θ L q 1 pR d`1 q .
Proof. By duality, we have

}J 2θ´1 x J ´θ t f } L q θ pR d`1 q " sup φ ˇˇˇij xpJ 2θ´1 x J ´θ t f qpx, tq, φpx, tqy dx dt ˇˇˇ, (4)
where the supremum is taken over all φ P SpR d`1 ; C N q normalized in L q 1 θ pR d`1 q. The idea of proof, coming from the Riesz-Thorin theorem, is to use a holomorphic parametrization of the duality pairing for fixed φ via functions defined on the strip S :" ta `ib : a P p0, 1q, b P Ru. More precisely, we define whenever z P S , Fpzq :" e pz´θq 2 J 2z´1

x J ´z t f, Gpzq :" |φ|

p1´zqq 1 θ q 1 0 `zq 1 θ q 1 1 φ |φ| ,
where the expression for Gpzq is interpreted as 0 on the set where φ vanishes.

We derive properties of F. For z " a `ib we have Fpzq " ´epa´θq 2 ´b2 e i2bpa´θq ¯J2a`2ib

x J p1´aq´ib t pJ ´1 x J ´1 t f q. (5)
Since J ´1

x J ´1 t f is a Schwartz function, Lemma 5 applied componentwise in combination with Fubini's theorem yields that F is qualitatively bounded on S with values in L 2 pR d`1 ; C N q. (The polynomial growth in b is compensated by the exponential function.) Again using Lemma 5, we have the following quantitative bounds on BS :

}Fpibq} L q 0 pR d`1 q ď Cpd, N, q 0 q}J ´1 x f } L q 0 pR d`1 q , }Fp1 `ibq} L q 1 pR d`1 q ď Cpd, N, q 1 q}J ´1 t J 1 x f } L q 1 pR d`1 q . (6)
Next, it follows from F f P SpR d`1 ; C N q and dominated convergence, that we have a continuous function

F F : S Ñ L 2 pR d`1 ; C N q, z Þ Ñ e pz´θq 2 p1 `|ξ| 2 q 1´2z 2 p1 `|τ| 2 q z 2 F f pτ, ξq.
Since the Fourier transform is isometric on L 2 pR d`1 ; C N q, the same follows for F. Finally, F is holomorphic in S . Indeed, for any ψ P SpR d`1 ; C N q we can use Parseval's formula to give

ij xFpzq, ψy dx dt " ij @ e pz´θq 2 p1 `|ξ| 2 q 1´2z 2 p1 `|τ| 2 q z 2 F f, F ψ D dξ dτ
and the integral in z along any triangle △ Ť S vanishes by Fubini's theorem and holomorphy of the integrand for fixed pτ, ξq.

The function G : S Ñ L 2 pR d`1 ; C N q enjoys the same kind of properties. Here, boundedness follows directly from φ P SpR d`1 ; C N q, continuity and holomorphy are obtained as before, and on BS we get from Hölder's inequality and the normalization of φ that

}Gpibq} L q 1 0 pR d`1 q ď 1, }Gp1 `ibq} L q 1 1 pR d`1 q ď 1. (7)
Now, define a scalar-valued function on S by Hpzq :" ij xFpzq, Gpzqy dx dt.

The L 2 pR d`1 ; C N q-valued properties for F and G above imply that H is bounded and continuous on S and holomorphic in S . (The inner product preserves continuity and holomorphy by nearly the same proof as for products of scalar functions.) On the boundary, we conclude from ( 6), ( 7) and Hölder's inequality that |Hpibq| ď Cpd, N, q 0 q}J ´1 x f } L q 0 pR d`1 q ": M 0 , |Hp1 `ibq| ď Cpd, N, q 1 q}J ´1 t J 1 x f } L q 1 pR d`1 q ": M 1 .

Hadamard's three lines theorem yields |Hpθq| ď M 1´θ 0 M θ 1 . This means that

ˇˇˇij xJ 2θ´1 x J ´θ t f, φ y dx dt ˇˇˇď C}J ´1 x f } 1´θ L q 0 pR d`1 q }J ´1 t J 1 x f } θ L q 1 pR d`1 q
and the claim follows since φ was arbitrary and normalized in L q 1 θ pR d`1 q.

Proof of Theorem 2

Let u be a weak solution to (1) in I ˆQ. We define nested intervals and cubes

I 2 Ť I 1 Ť I and Q 2 Ť Q 1 Ť Q and pick a smooth function 1 I 2 ˆQ2 ď χ ď 1 I 1 ˆQ1
. Define a localized version v " χu as in Lemma 3. Since the nested sets are arbitrary, it suffices to obtain the continuity statements on I 2 ˆQ2 .

Step 1: Hölder continuity with values in spatial L q . Lemma 3 justifies applying Proposition 6 to f " v ǫ with q 0 " p `δ and q 1 " p 1 . Thus, setting θ " 1 2 in Proposition 6, we have

}J ´1{2 t v ǫ } L q pR d`1 q ď C, ( 8
)
for some admissible C. The exponent q ą 2 is given by

1 q " 1 2 `1 2 ˆ1 p `δ ´1 p ˙. (9)
We have pJ ´1{2 t v ǫ qp¨, xq " J ´1{2 t pv ǫ p¨, xqq P L q pRq for every x P R d since v ǫ is a Schwartz function, but [START_REF] Stein | Singular Integrals and Differentiability Properties of Functions[END_REF] gives a quantitative bound. Fix x P R d . As q 1 ă 2, we obtain from Minkowski's inequality that G 1{2 t P L q 1 pRq. This Bessel kernel was defined in [START_REF] Auscher | On regularity of weak solutions to linear parabolic systems with measurable coefficients[END_REF]. Hence, we have by Young's convolution inequality that

}v ǫ p¨, xq} L 8 pRq ď }G 1{2 t } L q 1 pRq }J ´1{2 t v ǫ p¨, xq} L q pRq . (10)
Now, we appeal to a Fourier analytic characterization of Hölder continuity. This uses a smooth function ψ with support in the set t 1 2 ď t ď 4u with the property that ř jPZ ψ j ptq " 1 for all t ‰ 0, where ψ j ptq " ψp2 ´jtq. For one construction see Lemma 8.1 in [START_REF] Muscalu | Classical and multilinear harmonic analysis[END_REF].

Lemma 7 (Lemma 8.6 in [START_REF] Muscalu | Classical and multilinear harmonic analysis[END_REF]). Let f P SpRq and let 0 ă α ă 1. There is a constant C " Cpαq such that for all t ‰ s,

| f ptq ´f psq| |t ´s| α ď C sup jPZ 2 jα }F ´1 t pψ j F t f q} L 8 pRq .
We put f :" v ǫ p¨, xq. Since F ´1 t pψ j F t f q has a Fourier transform with support in p´2 j`2 , 2 j`2 q, Bernstein's inequality (Lemma 4.13 in [START_REF] Muscalu | Classical and multilinear harmonic analysis[END_REF]) yields

}F ´1 t pψ j F t f q} L 8 pRq ď c2 j{q }F ´1 t pψ j F t f q} L q pRq ď c2 j{q Cpp, ψq2 mint´j{2,1u }J ´1{2 t v ǫ p¨, xq} L q pRq ,
where c is a numerical constant and the second step is due to the Mihlin multiplier theorem applied to mpτq " ψ j pτqp1 `|τ| 2 q ´1{4 . The computation of the Mihlin norm is done verbatim as in the proof of Lemma 5, taking into account that on the support of ψ j we have 2 j´1 ď |τ| ď 2 j`2 in order to obtain the decay in j.

The assumptions of Theorem 4 are hence satisfied. Consequently, we can take α " 1 2 ´1 q " 1 2 p 1 p ´1 p`δ q in Lemma 7. In view of (10) we find for all t ‰ s and all

x P R d that |v ǫ pt, xq| `|v ǫ pt, xq ´vǫ ps, xq| |t ´s| α ď C}J ´1{2 t v ǫ p¨, xq} L q pRq . (11)
We fix a representative for v, a subsequence of ǫ and a set E of pd `1q-measure zero such that v ǫ pt, xq Ñ vpt, xq as ǫ Ñ 0, whenever pt, xq P R d`1 zE. Then we integrate the q-th power in x, use [START_REF] Stein | Singular Integrals and Differentiability Properties of Functions[END_REF] Since v " u on I 2 ˆQ2 , we obtain u P C α pI 2 ; L q pQ 2 qq as required.

Step 2: Hölder continuity on almost all segments in time. Since the Fourier transform turns convolutions into products, we obtain for all ϕ P SpR d`1 q that xJ ´1{2 t v ǫ , ϕy " xJ

´1{2 t v, ϕ ǫ y, (12)
where we use the duality pairing on S 1 pR d`1 q. In the limit as ǫ Ñ 0, we have ϕ ǫ Ñ ϕ in SpR d`1 q and therefore J ´1{2 t v ǫ Ñ J ´1{2 t v in S 1 pR d`1 q. On the other hand, this sequence is bounded in L q pR d`1 q by (8) and hence admits a weakly convergent subsequence. Identifying the limits, we get J ´1{2 t v P L q pR d`1 q. Now, we can use (12) to write J ´1{2 t v ǫ " pJ ´1{2 t vq ǫ and obtain strong convergence in L q pR d`1 q. This implies that }J ´1{2 t v ǫ } L q pRq Ñ }J ´1{2 t v} L q pRq in L q pR d q. Hence, we can pass to a subsequence such that for almost every x P R d , }J ´1{2 t v ǫ p¨, xq} L q pRq Ñ }J ´1{2 t vp¨, xq} L q pRq . Fix x with this property and such that E x :" tt : pt, xq P Eu has 1-measure zero, where E is as in Step 1. Passing to the limit in (11), we obtain that vp¨, xq satisfies the α-Hölder condition on E x . Hence, we can re-define vp¨, xq so that it is α-Hölder continuous. Since all modifications take place in E, we obtain a representative for v that is α-Hölder continuous on R ˆtxu for a.e. x P R d . We conclude again since v " u on I 2 ˆQ2 .

  1,p pQqq . , we have the Sobolev embedding W 1,p pQq Ď L 2 pQq. Thus, }φ ǫ } L p pI;L 2 pQqq À }φ ǫ } L p pI;W 1,p pQqq ď }φ} L p pR;W 1,p pR d qq " 1.

	Since p ě 2d d`2

Altogether, we have found that ˇˇˇij xv ǫ , φy dx dt ˇˇˇÀ }u} L 2 pIˆQq `}u} L p pI;W 1,p pQqq .

  ∇uqq ǫ , ∇φ i y `ij xA i pt, x, ∇uq, p∇χqpφ ǫ q i y ´ij B i pt, x, ∇uqpχφ ǫ q i ´ij xu, pB t χqφ ǫ y

": I `II ´III ´IV. Using Hölder's inequality and the upper bound for A, we have |I| À pc 4 `c1 }∇u} p´1 L p pIˆQq q}∇φ} L p pIˆQq ď c 4 `c1 }∇u} p´1 L p pIˆQq . Similarly, replacing ∇φ by φ, we get

  and pass to the limit via Fatou's lemma, to get

	sup tPR ˆżR d	|vpt, xq| q	˙1{q	`sup t‰s t,sPR	ˆżR d	|vpt, xq ´vps, xq| q |t ´s| αq

Simon Bortz, Department of Mathematics, University of Alabama, Tuscaloosa, AL, 35487, USA Email address: sbortz@ua.edu Moritz Egert, Université Paris-Saclay, CNRS, Laboratoire de mathématiques d'Orsay, 91405, Orsay, France Email address: moritz.egert@universite-paris-saclay.fr Olli Saari, Mathematical Institute, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany Email address: saari@math.uni-bonn.de