Optimal partial regularity of second-order parabolic systems under natural growth condition

Electrorheological fluids are special viscous liquids, that are characterized by their ability to undergo significant changes in their mechanical properties when an electric field is applied. This property can be exploited in technological applications, e.g., actuators, clutches, shock absorbers, and rehabilitation equipment to name a few [1].

where $\mathrm{\Omega }\subset R^n$ is a bounded domain and $T>0$, $z=(x,t)$ with $x\in \mathrm{\Omega }$, $0<t\le T$, denote a point in $Q_T=\mathrm{\Omega }\times (-T,0)$. Let $u(z)=(u^1(z),u^2(z),\dots ,u^N(z))$ be a vector-valued function defined in $Q_T$. Denote by Du the gradient of u, i.e., $Du={\{D_{\alpha }{u}^i\}}_{i=1,\dots ,N;\alpha =1,\dots ,n}$. $m>2$ is a real number.

In order to define the weak solution of (1.9), one needs to impose some regularity conditions and constructer conditions to $A_i^{\alpha }$ and $B_i$. For a vector field $A_i^{\alpha }:Q_T\times R^N\times R^{nN}$, we shall denote the coefficients by $A_i^{\alpha }(z,u,p)=A_i^{\alpha }(x,t,u,p)$ if $z=(x,t)$, $u\in R^N$ and $p\in R^{nN}$. We assume that the functions $(z,u,p)\mapsto A_i^{\alpha }(z,u,p)$; $(z,u,p)\mapsto \frac{\partial A_i^{\alpha }}{\partial p_{\beta }^j}(z,u,p)$ are continuous in $Q_T\times R^N\times R^{nN}$ and that the following growth and ellipticity conditions are satisfied:

where $\lambda >0$ and $1\le L<\mathrm{\infty }$. Now we shall specify the regularity assumptions on $A_i^{\alpha }(z,u,p)$ with respect to the ‘coefficient’ $(z,u)$ and assume that the function $(z,u)\mapsto \frac{A_i^{\alpha }(z,u,p)}{1+|p|}$ is Hölder continuous with respect to the parabolic metric $\sqrt{{|x-x_0|}^2+|t-t_0|}$ with Hölder exponent $\beta \in (0,1)$ but not necessarily uniformly Hölder continuous; namely we shall assume that:

for any $z=(x,t)$ and $z_0=(x_0,t_0)$ in $Q_T$. u and $u_0$ in $R^n$ and for all $p\in R^{nN}$, where $\theta (y,s)=min\{1,\tilde{K}(y)s^{\beta }\}$, $\tilde{K}:[0,\mathrm{\infty })\mapsto (1,\mathrm{\infty })$ is a given non-decreasing function. Note that θ is concave in the argument. This is the standard way to prescribe (non-uniform) Hölder continuity of the function $A_i^{\alpha }(z,u,p)$. We find it a bit difficult to handle, therefore, in many points of the paper, we shall use:

valid for any $z=(x,t)$ and $z_0=(x_0,t_0)$ in $Q_T$, u and $u_0$ in $R^n$ and $p\in R^{nN}$.

for any $z=(x,t)$ and $z_0=(x_0,t_0)$ in $Q_T$, any u, $u_0$ in $R^n$ and $p,p_0\in R^{nN}$ whenever $|u|+|p|+|u-u_0|+|p-p_0|\le M$.

for all $z\in Q_T$, $u\in R^N$ and $p,q\in R^{nN}$.

for all $\phi \in C_0^{\mathrm{\infty }}(Q_T,R^N)$.

In [4] Duzaar and Mingione considered the partial regularity of homogeneous systems of (1.9) with $m\equiv 2$ under the natural growth condition. In this paper, we extend their results to the case of $m>2$. We have to overcome the difficulty of $m>2$. Motivated by the works of Duzaar [4, 5], Chen and Tan [6–9] and Tan [10], we use the technique of ‘A-caloric approximation’ to establish the optimal partial regularity of nonlinear parabolic systems (1.9). In fact, the use of the ‘A-caloric approximation lemma’ allows optimal regularity, without the use of Reverse-Hölder inequalities and (parabolic) Gehring’s lemma. The method is based on an approximation result that we called the ‘A-caloric approximation lemma’. This is the parabolic analogue of the classical harmonic approximation lemma of De Giorgi [11, 12] and allows to approximate functions with solutions to parabolic systems with constant coefficients in a similar way as the classical harmonic approximation lemma does with harmonic functions. And we can obtain the following theorem.

At the end of the section, we summarize some notions which we will be used in this paper. For $x_0\in R^n$, $t_0\in R$, we denote $B(x_0,R)=\{x\in R^n:|x-x_0|<R\}$, $Q((x_0,t_0),R)=B(x_0,R)\times (t_0-R^2,t_0)$. If v is an integrable function in $Q(z_0,\rho )=Q_{\rho }(z_0)=B_{\rho }(x_0)\times (t_0-{\rho }^2,t_0)$, $z_0=(x_0,t_0)$, we will denote its average by , where ${\alpha }_n$ denotes the volume of the unit ball in $R^n$. We remark that in the following, when not crucial, the ‘center’ of the cylinder will be often unspecified, e.g., $Q_{\rho }(z_0)=Q_{\rho }$; the same convention will be adopted for balls in $R^n$ therefore denoting $B(x_0,\rho )=B_{\rho }(x_0)$. Finally, in the rest of the paper, the symbol C will denote a positive, finite constant that may vary from line to line; the relevant dependencies will be specified.

In this section we introduce the A-caloric approximation lemma [4] and some preliminaries. Recall a strongly elliptic bilinear form $A_i^{\alpha }$ on $R^{nN}$ with an ellipticity constant $\lambda >0$, and upper bound $\mathrm{\Lambda }>0$ means that $\lambda {|\tilde{p}|}^2\le A_i^{\alpha }(\tilde{p},\tilde{p})$, $A_i^{\alpha }(p,\tilde{p})\le \mathrm{\Lambda }|p||\tilde{p}|$, $\mathrm{\forall }p,\tilde{p}\in R^{nN}$, we define A-caloric approximation function.

Remark 2.2 Obviously, when $A(\tilde{p},\tilde{p})\equiv {|\tilde{p}|}^2$ for every $\tilde{p}\in R^{nN}$, then an A-caloric function is just a caloric function $h_t-\mathrm{△}h\equiv 0$.

Lemma 2.3 (A-caloric approximation lemma)

Actually, we could have directly applied Theorem 5 of [13] with the choice $X=W^{1,2}(B,R^N)$, $B=L^2(B,R^N)$, $R=W^{-l,2}(B,R^N)$, $F={(v_k)}_{k\in N}$, $p=2$ to conclude that ${(v_k)}_{k\in N}$ is relatively compact in $L^2(Q_T,R^N)=L^2(-1,0;L^2(B,R^N))$.

Lemma 2.4 There exists a positive function $\delta (n,N,\lambda ,\mathrm{\Lambda },\epsilon )\le 1$ with the following property: Whenever A is a bilinear form on $R^{nN}$ which is strongly ellipticity constant $\lambda >0$ and upper bound Λ, ε is a positive number, and $u\in L^2(t_0-{\rho }^2,t_0;W^{1,2}(B_{\rho }(x_0),R^N))$ with

(2.4)

is approximatively A-caloric in the sense that

(2.5)

then there exists $h\in L^2(t_0-{\rho }^2,t_0;W^{1,2}(B_{\rho }(x_0),R^N))$ A-caloric on $Q_{\rho }(z_0)$ such that

(2.6)

For $u\in L^2(Q_{\rho }(z_0),R^N)$ we denote by $l_{z_0,\rho }$ the unique affine function (in space) $l(z)=l(x)$ minimizing , amongst all affine functions $a(z)=a(x)$ which are independent of t. To get an explicit formula for $l_{z_0,\rho }$, we note that such a unique minimum point exists and takes the form $l_{z_0,\rho }(x)={\xi }_{z_0,\rho }+{\nu }_{z_0,\rho }(x-x_0)$, where ${\nu }_{z_0,\rho }\in R^{nN}$. A straightforward computation yields that , for any affine function $a(x)=\xi +\nu (x-x_0)$ with $\xi \in R^N$ and $\nu \in R^{nN}$. This implies in particular that and .

For convenience we recall from [14] the following.

Lemma 2.5 Let $u\in L^2(Q_{\rho }(z_0),R^N)$, $0<\theta <1$, and $l_{z_0,\rho }$ respectively $l_{z_0,\theta \rho }$ the unique affine functions minimizing respectively . Then there holds

Moreover, if $Du\in L^2(Q_{\rho }(z_0),R^{nN})$, we have

In this section we prove Caccioppoli’s second inequality.

Theorem 3.1 (Caccioppoli second inequality)

Let $u\in L^m(-T,0;W^{1,m}(\mathrm{\Omega },R^N))\cap L^{\mathrm{\infty }}(Q_T;R^N)$ be a weak solution to (1.9) under the assumptions (H1)-(H4) and the natural growth condition (H5). Then, for any $M>0$, any affine function $l(z)=l(x)$ independent of t and satisfying $|l(z_0)|+|Dl|\le M$, and any $Q_{\rho }(z_0)\subset \subset Q_T$ with $0<\rho <R\le 1$, we have

Then the desired result follows by taking the limit $\sigma \to 0$. □

The next lemma is a prerequisite for applying the A-caloric approximation technique.

Lemma 4.1 Let $u\in L^m(-T,0;W^{1,m}(\mathrm{\Omega },R^N))\cap L^{\mathrm{\infty }}(Q_T;R^N)$ be a weak solution to (1.9) under the assumptions (H1)-(H6). Then for any $M>0$, we have

for any $Q_{\rho }(z_0)\subset \subset Q_T$ and $\phi \in C_0^{\mathrm{\infty }}(Q_{\rho }(z_0),R^N)$ with $\rho \le 1$ and any affine function $l(z)=l(x)$ independent of time, satisfying $|l(z_0)|+|Dl|\le M$. Here $C_{\mathrm{Eu}}=C_{\mathrm{Eu}}(M,L,m)$ and we write

Proof Without loss of generality, we can assume that ${sup}_{Q_{\rho }(z_0)}|D\phi |\le 1$. From (1.12) and the fact that and , we deduce

and $s_1=Q_{\rho }(z_0)\cap \{z:|Du-Dl|\le 1\}$, $s_2=Q_{\rho }(z_0)\cap \{z:|Du-Dl|>1\}$.

We proceed estimating the two resulting pieces. As for $I_1$, using (H6), the fact that $s\mapsto {\omega }^m(t,s)$ is concave and Jensen’s inequality (note that $\frac{m-1}{m}>\frac{1}{2}$), we get

and therefore

Similarly, we also have

Using (H1), (H2) and the previous inequality, we then conclude the estimate of $I_2$ as follows:

For the remaining pieces, using (H4′), we deduce

Here we have used that $K\ge 1$ and the assumption that $\rho \le 1$. Using again (H4′) and Young’s inequality, we estimate

and

Noting the definition of H and combining the estimates just found for I, II, III and IV, we obtain

A simple scaling argument yields the result for general φ. □

The next lemma is a standard estimate for weak solutions to linear parabolic systems with constant coefficients [15], Lemma 5.1.

Lemma 4.2 Let $h\in L^2(t_0-{\rho }^2,t_0;W^{1,2}(B_{\rho }(x_0),R^N))$ be a weak solution in $Q_{\rho }(z_0)=B_{\rho }(x_0)\times (t_0-{\rho }^2,t_0)$ of the following linear parabolic system with constant coefficients:

Here we write

In the following we consider a weak solution u of the nonlinear parabolic system (1.9) on a fixed sub-cylinder $Q_{\rho }(z_0)\subset Q_T$ and $\rho \le 1$.

Proof Given $M>1$. And we shall always consider $\rho \le 1$. We first want to apply Lemma 4.1 on $Q_{\rho /2}(z_0)$ to $u-l$, where $l(z)=l(x)$ is an affine function independent of t satisfying $|l(z_0)|+|Dl|\le M$. We observe that Ψ has the following property:

(4.1)