SMOOTHNESS AND $\mathrm{D}\mathrm{I}\mathrm{S}\mathrm{C}\mathrm{O}\mathrm{N}\mathrm{T}\mathrm{I}1\backslash ^{\mathfrak{s}}$UITIES $\mathrm{O}\Gamma^{\wedge}\mathrm{W}\mathrm{E}\mathrm{A}1\backslash \prime \mathrm{S}\mathrm{O}\mathrm{L}\mathrm{U}\mathrm{T}\mathrm{I}\mathrm{O}1\backslash ^{\mathfrak{s}}\mathrm{S}$TO PARABOLIC SYSTEMS
J.
Star\’a,
O. JohnThe aim of this note is to $\mathrm{g}\mathrm{i}\backslash r\mathrm{e}$ a survey of several $1^{\cdot}\mathrm{e}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}$ results dealing with
regularity ofweak solutions to parabolic systems (existence of$L_{2}$-time derivative,
H\"older continuity in two dimensional space) and to illustrate that in three
dimen-sions parabolic systems do $.$
$\mathrm{n}\mathrm{o}\mathrm{t}$ conserve, in general, theregularizingproperty of the
heat equation.
In the first $\mathrm{p}\mathrm{a}1^{\cdot}\mathrm{t}_{J}.\backslash \nwarrow^{\gamma}\mathrm{e}$ give a comparison of results about elliptic and parabolic
systems and list some open problems. Second part is devoted to examples of
non-regular solutions and the third part contains partial positive results valid under
additional assumptions on coefficients.
1. $\mathrm{C}_{0111}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{n}$ of results for elliptic and parabolic systelns
$11^{\gamma}\mathrm{e}$ are interested in quasilinear elliptic systelns of the form
(1.1) $D_{\mathrm{o}}(A_{ij}^{0\beta}(x, u)D_{\beta}u^{j})=0,$$i=1,$
$\ldots,$
$\mathrm{J}I$ on $\Omega$;
and their parabolic counterpart
(1.2) $u_{t}^{i}=D_{\mathrm{o}}(A_{ij}^{\mathrm{o}\beta}(x, t, u)D_{\beta}u^{j})\wedge\cdot i=1,$
$\ldots,$
$\mathrm{J}I$ on $Q$.
(The sunlmation convention is used throughout thepaper.) Domain$\Omega$ is considered
to be a nonempty open subset of $\mathbb{R}^{n\iota},$ $Q=\Omega\cross(0, T)$ for a positive T. $A_{ij}^{\alpha\beta}(i,j=$
$1,$
$\ldots,$
$\mathrm{J}I,$$\alpha,$$\beta=1,$
$\ldots,$$?n)$ are uniformly bounded
$\mathrm{C}\mathrm{a}_{}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{h}\acute{\mathrm{e}}\mathrm{o}\mathrm{d}\mathrm{o}\mathrm{r}\mathrm{y}$ functions satisfying
in case (1.1) ellipticity condition and its
a.n
alogy1
(1.3) $\exists\lambda_{0_{}}.\lambda_{1}\in(0, \infty)\forall\xi\in \mathbb{R}^{m.M}\forall u\in \mathbb{R}^{M}$
$\mathrm{f}\mathrm{o}1^{\cdot}$ almost every $x\in \mathbb{R}^{m}$ for almost every $t\in(0_{i}T)$
$\lambda_{0}|\xi|^{\underline{9}}\leq\langle A(x_{J}.t, u)\xi, \xi\rangle\leq\lambda_{1}|\xi|^{2}$.
in $\mathfrak{t}_{1}\mathrm{h}\mathrm{e}$ parabolic case (1.2). For simplicity case we shall deal mainly with interior
regularity.
Elliptic systenls, linear case
If $A_{i_{1}}^{0\beta}$ depend only on $x$ and are continuous on their domain, then according to
classical results of$\cdot$
C. B.Morrey $[6]_{/}.$ A. Douglis, L. Nirenberg [2] $\mathrm{e}\backslash$’ery weaksolution
of (1.1) is locally H\"oldel continuous. The $\mathrm{p}_{\mathrm{l}\mathrm{O}()}\mathrm{f}$of Theoreln 3.1 in [3] indicates that
the continuity of coefficients in one point implies the H\"older continuity of any weak
solution in a neighbourhood of this point.
1$l\lambda^{\gamma}\mathrm{e}$ denol ed $\mathrm{b}.\backslash .\langle \mathrm{t}‘. \mathrm{t}|\rangle$ scalarproduct in any finite dilnellsional space$\mathbb{R}^{p},$$p\in \mathrm{N},$ $|v|=\langle u, v\rangle^{\frac{1}{2}}$.
On the other hand, for ? $\geq 3$ the discontinuity of coefficients in one point
can cause the discontinuity ($\mathrm{e}\backslash \prime \mathrm{e}\mathrm{n}$ unboundedness) of a solution (see [1]). The
counterexampleof J. Sou\v{c}ek $(\mathrm{s}\mathrm{e}\mathrm{e}[\overline{(}])\mathrm{g}\mathrm{i}\lambda^{r}\mathrm{e}\mathrm{s}$a solution of (1.1)
$\backslash \mathrm{v}\mathrm{h}\mathrm{i}\mathrm{c}\mathrm{h}$ is discontinuous
on a dense countable subset. Moreover, for $\mathrm{e}\backslash \prime \mathrm{e}\mathrm{r}\mathrm{y}$ set $F\subset \mathbb{R}^{m}$ of the type
$F_{\sigma}$ there
is a $\mathrm{s}_{3^{r\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m}}}(1.1)$ and its solution $u$ which is bounded. essentially discontinuous at
all points of $F$ and essentially continuous at all points of $\mathbb{R}^{m}-F$. (See [8].)
In any dimension we have a $W^{1,p}$ estimate of solutions for some (sufficiently
small) $p>2$ (see e.g. [5], [3]). If $m=2$, this estimate and embedding theorems
guarantee H\"older continuity of solutions.
Parabolic systenls, linear case
In this case, too, any weak solution (i.e., any locally square integrable function
with locally square integrable space gradient satisfying the system in the sense of
distributions) is H\"older continuous on a neighbourhood of $\mathrm{a}113^{r}$ point of continuity
ofcoefficients (see [12], [19]).
Any weak solution of an elliptic system (1.1) can be considered as a stationary
solution to a parabolic system (1.2). Thus elliptic examples can be interpreted
as stationary parabolic problems on $Q$. It would indicate singularities of solutions
appearing on cylindrical subsets of $\mathbb{R}^{m+1}$. It is more interesting to ask whether
a weak solution of a parabolic system call develop a singularity in the interior
of space-time cylider starting from smooth initial data. If $n’\iota\geq 3$ this situation
can occur (see [9] and part 2 of this paper) eventhough much less is known about
possible structure of a singular set (see [10]).
For parabolic systems $L_{p}$ estimates of space graldient for sufficiently small $p>2$
(see [11], [13], [15], part.3 of this paper) hold, too. $\mathrm{H}\mathrm{o}\mathrm{w}\mathrm{e}\backslash r\mathrm{e}\mathrm{r}$ even for $m=2$ they do
not imply H\"older continuity of solutions. As far as we know the question whether
for $m=2$ any weak solution to a linear parabolic systeln with $L_{\infty}$ coefficients is
locally H\"older continuous is open.
2. $\mathrm{E}_{\mathrm{X}\mathrm{a}\mathrm{n}1}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{s}$
Theorem 2.1 (see [9]) Let $n\geq 3,$$\kappa\in(0,2(n-1)(n-2))$;
for
$x\in \mathbb{R}_{\eta},$$t\in(-\infty, 1)$put
$u(x, t)= \frac{x}{\sqrt{\prime_{\mathrm{t}}’(1-t)+|x|^{2}}}$.
Then $u$ as $r\cdot eal$ analytic on $\mathbb{R}_{n}\cross(-\infty, 1)$ and solves a $quas\dot{\iota}linear$ parabolic system
(2. 1) $u_{t}^{i}=D_{\alpha}(A_{ij}^{0\beta}(u)D_{\beta}u^{j},$$i=1,$
$\ldots,$$n$
with real analytic $coeffic\dot{\iota}e\uparrow-\iota tsA_{\mathrm{i}}^{0\beta},(u)$ on a neighbourhood
of
$\overline{B(0,1)}$.The coefficients are given by the $\mathrm{f}\mathrm{o}\mathrm{l}\cdot \mathrm{m}\mathrm{u}\mathrm{l}\mathrm{a}$
with
(2.2)
For $\theta\in(0, n-9arrow-\frac{t\mathrm{i}}{\underline{9}(n-1)})$the expression under thesquareroot in the denominator
is positive on $\overline{B(0_{\backslash }1)}$. The coefficients are then real analytic on the same set and
satisfy ellipticity condition
$\lambda_{0}|\xi|^{2}\leq\langle A(u)\xi.‘\xi\rangle\leq\lambda_{1}|\xi|^{2}$
$\backslash \mathrm{v}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}$
$\lambda_{0}=\theta,$ $\lambda_{1}=\frac{(n-1)^{2}-\theta(n-2-\frac{\kappa}{2(n-1)})}{\uparrow\tau-2-\frac{\kappa}{2(n-1)}-\theta}$.
Denote $r>1$ radius of a ball on which the denominator in (2.2) is positive
and let $\Phi\in C^{\infty}(\mathbb{R})$ be any function such that $0\leq\Phi\leq 1$ on $\mathbb{R},$$\Phi(s)=0$ for
$s\geq r,$ $\Phi(s)=1$ for $|s|< \frac{1+r}{2}$. Put
$A_{?1}^{0\beta}.(u)=\sim\{$
$\theta\delta_{ij}\delta\alpha\beta+\Phi(|u|^{2})A_{i\mathrm{o}}(u)A_{j\beta}(u)$ $\mathrm{f}\mathrm{o}\mathrm{r}|u|<r$
$\theta\delta_{ij}\delta\alpha\beta$ otherwise.
Then $A_{ij}^{0\beta}\sim$ are infinitely diflerentiable on $\mathbb{R}^{n}$, system with these coefficients
sat-isfies ellipticity condition with the same $\lambda_{0},$$\lambda_{1}$ and admits the same solution $u$.
Inserting values of$u$in $\tilde{A}_{ij}^{\alpha\beta}(u)$ we see that $u$ solves also a linear para,bolic system
with coefficients $\backslash \mathrm{v}\mathrm{h}\mathrm{i}\mathrm{c}\mathrm{h}$ are bounded. real analytic on $(\mathbb{R}^{n}\backslash \{0\})\cross(-\infty., 1)$ and can beextended by different ways on $\mathbb{R}^{n+1}$ as bounded and$\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{u}\mathrm{l}\cdot \mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}$ functions. Thus
the discontinuity ofa solution can disappear for $t>1$ or can survive for any time
interval.
By analogous procedure as in the first quasilinear case we obtain examples of
$L_{\infty}$ blow up for linear parabolic system.
Theorenl 2.2 $(\mathrm{s}\mathrm{e}\mathrm{e}[9])Letn\geq 3,$ $\gamma^{J}\in(0,$$\min(\sqrt{\uparrow\tau-1}-1, \frac{1}{2}),$$\kappa\in(0,2(n-1)(n-$
$2-2\gamma)))$.
for
$x\in \mathbb{R}_{n},$$t\in(-\infty, 1)$ put$u(x, t)= \frac{x}{|x|\gamma\sqrt{\kappa^{\wedge}(1-t)+|x|^{2}}}$.
Then $u$ is H\"older
$\cdot$
continuous on $\mathbb{R}_{n}\cross(-\infty, 1)$ and it is a weak solution
of
alinca? ])$a\uparrow\cdot abolics\mathrm{t}/\backslash stem$,
$v_{t}^{i}=D_{\mathrm{o}}(A_{ij}^{0\beta}(x, t)D_{\beta}u^{j},$ $i=1,$
$\ldots,$$n$
with $A_{ij}^{0\beta}\in L_{\infty}(\mathbb{R}^{n}\cross(-\infty., 1)satisfy\dot{\iota}ng$
untform
$ell\dot{\iota}pticity$ condition$\exists/\backslash _{0},$$\lambda_{1}\in(0, \infty)\forall\xi\in \mathbb{R}^{n^{2}}\forall x\in \mathbb{R}^{n}\forall t\in(-\infty, 1)$
Nevertheless,
$\lim_{tarrow 1-}||u(., t)||_{L_{\infty}(\mathbb{R}^{\mathrm{n}})}=\infty$.
The question of how $\mathrm{u}j$
large” the sets ofsingular points of a solutionto nonsmooth
parabolic $\mathrm{s}]^{\gamma}\mathrm{S}\mathrm{t}\mathrm{e}\mathrm{n}1$ can be is not completely
$\mathrm{s}\mathrm{o}1)^{\gamma}$ed. Partial $\mathrm{a},1\mathrm{l}\mathrm{S}\backslash \backslash \cdot \mathrm{e}\mathrm{l}$. to this question is given in [10].
3. Regularity
In this part we concentrate nlainly on two points, i.e. the existence of time
derivative and H\"older continuity of solutions.
Ne\v{c}as and
\v{S}ver\’ak
in [13] considered a nonlinear system(3.1) $u_{t}^{i}=D_{\mathrm{o}}(a_{i}^{\mathrm{o}}(\nabla u),$$i=1,$
$\ldots.,$
$\mathbb{J}I$ on $Q$.
with continuously differentiable coefficients $a_{i}^{\mathrm{o}}$ and $\mathrm{p}\mathrm{r}\mathrm{o}1^{r}\mathrm{e}\mathrm{d}$ that $u\in C_{loc}^{1,\mu}(Q)$ if
$m=2$ and $u\in C_{loc}^{0,\mu}(Q)$ if$m\leq 4$.
In 1995, Gr\"oger and Rehberg (see [16]) considered the systeln
(3.2) $u_{t}^{i}-D_{\mathrm{o}}(\mathrm{A}_{ij}^{0\beta}(x, t, u)D_{\beta}u^{i})=f^{i}.,\dot{\iota}=1,$
$\ldots,$$M$ on $Q$.
They solved initial andboundaryvalue problem for this system for sufficiently small
time $T$ in a space, which for $m=2$ is elnbedded in $C^{0,\mathit{1}^{l}}(Q)$. $\mathrm{C}^{1}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}A_{ij}^{\alpha\beta}$ are
supposed to be uniformly continuous in $t$ and bounded and measurable in space
variables. Under these assumptions the time $\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{i}1^{\gamma}\mathrm{a}\mathrm{t}\mathrm{i}\lambda^{\gamma}\mathrm{e}$belongsto $L_{q}((0, T);W^{-1,p})$
for a $p>2$ .
In 1997 Naumann, $\backslash \lambda’\mathrm{o}1\mathrm{f}\mathrm{f}$ and Wolf in [17] proved that if$\backslash \backslash r\mathrm{e}$ suppose coefficients
in (3.2) to be $\mu$-H\"older continuous with $\mu,$ $> \frac{1}{2}$ (
$\mathrm{s}\mathrm{u}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{l}\backslash .’$near to 1) and $?7\mathit{1}=2$ then $u\in C^{0,\mu}(Q)$ and there is a$p>2$ such that $u_{t}\in L_{p}((0_{j}T)\cdot., L_{2}(\Omega))$.
In 1996 we proved in [15] that if coefficients $a_{i}^{\alpha}$ are Lipschitz continuous in $t$ and
bounded and measurable in space variables and $m=2_{\wedge}$. then all solutions of
(3.3) $u_{t}^{i}-D_{\mathrm{o}}(a_{i}^{\mathrm{o}}(x_{}.t, u, \nabla u)=f^{\tau},$$i=1,$
$\ldots,$$M$ on
$Q$.
are H\"older continuous and there is a $p>2$ such that $u_{t}\in L_{\infty}((\mathrm{O}, T);L_{p}(\Omega))$.
If we drop the assumption $\uparrow n=2$ we obtain a result which is a slight
generaliza-tion of [18]:
Theorelll 3.1(see [14]) Lit $f^{i}\in L_{2}(Q)_{i}coeffi,cie\uparrow \mathit{1},tso_{\dot{l}}^{\mathrm{o}}(x.t.u.q_{J}\prime\prime)$ be
Carathe’o-dory
functions
continuously $differcntiabl,ei?\mathit{1},$ $u,p$ andsatisf.
$|/O?’,$ $t_{J}he\dot{\iota}\uparrow\cdot domai\uparrow?,s$(i) growth conditions.$\cdot$
$|a_{i}^{\mathrm{o}}(x, t, u,p)|\leq l|l(1+|u|+|p|)j$
(ii) ellipticity $co?\tau d\dot{\iota}tion.\cdot$
$\frac{\partial \mathit{0}_{i}^{\mathrm{o}}}{\partial p_{\beta}^{j}}(x, t, u_{J}.p)\xi_{\mathrm{o}}^{i}\xi_{\beta}^{7}\geq/\backslash _{0}|\xi|^{2}’$.
$(ii\dot{\iota})$ H\"older $co?7,t\iota uity$ in, $t$:
for
$\gamma\in(\underline{\frac{1}{9}}’.1]$$|a_{i}^{\mathrm{o}}(x.t_{\mathrm{l}}, u_{J}.p)-a_{i}^{\zeta)}(x_{e}.t_{2}, u.p)|\leq L|t_{1}-t_{2}|^{\gamma}(1+|u|+|p|)$ .
$The\uparrow?$,
for
$ever\cdot y$ weak solution $u$ to (3.3) $u_{t}$ belongs to $L_{\underline{9}},\iota_{o\mathrm{c}}(Q)$.
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