188
MONOTONICITY ESTIMATE AND GLOBAL EXISTENCEFOR THE P‐HARMONIC FLOW

( p 調和写像流に対する単調性評価と大域存在)

Masashi Misawa (三沢 正史) (*)

mmisawa@kumamoto‐u ac jp

Department of Mathematics, Faculty of Sciences, Kumamoto University,
2‐39‐1 Kurokami, Kumamoto‐shi, Kumamoto 860‐8555, Japan
1

Introduction

Let \mathcal{N} be a n ‐dimensional smooth compact Riemannian manifold without boundary

and isometrically embedded in \Gamma t^{l}(l>n) . For a map u from B_{\infty}^{m} :=(0, \infty)x\mathbb{R}^{m} to IR^{l}we consider the p ‐harmonic flow

\begin{ar ay}{l} \partial_{t}u-div(|Du|^{p-2}Du)+|Du|^{p-2}A(u) (Du, Du)=0 u\in \mathcal{N} \end{ar ay}
l , is a vector‐valued function, defined forwhere p\geq 2 , u(t, x)=(u^{i}(t, x)) , i=1,
m, Du=(D_{\alpha}u^{i}) is the(t, x)\in B_{\infty}^{m} with values into \mathbb{R}^{l}. D_{\alpha}=\partial/\partial x_{\alpha}, \alpha=1,

spatial gradient of a map u, |Du|^{2}= \sum_{\alpha=1}^{m}\sum_{i=1}^{l}(D_{\alpha}u^{i})^{2} and \partial_{t}u is the derivative ontime t . The second fundamental form A(u) (Du, Du) of \mathcal{N}\subset JR^{l} is on the orthogonal
complement of the tangent space T_{u}\mathcal{N} (if necessary, the manifold \mathcal{N} is assumed to beorientable). Since u=u(t, x), (t, x)\in B_{\infty}^{m} , moves on the manifold \mathcal{N}, \partial_{t}u\in T_{u}\mathcal{N} , andthus, \partial_{t}u\cdot A(u)(Du, Du)=0 and, by multiplying the equation by \partial_{t}u and the divergence
theorem| \partial_{t}u|^{2}-div(|Du|^{p-2}Du\cdot\partial_{t}u)+\partial_{t}\frac{1} {p}|Du|^{p}=0,
E(u):= \int_{1R^{m} \frac{1}{p}|Du|^{p}dx, \frac{d}{dt}E(u(t) =- \Vert\partial_{t}u(t)\Vert_{2}^{2}
and thus, E(u(t))\searrow 0 and u(t) may converge to a constant map ab^{\backslash }t\nearrow\infty.

Theorem 1 (A global existence and reguıarity for the p ‐harmonic flow) Let p>2 andlet u_{0} be a smooth map defined on B^{M} with values to \mathcal{N}_{i} satisfying E(u_{0})<\infty . Then,
there exists a global weak solution u of the Cauchy problem for the p ‐harmonic flow withinitial data u_{0} , satisfying the energy inequality\Vert\partial_{t}u\Vert_{L^{2}(1R_{\infty}^{m})}^{2}+\sup_{0<t<\infty}E(u(t) \leq E(u_{0}) .
Moreover, the solution u is partial regular in the following sense: For any positive number\gamma_{0},2<\gamma_{0}<p , there exists a relatively closed set S in B_{\infty}^{m} such that u and its gradient Du
are locally in time‐space continuous in the complement B_{\infty}^{m}\backslash S , and the size of \mathcal{S} is also
estimated by the Hausdorff measure : The set S is of at most locally zer\cdot om ‐dimensional
Hausdorff measure with respect to the time‐space metric |t|^{1/\gamma_{0}}+|x| , and, furthermor e , forany positive time \tau<\infty , the (m-\gamma_{0}) ‐dimensional Hausdorff measure of \{\tau\}\cross S withr.espect to the usual Euclidean metric is locally zero.
Remark. The exponent \gamma_{0} can be as close to p as possible.

In this note we report on the global existence of a partial regular weak solution of the
Ca uchy problem for p- harmonic flow. We use the so‐called penalty approximating equa‐tion for the p- harmonic flow, and devise new monotonicity type formulas of a local scaled(*)

The work is partially supported by JSPS KAKENHI Grant number ı5K04962.

189
energy and establish a uniform local regularity estimate for regular solutions of thoseequation. The regularity criterion obtained is alrnost optimal, comparing with that of thecorresponding stationary case.

2

Penalty approximation

In this section we explain the approximation scheme for the p ‐harmonic flow. We willapproximate the p ‐harmonic flow by the solutions of the gradient flow for the so‐called

penalized functional, introduced in [3] for the harmonic flow case p=2.
Since the manifold \mathcal{N} is smooth and compact, there exists a tubular neighborhood\mathcal{O}_{2\delta_{N} with width 2\delta_{\mathcal{N} of \mathcal{N} in B^{\iota} such that any point u\in \mathcal{O}_{2\delta_{N} has a unique nearest
point \pi \mathcal{N}(u)\in \mathcal{N} satisfying dist (u, \mathcal{N})=|u-\pi \mathcal{N}(u)| for the Euclidean distance |\cdot| in B^{l},where the projection \pi \mathcal{N} : \mathcal{O}_{2\delta_{N} ar ow \mathcal{N} is smooth, since the manifold \mathcal{N} is smooth. The
distance function dist (u, \mathcal{N}) is Lipschitz continuous on u\in \mathcal{O}_{2\delta_{N} .

Let \chi be a smooth, non‐decreasing real‐valued function defined on [0, \infty ) such that\chi(s)=s for s\leq(\delta_{\mathcal{N} )^{2} and \chi(s)=2(\delta_{N})^{2} for s\geq 4(\delta_{\mathcal{N} )^{2} . Then, the function\chi(dist^{2}(u, \mathcal{N}))is smooth on u\in B^{l} (for the proof we refer to the recent study of the squared distancefunction to manifold, due to Ambrosio et al. [1, Theorem 2.1]). Its gradient at u\in \mathcal{O}_{2\delta_{N}is computed as

D_{u}\chi (dist2 (u, \mathcal{N}) ) =2\chi' (dist2 (u, \mathcal{N}) ) dist(u, \mathcal{N})D_{u}dist(u, \mathcal{N})D_{u} dist (鋭, \mathcal{N} );= \frac{u-\pi \mathcal{N}(u)}{|u-\pi \mathcal{N}(u)|}parallel to the vector field u-\pi \mathcal{N}(u) and orthogonal to T_{\pi(u)}\mathcal{N}N^{\cdot} We also have that, for
any u\in \mathcal{N} and any tangent vector \tau\in T_{u}\mathcal{N},|\tau^{i}\tau^{j}D_{u^{1} \cdot D_{u^{j.dist}(u, \mathcal{N})|\leq C(\mathcal{N})|\tau|^{2}(See [1, Theorem 2.2]).
For positive parameters 1\leq K\nearrow\infty and 1>\epsilon\searrow 0 , we consider the Cauchy problem
in B_{\infty}^{m} with initial data u_{0} for the gradient flow, called the penalized equation,(2.1)\{\begin{ar ay}{l} \partial_{t}u-\triangle_{p,\epsilon}u+C_{0}K\chi' (dist2(u, \mathcal{N}) dist (u, \mathcal{N})D_{u}dist(u, \mathcal{N})=0 u(0)=u_{0} \end{ar ay}

associated with the penalized functional, defined by(2.2)

F_{K,\epsilon}(u):=E_{\epsilon}(u)+C_{0} \frac{K}{2}\int_{1R^{m} . (dist2 (u, \mathcal{N}) )
\chi

dx,

where the positive constant C_{0} will be stipulated later, depending only on p, m and \mathcal{N} (See
Lemma 8). The partial differential operator \triangle_{p,\epsilon} and its corresponding energy, called theregularized p ‐Laplace operator and the regularized p ‐energy, respectively, are defined as(2.3)\triangle_{p,\epsilon}u:=div( \epsilon+|Du|^{2})^{\frac{p-2}{2} Du);

E_{\epsilon}(u):= \int_{\Gamma_{\iota}^{m} \frac{1}{p}(\epsilon+|Du|^{2})^{\frac{p}{2} dx.

We have the global existence for (2.1), by the usual Galerkin method and monotonicityof the p ‐Laplace operator (refer to [2]). The regularity of solutions are obtained from
Hölder regularity estimates for the evolutionary p ‐Laplace operator, with a boundednessof the derivative of the penalty term, the last term in (2.1).

190
Lemma 2 (Existence for the penalty approximation)
Let p>2 and let u_{0} be a smooth
map def_{07}\iota ed on IR^{m} with values to \mathcal{N} , satisfying E(u_{0})<\infty . For each positive numbersKand \epsilon , there exists a weak solution u=u_{K,\epsilon} of the Cauchy problem for the penalized

equation (2.1) such that u=u_{K,\epsilon} satisfies the energy inequality\Vert\partial_{t}u\Vert_{L^{2}(JR_{\infty\ovalbox{\t \smal REJECT} ^{m}) ^{2} +\sup_{0<t<\infty}F_{K,\epsilon}(u)\leq E_{\epsilon}(u_{0})(2.4)

and, that u , Du, \partial_{t}u and D^{2}u are locally (Hölder) continuous on time and space (withsome Hölder exponent) in IR_{\infty}^{m} and u satisfies the penalized equation everywhere in \mathbb{R}_{\infty}^{m}.
We will call a solution having the regularity properties as in Lemma 2, a regular solution.

3

Uniform regularity estimate
In this section we show some regularity estimates for solutions u=u_{K,\epsilon} of the penalized

equations (2.1).

Lemma 3 (Energy inequality)
Let u_{0} be a smooth map on B^{m} with values to \mathcal{N},satisfying E(u_{0})<\infty , and u=u_{K,\epsilon} be a regular solution of (2.1). Then, (2.4) holds.
Proof. The energy inequality (2.4) is shown to be valid in the proof of Lemma 2. However,as a priori estimates for regular solutions of (2.1), we naturally multiply (2.1) by \partial_{t}u and\squareintegrate by parts on space variable in B_{T}^{m} for any T>0.

Lemma 4 (Boundedness)

Let u=u_{K,\epsilon} be a regular solution of (2.1). Then it holds

that \sup_{IR_{\infty}^{m}}|u|\leq H , where the positive number Hi_{b}. so large that B(H)\supset \mathcal{O}_{2\delta_{N} (\mathcal{N}) in
B^{l} , where B(H)=B(H, 0)i_{6}. a ball in B^{l} of radius H with center of origin 0.

Proof. We multiply (2.1) by u(|u|^{2}-H^{2})_{+} and integrate in B_{\infty}^{m} , where (f)_{+} is the positive
part of a function f . Since the support of \chi' is in \mathcal{O}_{2\delta_{N} (\mathcal{N})\subset B(H), \chi'(dist^{2}(u, \mathcal{N})) iszero in IR^{l}\backslash B(H) . Also u_{0}\in \mathcal{N}\subset B(H) . Hence, we have\frac{1}{4}\int_{1R^{m} ㈹ |^{2}-H^{2})_{+}dx㌦. ( \epsilon+|Du|^{2})^{\frac{p-2}{2} (\frac{1}{2}|D(|u|^{2}-H^{2})_{+}|_{q}^{2}+ |Du|^{2}(|u|^{2}-H^{2})_{+})dz=0 ;㌔ (|u(t)|^{2}-H^{2})_{+}^{2}dx\leq 0

and thus, |u(t)|\leq H in 1R^{m} and any t\geq 0 .

口The partial regularity is based on the so‐called small energy regularity estimate (refer

to [9, Theorems 5.1, 0‐.3, 5.4 ; their proofs, pp. 491‐494]). The small energy regularity
estimate for the p ‐harmonic flow in the case p>2 has been recently established in [7, 8].
Our main assertion here is that the small energy regularity estimate holds uniformly forsolutions of the penalized equations.
Let us denote the penalized energy density for a map u by

(3.1)

e_{K},.(u):= \frac{1}{p}(\epsilon+|Du|^{2})^{\frac{p}{2} +\frac{K}{2}\chi (dist2 (u, \mathcal{N})

191
191
Theorem 5 (Small energy regularity estimate)

Let p>2 . Let \lambda_{0}, B_{0} and a_{0} be

positive numbers satisfying the conditions\frac{6p-4}{p+2}<\lambda_{0}=B_{0}<p ; \frac{\lambda_{0}-2}{p-2}<a_{0}\leq 1.

(3.2)

Let u=u_{K,\epsilon} be a regular solution of (2.1) on IR_{T}=(0, T)\cross B^{m} for a positive T<\infty,\cdot

satisfying the energy bound(3.3)\Vert\partial_{t}u\Vert_{L^{2}(1R_{T}^{\mathfrak{m} \cdot)}^{2}+\sup_{(<t<T}F_ {K,\epsilon}(u)\leq C

for a positive number C depending only on m, p and \mathcal{N} . Then, there exists a small positive
numeber R_{0}<1 , depending only on m, \mathcal{N}, p, B_{0} and a_{0} , and the following holds true :Let \gamma_{0} be any positive number satisfying

2< \gamma_{0}<\frac{B_{0}(p+2)-4p}{p-2}.
If, for some small positive R< \min\{R_{0}, T^{1/\lambda_{0}}\},

1 if 1\sup_{r\sear ow 0}r^{\gamma 0-m}\int_{\{t=T-R^{\lambda_{0} \}\cros B(7_{:^{0)} }.e_{K,\epsilon}(u(t, x) dx\leq 1(3.4)
then, ther e hold6(3.5)\sup e_{K,\epsilon}(u(t, x))\leq CR^{-a0p},(T-(R/4)^{\lambda_{0_{:}}}T)\cross B(R/4_{:}0)

where the positive constant C depends only on \gamma_{0}, \lambda_{0}, B_{0}, a_{0}, p,m

and \mathcal{N}.

Remark. The positive number \gamma_{0} can be as close to p as possible, if B_{0} is close to p.

The novelty here is a new monotonicity type estimate of a localized scaled energy, which
may be of its own interest. Let us define our localized scaled energy in the following way:
Let T\geq 0 and X\in B^{m} be given, and (t_{0}, x_{0}) in the parabolic like envelope\{(t, x)\in IR_{\infty}^{m} : t-T\geq|x-X|^{\lambda_{0}}\};\lambda_{0}>2.

Hereafter the notation of double sign correspondence is used. The localized scaled energy
is defined by

E_{\pm}(r)= \frac{1}{\Lambda^{p} \int_{\{t=t_{0}\pm\Lambda^{2-p}r^{2}\}\cros 1R^{0\eta} \overline{e}_{K,\epsilon}(u(t, x) \mathcal{B}_{\pm}(t_{(j}, x_{0};t, x)C^{q}(t, x)dx ;(3.6)\overline{e}_{K,\epsilon}(u):= \frac{1}{p}(\epsilon+|Du|^{2})^{\frac{p}{2} +C_{0}\frac{K}{2}\chi (dist2 (u, \mathcal{N})and \Lambda=\Lambda(r) is a function of a scale radius r , defined as

A=A(r)=r^{\frac{B0-2}{2-p}}(3.7)
for anyr>0 .

;B_{0}>\frac{6p-4}{p+2}

The forward or backward in time Barenblatt like function, denoted by \mathcal{B}+and \mathcal{B}_{-} , respectively, are defined by(3.8)\mathcal{B}_{\pm}(t_{0},x_{0};t,x)=\frac{1}(\mpt_{0}\pmt)^{\frac{m}{B_{(} }(1-\frac{|x- _{0}|{2(\mpt_{()}\pmt)^{\frac{1}I3_{0} )^{p-\imath} {\imath}\ovalbox{\t smal REJ CT})_{十}^{\frac{f^-1}{p-2},\mp t<\mp t_{0}.

192
The localized function \mathcal{C} is defined and used as(3.9)

C(t, x):=((t-T)^{1/\lambda_{0}}-|x-X|)_{十}

;

q>2.

We call E_{+}(r) and E_{-}(r) the forward and backward localized scaled p ‐energy, respec‐tively.

Our main ingredient is the following monotonicity type estimate of a scaled energy.

Lemma 6 (Monotonicity estimate for the backward localized scaled p ‐energy)Letp>2 and q>2 . Suppose that t_{0}-T\leq 1 . For any regular solution to (2.1) the following
estimate holds for all positive numbers

r, \rho,rnin\{1, (t_{0}-T)/2\},(3.10)

r^{B_{0}}=\Lambda(r)^{2-p}r^{2}<\rho^{B_{0}}=\Lambda(\rho)^{2-p}\rho^{2}\leq

E_{-}(\tau) \leq E_{-}(\rho)+C(\rho^{\mu}-r^{\mu})+C\int_{ 0}-\rho^{B_{0} ^{t_0}-\tau^{B_{0} \Vert\mathcal{C} ^{\overline{q}(t)\overline{ }_{K},.(ut)^{2}\Vert_{L^{\infty}(B(t_{0}-t) ^{1/B}0_{:}x_{0}) dt,

where \overline{q}=\min\{q-2, q(p-1)/p\}, B_{0} as in (3. 7), and the positive exponent \mu depends
only on \mathcal{N}, m, p and B_{0} , and the positive constant C depends only on the same ones as\mu and q.

Lemma 7 (Monotonicity estimate for the forward localized scaled p ‐energy)
Let p>2
and q>2 . Suppose that t_{0}-T\leq 1 . For any regular solution to (2.1) the following estimate
holds for all positive numbe\gamma\cdot s\cdot r, \rho, r^{B_{0}}=\Lambda(r)^{2-p}r^{2}<\rho^{B_{0}}=\Lambda(\rho)^{2-p}\rho^{2}\leq 1

E_{+}(\rho) \leq (1+r^{-c0^{B_{0}}})E_{+}(r)+C(\rho^{\mu}-r^{\mu})(3.11)+C\int_{ 0+r^{B}0 ^{t_0}+\rho^{B_{0} \Vert\mathcal{C}^{\overline{q}(t) \overline{ }_{K.\epsilon\propto(B(t-_{0})^{1/B_{0_{:} x_{0}) (ut)^{2} \Vert_{L}dt,

where c_{0} is a positive number satisfying c_{0}>2(p-B_{0})/B_{0}(p-2) , which can be as close
to 2 (p-B_{0})/B_{0}(p-2) as possible, \overline{q}=\min\{q-2, q(p-1)/p\}, B_{0} as in (3. 7)., and thepositive constants \mu and C have the same dependence as those in Lemma 6.

Remark. In Lemma 7, tl_{1}e positive number c_{0} can be as close to 0 as possible, if B_{0} isclose to p.

We need the so‐called Bochner type estimate for the penalized energy density. Herethe positive constant C_{0} in (2.1) is appropriately chosen.

Lemma 8 (Bochner type estimate)
Let p>2 and u=u_{K,\epsilon} be a r.egular solution to(2.1). For brevity, put e(u)=e_{K,\epsilon}(u) . Then, it holds in JR_{\infty}^{m} that
\cdot

\partial_{t}e(u)-\sum_{\alpha,\beta=1}^{\gam a\gam at}D_{\alpha}( \epsilon+ |Du|^{2})^{\frac{p-2}{2} \mathcal{A}^{\alpha\beta}D_{\beta}e(u)(3.12)+C_{1}(\epsilon+|Du|^{2})^{\frac{p-2}{2} |D^{2}u|^{2}+C_{2}|2^{-1}KD_{u}\chi (dist2 (u, \mathcal{N}) ) |^{2}\leq C_{3}(1+e(u)^{\frac{2}{p} )e(u)^{2(1-\frac{1}{p})},where\mathcal{A}^{\alpha\beta}:=\delta^{\alpha\beta}+(p-2)\frac{D_{\alpha}u\cdot D_ {\beta}u}{\epsilon+|Du|^{2} ,the positive constants C_{i}(i=1,2,3) depend on m, p and \mathcal{N}.

193
4

Passing to the limit

In this section we present the proof of Theorem 1, based on Theorem 5.
Let \{\epsilon_{k}\} and \{K_{k}\} be sequences such that \epsilon_{k}\searrow 0 and K_{k}\nearrow\infty as karrow\infty . Let uK_{k} , \epsilon_{k},k=1,2 ,
be a sequence of solutions of the Cauchy problem with initial data u_{0} for the
penalized equations (2.1) with approximating numbers \epsilon=\epsilon_{k} and K=K_{k} , obtained in
Lemma 2. Hereafter we put u_{k}=u_{K_{k},\epsilon_{k}}e_{k}(u_{k})=e_{K_{k},\epsilon_{k}}(u_{K_{k}, \epsilon_{k}}) , for brevity.

By the energy inequality (2.4), there exist a subsequence of \{u_{k}\} , denoted by the same

notation, and the limit map u such that, as karrow\infty,

(4.1)
(4.2)
(4.3)

(4.4)
(4.5)

*inL^{\infty}(0, \infty;W^{1_{:}p}(B^{m}, IR^{l})) ,\partial_{t}u_{k}arrow\partial_{t}u weakly in L^{2}(\Gamma t_{\infty}^{m}, IR^{l}) ,Du_{k}arrow Du weakly in L_{1oc}^{p}(\Gamma t_{\infty}^{m}, IR^{ml}) ,\chi(dist^{2}(u_{k}, \mathcal{N}))arrow 0 strongly in L_{1oc}^{2}(\Gamma t_{\infty}^{m}, B^{l}) ,u_{k}arrow ustrongly in L_{1oc}^{q}(B_{\infty}^{m}, B^{l}) for aIly q 1 \leq q<\frac{mp}{(m-p)_{+}},
u_{k}arrow u

weakly,where the strong convergence in (4.5) follows from (4.1) and (4.2) (see [2, Lemma 1.4, p.
28]). Thus , furthermore, for a subsequence \{u_{k}\} denoted by the same notation,(4.6)

u_{k}arrow u ,dist (u_{k}, \mathcal{N})arrow 0

almost everywhere in B_{\infty}^{m}.

We demonstrate that the limit map u is a partial regular weak solution of the p ‐harmonic

flow, as in the statement of Theorem 1. The proof is divided to several steps and proceeded.
Size estimate of the singular set
Let R_{0} be a sufficient small positive number,determined in Theorem 5. For \tau, 0<\tau<\infty , and R, 0<R< \min\{R_{0}, \tau^{1/\lambda_{0}}\} , we put
two subsets in B^{m} as(4.7)

S( \tau, R):=\{x_{0}\in \mathb {R}^{7n}:\lim_{kar ow}\sup_{\infty} (\lim_{r\sear ow()}\sup r^{\gamma 0-m}\int_{\{t=\tau-R^{\lambda}o\}\cros B(r, xo)}e_{k}(u_{k}(t, x) dx)\geq 1\}
T(\tau,R):=\bigcap_{l=1}^{\infty}\bigcup_{k=\prime}^{\infty}. \{x_{0}\in 1R^{\tau n}:\lim_{r\sear ow 0}\sup r^{\gamma_{0-\prime}n}\int_{\{t= \tau-R^{\lambda_{0} \}\cros B(r_{:}x_{0}) e_{k}(u_{k}(t, x) dx>1/2\}.

;

From the definition of limit supremum on k and (4.7), we see that, for every \tau, 0<\tau<\infty,
andR, 0<R< \min\{R_{0}, \tau^{1/\lambda_{0}}\},(4.8)\mathcal{S}(\tau, R)\subset T(\tau, R) .

Here we have the estimation of size (see [5, Theorem 2.2 ; its proof, pp. 101‐ı03] for the
proof) : It holds that, for every \tau, 0<\tau<\infty , and R, 0<R< \min\{R_{0}, \tau^{1/\lambda_{0}}\},\mathcal{H}^{m-\gamma_{0}}(T(\tau, R))=0

and so, by (4.8),\mathcal{H}^{m-\gamma_{0}}(\mathcal{S}(\tau, R))=0;\mathcal{H}^{7n-\gamma_{0} (\bigcap_{7^{1/\lambda_{0} 0<R<\min\{R_{0},\} \mathcal{S}(\tau, R) =0.

194
Let us define the singular bet as

(4.9)
where

\mathcal{S}=\bigotimes_{0<T<\infty}\bigcap_{0 <R<\min\{R_{0},\tau^{1/\lambda} \} .\mathcal{S}(\tau, R),means the direct product of sets on positive time \tau<\infty . Then, for any\otimes0<\tau<\infty

positive T<\infty and any open set K compactly contained in ] R^{M} , letting K_{T}=(0, T)\cross K,
with respect to the time‐space metric |t|^{1/\gamma 0}+|x|,\mathcal{H}^{m}(\mathcal{S}\cap K_{T})=\int_{0}^{T}\mathcal{H}^{m-\gam a 0} (\bigcap_{0<R.<R0}\mathcal{S}(\tau, R)\cap K)d\tau=0.
Regularity of the limit map
We now show the regula1ity of limit map u in theS
complement of . Let (t_{0}, x_{0}) be in the complement of S . Thus, there exist a positive
R< \min\{R_{0}, (t_{0})^{1/\lambda_{0}}\} and an infinite family \{u_{k}\} of regular solutions such that

\lim_{r\sear ow()}\sup r^{\gamma 0-M}\int_{\{t=t_{0}-R^{\lambda_{0} \}\cros B (r_{:}x_{0}) e_{k\wedge}(u_{k}(t, x) dx<

ı.

Then we can apply Theorem 5 for each u_{k} above to obtain(4.10)\sup e(u_{k})\leq CR^{-a_{0}p},(t_{0}-(R/4)^{\lambda}0_{:}t_{0})\cross B(R./4_{:}x_{0})where the positive constant C depends only on \lambda_{0}, B_{0}, m, p and \mathcal{N}.
Now we will show the uniform continuity of \{u_{k}\} in Q :=(t_{0}-(R/8)^{\lambda_{0} , t_{0})\crossB(R/8, x_{0}) . For this purpose we will have a local L^{2} estimate of derivative of the penalty
term. For any smooth function \phi of compact support in Q , we multiply the Bochner typeestimate (3.12) by \phi^{2} and integrate by parts in Q to have, letting K=K_{k}, u=u_{k} ande(u)=e_{k}(u_{k}),(4.11)\int_{Q}\phi^{2}(\frac{C_{ \imath} {2}(\epsilon+|Du|^{2})^{\frac{p-2}{2} |D^{2}u|^{2}+\frac{C_{2} {2}|\frac{K}{2}D_{u}\chi (dist2 (u, \mathcal{N})|^{2} )dz\leq\int_{Q}(\phi|\partial_{t}\phi|e(u)+|D\phi|^{2}(\frac{2p}{C_{1} e(u)+\frac {2}{C_{2} e(u)^{\frac{2}{p} )+C_{3}\phi 2(1+e(u)^{\simeq}p?)e(u)^{2(1-\frac{1} {p})})dz,

where we use the Cauchy inequality in the first inequality.
Let (t_{0}, x_{0})\subset Q be any point and r\leq R/8 be any positive number, and Q(r)=(t_{0}-r^{q}, t_{0})\cross B(r, x_{0}) with q>1 . In (4.11) we choose a smooth function \phi such that0\leq\phi\leq 1 , \phi=1 in Q(r) , \phi=0 outside Q(2r) , and |D\phi|\leq C/r and |\partial_{t}\phi|\leq C/r^{q} . Thus
we have, by (4.10),(4.12)\int_{Q(t)}(\frac{C_{1} {2}(\epsilon+|Du|^{2})^{\frac{p-2}{2} |D^{2}u|^{2}+ \frac{C_{2} {2}|\frac{K}{2}D_{u}\chi (dist2 (u, \mathcal{N})|^{2} )dz\leq C(r^{m}+r^{?n+q-2}+r^{\gamma n+q})\leq C\tau^{77\tau}.

We also need the Poincaré inequality of parabolic type (refer to [6]) : Let u=u_{k} . Thereexists a positive constant C , depending only on m and p_{i} such that, for any Q(r)\subset Q,(4.13)\Vert u-\overline{u}_{Q(r)}\Vert_{L^{2}(Q(r) }^{2}\leq C(r^{2}\Vert Du|\Vert_{L^{2}(Q(r) }^{2}+r^{-7n+q-2}\Vert(\epsilon+|Du|^{2})^{1/2}\Vert_{L^{p- 1}(Q(r) }^{2(p-{\imath})}+r^{2q}\Vert 2^{-1}KD_{u}\chi(^{2}(u, \mathcal{N}) \Vert_{L^{2}(Q(r) }^{2}) ,195
where \overline{u}_{Q(r)} is the integral mean of u in Q(r) .

Substituting (4.10) and (4.12) into (4.13), we have, for any (t_{0}, x_{0})\subset Q , any positiver\leq R/8 , and Q(r)=(t_{0}-r^{q}, t_{0})\cross B(r, x_{0}) ,\Vert u-\overline{u}_{Q(r)}\Vert_{L^{2}(Q(r))}^{2}\leq C(r^{7\Pi+q+2}+r^{m+3q- 2}+r^{\gamma n+2q})(4.14)and thus, choosing q>1 in (4.14), we obtain from Campanato’s isomorphism theorem(refer to [5]) that \{u_{k}\} is uniformly Hölder continuous in Q with exponent \min\{1, q-1, \frac{q}{2}\},
uniformly on u_{k} . Thus, we see that \{u_{k}\} is equicontinuous, and uniformly bounded in Q
by Lemma 4. Therefore, by Arzela‐Ascoli theorem we find for a subsequence denoted by
the same notation \{u_{k}\} and the limit map u that, as karrow\infty,(4.15)

uniformly in Q

u_{k}arrow uand that the limit map u is uniformly continuous in Q . From (4.10) and (4.15), we obtainthat, as karrow\infty,(4.16)\chi(dist^{2} (u_{k}, \mathcal{N}))\leq C/K_{k}arrow 0 uniformıy in Q

\Rightarrow u\in \mathcal{N}in Q

Now we will show that the limit map u satisfies the p ‐harmonic flow equation in Q.

From (4.10) and (4.11) we also see that \{(K_{k}/2)D_{u}\chi(dist^{2}(u, \mathcal{N})|_{71_{=}U}k } is bounded inL^{2}(Q, B^{l}) and thus, there exists a vector‐valued function \nu\in L^{2}(Q, B^{l}) such that, askarrow\infty,(4. 17)(K_{k}/2)D_{u}\chi (dist2 (u, \mathcal{N})|_{u=u_{A:}}arrow\nuweakıy in L^{2}(Q) .

By the continuity of u in Q the image u(Q) of Q is an open subset of \mathcal{N}. Let\mathcal{P}_{\mathcal{N}}(u(Q))
be a neighborhood of u(Q) in \mathcal{N} . Let \tau(v) be any smooth tangent vector field of \mathcal{N} on\mathcal{P}_{\mathcal{N} (u(Q) , \tau(v)\in T_{v}\mathcal{N} for any v\in \mathcal{P}_{\mathcal{N}}(u(Q)) . By (4.15), we can choose a sufficiently
large k_{0} such that, for any k\geq k_{0}, u_{k}\in \mathcal{O}_{\delta_{\Lambda^{(} } in Q , and \pi N(u_{k})\in \mathcal{P}_{\mathcal{N}}(u(Q))\subset \mathcal{N} and\tau(\pi \mathcal{N}(u_{k}))\in T_{\pi(u_{k})}\mathcal{N}N in Q , where \mathcal{O}_{\delta_{N} is a tubular neighborhood of \mathcal{N} with width \delta_{\mathcal{N} ,and \pi \mathcal{N} is the nearest point projection to \mathcal{N} from the tubular neighborhood of \mathcal{N} . Thus,
we have that

D_{u}\chi (dist2 (u, \mathcal{N}) ) |_{u=u_{k} . \tau(\pi \mathcal{N}(u_{k}))==2\chi'dist(u_{k}, \mathcal{N})D_{u}dist(u, \mathcal{N})|_{u=u_{k}} . \tau(\pi \mathcal{N}(u_{k}))
0

in

Q,

because D_{u} dist (u, \mathcal{N}) 沖 u_{k} is orthogonal to T_{\pi(u_{k}))}\mathcal{N}N for any z\in Q , and thus,(4.18)\int_{Q}\frac{K_{k} {2}D_{u}\chi (dist2 (u, \mathcal{N}) ) |_{u=u}A\wedge. \cdot\tau(\pi \mathcal{N}(u_{k}))dz=0.

By (4.15) and (4.17), we can take the limit as

karrow\infty in(4.18) to have, for any smoothtangent vector field \tau(v) of \mathcal{N} on \mathcal{P}_{N}(u(Q))\subset \mathcal{N} , as karrow\infty,(4.19)

0= \int_{Q}\frac{K_{k} {2}D_{u}\chi (dist2 (u, \mathcal{N}) ) |_{u=u_{k} . \tau(\pi N(u_{k}) dzarrow\int_{Q}\nu\cdot\tau(u)dz\Rightar ow\int_{Q}\nu\cdot\tau(u)dz=0\Leftrightarrow\nu(z)\perp T_{u(z)}\mathcal{N} for any z\in Q.

and, thus, \nu(z) is a normal vector field along u(z) for any z\in Q . In the weak form of(2.1), for any smooth map \phi with compact support in Q,\int_{Q} ( \partial_{t}u_{k}\cdot\phi+(\epsilon_{k}+|Du_{k}|^{2})^{\frac{P-2}{2} Du_{k} \cdot D\phi+\frac{K_{k\wedge} {2}D_{u}\chi(dist^{2}(u, \mathcal{N}) |_{u=u_{k} . \cdot\phi)dz=0,

196karrow\infty to find that the limit map

we pass to the limit as(4.20)

satisfiesu\int_{Q}(\partial_{t}u\cdot\phi+|Du|^{p-2}Du\cdot D\phi+\nu\cdot\phi)dz=0,

where we use the convergence in the 1st line of (4.19) and, the strong convergence of
gradients \{Du_{k}\} , obtained from (2.1) with the convergence (4.1), (4.2) and (4.17) (see [2,Theorem 2.1, pp. 31‐33]). Therefore, we have that
(4.21)\partial_{t}u-\triangle_{p}u+\nu=0

almost everywhere in Q as L^{2}(Q) ‐map.

We now observe that

(4.22)|\nu(z)|=-|Du(z)|^{p-2}Du(z)\cdot(Du(z)\cdot D_{u}\gamma(u)|_{u=u(z)})

ahnost every z\in Q.

Let \overline{z}=(\overline{t},\overline{x})\in Q be arbitrarily taken and fixed. Let \gamma(v) be a smooth unit normal
vector field of \mathcal{N} in u(Q)\subset \mathcal{N} such that \gamma(v)\in(T_{v}\mathcal{N})^{\perp}, |\gamma(v)|=1 for any v\in u(Q) and\gamma(u(\overline{z}))=\nu(\overline{z})/|\nu(\overline{z})| . We take the composite map \gamma(u) of \gamma(\cdot) and the limit map u , anduse a test function \gamma(u)\eta for any smooth real‐valued function \eta with compact support in
Q to have\int_{Q}(\partial_{t}u\cdot\gamma(u)\eta+|Du|^{p-2}Du\cdot(D\gamma(u)\eta+ \gamma(u)D\eta)+\nu\cdot\gamma(u)\eta)dz=0\int_{Q}(|Du|^{p-2}Du\cdot D\gamma(u)+\nu\cdot\gamma(u) \eta dz=0,\Rightarrow\nu\cdot\gamma(u)=-|Du|^{p-2}Du\cdot D\gamma(u)almost everywhere in Q,

where, in the 2nd line, we use that \partial_{t}u, D_{\alpha}u\in T_{u}\mathcal{N}, \alpha=1 , .
Q . The last line yields, at z=\overline{z},|\nu(\overline{z})|=\dashv Du(\overline{z})|^{p-2}Du(\overline{z})

.

;

m

, and \gamma(u)\in(T_{u}\mathcal{N})^{\perp} in(Du(\overline{z}) . D_{u}\gamma(u)|_{u=u(\overline{z})}).

Furthermore. we find that, for a positive constant C depending only on bounds of curva‐
ture of \mathcal{N},(4.23)|\nu|\leq C|Du|^{p}almost everywhere in Q.

In fact, from (4.22) we obtain| \nu(z)|\leq C\max_{v\in t\iota(Q)}|D_{v}\gamma(v)||Du(z)|^{p}for almost every z\in Q.

Finally, we have by (4.23) and (4.10) that\partial_{t}u-\triangle_{p}u=-\nu\in L^{\infty}(Q)(4.24)

almost everywhere in Q

\Rightarrow Du is locally Hölder continuous inQ,

where, for the last statement of gradient continuity, we refer to [4, Theorem 1.1, p. 245 ;Sect 4, p. 291 ; Sect. 1 ‐ (i)_{\dot{\ovalbox{\t smal REJ CT} pp. 217‐218]. The use of convergence (4.3) and (4.2) in theenergy boundedness (2.4) for u_{k} also yields(4.25)\Vert\partial_{t}u\Vert_{L^{2}(IR_{\infty}^{m})}^{2}+ \sup E(u(t))\leq E(u_{0} ) .
0<t<\infty

197
S is actually closed set in \mathcal{M}_{\infty} . For any z_{0}=(t_{0}, x_{0}) in the
Closedness of S
complement of S , we can take a positive R\leq R_{0} and an neighborhood of z_{0}, Q' :=(t_{0}-(R/4)^{\lambda_{0}}, t_{0})xB(R/4)(x_{0}) , and an infinite family \{u_{k}\} of regular solutions of (2.1),
and have the uniform boundedness in Q' of gradients as in (4.10). Thus, we have that,for any solution u_{k} , and any z'=(t', x') in Q :=(t_{0}-(R/8)^{\lambda_{0}}, t_{0})\cross B(R/8)(x_{0}) and all
small positive r<R/8,r^{\gamma_{0}-m} \int_{\{t=t'-(R/8)^{\lambda}0\}\cros B(r_{\wedge}.x')}e(u_{k} (t, x) dx\leq CR^{-pa_{0} r^{\gamma_{0}(4.26)and thus, for any z'=(t', x') in Q,\lim_{kar ow}\sup_{\infty}(\lim_{r\sear ow()}\sup_{\{t=t'-R^{\lambda} r^{\gamma 0-m}\int_{0\}\cros B(r_{:}x')}e(u_{k}(t, x) dx)=0,
which implies that Q is a subset of the complement ofcomplement of \mathcal{S} is open and thus, S is closed.

S.

Thet.efore, we see that theWeak solution of the p ‐harmonic flow
The proof is based on the size estimate of
singular set S above. A covering argument is applied for the singular set \mathcal{S} , by use of afamily of parabolic cylinders under an intrinsic scaling, depending on a size of gradient ofsolution. For the details see [8].

\squareAcknowledgments : The authour would like to express his sincere gratitude to Professor
Katsuo Matsuoka for supporting and giving him an opportunity to talk at RIMS.

References[1] L. Ambrosio, C. Mantegazza, Curvature and distance function from a manifold, J.
Geom. Anal. 8, no. 5 (1998), 723‐748, Dedicated to the memory of Fred Almgren.

[2] Y.‐M. Chen, M.‐C. Hong, N. Hungerbuhler, Heat flow of p ‐harmonic maps withvalues into spheres, Math. Z. 215 (1994) 25‐35.
[3] Y.‐M. Chen, M. Struwe, Existence and partial regularity results for the heat flow forharmonic maps, Math. Z. 201 (ı989) 83‐103.
[4] E. DiBenedetto, Degenerate Parabolic Equations ,Springer‐Verlag. xv, 387 (1993).

Universitext, New York, NY:[5] M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic
systems, Ann. of Math. Stud. 105, Princeton University Press, Princeton NJ, 1983.

[6] C. Leone, M. Misawa, A. Verde, The regularity for nonlinear parabolic systems ofp ‐Laplacian type with critical growth, J. Differential Equations 256 (2014), 2807‐
2845.

[7] M. Misawa, Regularity for the evolution of p ‐harmonic maps, J. Differential Equa‐tions 264 (20ı8), 1716‐1749.
[8] M. Misawa, Local regularity and compactness for the p ‐harmonic map heat flows,
Adv. Calc. Var. to appear (accepted at 2017 for publication).
[9] M. Struwe, On the evolution of harmonic maps in higher dimensions, J. Differential
Geometry 28, (1988) 485‐502.