数理解析研究所講究録
1197 巻 2001 年 73-83

73

Hamilton-Jacobi equations in Hilbert spaces
with applications to Navier-Stokes equations
東京都立大学大学院理学研究科数学専攻
嶋野 和史 (Kazufumi Shimano)

Department of Mathematics, Tokyo Metropolitan University

Introduction

1and norm
Let $H$ be a real Hilbert space with inner productWe consider the following stationary Hamilton-Jacobi equation$\langle\cdot, \cdot\rangle$

in$\lambda u+\langle\partial\varphi(x)+B(x), Du\rangle+F(x, Du)=0$

$(E)$

$S$

$|\cdot|$.

.

Here : $Harrow[0, \infty]$ is a proper lower semicontinuous convex funcdenotes the subdifferential of ,tion, is a positive constant,, and moreover,is a closed subset of $H$ which satisfies
is dense in S. $u(x)$ represents a real unknown function
on , and $Du$ denotes the Fr\’echet derivative in $H$ of . $F$ is a given
is a nonlinear (multivalued) operator
H.
real function on.
from $D(B)\subset H$ into $H$ such that, the existence and comparison
When $B=0$ andand Crandalltheorems for (E) were proved in
Lions [2].
We introduce the existence and comparison theorem for (E) in
the case that is an unbounded (multivalued) operator.
$\varphi$$\lambda$$\partial\varphi$$\varphi$$S\subset\overline{D(\partial\varphi)}$$S$$S\cap D(\partial\varphi)$

$S$

$u$

$S\cross$$B$$D(\partial\varphi)\subset D(B)$$S=\overline{D(\partial\varphi)}$$\mathrm{I}\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{i}[3],$$\mathrm{T}\mathrm{a}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{u}[7]$$B$

We give the assumptions on$(\mathrm{A}1)_{\nu}$$\partial\varphi,$$B$

There exist positive constants

and $F$ . Let$C_{1}$$l\text{ノ}\in(0,2]$.

and $0<k<1$ such that$\langle b_{1}, \xi_{1}\rangle\geq-k|(\partial\varphi)^{0}(x1)|^{2}-C_{1}$,74$|b_{1}|\leq C_{1}(|(\partial\varphi)^{0}(X_{1})|^{\mathcal{U}}+1)$

and. $\langle(b_{1}+\xi 1)-(b2+\xi 2), x1-x_{2}\rangle\geq-C_{1}|x_{1}-x_{2}|^{2}$for all$x_{1},$$x_{2}\in S\cap D(\partial\varphi),$$b_{2}\in B(X_{2})$(A2)$\xi_{1}\in\partial\varphi(X_{1}),$$F\in C(S\cross H)$$p,$$|x|\leq R,$,$|p-q|\leq r\}\leq 0$.

There is a constant $C_{2}>0$ such that

C2 $(|p|^{2}+1)$

for all

There is a continuous function$\omega(0)=0$ for which$\omega$$(x,p)\in S\cross H$.

: $[0, \infty)arrow[0, \infty)$ satisfying$F(y, \alpha(x-y))-F(X, \alpha(x-y))\leq\omega(\alpha|x-y|^{2}+\frac{1}{\alpha})$for all(A5),. Moreover for each $R>0$ ,$q\in B(0, R),$$|F(x,p)|\leq$(A4)$b_{1}\in B(x_{1})$.

$\lim_{r\searrow 0}\sup\{F(_{X},p)-F(x, q)|x\in S$(A3)$\xi 2\in\partial\varphi(x2),$$x,$$y\in S\cap D(\partial\varphi)$

and $\alpha>0$ .

If $\theta>0,$ $\psi\in C^{1}(S)$ , andattains a minimum value at, then there are sequences
andsuchthat the following three properties are satisfied:$\theta\varphi+\psi$$\hat{x}\in S$

(i)
(ii)

(iii)

$\{x_{n}\},$$x_{n}arrow\hat{x}$$\{\xi_{n}\}$$\{\zeta_{n}\}\subset H$

in $H$ as $narrow\infty$ .

$x_{n}\in S\cap D(\partial\varphi),$$\xi_{n}\in\partial\varphi(x_{n})$, and$\zeta_{n}\in B(x_{n})$for $n\in$ N.

The inequality$\langle\xi_{n}+\zeta_{n},p+D\psi(X_{n})+\theta\xi n\rangle+F(x_{n},p+D\psi(x_{n})+\theta\xi_{n})$$\leq$$\langle\xi_{n}+\zeta n’ p\rangle+F(_{X_{n}},p)$holds for all $p\in H$ .

75

$F(x, 0)$ is bounded on $S$ .

(A6)

Our main result in this paper is stated as follows.
Theorem 1. Assume that
hold for some $\nu\in(0,2)$ .
Then there is a unique viscosity solution $u\in BUC(S)$ of (E).
$(\mathrm{A}1)_{\mathcal{U}^{-}}(\mathrm{A}6)$

The above theorem can be applied to characterizing the valuefunctions associated with optimal control of systems governed bypartial differential equations of parabolic type. In particular, it
is important to be able to deal with nonstationary Navier-Stokes
equations in bounded domains.

2

Viscosity solutions and comparison result

Let : $Harrow[0, \infty]$ be a proper lower semicontinuous convex function.
$\varphi$

We use the standard notation:

Notation.
$\partial\varphi(X)=${for all $y\in H$ },$p\in H|\varphi(y)\geq\varphi(x)+\langle p,$ $y-x\rangle$$D(\varphi)=\{x\in H|\varphi(x)<\infty\}$,$D(\partial\varphi)=\{x\in H|\partial\varphi(x)\neq\emptyset\}$.

Then it is well known thatis a (multivalued) maximal monotone$H$
operator on , i.e., the following (i) and (ii) hold.
$\partial\varphi$(i) (Monotonicity) For$x,$ $y\in D(\partial\varphi)$and$p\in\partial\varphi(x),$$\langle p-q, x-y\rangle\geq 0$(ii) (Solvability)$R(I+\partial\varphi)=H$$q\in\partial\varphi(y)$

,

.

.

We refer the reader tofor the proof of maximal monotonicity of. By the maximal monotonicity we see that.
$\mathrm{B}\mathrm{r}\mathrm{e}\mathrm{z}\mathrm{i}\mathrm{s}[1]$$\partial\varphi$$\overline{D(\varphi)}$$\overline{D(\partial\varphi)}=$

76

denotes the minimal section ofas the unique element of least norm in$(\partial\varphi)^{0}$$\partial\varphi$, i.e.,$\partial\varphi(x)$is defined$(\partial\varphi)^{0}(x)$.

Definition 1. Let be a relatively open subset of . A function
is called a classical subsolution (resp., supersolution andsolution) of (E) in if$S$$\mathcal{O}$$u\in C^{1}(\mathcal{O})$$\mathcal{O}$$\lambda u(x)+\langle\xi+\zeta, Du(x)\rangle+F(x, Du(x))\leq 0$

for all$x\in \mathcal{O},$$\xi\in\partial\varphi(x)$(resp.,$\geq 0$and $=0$ )and $\zeta\in B(x)$ .

Before we give the difinition of a viscosity solution, we introduce
the notation. $B(x, r)$ denotes the closed ball of radius with center
wein $H$ . Let be a relatively open subset of . For
define$r$$S$$\mathcal{O}$$x$$u:\mathcal{O}arrow \mathrm{R}$,$u^{*}(x)= \lim_{r\searrow 0}\sup\{u(y)||y-x|\leq r, y\in \mathcal{O}\}$$u_{*}(x)= \lim_{r\searrow 0}\inf\{u(y)||y-x|\leq r, y\in \mathcal{O}\}$.

It is clear that $u_{*}\leq u\leq u^{*}$ and $u_{*}=-(-u)^{*}$ Also, if
bounded in , then$u$is locally$\mathcal{O}$$u_{*}\in LSC(\mathcal{O})$and$u^{*}\in USC(\mathcal{O})$.

Here we write$USC(\mathcal{O})=$

{

$f$

:

$\mathcal{O}arrow \mathrm{R}\mathrm{U}\{\pm\infty\}$

is upper semicontinuous},$LSC(\mathcal{O})=$

{

$f$

:

$\mathcal{O}arrow \mathrm{R}\mathrm{U}\{\pm\infty\}$

is lower semicontinuous}.

Definition 2. Let be a relatively open subset of . A function:
is called a viscosity subsolution (resp., viscosity superif it is locally bounded insolution) of (E) in
and if there is a
, andsuch that whenever) attains a maximum (resp., minimum) value(resp.,$S$$\mathcal{O}$$u$$\mathcal{O}arrow \mathrm{R}$$\mathcal{O}$$\mathcal{O}$$\psi\in C^{1}(\mathcal{O}),$$\theta_{0}>0$$(u+\theta\varphi)_{*}-\psi$$0<\theta<\theta_{0}$$(u-\theta\varphi)*-\psi$77

at a point$m$$\hat{x}\in \mathcal{O}$, then$\lambda(m+\psi(\hat{x})+\theta\varphi(\hat{X}))$$+ \lim_{r\searrow 0}\inf\{\langle\xi+\zeta, D\psi(x)+\theta\xi\rangle+F(x, D\psi(X)+\theta\xi)|$$x\in B(\hat{x}, r)\cap S\cap D(\partial\varphi),$ $\xi\in\partial\varphi(X),$ $\zeta\in B(x)\}\leq 0$(resp.,$\lambda(m+\psi(\hat{X})-\theta\varphi(\hat{x}))$$+ \lim_{r\searrow 0}\sup\{\langle\xi+\zeta, D\psi(x)-\theta\xi\rangle+F(x, D\psi(X)-\theta\xi)|$$x\in B(\hat{x}, r)\cap S\cap D(\partial\varphi),$ $\xi\in\partial\varphi(X),$ $\zeta\in B(x)\}\geq 0)$.

A function :is called a viscosity solution of (E) if it isa viscosity subsolution and a viscosity supersolution of (E) in$\mathcal{O}arrow \mathrm{R}$$u$$\mathrm{b}\mathrm{o}\mathrm{t}\mathrm{h}\mathcal{O}$.

A comparison result for (E) is the following.
Theorem 2. Assume that
hold. Let and be a viscosity subsolution and a viscosity supersolution of (E), respectively.
Assume that and-v are bounded above on. Then $u\leq v$
in $D(\varphi)\cap S$ . Moreover the inequality$(\mathrm{A}1)_{2^{-(\mathrm{A}}}4)$

$u$

$u$

$v$

$D(\varphi)\cap S$$\lim_{r\searrow 0}\sup\{u(X)-v(y)|x, y\in D(\varphi)\cap S, |x-y|\leq r\}\leq 0$.

See [6] for the proof of Theorem 2.

3

Existence result

In this section, we introduce the existence theorem for viscosity
solutions of (E). Before we state the existence theorem, we give thefollowing proposition which is called Perron’s method.

78

Proposition 1 (Perron’s method). Assume that
hold for some
. Let
and be a viscosity subsolution
and a viscosity supersolution of (E) in , respectively. Assume that$f\leq g$ in S.
Define,$l\text{ノ}\in(0,2)$$f$$(\mathrm{A}1)_{\mathcal{U}^{-}}(\mathrm{A}5)$

$g$

$S$

$u(x)= \sup\{v(x)|v$ is a viscosity subsolution of (E) in $S$

and $f\leq v\leq g$ in
Then

$u$

$S$

}

for

is a viscosity solution of (E) in$x\in S$

$S$

.

.

We need the following two lemmas in order to prove Proposiotion
1. See e.g., [3], [6] for details.
Lemma 1. Assume that
hold for some. Let
be a nonempty set of viscosity subsolutions of (E) in and set$u(x)= \sup\{v(x)|v\in \mathcal{F}\}$for $(x)\in S$ . Then, if is locally bounded
then is a viscosity subsolution of (E) in .
in$l\text{ノ}\in(0,2)$$(\mathrm{A}1)_{\mathcal{U}^{-}}(\mathrm{A}4)$$S$$\mathcal{F}$$u$$S_{f}$$S$$u$

See [3] for the proof of the above lemma.
Lemma 2. Assume that (A5) holds. Let
Set $u(x)=v(x)-\delta\varphi(X)$ and suppose that$\delta\geq 0$

and $v\in C^{1}(S)$ .

$\lambda u(x)+\langle\xi+\zeta, Dv(X)-\delta\xi\rangle+F(x, Dv(X)-\delta\xi)\leq 0$

holds for all
and $\zeta\in B(x)$ . Then is aviscosity subsolution of (E) in S. Similarly, if $u(x)=v(x)+\delta\varphi(x)$and the inequality$x\in S\cap D(\partial\varphi),$$\xi\in\partial\varphi(x)$$u$$\lambda u(x)+\langle\xi+\zeta, Dv(x)+\delta\xi\rangle+F(x, Dv(X)+\delta\xi)\geq 0$

holds for all
viscosity supersolution of (E) in$x\in S\cap D(\partial\varphi),$$\xi\in\partial\varphi(x)$$S$

and $\zeta\in B(x)$ . Then

$u$

is a

.

The proof of this lemma is easy and standard, and is left to thereader.

79

We obtain the following result by Theorem 2, Proposition 1 andLemma 2. See [6] for the proof for Theorem 3.
Theorem 3. Assume that
hold for some
There is a viscosity solution $u\in BUC(S)$ of $(E)$ .
$(\mathrm{A}1)_{\nu}-(\mathrm{A}6)$$l\text{ノ}\in(0,2)$.

We obtain Theorem 1 by Theorem 2 and Theorem 3.

4

An optimal control problem of systems governed by the three-dimensional Navier-Stokes
equations

Let be an arbitrary bounded domain inwith smooth boundary. In the usual way we define
and the Sobolev spacecomposed of three-dimensional vector nctions. We use the following spaces of solenoidal vector functions.
$\mathrm{R}^{3}$$\Omega$$\mathrm{L}^{2}(\Omega)$$\mathrm{H}^{m}(\Omega)$$\mathrm{f}.\mathrm{u}$$\mathrm{c}_{0,\sigma}^{\infty}(\Omega)=\{\mathrm{u}=(u^{1}, u^{2}, u^{3}) :$\mathrm{H}_{\sigma}(\Omega)=\mathrm{t}\mathrm{h}\mathrm{e}$completion of$\mathrm{H}_{\sigma}^{1}(\Omega)=\mathrm{t}\mathrm{h}\mathrm{e}$completion ofu^{i}\in C_{0}^{\infty}(\Omega)(i=1,2,3), \mathrm{d}\mathrm{i}\mathrm{v}\mathrm{u}=0\}$$\mathrm{C}_{0,\sigma}^{\infty}(\Omega)$$\mathrm{C}_{0,\sigma}^{\infty}(\Omega)$

under the$\mathrm{L}^{2}(\Omega)$–norm,

under the$\mathrm{H}^{1}(\Omega)$–norm.

We denote by $P$ the orthogonal projection operator from.
We define a proper lower semicontinuous convex function$\mathrm{L}^{2}(\Omega)$

onto$\mathrm{H}_{\sigma}(\Omega)$$\varphi(\mathrm{u})=$where$| \nabla \mathrm{u}|^{2}=\sum_{1i,,j=}^{3}|\frac{\partial u^{j}}{\partial x_{i}}|^{2}$$\varphi$

,

by

80

Then it can be shown thatdefined by$\partial\varphi$

coincides with the Stokes operator$D(A)=\mathrm{H}^{2}(\Omega)\cap \mathrm{H}_{\sigma}^{1}(\Omega)$$\{$$A(\mathrm{u})$for$=-P\triangle \mathrm{u}$We define a nonlinear operator$B$$D(^{\beta})=D(\partial\varphi)$

in

$A$

,

$\mathrm{u}\in D(A)$.

by$\mathrm{H}_{\sigma}(\Omega)$

,

$\{$

for

$B(\mathrm{u})=P(\mathrm{u}\cdot\nabla)\mathrm{u}$$\mathrm{u}\in D(B)$.

We consider the following initial value problem(IVP)$\frac{dX}{dt}(t)+\partial\varphi(X(t))+B(X(t))\ni g(X(t), t)$,$t\in(0, \infty)$

$\{$

,

$X(0)=x$ ,wherefor eachis assumed to satisfy, the function $g(u, t)$ of is measurable;$g:\mathrm{H}_{\sigma}(\Omega)\cross[0, \infty)arrow \mathrm{H}_{\sigma}(\Omega)$$u\in \mathrm{H}_{\sigma}(\Omega)$$t$

there is a constant $M>0$ such that$||g(u, t)-g(v, t)||_{\mathrm{L}(\Omega}2)\leq M||u-v||_{\mathrm{L}()}2\Omega$for all$u,$$v\in \mathrm{H}_{\sigma}(\Omega)$

and $0\leq t<\infty$ .

Proposition 2. Under the above assumptions, there exists a positive number such that if$\delta_{0}$$||x||\mathrm{L}^{2}(\Omega)+\varphi(x)\leq\delta_{0}$and$||g||L^{\infty}(\mathrm{H}_{\sigma}(\Omega)\cross(0,\infty);\mathrm{H}_{\sigma}(\Omega))\leq\delta_{0}$(IVP) has a unique -strong solution $X(t)$ in
the following (i) and (ii).
$s$(i) There exists a constant$C(\delta_{0})$$C(\delta_{0})arrow 0$$[0, \infty)f$

and $X$ satisfies

such that(as$\delta_{0}arrow 0$

,

),

and
$||X||L^{\infty}(0_{\infty},; \mathrm{H}_{\sigma}(\Omega))+\sup_{t>0}\varphi(X(t))\leq C(\delta_{0})$.

81(ii) Let $Y(t)$ be an -strong solution of (IVP) in
place of . If

with$[0, \infty)$

$s$

$y$

in

$x$

$||y||\mathrm{L}^{2}(\Omega)+\varphi(y)\leq\delta_{0}$then there is a positive constant$m_{0}$,

such thatfor$||X(t)-Y(t)||\mathrm{L}\underline{9}(\Omega)\leq e^{m_{0}t}||x-y||\mathrm{L}2(\Omega)$$t\geq 0$.

See e.g., [4], [5] for the proof of Proposition 2.
Next we consider the following control system:$(\mathrm{N}\mathrm{S})\{$$\frac{dX}{dt}(t)+\partial\varphi(X(t))+B(X(t))\ni g(X(t), \alpha(t))$,$t\in(0, \infty)$,$X(0)=x$ .
Let functions :andbe given and satisfy the following condition (A7).
$f$$g_{i}\mathrm{H}_{\sigma}(\Omega)\mathrm{x}\mathrm{H}_{\sigma}(\Omega)arrow$$\mathrm{H}_{\sigma}(\Omega)\cross \mathrm{H}_{\sigma}(\Omega)arrow \mathrm{R}$$\mathrm{H}_{\sigma}(\Omega)$(A7)$f\in C(\mathrm{H}_{\sigma}(\Omega)\cross \mathrm{H}_{\sigma}(\Omega);\mathrm{R}),$$g\in C(\mathrm{H}_{\sigma}(\Omega.)\cross \mathrm{H}_{\sigma}(\Omega);\mathrm{H}(\sigma\Omega))$

andthere is a constant $K>0$ such that$|f(x, z)|\leq K$,$|f(x, z)-f(y, Z)|\leq K||x-y||_{\mathrm{L}^{2}()}\Omega$$||g(x, z)||\mathrm{L}2(\Omega)\leq\delta_{0}$,$||g(_{X}, z)-g(y, Z)||_{\mathrm{L}^{2}(\Omega})\leq K||x-y||_{\mathrm{L}^{2}()}\Omega$for all$x,$ $y,$$z\in \mathrm{H}_{\sigma}(\Omega)$.

We define$X_{\delta}=\{x\in \mathrm{L}^{2}(\Omega)|||x||\mathrm{L}^{2}(\Omega)+\varphi(x)\leq\delta\}(0<\delta\leq\delta_{0})$

$C=$

{ :
$\alpha$$[0,$$\infty)arrow \mathrm{H}_{\sigma}(\Omega)|\alpha(\cdot)$,is measurable},$Y=Y(\delta)=\{X(t;X, \alpha)|t\geq 0, x\in X_{\delta}, \alpha\in C\}$.

. If is sufficiently
Here $X(t;X, \alpha)$ is the -strong solution ofis invariant, i.e., the following property issmall, then the set
satisfied:$(\mathrm{N}\mathrm{S})$$\mathrm{s}$$\delta$$\overline{Y}$$X(t;X, \alpha)\in\overline{Y}$for all$x\in\overline{Y},$$\alpha\in C$

and $0\leq t<\infty$ .

82

Let
functional$x\in\overline{Y}$and consider the problem of minimizing the cost$J(x, \alpha)=\int_{0}^{\infty}e^{-\lambda t}f(x(t), \alpha(t))dt$, whereover all-strong solution of$\alpha\in C$

is a given constant and $X$ is the uniquein. We define the value function$\lambda>0$$(\mathrm{N}\mathrm{S})$$\mathrm{s}$,$V$$[0, \infty)$by(1)$V(x)= \inf\alpha\in CJ(x, \alpha)$for$x\in\overline{Y}$.

Proposition 3.
Assume that (A7) holds. Then the value function
defined in (1) is bounded and uniformly continuous in .
$V$$\overline{Y}$The proof of this lemma is left to the reader. Also, see, e.g., [3].
The following theorem is proved by the method of dynamic progamming (cf. [3]).

Theorem 4. Assume that (A7) holds. Then the value function
defined in (1) is a unique viscosity solution of(2)$\lambda u+\langle\partial\varphi(x)+B(x), Du\rangle_{\mathrm{L}^{2}}(\Omega)+F(x, Du)=0$

where the function
(3)

$F$

in

$\overline{Y}$$V$,

is given by.

$F(x,p)= \sup_{z\in \mathrm{H}\sigma(\Omega)}\{-\langle g(x, Z),p\rangle_{\mathrm{L}(}2\Omega)-f(x, z)\}$Remark. In case of, the above function $F$ satisfies
and (A4), and moreover, we see that
holds. By Theorem 2.
it follows that is the unique viscosity solution of (2).
$S=\overline{Y}$$(\mathrm{A}2),(\mathrm{A}3)$$(\mathrm{A}1)_{2}$$V$

We need the next lemma to prove Theorem 4.
Lemma 3. (Dynamic Programming Principle). Assume that
(A7) holds. For any
and $\tau>0$ , the equality$x\in\overline{Y}$(4)$V(x)= \inf_{\alpha\in C}\{\int_{0}^{\mathcal{T}}e-\lambda tf(x(t), \alpha(t))dt+e^{-\lambda\tau}V(x(\mathcal{T}))\}$

83

holds, where $X(t)$ denotes the -strong solution of$s$$(\mathrm{N}\mathrm{S})$in$[0, \infty)$.

The above discussion can be also applied to the two-dimensional
Navier-Stokes equations.

References[1] H. Brezis, Op\’eratuers Maximaux Monotones et Semi-groupes
de Contractions dans les Espaces de Hilbert, North-Holland,
Amsterdam, 1973.
[2] M. G. Crandall and P. L. Lions, Hamilton-Jacobi Equations in
Infinite Dimensions, Part VI: Nonlinear A and Tataru’s Method
Refined, in Evolution Equations, Control Theory, and Biomathematics, P. Cl\’ement and G. Lumer, eds., Lecture Notes in Pureand Appl. Math. 155, Marcel Dekker, New York, 1994, pp. 51-

89.
[3] H. Ishii, Viscosity Solutions for a Class of Hamilton-Jacobi Equations in Hilbert spaces, J. Func. Anal. 105 (1992), pp. 301-341.
[4] M. Otani, Nonmonotone Perturbations for Nonlinear Parabolic
Equations Associated with Subdifferential Operators, Cauchy
Problems, J. Differential Equations 46 (1982), pp. 268-299.
[5] M. Otani and Y. Yamada, On the Navier-Stokes equations innoncylidrical domains: An approach by the subdifferential theory, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 25 (1978), pp.

185-204.
[6] K. Shimano, A class of Hamilton-Jacobi equations with unbounded coefficients in Hilbert spaces, preprint.
[7] D. Tataru, Viscosity solutions for Hamilton-Jacobi equations
with unbounded nonlinear terms, J. Math. Anal. Appl. 163(1992), pp. 345-392.