数理解析研究所講究録
第 824 巻 1993 年 1-10

1A Bound for the Pressure Integral in a Plasma Equilibrium

Yoshikazu Giga*
北大・理儀我美一
Department ofMathematics, Hokkaido University, Sapporo
Zensho Yoshida

東大・工吉田善昭
Department ofNuclear Engineering, University of Tokyo, TokyoAbstract. An interpolation inequality for the total variation of the gradient of a composite function has been derived by applying the coarea formula. The interpolation
inequality has been applied to the study of a bound for the pressure integral concern-ing a solution of the Grad-Shafranov equation of plasma equilibrium. A weak formu-lation of the Grad-Shafranov equation has been given to include singular current
profiles.

1. Introduction
A simple but essential question in the fusion plasma research is how large plasmaenergy can be confined by a given magnitude of plasma current.1-7 In a magnetohydrodynamic equilibrium of a plasma, the thermal pressure force

magnetic $stressj\cross B$, where$B$

is the magnetic flux density,$\nabla p$is balanced by the$j=\nabla\cross B/[l0$is the current

density in the plasma and [ is the vacuum perneability. The plasma equilibrium$\downarrow 0$

equation$\nabla p=j\cross B$

thus relates the pressure and the current. We want to estimate themaximum of the total pressure with respect to a fixed total current. Mathematicallythis problem reduces to an a priori estimate for the pressure integral with respect to asolution of the equilibrium equation with a given magnitude of current.
Here we assume a simple two dimensional plasma equilibrium. Let
bounded domain. We consider an infinitely long plasma column;*The first author is partly supported by the Inamori Foundation.

$\Omega$$\Omega\subset R^{2}$

be acorresponds to2
the cross section of a column containing the plasma. If there is no longitudinal mag-netic field, the equilibrium equations are$-\Delta\psi=P’(\psi)$$\psi=c$$\int_{\Omega}(-\Delta\psi)d\kappa=\mu_{0}I$

where$\psi$.

(1.1)$on\Re$

(1.2)

(1.3)

,

is the flux function,$=dP(t)/dt,$ $I$$\geq 0$$inq$

$P=[\downarrow 0p,$

$P(t)$

is a nonnegative function from$R$

to $RP’$is a given positive constant and is an unknown constant. We assume$c$$Since-\Delta\psi/\mu_{0}$$P’$parallels the current density, represents the total plasma current.
$I$

In this paper we study a bound for the total variation of the gradient $ofP(\psi)$ in$\Omega$.

A crucial step is to establish an interpolation inequality to estimate the total variationof the gradient $ofP(\psi)$ in$\Omega$. Our estimate reads$\int_{\Omega}|\nabla P(\psi(x))|\ \leq 2(P_{ma\kappa}\int_{\Omega}-\Delta\psi d\kappa)^{1\Omega}(\int_{\Omega}P’(\psi(x))d\kappa)^{1\Omega}$

provided that $A\psi\geq 0$ ina constant and$P_{\max}$$\Omega$and $\psi=c$ onis the maximum of$\partial\Omega$(1.4), and that $P’\geq 0$ with $P(c)=0$ , where$P(\psi)$

$c$

is

over . We prove this estimate by using$\Omega$

the coarea formula.8,9 In section 2 we prove (1.4) and extend it for discontinuous .
$P$

In this case the meaning of the $equation-\Delta\psi=P’(\psi)$ is not clear. We $shaU$ give ameaning for discontinuous in section 3.
$P$

2. An interpolation inequality
Our goal in this section is to estimate the total variation
valued measure), wheresmooth$\psi$

is monotone$and-\Delta\psi\geq 0$(as a vector-. We first derive the estimate for.

Theorem 2.1. Let$P\in C^{1}(R)$$P$$of\nabla(P(\psi))$$\Omega$

be a bounded domain in R and be a constant. Suppose thatwith $P’\geq 0$ and $P(c)=0$ , and that$c$$\psi\in C^{m}(\Omega)\cap C^{0}(\Omega)$with3(2.1)$in\zeta 1$$-\Delta\psi\geq 0$$on\partial C1$$\psi=c$where $m\geq 2$ and $m\geq n$ . Let$P_{ma\mathfrak{r}}= \sup_{x\epsilon\Omega}P(\psi(x))$$P_{\max}$

denote.

(2.2)

Then(2.3)$\int_{\Omega}|\nabla P(\psi(x))|dx\leq 2(P_{ma\mathfrak{r}}\int_{\Omega}(-\Delta\psi)dx)^{1/2}(\int_{\Omega}P’(\psi(x))dx)^{1/2}$, thenon , so (2.3) holds with zero for both sides. $IfP’(\psi)$
Proof.
or. Again (2.3) holds in this case, soon or $P_{\max}=0$, then either$If-\Delta\psi\cong O$$\equiv 0$$\psi\equiv c$$\Omega$$\Omega$$\psi\equiv c$$P\equiv 0$

we may assume that both integrals in the right hand side of (2.3) is nonzero. We may
is finite.
also assume that the norm$L^{1}$$of-\Delta\psi$

For $K>0$ denote the set $ofx\in\Omega$ for which $|\nabla\eta(x)|>K$ by . Let denote the$E$$D$

complement of$D$in . From the definition it follows that$\Omega$$\int_{E}|\nabla P(\psi(x))|dx=\int_{E}P’(\psi)|\nabla\psi|d\kappa$$\leq K\int_{E}P(\psi)dx\leq K\int_{\Omega}P(\psi)d\kappa$

(2.4)

,

since $P’\geq 0$ .
By the maximum principle to (2.1), we observe that$\leq P_{\max}$$\psi\geq c$on$\Omega$so $0=P(c)\leq P(\psi)$on . Applying the coarea formula (see e.g. Ref. 8 and 9) yields$\Omega$$\int_{D}|\nabla P(\psi)|dx=\int_{r}^{+\infty}lt^{larrow 1}(S_{t})P(t)dt=\int_{c^{\psi_{R}}}lt^{\hslash-1}(S_{t})P(t)dt$

(2.5)

4
with
$S_{t}=D\cap L_{t}$where$L_{t}=\{x\in\Omega;\psi(x)=t\}$,$\psi_{ma\kappa}=\sup_{x\epsilon\Omega}\psi(x)$,denotes the $n-1$ dimensional Hausdorff measure. Since$lt^{n- 1}$$|\nabla\psi|>K$

on it
$D$

follows that$\S t^{n-1}(S_{t})=\int_{S_{t}}|\nabla\psi||\nabla\psi|^{-1}ffi^{\prime\vdash 1}$$\leq K^{-1}\int_{L_{t}}|\nabla\psi|ffi^{\prime\vdash 1}$

Since$\psi\in C^{n}(\Omega)$, Sard’s $theorem^{1}$ implies thatevery (a.e. ). Note that $\psi>c$ in$t$$t\}$$t$

we observe

level set of$\psi,$$\overline{U_{t}}\subset\Omega$$\Omega$is$C^{n}$

submanifold in$L_{t}$

is$C^{n}$

for almost

boundary of . Since$U_{t}$$L_{t}$

is t-

is a unit normal vector field. Applying Green’s formulayields$\int_{L_{t}}|\nabla\psi|ffi^{\prime\vdash 1}=\int_{L_{t}}\nabla\psi\cdot nffi^{\prime\vdash 1}=\int_{U_{t}}(-\Delta\psi)d\kappa,$$From-\Delta\psi\geq 0$$\Omega$and $\psi=c$ on ffl. Thus for $U_{t}=\{x\in\Omega;\psi(x)>$for $t>c$ . For a.e. $t>c,$$n=\nabla\psi/|\nabla\psi|$$L_{t}$

it now follows that$\int_{L_{t}}|\nabla\psi|ffi^{l\vdash 1}\leq\int_{\Omega}(-\Delta\psi)d\kappa$.

Wrapping up these two estimates we obtain$l t^{larrow 1}(S_{t})\leq K^{-1}\int_{\Omega}(-\Delta\psi)dx$.

Applying this estimate to (2.5) yields$t>c$.

5$\int_{D}|\nabla P(\psi)|d\kappa\leq K^{-1}P_{mar}\int_{\Omega}(-\Delta\psi)dx$

where$P_{\max}$

(2.6)

,

is defined in (2.2). Summing (2.4) and (2.6) we obtain(2.7)$\int_{\Omega}|\nabla P(\psi)|d\kappa\leq K\int_{\Omega}P’(\psi)dx+K^{-1}P_{\max}\int_{\Omega}(-\Delta\psi)d\kappa$

for arbitrary $K>0$ . Taking$K=[P_{\max} \int_{\Omega}(-\Delta\psi)d\kappa/\int_{\Omega}P(\psi)d\kappa]^{112}$in (2.7) yields (2.3).

$If\psi$

is not$C^{2}$

Q.E.D.

, one should interpret$-\Delta\psi\geq 0$

in the distribution sense. As wellknownl1 a nonnegative distribution is a nomegative Radon measure. Let be a finite$\mu$Radon measure on a bounded domain$\Omega$

in$R^{n}$. The unique solvability of the Dirichletproblem$-\Delta\psi=\mu$$\psi=c$in ,(2.8a)$\Omega$on$\infty$( : constant)(2.8b)$c$is now well known for smooth boundary$\partial\Omega$. We solve this problem by using a resultof Simader12 when the boundary is$C^{1}$order one $(1 <q<\infty)$ . Let

be the subspace $\{u\in W^{1,q}(\Omega);u=0on\partial\Omega\}$ . We

denote by$W^{1,q}(\Omega)$$W_{0}^{1,q}(\Omega)$the dual space of. Let$W_{0}^{1,\zeta}(\Omega)$

Lemma 2.2 (Theorem 4.6 of $Simader^{12}$ ).

boundary in$R^{n}$$W^{1,q}(\Omega)$bounded linear from

$W_{0}^{1,q}(\Omega)$

to$\Omega$$L^{q}$

Sobolev space of

where $1/q=1-1/q’$.

$Let\Omega$

be a bounded domain with. Assume that $1<q<\infty$. For eachsolution $\Phi\in W_{0}^{1,q}(\Omega)for-\Delta\Phi=f$ indenote the$f\in W^{1,q}(\Omega)$

there is a unique. Moreover the mapping from to$\dagger\Gamma^{1,q}(\Omega),$$f$$i.e.$,$C^{1}$$\Phi$is6(2.9)$\Vert\Phi\Vert_{1,q}\leq C||f|\llcorner_{1,q}$with a constant $C=C(f1q, n)$ .

Corollary 2.3. Let

Radon measure$\Omega$

be a bounded domain with$\mu on\Omega$$C^{1}$boundary in$R^{n}$. For a finite

there is a unique $solution\psi$ of $(2.8a, b)$ such $that\psi\in W^{1,r}(\Omega)$

for $1<r<n/(n-1)$ .
Proof. Observe that $r’>n$ implies$W_{0}^{1,r’}(\Omega)\subset C(\Omega)-$by the Sobolev inequality. This

yields $\mu\in W^{1,r}(\Omega)by$ a duality, where $1/r=1-1/r’$. Applying Lemma 2.2 with $f=$$\mu$

obtains a unique solution

Theorem 2.4.

$\psi$

by $\psi=\Phi+c$ .

Q.E.D.

be a bounded domain with$Let\Omega$$C^{1}$

boundary in$R^{n}$. Let be a con$c$

stant. Suppose that $P\in C^{1}(R)$ with $P’\geq 0$ and $P(c)=0$ . Suppose $that\psi\in$$W^{1,r}(\Omega)$

for some such that$r$$-\Delta\psi\geq 0$$\psi=C$Let$\psi\max$

on [$c,$, and that$\psi$

satisfies$in\Omega(inthedistributionsense)$ ,on ffl.

be the essential supremum $of\psi over\Omega$. Assume that and$P$$\psi_{m}\theta\cdot$$P’$

are boundedThen$\int_{\Omega}|\nabla P(\psi(x))$

where$1<r<\mathcal{N}(n-1)$

I$dx \leq 2(P_{ma\kappa}II-\Delta_{\psi}\Vert_{1})^{1\Omega}(\int_{\Omega}P(\psi(x))d\kappa)^{1a}$$P_{\max}= \sup\{P(\sigma);c\leq\sigma\leq\psi_{ma\kappa}\}$and$\Vert\cdot\Vert_{1}$,(2.10)

denotes the total variation of ameasure on .
$\Omega$

For the proof of this Theorem, the reader is referred to Ref. 13.
We next extend the inequality (2.9) when a nondecreasing function is not neces$P$

sarily continuous. Let us give an interpretation of each integral appeared in (2.9).

Instead of the integral) $dx$ , we consider$\int {}_{\Omega}Pt\psi$7$[P( \psi)]=\inf\varliminf_{larrow\infty}\int_{\Omega}P_{l’}(\psi)dx$.

Here the infimum is taken over $aU$ sequence

for someas$P_{l}\in C^{1}(R)$withsuch that$P_{l’}\geq 0$$P(\psi)$. We$arrow P(\psi)$insay

is an admissible $aDDroximation$ of if these properties hold. If is itself$\{P\iota\}$$L^{S}(\Omega)$$1\leq s<\infty$$larrow\infty$

and that ($Pb_{na\kappa} arrow ess\sup_{\Omega}P(\psi)$

$P$

$P$

and satisfies the assumptions in Theorem 2.4,

$P$$C^{1}$

itself is an admissible approximation

so for such a we have$P$$[P’( \psi)]\leq\int_{\Omega}P’(\psi)d\kappa$

Since$\int_{\Omega}|\nabla P(\psi)|dx$.

is the total variation$of\nabla P(\psi)$on$\sigma\iota$

i.e.

$\Vert\nabla P(\psi)\Vert_{1}=\int_{\Omega}|\nabla P(\psi(x)\lambda d\kappa$$:= \sup${$\int_{\Omega}P(\psi(x))\nabla\cdot\propto x)dx;\varphi\in c_{0}^{\iota_{(\Omega),|\varphi(\kappa)|\leq 1}}$on$\Omega$},

it is easy to see$\Vert\nabla P(\psi)\Vert_{1}\leq\varliminf_{larrow\infty}\int_{\Omega}|\nabla P_{l}(\psi)|d\kappa$

for any admissible approximation $\{P_{l}\}ofP$ since$\sup\varliminf\leq\varliminf$sup. We have thus

proved the following assertion.
Theorem 2.5. Assume the hypotheses of Theorem 2.4 conceming
be a nondecreasingfimction on$R$$c,$$\Omega$

and$\psi$. Let$P$

with $P(c)=0$ . Then$\Vert\nabla P(\psi)\Vert_{1}\leq 2(P_{mar}\Vert-\Delta\psi\Vert_{1})^{1\Omega}[P’(\psi)]^{1/2}$

provided that $P_{mx}= ess\sup_{\Omega}P(\psi)$ is finite.

(2.11)

8
Remark 2.6. $IfP(0)=0$, the inequality (2.10) is an interpolation inequality,$\Vert\nabla\psi\Vert_{1}\leq 2(P_{ma\mathfrak{r}}\Vert-\Delta\psi\Vert_{1})^{1\Omega}|\Omega|^{\iota a}$$where|\Omega|denotes$

the Lebesgue measure of .
$\Omega$

3. Weak solution of the Grad-Shafranov equation
We shall give a meaning $of-\Delta\psi=P(\psi)$ when a nondecreasing hnction is not$P$continuous and is not smooth.
$\psi$

Defmition 3.1. Suppose thatcreasing. We say$\psi$$\psi\in W^{1,r}(\Omega)$

for some$r,$$1<r<\infty$

and that is nonde$P$

and satisfy$-\Delta\psi=P’(\psi)$$P$

in$\Omega$

if the following properties hold.
(i)$-\Delta\psi\geq 0$

on in the distribution sense.
$\Omega$(ii) There is an admissible sequence$\{P_{l}\}$such that$\lim_{larrow\infty}\int_{\Omega}(-\Delta\psi-P_{l}’(\psi))\varphi dx=0$

for all$\varphi\in C(\Omega)$.

Theorem 3.2. Let$\Omega$stant. Assume thatsome$r,$$P$$C^{1}$

boundary in$\psi=C$$R^{n}$. Let be a con$c$

is a nondecreasingfimction on R. Assume that $\psi\in W^{1,r}(\Omega)for$$1<r<’\sqrt{}(n-1)andthat\psi satisfies$$-\Delta\psi=P’(\psi)$Thenbe a bounded domain with

in$\Omega$(in the sense ofDefmition 3.1)

on ffl.

9$\Vert\nabla P(\psi)\Vert_{1}\leq 2P_{\max^{2}[!0}^{1\prime}I$

,

(3.1)

where

$I= U_{0}^{-1}\int_{\Omega}(-\Delta\psi)d\kappa=U_{0}^{-1}\Vert-\Delta\psi\Vert_{1}$Proof. We may assume$P_{ma\mathfrak{r}}<\infty$.

. By Definition 3.1 (ii) with$\varphi\equiv 1$

we observe that$[P( \psi)]\leq\lim_{larrow\infty}\int_{\Omega}P_{l’}(\psi)dx=\int_{\Omega}(-\Delta\psi)dx=\Vert-\Delta\psi\Vert_{1}$$since-\Delta\psi\geq 0$. The inequality (2.11) yields (3.1).

Q.E.D.

4. Discussions
In plasma physics, the poloidal beta ratio, which is define by$\beta=\int_{\Omega}pdx/(1^{2}\mu\sqrt 8\pi)=8\pi\int_{\Omega}P(\psi)d\kappa/(\int_{\Omega}(-\Delta\psi)dx)^{2}$,is an important quantity to characterize a plasma equilibrium. In the case of the spacedimension $n=2$ , the Payne-Rayner inequality14 applies to the estimate of , and one$\beta$finds$\beta\leq 1$. A general toroidal equilibrium problem includes two different effects; In

the equilibrium equation (1.1),$-\Delta\psi$should be replaced by a more complicated term

including the toroidal curvature effect, and a new term should be added on the right-

hand side, which represents the diamagnetic effect of the longitudinal magnetic field.

Limitation of in such a situation has been discussed by many authors, while no rig$\beta$

orous estimate of the bound have been given. Extension of the Payne-Rayner
inequdity $wm$ be discussed elsewhere to estimate the bound for .
$\beta$10REFERENCE

1.

Yoshikawa, S.: Toroidal equilibrium of current-carrying plasmas. Phys. Fluids

17, 178-180 (1974).

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Clarke, J.F., Sigmar, D.J.: High-pressure flux-conserving tokamak equilibria.

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Dory, R.A., Peng, Y.-K. M.: High-pressure flux-conserving tokamak equilibria.
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Mauel, M.E. et aL: Operation at the tokamak equilibrium poloidal beta limit inTFTR. Nucl. Fusion32, 1468-1473 (1992)

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Freidberg, J.P.: Ideal Magnetohydrodynamics. New York London: Plenum 1987.

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White, R.B.: Theory of Tokamak Plasmas. Amsterdam Oxford New York Tokyo:

North-Holland 1989.

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Federer, H.: Geometric measure theory. New York: Springer-Verlag 1969.

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Morgan, F.: Geometric measure theory, a beginner’s guide. Boston: AcademicPress 1988.

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