数理解析研究所講究録983 巻 1997 年 133-141

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Formal power series solutions of non-linear
partial differential equationsof the first order
Hidetoshi TAHARA(Department of Mathematics, Sophia University)

In 1903, Maillet [1] proved that if an algebraic ordinary differential equation has a formal power series solution then it is in some formal Gevrey class.
Later, this result was extended to general analytic ordinary differential equations by G\’erard [2] and Malgrange [3].
In this note, I will try to generalize this result to partial differential equations.

\S 1. Maillet’s theorem
First, let us recall the general form of Maillet’s theorem.
Denote by $C[[t]]$ the ring of formal power series in , and by $C\{t\}$ thesubring of $C[[t]]$ of convergent power series in .
$t$

$t$

Definition 1. If a formal power series$\Sigma_{i\geq 0}a_{i}t^{i}\in C[[t]]$satisfies$\sum_{i\geq 0}\frac{a_{i}}{i!^{s-1}}t^{i}\in C\{t\}$for some $s\geq 1$ , we say that$u$is in the formal Gevrey class$C\{t\}_{s}$.

Let $n\in N(=\{0,1,2\ldots\})$ , let $G(t, X_{0,1}X, \ldots, X_{n})$ be a holomorphic
function defined in a neighborhood of the origin of $C^{n+2}$ satisfying$A_{1})$$G(t, X_{0,1}X, \ldots, X_{n})\not\equiv 0$$A_{2})$$c(\mathrm{O}, \mathrm{O}, \ldots, \mathrm{O})=0$

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and let us consider the following ordinary differential equation$G(t, u, \theta u, \ldots, \theta nu)=0$$(e_{1})$

wherehave$\theta=t\frac{d}{dt}$,and $u=u(t)$ is an unknown function. UnderTheorem A ([2], [3]). If$(e_{1})$$A_{1}$) and$A_{2}$) we

has a formal power series solution $u(t)\in$$C[[t]]$ satisfying $u(\mathrm{O})=0$ then $u(t)$ belongs to the formal Gevrey class $C\{t\}_{s}$for some $s\geq 1$ .

The general form of Maillet’s theorem is formulated as follows:
Let $F(t, X_{0}, X_{1}, \ldots, X_{n})$ be a holomorphic function defined in a neighborhood ofsatisfying$(\mathrm{o}, a_{0}, a_{1}, \ldots, a_{n})\in C^{n+2}$$B_{1})$$F(t, X_{0}, X_{1}, \ldots, X_{n})\not\equiv 0$$B_{2})$$F(0, a_{0,1,\ldots,n}aa)=0$,and let us consider the following ordinary differential equation$F(t,$ $u,$ $u’,$ $u^{(2)},$ $\ldots,$ $u^{()})n=0$$(e_{2})$with an unknown function $u=u(t)$ .
Maillet’s Theorem. If
has a formal power series solution $u(t)\in$$C[[t]]$ satisfying $u^{(p)}(0)=a_{p}$ for $p=0,1,$
then $u(t)$ belongs to theformal Gevrey classfor some $s\geq 1$ .
$(e_{2})$$\ldots,$$n$$C\{t\}_{s}$From Theorem to Maillet’s Theorem. Let us explain here how to reduce. Let $u(t)= \sum_{i\geq 0}u_{i}t^{i}\in C[[t]]$ be a formal solution ofto. Put$A$$(e_{2})$$(e_{1})$$(e_{2})$$u=\varphi+t^{n}w$

Then the equation
respect to :$(e_{2})$

,

where

$\varphi=\sum_{i\geq 0}u_{i}t^{i},$with respect to$u$$w= \sum_{i\geq 1}w_{n+}it^{i}$is rewritten into an equation with$w$$(*)$.

$F(t,$ $\varphi+t^{n}w,$ $\varphi^{;n-}+t11w,$ $\ldots,$ $\varphi+(n)nw)=0$

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where$_{1}$$=$$(\theta+n)$$_{2}$$=$$(\theta+(n-1))(\theta+n)$$_{n}$$=$$(\theta+1)\cdots(\theta+(n-1))(\theta+n)$.

By applying Theorem A to $(*)$ we can get $w\in C\{t\}_{s}$ for some $s\geq 1$ ; thisimplies $u\in C\{t\}_{s}$ .

\S 2. Formulation in PDEFollowing$(e_{1})$, let us consider the non-linear partial differential equation:$G(t,$ $x,$ $u,$$(E)$$t \frac{\partial u}{\partial t},$$\frac{\partial u}{\partial x}\mathrm{I}=0$,

where $(t, x)\in C^{2},$ $u=u(t, X)$ is an unknown function and $G(t, x, x_{1}, X_{2}, x_{3})$is a holomorphic function defined in a neighborhood of the origin ofsatisfying$C^{5}$$C_{1})$$G(t, x, X_{1,2}X, x_{3})\not\equiv 0$$C_{2})$$G(\mathrm{O}, x, 0,0,0)\equiv 0$,(near $x=0$ ).

Denote by $C[[t, x]]$ the ring of formal power series in $(t, x)$ , and by $C\{t, x\}$the subring of $C[[t, x]]$ of convergent power series in $(t, x)$ .

Definition 2. If a formal power series $\sum a_{i,j}t^{ij}x\in C[[t, x]]$ satisfies$\sum\frac{a_{i,j}}{\dot{i}!s-1j!\sigma-1}t^{i}x^{j}\in C\{t, X\}$

for some $s\geq 1$ and.

$\sigma\geq 1$, we say that$u$is in the formal Gevrey class$C\{t, x\}(s,\sigma)$

The following is a direct analogy of the case in \S 1 to PDE:

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Problem 1. Under what condition is the following assertion (M) valid ?
(M) If (E) has a formal power series solution then it is in some formal
Gevrey class.
Example 1. Let us consider$(t \frac{\partial u}{\partial t})^{2}-u=a2(x)t3+(\frac{\partial u}{\partial x})^{3}$

(2.1)

,

where $a(x)\in C\{x\}$ . It is easy to see:
1) For any $\phi(x)\in C[[x]]$ satisfying $\phi(0)\neq 0$ the equation (2.1) has aunique formal power series solution $u(t, x)\in C[[t, x]]$ of the form$u(t, x)= \phi(x)t+k\sum\phi\geq 2k(_{X})t^{k}$.

(We denote this by $U(\phi)\in C[[t,$ $x]].$ )2) If we choose
being out of formal Gevrey classes, then $U(\phi)$ does
not belong to any formal Gevrey class.
3) This implies that (2.1) does not satisfy (M).
$\phi(x)$Usually, as is seen above, in the case of PDE some data (for example,
initial data or boundary data,) can be given freely and therefore bychoosing this free data as a formal power series out of formal Gevrey classes
we can easily conclude that (M) is not valid.
In order to include such cases as Example 1, instead of (M) let us consider
the following problem:$\mathrm{e}\mathrm{c}\mathrm{t}$?

Problem 2. Under what condition is the following assertion (AM) valid(AM) If (E) has a formal power series solution then (E) has a formalpower series solution in some formal Gevrey class.

Of course, (AM) is valid for the equation (2.1).

\S 3. Some definitions
A formal power series $f(t, x)\in C[[t, x]]$ is expressed in the form
$f(t, x)= \sum_{0i\geq}f_{i}(x)t^{i}$,$f_{i}(x)\in C[[_{X}]]$.

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of $f(t, x)$ in is defined by the followDefinition 3. The valuationing:$v_{t}(f)--\infty$ ;(1) if$v_{t}(f)= \inf\{i;f_{i}(x)\not\equiv 0\}$ .
(2) if$v_{t}(f)$$t$$f\equiv 0,$$f\not\equiv \mathrm{O},$Let $G(t, X, X_{1,2}X, x_{3})$ be a holomorphic function defined in a neighborsatisfying ) and ), and let us consider
hood of the origin of$C^{5}$$G(t,$ $x,$ $u,$$(E)$$C_{2}$$C_{1}$$t \frac{\partial u}{\partial t},$$\frac{\partial u}{\partial x}\mathrm{I}=0$.

For simplicity we write$Du$$=$$(u,$$Du(t, 0)$$=$$(u,$$t \frac{\partial u}{\partial t},$$t \frac{\partial u}{\partial t},$$\frac{\partial u}{\partial x})$$\frac{\partial u}{\partial x})|_{x=0}$

For $u(t, x)\in C[[t, x]]$ with $v_{t}(u)\geq 1$ we put,$i=1,2,3$ ;$q_{i}(u)$$=$$v_{t}( \frac{\partial G}{\partial X_{i}}(t, x, Du))$$q_{i}(u;0)$$=$$v_{t}( \frac{\partial G}{\partial X_{i}}(t, 0, Du(t, 0))\mathrm{I}$$p(u;t)$$=$$- \frac{\frac{\partial G}{\partial X_{1}}(t,\mathrm{o},Du(t,0))}{\frac{\partial G}{\partial X_{2}}(t,\mathrm{o},Du(t,0))}$Introduce new variables$\xi_{1},$$\mathcal{X}=\sum_{i\geq 1}\xi it^{i}$$\xi_{2},$$\ldots,$,$\eta_{1},$$\ldots,$,$\dot{i}=1,2,3$;.

$\zeta_{1},$$\eta_{2},$$\mathcal{Y}=\sum_{i\geq 1}\eta_{i}t^{i}$,$\zeta_{2},$$\ldots$, and put$Z= \sum_{1i\geq}\zeta_{i}t^{i}$.

We define $d_{1}(G),$ $d_{2}(G),$ $d_{3}(G)$ by the following:$d_{1}(G)=v_{t}(G(t, x, \mathcal{X}, y, z)-c(t, X, \mathrm{o}, \mathcal{Y}, z))$,$d_{2}(G)=v_{t}(G(t, x, \mathcal{X}, y, Z)-^{c}(t, X, \mathcal{X}, \mathrm{o}, \mathcal{Z}))$

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$d_{3}(G)=v_{t}(G(t, x, \mathcal{X}, \mathcal{Y}, z)-c(t, x, \mathcal{X}, y, 0))$.

By the definition it is easy to see:Lemma 1. (1) $d_{i}(G)\geq 1(\dot{i}=1,2,3)$ .
(2) $q_{i}(u)=0$ is equivalent to(3) $q_{i}(u)\geq 1$ implies $d_{i}(G)\geq 2$ .

$(\partial G/\partial X_{i})(\mathrm{o}, x, \mathrm{o}, 0,0)\not\equiv 0$.

\S 4. Results.
Let $u(t, x)\in C[[t, x]]$ be a formal power series solution of (E) satisfying. Put$u(\mathrm{O}, x)\equiv 0$$q(u)= \min\{q_{1}(u), q2(u), q_{3}(u)\}$ ,$l(u)= \max\{\dot{i} ; q_{i}(u)=q(u)\}$ .

It is clear that $q_{l(u)}(u)=q(u)$ holds. Assume:a-l)a-2)a-3)$q(u)<\infty$,$q_{l(u)}(u;0)=q(u)$ ,$\rho(u;\mathrm{O})-q(u)\not\in\{1,2\ldots\}$, if $l(u)=2$ .

Note that in case $l(u)=2$ we have $q_{2}(u;0)=q_{2}(u)\leq q_{1}(u)\leq q_{1}(u;\mathrm{o})$ andtherefore $\rho(u;0)=\rho(u;t)|_{t_{-arrow}0}$ is well-defined.
Theorem 1. If (E) has a formal power series solution $u(t, x)\in C[[t, x]]$satisfying
and if the conditions a-l), a-2), a-3) and$u(\mathrm{O}, x)\equiv 0$$q(u) \leq\frac{d_{3}(c)-1}{2}$(4.1)

are valid, then (E) has a formal power series solution $w(t, x)$ such that$C\{t, x\}_{(2},1)$, when $l(u)=1$ ,$C\{t, x\}_{(1},1)$, when $l(u)=2$ or 3.

$w(t, x)\in\{$

Remark 1. (1) In case $l(u)=2$ or 3, the conclusion of Theorem 1 asserts
the existence of a holomorphic solution.

139(2) In case $q(u)–0$ and $l(u)=3$ , the above result is nothing but theCauchy-Kowalewski theorem.
(3) This result was extended to general non-linear higher order partial
differential equations in $[4],[5]$ .
The following theorem fills the case where the condition (4.1) is not satisfied.
Theorem 2. If (E) has a formal power series solution $u(t, x)\in C[[t, x]]$and if the conditions a-l), a-2), a-3) andsatisfying$u(\mathrm{O}, x)\equiv 0$(4.2)$q(u) \leq\max\{d_{1}(c), d2(G)\}$

are valid, then (E) has a formal power series solution $w(t, x)$ such that$w(t, x)\in\{$$C\{t, X\}_{(3,2)}$, when $l(u)=1$ ,$C\{t, x\}_{(2},2)$, when $l(u)=2$ ,$C\{t, x\}_{(1},2)$, when $l(u\mathrm{I}=3$ .

\S 5. Examples.
Example 1. Let us consider$(t \frac{\partial u}{\partial t})^{2}-u^{2}=a(X)t^{3}+(\frac{\partial u}{\partial x})^{3}$(5.1)

where $a(x)\in C\{x\}$ .
1) For any $\phi(x)\in C[[x]]$ satisfying $\phi(0)\neq 0$ the equation (5.1) has aunique formal power series solution $u(t, x)\in C[[t, x]]$ of the form$u(t, x)= \phi(x)t+k\sum_{\geq 2}\phi_{k}(x)t^{k}$.

2) In this case we have $q_{1}(u)=1,$ $q_{2}(u)=1,$ $q_{3}(u)\geq 2,$ $q_{1}(u;0)=1$ ,$q_{2}(u;0)=1,$ $q_{3}(u;0)\geq 2,$ $d_{1}(G)=2,$ $d_{2}(G)=2,$ $d_{3}(G)=3$ . Therefore,

140$q(u)=1,$ $l(u)=2$ and $\rho(u;0)=1$ . Since the condition (4.1) is valid, we canapply Theorem 1 to the equation (5.1).
3) Conclusion. The equation (5.1) has a holomorphic solution $w(t, x)$satisfying.
$w(\mathrm{O}, x)\equiv 0$

Example 2. Let us consider$(t \frac{\partial u}{\partial t})^{2}-u2=(x^{2}\frac{\partial u}{\partial x}-u+(1+x)t)^{2}+a(X)t3$(5.2)

where $a(x)\in C\{x\}$ .
1) By a calculation we see that the equation (5.2) has a unique formalsolution $u(t, x)$ of the form$u(t, x)=(1+X+ \sum_{p\geq 2}(p-1)!X^{p})t+\sum_{k\geq 2}\phi k(_{X})t^{k}$.

case we2) In thishave $q_{1}(u)=1,$ $q_{2}(u)=1,$ $q_{3}(u)\geq 2,$ $q_{1}(u;0)=1$ ,$q_{2}(u;0)=1,$ $q_{3}(u;0)=\infty,$ $d_{1}(G)=2,$ $d_{2}(G)=2,$ $d_{3}(G)=2$ . Therefore,$q(u)=1,$ $l(u)=2$ and $\rho(u;0)=1$ . Since the condition (4.2) is valid, we canapply Theorem 2 to the equation (5.2).
3) Conclusion. The equation (5.2) has a unique formal power series solution $u(t, x)$ satisfying
and it belongs to the formal Gevrey class$u(\mathrm{O}, x)\equiv 0$$C\{t, x\}_{(2},2)$.

4) Note that (5.2) does not satisfy the condition (4.1) and we can not
apply Theorem 1 to this case.

References[1] E. Maillet: Sur les s\’eries divergentes et les \’equations diff\’erentielles, Ann.
Ecole Normale, Ser. 3, 20 (1903), 487-518.

[2] R. G\’erard: Sur le Th\’eorem\‘e de Maillet, Funkcial. Ekvac. 34 (1991),117-125.

[3] B. Malgrange: Sur le Th\’eorem\‘e de Maillet, Asymptotic Analysis 2(1989), 1-4.

141[4] R. G\’erard and H. Tahara: On the existence of holomorphic solutions of
the Cauchy problem for non linear first order partial differential equations, J. Fac. Sci. Univ. Tokyo, 40 (1993), 549-560.

[5] R. G\’erard and H. Tahara: On the existence of holomorphic solutions of
the Cauchy problem for non linear partial differential equations, Funkcial. Ekvac., 37 (1994), 317-327.
[6] R. G\’erard and H. Tahara: Singular nonlinear partial differential equations, Vieweg-Verlag, 1996.