Annales de la Faculté des Sciences de Toulouse

Guy Casale

$\mathcal {D}$-enveloppe d’un difféomorphisme de $(\mathbb {C},0)$

Manfred Stingl

Construction of coupling-resurgent amplitudes in quantum field theory

Daniel Bertrand

Vladimir Chirskii

Johan Yebbou

Effective estimates for global relations on Euler-type series

Mireille Canalis-Durand

Reinhard Schäfke

Divergence and summability of normal forms of systems of differential equations with nilpotent linear part

Rodica D. Costin

Power and exponential-power series solutions of evolution equations

Ovidiu Costin

Saleh Tanveer

ANNALES DE LA FACULTÉ DES SCIENCES DE TOULOUSE
OVIDIU COSTIN
SALEH TANVEER
Analyzability in the context of PDEs and applications
Annales de la faculté des sciences de Toulouse 6e série, tome 13,
no 4 (2004), p. 539-549
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539
Analyzability in the context of PDEs and applications
OVIDIU COSTIN (1), SALEH TANVEER (2)
Annales de la Faculté des Sciences de Toulouse Vol. XIII, n° 4, 2004
ABSTRACT. - We discuss the notions of resurgence, formalizability, and
formation of singularities in the context of partial differential equations.
The results show that Ecalle’s how analyzability theory extends naturally
to PDEs.
RÉSUMÉ. - Nous discutons les notions de résurgence, formalisabilité,
et formation des singularités dans le contexte des équations aux dérivées
partielles. Les résultats montrent que la théorie des fonctions analysables
d’Ecalle s’étend naturellement aux équations aux dérivées partielles.
1. Introduction
The study of nonlinear partial differential equations in the complex do-
main and especially formation of spontaneous singularities of their solutions
is not a well understood subject. The theory of Ecalle’s analyzable functions,
originally developed (mainly) for functions of one variable, provides a set of
tools which are well suited to address some of these issues, but the extension
to several variables is not immediate.
In the case of linear ODEs under suitable assumptions, there is a com-
plete system of formal solutions as transseries, [.10], and these are generalized
Borel (multi)summable to a fundamental system of actual solutions of the
system; sufficiently powerful results of a similar nature have been shown
(*) Reçu le 29 août 2003, accepté le 14 janvier 2004
(1) Rutgers University, Dept. of Mathematics, Hill Center, 110 Frelinghuysen Rd Rut-
gers, State University of New Jersey, 08854-80 Piscataway, USA.
E-mail: costin@math.rutgers.edu
(2 ) The Ohio State University, 231 West 18th Avenue Columbus, OH 43210-1174.
E-mail: tanveer@math.ohio-state.edu
- 540 -
in the nonlinear case as well [10], [12], [14], [15]. This is true for difference
equations as well, [6, 8].
While transseries solutions can be considered and their Borel summation
shown in the context of PDEs there are a number of difficulties specific to
several variables. We first discuss a number of specific obstacles to extending
the theory in a straightforward way, and then refer to what we expect to
be a general approach to many problems and overview a number of recent
results utilizing this approach.
2. Difficulties of formalizability and analyzability in PDEs
2.1. Insufficiency of formal representations
For PDEs even the notion a general formal solution appears to elude a
definition that is reasonably simple and useful.
Example 1. The equation ft + fx = 0 has the general solution
with F any differentiable function; it is not clear to us that a worthwhile
definition can be associated to the description "general formal expression
in t - x" ; on the other hand, restricting ourselves to special, well defined,
combinations in t - x would correspondingly limit the number of associated
actual functions, precluding a complete solution of the original PDE. In this
example, actual solutions outnumber by far formalizable ones.
The (apparently) opposite situation is possible as well.
2.2. Absence of général summation procedures
Let now fo E C~ (a, b). The initial value problem
always has formal solution as t ~ 0:
- 541 -
but it has no actual solution, if fo is real-valued non-analytic (cf. the proof
of next proposition).
There is no nontrivial summation procedure of formal Taylor series
over an interval nor a more restricted one that would associate actual solu-
tions to (2.1) as the following proposition shows. (See also Remark 3.1).
PROPOSITION 2.1. - Let S be a summation procedure defined on a dif-
ferential algebra Ds of formal series of the form
where S is assumed to have the following (natural) properties:
2022 S is linear.
2022 S commutes with differentiation.
2022 S’() ~ g0(x) as t ~ 0.
2022 If f E Ds then Sf : (a, b) x (0, E) ~ C (where E is allowed to depend
on f).
Assume if fo is real valued. Then f in (2.2) is in Ds iff f is convergent (in
the usual sense).
Proof. Assume f E Ds. Then, by the properties of S, the function
f - Sf is a differentiable function, and it is a solution of (2.1) in a domain
D = (a, b) x (0, E). If we write f = u + iv we see that the pair (u, v) satîsfies
the Cauchy-Riemann equations in D and thus f is analytic in z = t+ix with
(x, t) E D. The third property of S shows that f o is the limit as z approaches
the interval (a, b) from the upper-half plane. Since fo is real valued, then
by the Schwarz reflection principle f extends analytically through (a, b) to
a neighborhood of (a, b) in the lower half plane; in particular f is analytic
on (a, b). But then (2.1), which is the Taylor series of f at points on (a, b)
is convergent. D 
2.3. Obstacles to determining the formal solutions
We now contrast formal analysis of ODEs and PDEs.
1. Consider the Painlevé P1 equation
- 542 -
a rather nontrivial example of a second order nonlinear ODE. A de-
tailed analysis of transseries and their Borel summability in this ex-
ample are discussed in [14]. We only mention a few aspects relevant
to the present discussion.
Finding formal solutions of (2.3) is quite straightforward. Searching
first for algebraic behavior, dominant balance shows that 6y2 rv -x,
say y - 6-1 2ix1 2, and then, consistent with this, y" - o(x). It follows
that a formal series expansion can be gotten by taking yo - 0 and
then, for n E N, iterating the recurrence
A power series solution is readily obtained in this way,
which is not classically convergent but is Borel summable to an actual
solution [15]; the complete transseries can be calculated and Borel
summed in a similar way, [15]. The possibility (and convenience) of
the formal calculation is partly due to asymptotic simplification, re-
sulting in a dominant balance equation,
which can be solved exactly, from which a complete solution of the
full problem follows by appropriate perturbation theory.
2. Compare this problem to the periodically forced Schrôdinger equation
Under physically reasonable assumptions 03C8 is transseriable [4]:
THEOREM 2.2 ([4]). - Assume 03A9, V are compactly supported
and continuous, and 03A9 &#x3E; 0 throughout the support of V. For t &#x3E; 0
there exist N E N and {0393k}kN, {F03C9;k(t, x)}kN, 203C0/03C9-periodic func-
tions of t, such that
- 543 -
with R0393k &#x3E; 0 for alll k  N, and hk(t,x) = O (t-3 2,|j|!-1 2) have
Borel summable power series in t,
The operator 8 (Laplace-Borel) stands for generalized Borel sum-
mation [15].
Insofar as a formai analysis would be concerned, it is to be noted that
there is no small parameter in (2.5) and largeness of t does not make
any term negligibly small; a posteriori, knowledge of the transseries
(2.6) confirms this. In this sense, (2.5) admits no further simplifica-
tion. There is, to thé knowledge of the authors, no straightforward
formai way based on (2.5) to détermine whether 03C8 ~ 0, let alone its
asymptotic expansion.
3. Overcoming thèse difficulties : thé approach
of asymptotic regularization
First note an implication of Ecalle’s analyzability techniques [5]: a wide
class of problems can be regularized by suitable Borel transforms. Summa-
bility of general solutions of ODEs or difference equations, [5, 12, 13, 15]
shows that, under appropriate transformations, the resulting equations ad-
mit convergent solutions, an indication of the regularity of the associated
equation.
Transseries are obtained, by suitable inverse transforms, from these reg-
ularized solutions. 
In the case of PDEs it appears that regularizing the equation is in many
cases the adequate approach. The result (2.6) is obtained in this way.
3.1. Elementary illustration: regularizing the heat equation
Since (3.1) is parabolic, power series solutions
- 544 -
are divergent even if Fo is analytic (but not entire). Nevertheless, under
suitable assumptions, Borel summability results of such formal solutions
have been shown by Lutz, Miyake, and Schâfke [9] and more general results
of multisummability of linear PDEs have been obtained by Balser [7].
The heat equation can be regularized by a suitable Borel summation.
The divergence implied, under analyticity assumptions, by (3.2) is Fk =
0(k!) which indicates Borel summation with respect to t- l . Indeed, the
substitution
yields
which becomes after formal inverse Laplace transform (Borel transform)
in T,
which is brought, by the substitution (p, x) = P - ÎU(X 1 2p1 2); y = 2p1 2, tc
the wave equation, which is hyperbolic, thus regular
Existence and uniqueness of solutions to regular equations is guaranteed by
Cauchy-Kowalevsky theory. For this simple equation the general solution is
certainly available in explicit form: u = u- (x - y) + u+ (x + y) with u-, u+
arbitrary C2 functions. Since the solution of (3.5) is related to a solution of
(3.1) through (3.3), to ensure that we do get a solution it is easy to check
that we need to choose u- = u+ = uo (up to an irrelevant additive constant
which can be absorbed into u_) which yields,
which, after splitting the integral and making the substitutions x ± 2 y 2 = s
is transformed into the usual heat kernel solution,
In conclusion although there is perhaps no systematic way to formal-
ize the general solution of the heat equation, appropriate inverse Laplace
transforms allow us a complete solution of the problem (in an appropriate
class of initial conditions which ensure convergence of the integrals).
- 545 -
Remark 3.1. - Proposition 2.1 can be also understood in the following
way. Equation (2.1) is already regular. Any actual solution, if it exists with
the initial condition given in the Proposition, is trivially formalizable since it
is then analytic. It is thus natural that no further summable formal solutions
exist.
3.2. Nonlinear équations : regularization
by Inverse Laplace Transform
In this section we briefly mention a number of our results of that sub-
stantiate regularizability.
3.2.1 Consider the third order scalar évolution PDE:
Formal Inverse Laplace Transform with respect to y givesl
where convolution is the Laplace one, ( f * g) (p) = ~p0 f (s)g(p - s)ds. This
equation is regular in that formal power series in p converges, since the
coefficients in the equation are analytic.
Multiplying by the integrating factor of the l.h.s. and integrating yields
The regularity of this equation plays a crucial role in the proofs in [1] where
we find the actual solutions of equation (3.8).
3.2.2 Similar methods were later extended [3] to equations of the
form
(1) For technical convenience, in [1] we used oversummation. The paper [3] shows
that in fact Borel summability holds in the correct variable, in the more general setting
decribed in the next section.
- 546 -
with u E Cr, for t E (0, T) and large Ixl in a poly-sector S in C d (~jx =
~j1x1~j2x2··· ~jdxd and jl + -+ jd  n). The principal part of the constant
coefficient n-th order differential operator P is subject to a cone condition.
The nonlinearity g and the functions uI and u satisfy analyticity and decay
assumptions in S.
The paper [3] shows existence and uniqueness of the solution of this
problem and finds its asymptotic behavior for large |x|.
Under further regularity conditions on g and uI which ensure the exis-
tence of a formal asymptotic series solution for large Ixl to the problem, we
prove its Borel summability to the actual solution u.
In special cases motivated by applications we show how the method
can be adapted to obtain short-time existence, uniqueness and asymptotic
behavior for small t, of sectorially analytic solutions, without size restriction
on the space variable.
3.3. Nonlinear Stokes phenomena and movable-singularities
In the context of ODEs it was shown [14], under fairly general assump-
tions, that the information contained in the regularized problem (equiva-
lently, in the transseries) can be used to determine more global behavior of
solutions of nonlinear equations, in particular the fact that they form spon-
taneous singularity close to anti-Stokes lines. The method, transasymptotic
matching, was extended to difference equations [8, 6].
In nonlinear partial differential equations, formation of singularities is a
very important phenomenon but no general methods to address this issue
existed.
The method of regularization that we described provides such a method.
We briefly discuss the main points of [2].
At present our methods apply to nonlinear evolution PDEs with one
space variable; even for these, substantial new difficulties arise with respect
to [14].
Consider the modified Harry Dym equation (arising in Hele-Shaw dy-
namics)
in an appropriate sector.
- 547 -
Small time behavior. From [1] it follows that there exists a unique solutio]
to above problem, and it has Borel summable series for small t and smal
y - x - t:
Singularity manifolds near anti-Stokes lines. To apply the method oj
transasymptotic matching, we look on a scale where the asymptotic expan-
sion becomes formally invalid: y = x - t = O(t2/9). The transition variable
is thus 
Substituting into (3.10), we obtain the following equivalent equation
The natural formal expansion solution in this regime is
with matching conditions at large ~, to ensure the solution agrees with the
one obtained in [1]:
We show that the series (3.11) is actually convergent and equals H(x,t) in
the Borel summed region (the radius of convergence shrinks however with
~). The convergence problem is subtle and required a rather delicate con-
struction of suitable invariant domains. Having shown that, it is intuitively
clear (and not difficult to prove) that if Go is singular, then H is singular.
The leading order solution Go satisfies
while for k  1,
where
- 548 -
and the right hand side Rk is given by
The nonlinear ODE of Go with asymptotic condition has been studied in
[16] and computational evidence suggested clusters of singularities ~s, were
For a rigorous singularity analysis of (3.12) we now used transasymptotic
matching as developed for ODEs [14]. The asymptotic behavior of Go is of
the form
where
(where C is the Stokes constant) and U(03B6) satisfies algebraic equation
Singularities of U(03B6) occur at 03B6s = In 4 - 2 - i7r, corresponding for n E N
large to
The Theorem that we prove in [2] is that For a singularity s of U 2
there exists a domain D around the singularity 3 such that the expansion is
convergent for small T.
In particular, for small T, the singularity of G(~, T) =- H(x, t) approaches
the singularity of U and is, to the leading order, of the same type, (~-~s)2/3.
(2) with |s| large enough and with arg s close to the anti-Stokes line arg ~ = - 403C0 9
(3) that extends to oo with arg 1] ~ (-203C0 9 + 03B4, 203C0 9 - 8) for some 9 &#x3E; 03B4 &#x3E; 0, and
includes a region S around the the singularity ils but excludes an open neighborhood.
- 549 -
Bibliography
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higher-order partial differential equations in the complex plane, Comm. Pure Appl.
Math, Vol. LIII, p. 1092-1117 (2000).4
[2] COSTIN (O.), TANVEER (S.). - Complex Singularity Analysis for a nonlinear PDE
(in preparation)4.
[3] COSTIN (O.), TANVEER (S.). 2014 Nonlinear evolution PDEs in R+  Cd: existence
and uniqueness of solutions, asymptotic and Borel summability (submitted).4
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(4) http://math.rutgers.edu/~costin/


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$\mathcal {D}$ -enveloppe d’un difféomorphisme de $(\mathbb {C},0)$