On complex singularity analysis for some linear partial differential equations in $\mathbb{C}^3$

We investigate the existence of local holomorphic solutions $Y$ of linear partial differential equations in three complex variables whose coefficients are singular along an analytic variety $\Theta$ in $\mathbb{C}^{2}$. The coefficients are written as linear combinations of powers of a solution $X$ of some first order nonlinear partial differential equation following an idea we have initiated in a previous work \cite{mast}. The solutions $Y$ are shown to develop singularities along $\Theta$ with estimates of exponential type depending on the growth's rate of $X$ near the singular variety. We construct these solutions with the help of series of functions with infinitely many variables which involve derivatives of all orders of $X$ in one variable. Convergence and bounds estimates of these series are studied using a majorant series method which leads to an auxiliary functional equation that contains differential operators in infinitely many variables. Using a fixed point argument, we show that these functional equations actually have solutions in some Banach spaces of formal power series.


Introduction
In this paper, we study a family of linear partial differential equations of the form (1) ∂ S w Y (t, z, w) = k∈S (a 1,k (t, z, w)∂ t ∂ k w Y (t, z, w) + a 2,k (t, z, w)∂ z ∂ k w Y (t, z, w) + a 3,k (t, z, w)∂ k w Y (t, z, w)) for given initial data ∂ j w Y (t, z, 0) = ϕ j (t, z), 0 ≤ j ≤ S − 1, where S is a subset of N 2 and S is an integer which satisfy the constraints (116). The coefficients a m,k (t, z, w) are holomorphic functions on some domain (D(0, r) 2 \ Θ) × D(0,w) where Θ is some analytic variety of D(0, r) 2 (where D(0, δ) denotes the disc centered at 0 in C with radius δ > 0) and the initial data ϕ j (t, z) are assumed to be holomorphic functions on the polydisc D(0, r) 2 .
In order to avoid cumbersome statements and tedious computations, the authors have chosen to restrict their study to equations (1) that involve at most first order derivatives with respect to t and z but the method proposed in this work can also be extended to higher order derivatives too.
In this work, we plan to construct holomorphic solutions of the problem (1) on (D(0, r) 2 \ Θ) × D(0,w) and we will give precise growth estimates for these solutions near the singular variety Θ of the coefficients a m,k (t, z, w) (Theorem 1).
There exists a huge literature on the study of complex singularities and analytic continuation of solutions to linear partial differential equations starting from the fundamental contributions of J. Leray in [12]. Many important results are known for singular initial data and concern equations with bounded holomorphic coefficients. In that context, the singularities of the solution are generally contained in characteristic hypersurfaces issued from the singular locus of the initial conditions. For meromorphic initial data, we may refer to [3], [8], [20], [21] and for more general ramified multivalued initial data, we may cite [9], [10], [22], [24]. In our framework, the initial data are assumed to be non singular and the coefficients of the equation now carry the singularities. To the best knowledge of the authors, few results have been worked out in that case. For instance, the research of so-called fuchsian singularities in the context of partial differential equations is widely developed, we provide [1], [2], [7], [18] as examples of references in this direction. It turns out that the situation we consider is actually close to a singular perturbation problem since the nature of the equation changes nearby the singular locus of it's coefficients.
This work is a continuation of our previous study [16]. In the paper [16], the authors focused on linear partial differential equations in C 2 . They have constructed local holomorphic solutions with a careful study of their asymptotic behaviour near the singular locus of the initial data. These initial data were chosen to be polynomial in t,z and a function u(t) satisfying some nonlinear differential equation of first order on some punctured disc D(t 0 , r) \ {t 0 } ⊂ C and owning an isolated singularity at t 0 which is either a pole or an algebraic branch point according to a result of P. Painlevé. Inspired by the classical tanh method introduced in [17], they have considered formal series solutions of the form (2) u(t, z) = l≥0 u l (t, z)(u(t)) l where u l are holomorphic functions on D(t 0 , r) × D where D ⊂ C is a small disc centered at 0. They have given suitable conditions for these series to be well defined and holomorphic for t in a sector S with vertex t 0 and moreover as t tends to t 0 the solutions u(t, z) are shown to carry at most exponential bounds estimates of the form C exp(M |t − t 0 | −µ ) for some constants C, M, µ > 0. In this work, the coefficients a m,k (t, z, w) are constructed as polynomials in some function X(t, z) with holomorphic coefficients in (t, z, w), where X(t, z) is now assumed to solve some nonlinear partial differential equation of first order and is asked to be holomorphic on a domain D(0, r) 2 \ Θ and to be singular along the analytic variety Θ. For some specific choice of X(t, z), one can build the coefficients a m,k (t, z, w) for instance to be some rational functions of (t, z) (see Example 1 of Section 2.1).
In our setting, one cannot achieve the goal only dealing with formal expansions involving the function X(t, z) like (2) since the derivatives of X(t, z) with respect to t or z cannot be expressed only in term of X(t, z). In order to get suitable recursion formulas, it turns out that we need to deal with series expansions that take into account all the derivatives of X(t, z) with respect to z. For this reason, the construction of the solutions will follow the one introduced in a recent work of H. Tahara and will involve Banach spaces of holomorphic functions with infinitely many variables.
In the paper [23], H. Tahara introduced a new equivalence problem connecting two given nonlinear partial differential equations of first order in the complex domain. He showed that the equivalence maps have to satisfy so called coupling equations which are nonlinear partial differential equations of first order but with infinitely many variables. It is worthwhile saying that within the framework of mathematical physics, spaces of functions of infinitely many variables play a fundamental role in the study of nonlinear integrable partial differential equations known as solitons equations as described in the theory of M. Sato. See [19] for an introduction.
The layout of the paper is a follows. In a first step described in Section 2.2, we construct formal series of the form solutions of some auxiliary non-homogeneous integro-differential equation (11) with polynomial coefficients in X(t, z). The coefficients φ α , α ≥ 0, are holomorphic functions on some polydisc in C α+3 that satisfy some differential recursion (Proposition 1). In Section 2.3, we establish a sequence of inequalities for the modulus of the differentials of arbitrary order of the functions φ α denoted ϕ α,n 0 ,n 1 ,(l h ) 0≤h≤α for all non-negative integers α, n 0 , n 1 , l h with 0 ≤ h ≤ α (Proposition 2). In the next section, we construct a sequence of coefficients ψ α,n 0 ,n 1 ,(l h ) 0≤h≤α which is larger than the latter sequence ϕ α,n 0 ,n 1 ,(l h ) 0≤h≤α ≤ ψ α,n 0 ,n 1 ,(l h ) 0≤h≤α for any non-negative integers α, n 0 , n 1 , l h with 0 ≤ h ≤ α and whose generating formal series satisfies some integro-differential functional equation (35) that involves differential operators with infinitely many variables (Propositions 3 and 4). The idea of considering recursions over the complete family of derivatives and the use of majorant series which lead to auxiliary Cauchy problems were already applied in former papers by the authors of this work, see [11], [13], [14], [15], [16].
In Section 3, we solve the functional equation (35) by applying a fixed point argument in some Banach space of formal series with infinitely many variables (Proposition 10). The definition of these Banach spaces (Definition 2) is inspired from formal series spaces introduced in our previous work [16]. The core of the proof is based on continuity properties of linear integrodifferential operators in infinitely many variables explained in Section 3.1 and constitutes the most technical part of the paper.
Finally, in Section 4, we prove the main result of our work. Namely, we construct analytic functions Y (t, z, w), solutions of (1) for the prescribed initial data, defined on sets K × D(0,w) for any compact set K ⊂ D(0, r) 2 \ Θ with precise bounds of exponential type in term of the maximum value of |X(t, z)| over K (Theorem 1). The proof puts together all the constructions performed in the previous sections. More precisely, for some specific choice of the non-homogeneous term in the equation (11), a formal solution (3) of (11) gives rise to a formal solution Y (t, z, w) of (1) with the given initial data that can be written as the sum of the integral ∂ −S w U (t, z, w) and a polynomial in w having the initial data ϕ j as coefficients. Owing to the fact that the generating series of the sequence ψ α,n 0 ,n 1 ,(l h ) 0≤h≤α , solution of (35), belongs to the Banach spaces mentioned above, we get estimates for the holomorphic functions φ α with precise bounds of exponential type in term of the radii of the polydiscs where they are defined, see (134). As a result, the formal solution U (t, z, w) is actually convergent for w near the origin and for (t, z) belonging to any compact set of D(0, r) \ Θ. Moreover, exponential bounds are achieved, see (135). The same properties then hold for Y (t, z, w).
2 Formal series solutions of linear integro-differential equations 2.1 Some nonlinear partial differential equation We consider the following nonlinear partial differential equation where d ≥ 2 is some integer, the coefficients a(t, z), a p (t, z) are holomorphic functions on some polydisc D(0, R ′ ) 2 ⊂ C 2 such that a d (t, z) is not identically equal to zero on D(0, R ′ ) 2 and the initial data f (z) is holomorphic on D(0, R ′ ) 2 . Notice that the problem (4) can be solved by using the classical method of characteristics which is described in some classical textbooks like [4], p. 118 or [6], p. 100. We make the assumption that (4) has a holomorphic solution X(t, z) on D(0, R ′ ) 2 \ Θ where Θ is some analytic variety of D(0, R ′ ) 2 (i.e. for any point M ∈ D(0, R ′ ) 2 , there exists a neighborhood U of M in D(0, R ′ ) 2 such that U ∩ Θ is the common zero locus of a finite set of holomorphic functions {f 1 , . . . , f m } on U for some integer m ≥ 1).

Example 1:
The solution of the problem where f (z) is some polynomial on C, writes Therefore, where f (z) is an holomorphic function on C, writes The solution X(t, z) is holomorphic on C 2 \ Θ where the singular variety Θ is the zero locus of one analytic function Θ = {(t, z) ∈ C 2 /1 − tf (exp(t)z) = 0}.

Composition series
Let X be as in the previous subsection. In the following, we choose a compact subset K 0 with non-empty interior of D(0, R) 2 \ Θ for some R < R ′ and we consider a real number ρ > 1 such that sup Let K K 0 be a compact set with non-empty interior Int(K). From the Cauchy formula, there exists a real number ν > 0 such that for all integers h ≥ 0. For all integers α ≥ 0, we denote I(α) = {0, . . . , α}. We consider a sequence of functions φ α (v 0 , v 1 , (u h ) h∈I(α) ) which are holomorphic and bounded on the polydisc D(0, R) 2 Π h∈I(α) D(0, ρ), for all α ≥ 0. We define the formal series in the w variable, For all α ≥ 0, we consider a holomorphic and bounded functionω α (v 0 , v 1 , (u h ) h∈I(α) ) on the product D(0, R ′ ) 2 Π h∈I(α) D(0, ρ). We define the formal series Let S be a finite subset of N and let S ≥ 1 be an integer which satisfies the property that for all k ∈ S. For all k ∈ S, m = 1, 2, 3, we consider holomorphic functions , for somew > 0, which are moreover polynomial with respect to u 0 of degree d m,k ≥ 0.

Majorant series and a functional equation with infinitely many variables
We keep the notations of the previous section and we introduce the following formal series: We also introduce the following linear operators acting on We stress the fact that although these operators act on their image does not have to belong to this space.

The lemma follows. ✷
We get that the inequality (49) follows from (52) together with (53). Finally, using similar arguments, one gets also the inequalities (50) and (51). ✷ In the next two propositions, we study norm estimates for linear operators acting on the Banach space G (ρ,V 0 ,V 1 ,(Ū h ) h≥0 ,W ) .

Proposition 7 Let a formal series
,W ) and the following inequality By definition, we get

Lemma 3 We have
By remembering (43) of Proposition 5, we deduce that
Then, there exists a constant C 8.1 > 0 (which is independent of ρ > 1) such that Then, there exists a constant C 8.2 > 0 (which is independent of ρ > 1) such that Proof 1) We show the first inequality (61). We expand By definition, we have Now, using Lemma 3, we deduce that In the next lemma, we give estimates for the coefficients of the series A j,α and |B 1,k,α |.
2) Now, we turn towards the estimates (63) which will follow from the same arguments as in 1). Using Lemma 3, we get that In the next lemma, we give estimates for the coefficients of the series B j,α and |B 2,k,α |.
Using the estimates (80), we deduce that .
Using the estimates (80), we deduce that .
In the next proposition, we solve a functional fixed point equation within the Banach spaces of formal series introduced in the previous subsection.
We are now in position to state the main result of our work.
Theorem 1 Let b m,k (t, z, u 0 , w) be the functions defined in (9) for m = 1, 2, 3 and k ∈ S. Let us assume that there exists b > 1 such that for all k ∈ S. For all 0 ≤ j ≤ S − 1, we consider functions ω j (t, z) which are assumed to be holomorphic and bounded on the product D(0, R ′ ) 2 . Then, there exist constants σ,W , C 12 > 0 such that the problem with initial data has a solution Y (t, z, w) which is holomorphic on Int(K) × D(0,W /2) and which fulfills the following estimates where ζ(b) = n≥0 1/(n + 1) b , for any compact set K ⊂ D(0, R) 2 \ Θ with non-empty interior Int(K) for some R < R ′ and any ρ > 1 which satisfies (5). We stress that the constants σ,W , C 12 > 0 do not depend neither on K nor on ρ > 1.