Asymptotic expansions and summability. Application to partial differential equations (Q6553921)

The book under review describes main results on the theory of asymptotic expansions and summability of formal solutions to partial differential equations in the complex domain. It is divided in three parts:\N\NPart I: In this first part, the author describes the formal power series which appear as formal solutions to partial differential equations. More precisely, formal power series in \(t\) variable with coefficients being holomorphic functions on a polydisc at the origin are considered.\N\NThe classical notion of Gevrey formal power series and Gevrey asymptotics is adapted to this framework, together with the main algebraic properties of the set of such formal power series, some important tools such as Newton polygon, and their natural appearance when solving partial differential equations.\N\NRegarding Gevrey asymptotics, the main classical properties are provided: characterization in terms of derivatives, algebraic structure, flat functions, Borel-Ritt-Gevrey theorem and Watson's lemma.\N\NPart II: The second part of the work is devoted to the \(k\)-summability of formal solutions to partial differential equations in the complex domain. After recalling the main algebraic properties associated to this notion, the author provides two characterizations to the \(k\)-summability of formal power series (as considered in the first part of the book). The first characterization is related to the successive derivatives of a function, whereas the second characterization is given in terms of the Borel-Laplace method. In the latter case, an adapted version of Nevanlinna's Theorem is provided.\N\NIn both cases, the application of the theory to different problems is shown.\N\NPart III: The theory described in the previous parts is generalized to the so-called moment partial differential equations, generalizing partial differential equations. Summability results are described in terms of kernel functions of some Gevrey order, and associated (formal and analytic) Borel and Laplace operators. The concrete application of the theory to concrete equations is described in detail.\N\N\NThe results described in the text are illustrated with many applications to concrete partial differential equations such as the heat equation, the Burgers equation, etc. and also to applications in physics. Moreover, the author provides references of the main results and applications for a deepened reading.