Gevrey regularity and summability of the formal power series solutions of the inhomogeneous generalized Boussinesq equations

Pascal Remy (1)

1 LMV - Laboratoire de Mathématiques de Versailles

In this article, we investigate Gevrey and summability properties of the formal power series solutions of the inhomogeneous generalized Boussinesq equations. We first prove that the inhomogeneity and the formal solutions are together $s$-Gevrey for any $s\geq1$, and that the formal solutions are generically $1$-Gevrey while the inhomogeneity is $s$-Gevrey with $s<1$. In the latter case, we give in particular an explicit example in which the formal solution is $s'$-Gevrey for no $s'<1$, that is exactly $1$-Gevrey. Then, we give a necessary and sufficient condition under which the formal solutions are $1$-summable in a given direction $\arg(t)=\theta$. In addition, we present some technical results on the generalized binomial and multinomial coefficients, which are needed for the proofs of our various results.

Mots clés: Gevrey regularity; Summability; Inhomogeneous partial differential equation; Nonlinear partial differential equation; Formal power series; Divergent power series; Generalized Boussinesq equation

Domaines: Systèmes dynamiques [math.DS]

Pascal Remy. Gevrey regularity and summability of the formal power series solutions of the inhomogeneous generalized Boussinesq equations. Asymptotic Analysis, 2022, 131 (1), pp.1-32. ⟨10.3233/ASY-221764⟩. ⟨hal-03445913⟩