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

Abstract

In this article, we investigate Gevrey and summability properties of the formal power series solutions of the inhomogeneous generalized Boussinesq equations. Even if the case that really matters physically is an analytic inhomogeneity, we systematically examine here the cases where the inhomogeneity is s-Gevrey for any s ⩾ 0, in order to carefully distinguish the influence of the data (and their degree of regularity) from that of the equation (and its structure). We thus prove that we have a noteworthy dichotomy: for any s ⩾ 1, the formal solutions and the inhomogeneity are simultaneously s-Gevrey; for any s < 1, the formal solutions are generically 1-Gevrey. 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 ) = θ. In addition, we present some technical results on the generalized binomial and multinomial coefficients, which are needed for the proofs of our various results.