Gevrey micro-regularity for solutions to first order nonlinear PDE
In this interesting paper, the authors study the microlocal Gevrey regularity of \(C^2\) solutions to a first-order nonlinear PDE. More precisely, let \(f=f(x,t,\zeta_0,\zeta)\) be Gevrey of order \(s>1\) and holomorphic in the variables \((\zeta_0,\zeta)\). Consider a solution \(u\in C^2\) to \(u_t=f(x,t,u,u_x)\). Let \(L^u\) be the linearized operator \[ L^u=\frac{\partial}{\partial t}-\sum_{j=1}^m\frac{\partial f}{\partial\zeta_j} (x,t,u,u_x)\frac{\partial}{\partial x_j}. \] It is known after \textit{N. Hanges} and \textit{F. Trèves} [Trans. Am. Math. Soc. 331, No. 2, 627--638 (1992; Zbl 0758.35018)], that the analytic wave-front set of such \(u\) is contained in the characteristic set \(\mathrm{Char}(L^u)\) of \(L^u\). More recently, \textit{S. Berhanu} [Ann. Inst. Fourier 59, No. 4, 1267--1290 (2009; Zbl 1195.35011)] has given a precise result about the location in \(\mathrm{Char}(L^u)\) of the analytic wave-front set of a \(C^2\) (and \(C^3\)) solution \(u\). The authors show here that in the Gevrey case, any given \(C^2\) solution has Gevrey wave-front set contained in \(\mathrm{Char}(L^u)\). They achieve this result through the study of the notion of \textit{Gevrey approximate solutions}, a tool that very likely will be useful to deal with a broader range of situations.
scientific article; zbMATH DE number 5608032