On the analyticity of solutions of first-order nonlinear PDE

Abstract

Abstract.Let (x, r) e Rm x R and u e C2(Rm x R). We discuss local and microlocal analyticity for solutions u to the nonlinear equation ut = f(x, t, u, ux).Here f(x, /, fo . 0 's complex valued and analytic in all arguments. We also assume / to be holomorphic in (Co, C) € C x Cm . In particular we show that WF^ u c Char(/_") where WF^ denotes the analytic wave-front set and Char(L") is the characteristic set of the linearized operatorIf we assume u 6 C3(Äm x R) then we show that the analyticity of u propagates along the elliptic submanifolds of Lu .

Système d'Euler incompressible et régularité microlocale analytique

“…Et on retrouve dans ce cas, de manière simple, le résultat d'Hanges et Trêves (voir [8]). .^-<^,,)(a^)).,.…”

On the 𝐶^{∞} wave-front set of solutions of first-order nonlinear PDEs

“…In this article we present a simple proof of a result found in Chemin [ 1 ] that, under the hypothesis above, the C°° wave-front set of u is contained in the characteristic set of the linearized operator LU = m-^{céJ){x^'u^Ux)dx--In Hanges and Trêves [2] it is shown that under the additional hypothesis that / be analytic in the variables (x, /) the analytic wave-front set of u is contained in the characteristic set of the linearized operator Lu. Our proof follows the general lines of Hanges and Trêves [2] and can be used to simplify their argument as well.…”

“…In this section we will follow very closely §2 of Hanges and Trêves [2]. Let Q C Rm+X be a neighborhood of the origin and suppose u £ C2(Q) is a solution Consider now the principal part of the holomorphic Hamiltonian of (7) We have that (7) implies (cf.…”

“…Our proof follows the general lines of Hanges and Trêves [2] and can be used to simplify their argument as well.…”

On microlocal analyticity of solutions of first-order nonlinear PDE

“…The variable x varies in an open subset of R m , t in an interval in R, and (ζ 0 , ζ) varies in an open subset of C m+1 . When u is a C 2 solution of (1.1), it was proved in [7] that the analytic wave-front set of u is contained in the characteristic set of the linearized operator…”