Approximate solutions of vector fields and an application to Denjoy-Carleman regularity of solutions of a nonlinear PDE

2018/09/18 by Nicholas Braun Rodrigues, Rodrigues, Nicholas Braun, Antonio V. da Silva +1 · 1 citation
Mathematics · #Advanced Banach Space Theory #Analysis of PDEs (math.AP) #FOS: Mathematics #Holomorphic and Operator Theory #Nonlinear Differential Equations Analysis

paper · pdf · doi:10.48550/arxiv.1809.06803

openalex publication_date 2018/09/18 · openalex created_date 2018/09/27 · openalex updated_date 2026/07/28

Abstract

In this paper we study microlocal regularity of a C² solution u of the equation u_t = f(x,t,u,u_x), where f(x,t,ζ₀, ζ) is ultradifferentiable in the variables (x,t)∈ ℝ^N × ℝ and holomorphic in the variables (ζ₀,ζ) ∈ ℂ × ℂ^N. We proved that if C^M is a regular Denjoy-Carleman class (including the quasianalytic case) then: WF_M (u)⊂ Char(L^u), where WF_M(u) is the Denjoy-Carleman wave-front set of u and Char(L^u) is the characteristic set of the linearized operator L^u: L^u = \dfrac∂∂ t - ∑_j=1^N(∂ f)/(∂ζ_j)(x,t,u,u_x)\dfrac∂∂ x_j.

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• Regularity of infinitesimal automorphisms of involutive structures

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