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Approximate solutions of vector fields and an application to Denjoy–Carleman regularity of solutions of a nonlinear PDE
Mathematische Nachrichten ( IF 1 ) Pub Date : 2021-07-31 , DOI: 10.1002/mana.201800516
Nicholas Braun Rodrigues 1 , Antonio Victor da Silva 2
Affiliation  

In this paper we study microlocal regularity of a C 2 -solution u of the equation
u t = f ( x , t , u , u x ) ,
where f ( x , t , ζ 0 , ζ ) is ultradifferentiable in the variables ( x , t ) R N × R and holomorphic in the variables ( ζ 0 , 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 = t j = 1 N f ζ j ( x , t , u , u x ) x j .


中文翻译:

矢量场的近似解及其在非线性偏微分方程解的 Denjoy-Carleman 正则性中的应用

在本文中,我们研究了一个 C 2 -方程的解u
= F ( X , , , X ) ,
在哪里 F ( X , , ζ 0 , ζ ) 在变量中是超微分的 ( X , ) 电阻 N × 电阻 和全纯变量 ( ζ 0 , ζ ) C × C N . 我们证明了如果 C 是常规的 Denjoy-Carleman 类(包括拟分析情况),则:
WF ( ) 字符 ( ) ,
在哪里 WF ( ) u和的 Denjoy-Carleman 波前集 字符 ( ) 是线性化算子的特征集
= - j = 1 N F ζ j ( X , , , X ) X j .
更新日期:2021-09-16
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