On Microlocal Smoothness of Solutions of First Order Nonlinear PDE

We study the microlocal smoothness of $C^2$ solutions $u$ of the first-order nonlinear partial differential equation\[u_t = f(x, t, u, u_x)\]where $f(x, t, \zeta_0, \zeta)$ is a complex-valued function which is $C^{\infty}$ in all the variables $(x, t, \zeta_0, \zeta)$ and holomorphic in the variables $(\zeta_0, \zeta)$. If the solution $u$ is $C^2 , \sigma \in \mathrm{Char}(L^u)$ and $\frac{1}{\sqrt{-1}} \sigma ([L^u , L^{\overline{u}}]) \lt 0$, then we show that $\sigma \notin WF(u)$. Here $WF(u)$ denotes the $C^{\infty}$ wave front set of $u$ and $\mathrm{Char}(L^u)$ denotes the characteristic set of the linearized operator\[L^u = \frac{\partial}{\partial t} - \sum^{m}_{j=1} \frac{\partial f}{\partial \zeta_j} (x, t, u, u_x) \frac{\partial}{\partial x_j} \textrm{ .}\]

Keywords

$C^{\infty}$ wave front set, linearized operator

2010 Mathematics Subject Classification

35A18, 35A21, 35A22, 42B10

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The author’s work was supported in part by ISP of Sweden.

Received 24 May 2016

Accepted 9 March 2017

Published 17 January 2018