On microlocal analyticity of solutions of first-order nonlinear PDE

Berhanu, Shif. "On microlocal analyticity of solutions of first-order nonlinear PDE." Annales de l’institut Fourier 59.4 (2009): 1267-1290. <http://eudml.org/doc/10427>.

@article{Berhanu2009,
abstract = {We study the microlocal analyticity of solutions $u$ of the nonlinear equation\[ u\_\{t\} = f(x, t, u, u\_\{x\}) \]where $f(x, t,\zeta _\{0\}, \zeta )$ is complex-valued, real analytic in all its arguments and holomorphic in $(\zeta _\{0\}, \zeta )$. We show that if the function $u$ is a $C^2$ solution, $\sigma \in \text\{Char\}\,L^\{u\}$ and $\frac\{1\}\{i\}\sigma ([L^\{u\}, \overline\{L^u\}]) &lt; 0$ or if $u$ is a $C^3$ solution, $\sigma \in \text\{Char\}\,L^\{u\}$, $\sigma ([L^\{u\}, \overline\{L^u\}]) = 0$, and $ \sigma ([L^u,[L^\{u\},\overline\{L^u\}]]) \ne 0$, then $\sigma \notin WF_\{a\}u$. Here $WF_\{a\}u $ denotes the analytic wave-front set of $u$ and Char$\,L^u$ is the characteristic set of the linearized operator. When $m=1$, we prove a more general result involving the repeated brackets of $L^u$ and $\overline\{L^u\}$ of any order.},
affiliation = {Temple University Department of Mathematics Philadelphia, PA 19122 (USA)},
author = {Berhanu, Shif},
journal = {Annales de l’institut Fourier},
keywords = {Analytic wave-front set; linearized operator; analytic wave-front set},
language = {eng},
number = {4},
pages = {1267-1290},
publisher = {Association des Annales de l’institut Fourier},
title = {On microlocal analyticity of solutions of first-order nonlinear PDE},
url = {http://eudml.org/doc/10427},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Berhanu, Shif
TI - On microlocal analyticity of solutions of first-order nonlinear PDE
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 4
SP - 1267
EP - 1290
AB - We study the microlocal analyticity of solutions $u$ of the nonlinear equation\[ u_{t} = f(x, t, u, u_{x}) \]where $f(x, t,\zeta _{0}, \zeta )$ is complex-valued, real analytic in all its arguments and holomorphic in $(\zeta _{0}, \zeta )$. We show that if the function $u$ is a $C^2$ solution, $\sigma \in \text{Char}\,L^{u}$ and $\frac{1}{i}\sigma ([L^{u}, \overline{L^u}]) &lt; 0$ or if $u$ is a $C^3$ solution, $\sigma \in \text{Char}\,L^{u}$, $\sigma ([L^{u}, \overline{L^u}]) = 0$, and $ \sigma ([L^u,[L^{u},\overline{L^u}]]) \ne 0$, then $\sigma \notin WF_{a}u$. Here $WF_{a}u $ denotes the analytic wave-front set of $u$ and Char$\,L^u$ is the characteristic set of the linearized operator. When $m=1$, we prove a more general result involving the repeated brackets of $L^u$ and $\overline{L^u}$ of any order.
LA - eng
KW - Analytic wave-front set; linearized operator; analytic wave-front set
UR - http://eudml.org/doc/10427
ER -