| Contents |
| When he derived the equations
$$
\partial_t F + v\cdot \nabla_x F =\mathcal B(F,F)
$$
Boltzmann identified special solutions which are both Maxwellian and which are solution of the advection equation:
$$
F= F(v,x-vt)\Rightarrow \mathcal B (F,F)=0 \quad \hbox{ and } \partial_t F+v\cdot \nabla F=0
$$
Dave Levermore dubbed these l solutions ``global Maxwellian" and identified them by their global moments.
In this talk I want to show that the global Maxwellian generate global solutions both of the compressible Euler and Navier Stokes equations.
Their stability and scattering properties can be studied following classical contributions of Kaniel-Shinbrot \cite{KS} or Wei and Zhang \cite{WZ}.
And eventually this perturbation analysis produces eternall solutions of the Boltzmann equation which do not coincide with global Maxwellian
\begin{thebibliography}{}
\bibitem{KS} S. Kaniel and M. Shinbrot,
{\em The Boltzmann Equation I: Uniqueness and Local Existence},
Commun. Math. Phys. {\bf 58} (1978), 65--84.
\bibitem{WZ}J. Wei and X. Zang
{\em Global Solution of the Initial Value Problem for the}
{\em Boltzmann Equation near a Local Maxwellian},
Arch. Rational Mech \& Anal. {\bf 102} (1988), 231--241.
\end{thebibliography}{} |
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