A nonlinear Fokker-Planck equation modelling the approach to thermal equilibrium in a homogeneous plasma

Abstract:

This work deals with the problem consisting in the equation \begin{equation*} \tag *{(1)} \frac {\partial f}{\partial t} = \frac {1}{x^{2}} \frac {\partial }{\partial x} [x^4 (\frac {\partial f}{\partial x} + f + f^2)], \quad \mathrm {when}\quad x\in (0,\infty ), t>0, \end{equation*} together with no-flux conditions at $x=0$ and $x=+\infty$, i.e. \begin{equation*}\tag *{(2)} x^4(\frac {\partial f}{\partial x} + f + f^2) = 0 \quad \mathrm {as}\quad x \longrightarrow 0 \quad \mathrm {or}\quad x \longrightarrow + \infty . \end{equation*} Such a problem arises as a kinetic approximation to describe the evolution of the radiation distribution $f(x, t)$ in a homogeneous plasma when radiation interacts with matter via Compton scattering. We shall prove that there exist solutions of $(1)$, $(2)$ which develop singularities near $x=0$ in a finite time, regardless of how small the initial number of photons $N(0)=\int _{0}^{+\infty }x^{2}f(x, 0)dx$ is. The nature of such singularities is then analyzed in detail. In particular, we show that the flux condition $(2)$ is lost at $x=0$ when the singularity unfolds. The corresponding blow-up pattern is shown to be asymptotically of a shock wave type. In rescaled variables, it consists in an imploding travelling wave solution of the Burgers equation near $x=0$, that matches a suitable diffusive profile away from the shock. Finally, we also show that, on replacing $(2)$ near $x=0$ as determined by the manner of blow-up, such solutions can be continued for all times after the onset of the singularity.