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

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- by M. Escobedo, M. A. Herrero and J. J. L. Velazquez
- Trans. Amer. Math. Soc.
**350**(1998), 3837-3901 - DOI: https://doi.org/10.1090/S0002-9947-98-02279-X
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## 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.## References

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## Bibliographic Information

**M. Escobedo**- Affiliation: Departamento de Matemáticas, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain
- Email: mtpesmam@lg.ehu.e
**M. A. Herrero**- Affiliation: Departamento de Matemáticas, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain
- Email: herrero@sunma4.mat.ucm.es
**J. J. L. Velazquez**- Affiliation: Departamento de Matemáticas, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain; Departamento de Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense, 28040 Madrid, Spain
- MR Author ID: 289301
- Email: velazque@sunma4.mat.ucm.es, velazque@sunma4.mat.ucm.es
- Received by editor(s): October 15, 1996
- Additional Notes: The first author was partially supported by DGICYT Grant PB93-1203 and EEC Contract ERB 4061 PL 95-0545

The second and third authors were partially supported by DGICYT Grant PB93-0438 and EEC Contract CHRX-CT-0413 - © Copyright 1998 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**350**(1998), 3837-3901 - MSC (1991): Primary 35K55, 35B40
- DOI: https://doi.org/10.1090/S0002-9947-98-02279-X
- MathSciNet review: 1491861