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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: M. Escobedo, M. A. Herrero and J. J. L. Velazquez
Journal: Trans. Amer. Math. Soc. 350 (1998), 3837-3901
MSC (1991): Primary 35K55, 35B40
MathSciNet review: 1491861
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Abstract: This work deals with the problem consisting in the equation

\begin{equation*}{\frac{\partial f}{\partial t}} ={\frac{1}{x^{2}}}{\frac{\partial }{\partial x}} [x^{4}({\frac{\partial f}{\partial x}}+f+f^{2})], \quad \hbox {when}\quad x\in (0,\infty ), t>0, \tag*{(1)}\end{equation*}

together with no-flux conditions at $x=0$ and $x=+\infty $, i.e.

\begin{equation*}x^{4}({\frac{\partial f}{\partial x}}+f+f^{2})=0\quad \hbox {as}\hskip 0.2cm x\mathop{\longrightarrow } 0 \hskip 0.2cm\hbox {or}\hskip 0.2cmx\mathop{\longrightarrow } +\infty . \tag*{(2)}\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.


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Additional 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
Email: velazque@sunma4.mat.ucm.es, velazque@sunma4.mat.ucm.es

DOI: http://dx.doi.org/10.1090/S0002-9947-98-02279-X
PII: S 0002-9947(98)02279-X
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
Article copyright: © Copyright 1998 American Mathematical Society