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

DOI:
https://doi.org/10.1090/S0002-9947-98-02279-X

MathSciNet review:
1491861

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Abstract | References | Similar Articles | Additional Information

Abstract: This work deals with the problem consisting in the equation

together with no-flux conditions at and , i.e.

Such a problem arises as a kinetic approximation to describe the evolution of the radiation distribution in a homogeneous plasma when radiation interacts with matter via Compton scattering. We shall prove that there exist solutions of , which develop singularities near in a finite time, regardless of how small the initial number of photons is. The nature of such singularities is then analyzed in detail. In particular, we show that the flux condition is lost at 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 , that matches a suitable diffusive profile away from the shock. Finally, we also show that, on replacing near 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:
https://doi.org/10.1090/S0002-9947-98-02279-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