Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-98-02279-X
MathSciNet review: 1491861
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [AE] J. Aguirre and M. Escobedo, On the blow-up of solutions of a convective reaction-diffusion equation, Proc. Royal Soc. Edinburgh 123A, (1993), pp. 433-460. MR 94d:35076
  • [A] D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Normale Sup. Pisa (3) 22 (1968), pp. 607-694. MR 55:8553, MR 55:8554
  • [CL] R. E. Caflisch and C. D. Levermore, Equilibrium for radiation in a homogeneous plasma, Phys. Fluids 29 (1986), pp. 748-752. MR 86c:76140 .
  • [HV1] M. A. Herrero and J. J. L. Velazquez, Blow-up behaviour of one-dimensional semilinear parabolic problems, Ann. Inst. Henri Poincaré, 10 (1993), pp. 131-189. MR 94g:35030
  • [HV2] M. A. Herrero and J. J. L. Velazquez, Generic behaviour of one-dimensional blow-up patterns, Ann. Scuola Normale Sup. Pisa (4) 19 (1992), pp. 381-450. MR 94b:35048
  • [HV3] M. A. Herrero and J. J. L. Velazquez, On the melting of ice balls, SIAM J. Math. Analysis 28 (1997), 1-32. MR 97m:35287
  • [K] A. S. Kompaneets, The establishment of thermal equilibrium between quanta and electrons, Soviet Physics JETP, 4, (1957), pp. 730-737.
  • [KL] O. Kavian and C. D. Levermore, On the Kompaneets Equation, a singular semi-linear parabolic equation with blow-up. In preparation.
  • [NT] R. Natalini and A. Tesei, Blow-up of solutions for a class of balance laws, Comm. Part. Diff. Eq., 19 (1994), pp. 417-453. MR 95a:35087
  • [V1] J. J. L. Velazquez, Classification of singularities for blowing up solutions in higher dimensions, Trans. Amer. Math. Soc. 338 (1993), pp. 441-464. MR 93j:35101
  • [V2] J. J. L. Velazquez, Curvature blow-up in perturbations of minimal cones evolving by mean curvature flow, Ann. Scuola Normale Sup. Pisa (4) 21 (1994), pp. 595-628. MR 96f:53018

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 35K55, 35B40

Retrieve articles in all journals with MSC (1991): 35K55, 35B40


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

American Mathematical Society