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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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 PDF
Trans. Amer. Math. Soc. 350 (1998), 3837-3901 Request permission

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.
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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
  • 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