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

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Escobedo, M. and Herrero, Miguel A. and Velázquez, J.J. L. (1998) A nonlinear Fokker-Planck equation modelling the approach to thermal equilibrium in a homogeneous plasma. Transactions of the American Mathematical Society, 350 (10). pp. 3837-3901. ISSN 0002-9947

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Official URL: http://www.ams.org/journals/tran/1998-350-10/S0002-9947-98-02279-X/S0002-9947-98-02279-X.pdf


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Abstract

This work deals with the problem consisting in the equation (1) partial derivative f/partial derivative t = 1/x(2) partial derivative/partial derivative x [x(4)(partial derivative f/partial derivative x + f + f(2))], when x is an element of (0, infinity), t > 0, together with no-flux conditions at x = 0 and x = +infinity, i.e. (2) x(4)( partial derivative f/partial derivative x + f + f(2))=0 as x --> 0 or x --> +infinity. 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) = integral(0)(+infinity) 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.


Item Type:Article
Uncontrolled Keywords:Blow-up; Bose-Einstein distribution; flux condition
Subjects:Sciences > Mathematics > Differential equations
ID Code:16863
Deposited On:25 Oct 2012 08:30
Last Modified:18 Feb 2019 13:05

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