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

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.