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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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.
|Uncontrolled Keywords:||Blow-up; Bose-Einstein distribution; flux condition|
|Subjects:||Sciences > Mathematics > Differential equations|
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.
D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Normale Sup. Pisa (3) 22 (1968), pp. 607-694.
R. E. Caflisch and C. D. Levermore, Equilibrium for radiation in a homogeneous plasma, Phys. Fluids 29 (1986), pp. 748-752.
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.
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.
M. A. Herrero and J. J. L. Velazquez, On the melting of ice balls, SIAM J. Math. Analysis 28 (1997), 1-32.
A. S. Kompaneets, The establishment of thermal equilibrium between quanta and electrons, Soviet Physics JETP, 4,(1957), pp. 730-737.
O. Kavian and C. D. Levermore, On the Kompaneets Equation, a singular semi-linear parabolic equation with blow-up. In preparation.
R. Natalini and A. Tesei, Blow-up of solutions for a class of balance laws, Comm. Part. Diff. Eq., 19 (1994), pp. 417-453.
J. J. L. Velazquez, Classiffication of singularities for blowing up solutions in higher dimensions, Trans. Amer. Math. Soc. 338 (1993), pp. 441-464.
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.
|Deposited On:||25 Oct 2012 10:30|
|Last Modified:||22 Nov 2013 19:27|
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