A nonlinear FokkerPlanck 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), 38373901
MSC (1991):
Primary 35K55, 35B40
MathSciNet review:
1491861
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: This work deals with the problem consisting in the equation together with noflux conditions at and , i.e. Such a problem arises as a kinetic approximation to describe the evolution of the radiation distribution in a homogeneous plasma when radiation interacts with matter via Compton scattering. We shall prove that there exist solutions of , which develop singularities near in a finite time, regardless of how small the initial number of photons is. The nature of such singularities is then analyzed in detail. In particular, we show that the flux condition is lost at when the singularity unfolds. The corresponding blowup 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 , that matches a suitable diffusive profile away from the shock. Finally, we also show that, on replacing near as determined by the manner of blowup, such solutions can be continued for all times after the onset of the singularity.
 [AE]
J.
Aguirre and M.
Escobedo, On the blowup of solutions of a convective reaction
diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A
123 (1993), no. 3, 433–460. MR 1226611
(94d:35076), http://dx.doi.org/10.1017/S0308210500025828
 [A]
D.
G. Aronson, Nonnegative solutions of linear parabolic
equations, Ann. Scuola Norm. Sup. Pisa (3) 22 (1968),
607–694. MR 0435594
(55 #8553)
D.
G. Aronson, Addendum: “Nonnegative solutions of linear
parabolic equations”\ (Ann. Scuola Norm. Sup. Pisa (3) 22 (1968),
607–694), Ann. Scuola Norm. Sup. Pisa (3) 25
(1971), 221–228. MR 0435595
(55 #8554)
 [CL]
R. E. Caflisch and C. D. Levermore, Equilibrium for radiation in a homogeneous plasma, Phys. Fluids 29 (1986), pp. 748752. MR 86c:76140 .
 [HV1]
M.
A. Herrero and J.
J. L. Velázquez, Blowup behaviour of onedimensional
semilinear parabolic equations, Ann. Inst. H. Poincaré Anal.
Non Linéaire 10 (1993), no. 2, 131–189
(English, with English and French summaries). MR 1220032
(94g:35030)
 [HV2]
M.
A. Herrero and J.
J. L. Velázquez, Generic behaviour of onedimensional blow
up patterns, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)
19 (1992), no. 3, 381–450. MR 1205406
(94b:35048)
 [HV3]
Miguel
A. Herrero and Juan
J. L. Velázquez, On the melting of ice balls, SIAM J.
Math. Anal. 28 (1997), no. 1, 1–32. MR 1427725
(97m:35287), http://dx.doi.org/10.1137/S0036141095282152
 [K]
A. S. Kompaneets, The establishment of thermal equilibrium between quanta and electrons, Soviet Physics JETP, 4, (1957), pp. 730737.
 [KL]
O. Kavian and C. D. Levermore, On the Kompaneets Equation, a singular semilinear parabolic equation with blowup. In preparation.
 [NT]
R.
Natalini and A.
Tesei, Blowup of solutions for a class of balance laws, Comm.
Partial Differential Equations 19 (1994), no. 34,
417–453. MR 1265806
(95a:35087), http://dx.doi.org/10.1080/03605309408821023
 [V1]
J.
J. L. Velázquez, Classification of singularities for
blowing up solutions in higher dimensions, Trans. Amer. Math. Soc. 338 (1993), no. 1, 441–464. MR 1134760
(93j:35101), http://dx.doi.org/10.1090/S00029947199311347602
 [V2]
J.
J. L. Velázquez, Curvature blowup in perturbations of
minimal cones evolving by mean curvature flow, Ann. Scuola Norm. Sup.
Pisa Cl. Sci. (4) 21 (1994), no. 4, 595–628. MR 1318773
(96f:53018)
 [AE]
 J. Aguirre and M. Escobedo, On the blowup of solutions of a convective reactiondiffusion equation, Proc. Royal Soc. Edinburgh 123A, (1993), pp. 433460. MR 94d:35076
 [A]
 D. G. Aronson, Nonnegative solutions of linear parabolic equations, Ann. Scuola Normale Sup. Pisa (3) 22 (1968), pp. 607694. 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. 748752. MR 86c:76140 .
 [HV1]
 M. A. Herrero and J. J. L. Velazquez, Blowup behaviour of onedimensional semilinear parabolic problems, Ann. Inst. Henri Poincaré, 10 (1993), pp. 131189. MR 94g:35030
 [HV2]
 M. A. Herrero and J. J. L. Velazquez, Generic behaviour of onedimensional blowup patterns, Ann. Scuola Normale Sup. Pisa (4) 19 (1992), pp. 381450. MR 94b:35048
 [HV3]
 M. A. Herrero and J. J. L. Velazquez, On the melting of ice balls, SIAM J. Math. Analysis 28 (1997), 132. MR 97m:35287
 [K]
 A. S. Kompaneets, The establishment of thermal equilibrium between quanta and electrons, Soviet Physics JETP, 4, (1957), pp. 730737.
 [KL]
 O. Kavian and C. D. Levermore, On the Kompaneets Equation, a singular semilinear parabolic equation with blowup. In preparation.
 [NT]
 R. Natalini and A. Tesei, Blowup of solutions for a class of balance laws, Comm. Part. Diff. Eq., 19 (1994), pp. 417453. 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. 441464. MR 93j:35101
 [V2]
 J. J. L. Velazquez, Curvature blowup in perturbations of minimal cones evolving by mean curvature flow, Ann. Scuola Normale Sup. Pisa (4) 21 (1994), pp. 595628. 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:
http://dx.doi.org/10.1090/S000299479802279X
PII:
S 00029947(98)02279X
Received by editor(s):
October 15, 1996
Additional Notes:
The first author was partially supported by DGICYT Grant PB931203 and EEC Contract ERB 4061 PL 950545
The second and third authors were partially supported by DGICYT Grant PB930438 and EEC Contract CHRXCT0413
Article copyright:
© Copyright 1998
American Mathematical Society
