[The local Cauchy problem for dissipative plasmas]
We investigate a system of partial differential equations modeling dissipative plasmas. Transport fluxes are anisotropic linear combinations of gradients and also include zeroth order contributions due to electromagnetic forces. There are also source terms depending on the solution gradient. By using entropic variables, we first recast the system in a partially symmetric form and next in the form of a quasilinear partially symmetric hyperbolic-parabolic system. Using a result of Vol'Pert and Hudjaev, we prove local existence and uniqueness of a bounded smooth solution to the Cauchy problem.
On étudie un système d'équations aux dérivées partielles modélisant les plasmas réactifs dissipatifs. Les flux de transport comprennent des combinaisons linéaires anisotropes des gradients et des termes d'ordre zéro dus au champ électromagnétique et les termes sources dépendent des gradients. En utilisant les variables entropiques, on récrit le système de lois de conservation sous une forme partiellement symétrique, puis sous la forme d'un système quasi-linéaire partiellement symétrique hyperbolique-parabolique. En utilisant un résultats de Vol'Pert et Hudjaev, on démontre un théorème local d'existence et d'unicité d'une solution bornée et régulière pour le problème de Cauchy.
Accepted:
Published online:
Vincent Giovangigli 1; Benjamin Graille 1
@article{CRMATH_2005__340_2_119_0, author = {Vincent Giovangigli and Benjamin Graille}, title = {Le probl\`eme de {Cauchy} local pour les plasmas dissipatifs}, journal = {Comptes Rendus. Math\'ematique}, pages = {119--124}, publisher = {Elsevier}, volume = {340}, number = {2}, year = {2005}, doi = {10.1016/j.crma.2004.12.009}, language = {fr}, }
Vincent Giovangigli; Benjamin Graille. Le problème de Cauchy local pour les plasmas dissipatifs. Comptes Rendus. Mathématique, Volume 340 (2005) no. 2, pp. 119-124. doi : 10.1016/j.crma.2004.12.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.12.009/
[1] Mathematical Theory of Transport Processes in Gases, North-Holland, 1972
[2] Multicomponent Flow Modeling, Birkhäuser, 1999
[3] Kinetic theory of partially ionized reactive gas mixtures, Physica A, Volume 327 (2003), pp. 313-348
[4] Asymptotic stability of equilibrium states for ambipolar plasmas, Math. Mod. Meth. Appl. Sci., Volume 14 (2004) no. 9, pp. 1361-1399
[5] V. Giovangigli, B. Graille, The local Cauchy problem for ionized magnetized reactive gas mixtures, Internal Report 532, Centre de Mathématiques Appliquées, Ecole Polytechnique, 2004
[6] The local Cauchy problem for multicomponent reactive flows in full vibrational non-equilibrium, Math. Meth. Appl. Sci., Volume 21 (1998), pp. 1415-1469
[7] Asymptotic stability of equilibrium states for multicomponent reactive flows, Math. Mod. Meth. Appl. Sci., Volume 8 (1998), pp. 251-297
[8] B. Graille, Modélisation de mélanges gazeux réactifs ionisés dissipatifs, Doctoral Thesis, Ecole Polytechnique, 2004
[9] S. Kawashima, Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics, Doctoral Thesis, Kyoto University, 1984
[10] On the normal form of the symmetric hyperbolic-parabolic systems associated with the conservation laws, Tôhoku Math. J., Volume 40 (1988), pp. 449-464
[11] On the Cauchy problem for composite systems of nonlinear differential equations, Math. USSR-Sb., Volume 16 (1972), pp. 517-544
Cited by Sources:
Comments - Policy