Analysis of bifurcation points and nontrivial branches of solutions to the stationary Vlasov--Maxwell system

N. A. Sidorov ; A. V. Sinitsyn

Matematičeskie zametki, Tome 62 (1997) no. 2, pp. 268-292.

Résumé

For the Vlasov–Maxwell system, sufficient conditions are obtained for the existence of bifurcation points $\lambda _0\in \mathbb R^+$ corresponding to distribution functions of the form $$ f_i(r,v) =\lambda\widehat f_i\bigl(-\alpha_iv^2+\varphi_i(r), vd_i+\psi_i(r)\bigr). $$ It is assumed that the values of the scalar and vector potentials of the electromagnetic field are prescribed at the boundary of the domain $D\subset\mathbb R^3$ in the form $\rho|_{\partial D}=0$, $j|_{\partial D}=0$, where $\rho$ is the charge density and $j$ is the current density. The bifurcation equation is derived and studied for the solutions. The asymptotics of nontrivial solutions of the Vlasov–Maxwell system is constructed in a neighborhood of the bifurcation point.

Bibliographie

[1] Glassey R., Strauss W., “Singularity formation in a collisionless plasma could occur only at high velocities”, Arch. Rational Mech. Anal., 92 (1986), 59–90 | DOI | MR | Zbl

[2] Schaeffer J., “The classical limit of the relativistic Vlasov–Maxwell system”, Comm. Math. Phys., 104 (1986), 403–421 | DOI | MR | Zbl

[3] Glassey R., Strauss W., “Absence of shocks in an initially dilute collisionless plasma”, Comm. Math. Phys., 113 (1987), 191–208 | DOI | MR | Zbl

[4] Glassey R., Schaeffer J., “Global existence for the relativistic Vlasov–Maxwell system with nearly neutral initial data”, Comm. Math. Phys., 119 (1988), 353–384 | DOI | MR | Zbl

[5] Rein G., “Generic global solutions of the relativistic Vlasov–Maxwell system of plasma physics”, Comm. Math. Phys., 135 (1990), 41–78 | DOI | MR | Zbl

[6] Guo Y., “Global weak solutions of the Vlasov–Maxwell system with boundary conditions”, Comm. Math. Phys., 154 (1993), 245–263 | DOI | MR | Zbl

[7] Abdallah N. B., “Weak solutions of the initial-boundary value problem for the Vlasov–Poisson system”, Math. Methods Appl. Sci., 17 (1994), 451–476 | DOI | MR | Zbl

[8] Vedenyapin V. V., “O klassifikatsii statsionarnykh reshenii uravneniya Vlasova na tore i kraevaya zadacha”, Dokl. RAN, 323:6 (1992), 1004–1006 | MR | Zbl

[9] Dolbeault J., “Classification des solutions stationnaires axisymétriques régulières du système de Vlasov–Maxwell”, Math. Probl. Mech., 1995 (to appear)

[10] Greengard C., Raviart P. A., “A boundary-value problem for the stationary Vlasov–Poisson equations. The plane diode”, Comm. Pure Appl. Math., 43 (1990), 473–507 | DOI | MR | Zbl

[11] Gogny D., Lions P. L., “Sur les états d'équilibre pour les densités éléctroniques dans les plasmas”, Math. Model. Numer. Anal., 23:1 (1989), 137–153 | MR | Zbl

[12] Degond P., The Child–Langmuir flow in the kinetic theory of charged particles. 1: Electron flows in vacuum, Prépubl. Mathématiques pour l'Industrie et la Physique C.N.R.S. UFR MIG, Univ. Paul Sabatier, France, 1994 | Zbl

[13] Weckler J., “The Vlasov–Poisson system on a bounded domain”, International Conf. “Nonlinear Equations in Many-Particle Systems”, Abstr., Mathematisches Forschungsinstitut, Oberwolfach, 1993, 12

[14] Maslov V. P., “Ob integralnom uravnenii vida $u(x)=F(x)+\int G(x,\xi)u^{k/2}_+(\xi)\,d\xi\big/\int u^{k/2}_+(\xi)\,d\xi$”, Funktsion. analiz i ego prilozh., 28:1 (1994), 41–50 | MR | Zbl

[15] Trenogin V. A., Funktsionalnyi analiz, Nauka, M., 1980 | Zbl

[16] Vainberg M. M., Trenogin V. A., Teoriya vetvleniya reshenii nelineinykh uravnenii, Nauka, M., 1986

[17] Holloway J. P., Dorning J. J., “Nonlinear but small amplitude longitudinal plasma waves”, Oper. Theory Adv. Appl., 51 (1991), 155–179 | MR | Zbl

[18] Holloway J. P., Longitudinal travelling waves bifurcating from Vlasov plasma equilibria, Ph. D. Dissertation in Engineering Physics, Univ. of Virginia, Charlottesville, 1989

[19] Hesse M., Schindler K., “Bifurcation of current sheets in plasmas”, Phys. Fluids, 29:8 (1986), 2484–2492 | DOI | MR | Zbl

[20] Kudryavtsev L. D., Kurs matematicheskogo analiza, T. 2, Vysshaya shkola, M., 1981

[21] Ladyzhenskaya O. A., Uraltseva N. N., Lineinye i nelineinye uravneniya ellipticheskogo tipa, Nauka, M., 1964

[22] Trenogin V. A., Sidorov N. A., “Issledovanie tochek bifurkatsii i nepreryvnykh vetvei reshenii nelineinykh uravnenii”, Differentsialnye i integralnye uravneniya, 1, IGU, Irkutsk, 1972, 216–248 | MR

[23] Markov Y., Rudykh G., Sidorov N., Sinitsyn A., Tolstonogov D., “Steady-state solutions of the Vlasov–Maxwell system and their stability”, Acta Appl. Math., 28 (1992), 253–293 | DOI | MR | Zbl

[24] Sidorov N. A., Trenogin V. A., Tochki i poverkhnosti bifurkatsii nelineinykh operatorov s potentsialnymi sistemami razvetvleniya, Preprint No 4, IrVTs SO AN SSSR, Irkutsk, 1991

[25] Dubrovin B. A., Novikov S. P., Fomenko A. T., Sovremennaya geometriya, Nauka, M., 1979

[26] Kronecker L., “Über Systeme von Functionen mehrerer Variabels”, Monatsberichte de l'Academie, Berlin, 1869, 159–198

[27] Conley C. C., Isolated invariant sets and the Morse index, CBMS Regional Conf. Ser. in Math., 38, Conf. Board Math. Sci., Washington, 1978 | MR | Zbl

[28] Rothe E., “Type numbers and critical points”, Math. Nachr., 4 (1950–1951), 12–27 | MR

[29] Sidorov N. A., Obschie voprosy regulyarizatsii v zadachakh teorii vetvleniya, IGU, Irkutsk, 1982

[30] Loginov B. V., Sidorov N. A., “Gruppovaya simmetriya razvetvlyayuschikhsya uravnenii Lyapunova–Shmidta i iteratsionnye metody v zadache o tochke vetvleniya”, Matem. sb., 182:5 (1991), 681–691 | Zbl