On a nonlocal stationary free-boundary problem arising in the confinement of a plasma in a stellarator geometry
| dc.contributor.author | Díaz Díaz, Jesús Ildefonso | |
| dc.contributor.author | Rakotoson, Jean Michel Theresien | |
| dc.date.accessioned | 2023-06-20T16:55:20Z | |
| dc.date.available | 2023-06-20T16:55:20Z | |
| dc.date.issued | 1996-03 | |
| dc.description.abstract | We prove the existence and some qualitative properties of the solution to a two-dimensional free-boundary problem modeling the magnetic confinement of a plasma in a Stellarator configuration. The nonlinear elliptic partial differential equation on the plasma region was obtained from the three-dimensional magnetohydrodynamic system by HENDER & CARRERAS in 1984 by using averaging arguments and a suitable system of coordinates (Boozer's vacuum coordinates). The free boundary represents the separation between the plasma and vacuum regions, and the model is described by an inverse-type problem (some nonlinear terms of the equation are unknown). Using the zero net current condition for the Stellarator configurations, we reformulate the problem with the help of the notion of relative rearrangement, leading to a new problem involving nonlocal terms in the equation. We use an iterative algorithm and establish some new properties on the relative rearrangement in order to prove the convergence of the algorithm and then the existence of a solution. | |
| dc.description.department | Depto. de Análisis Matemático y Matemática Aplicada | |
| dc.description.faculty | Fac. de Ciencias Matemáticas | |
| dc.description.refereed | TRUE | |
| dc.description.sponsorship | EURATOMCIEMAT Association | |
| dc.description.status | pub | |
| dc.eprint.id | https://eprints.ucm.es/id/eprint/15896 | |
| dc.identifier.doi | 10.1007/BF00376255 | |
| dc.identifier.issn | 0003-9527 | |
| dc.identifier.officialurl | http://www.springerlink.com/content/m3gr085758471jpp/ | |
| dc.identifier.relatedurl | http://www.springerlink.com/ | |
| dc.identifier.uri | https://hdl.handle.net/20.500.14352/57430 | |
| dc.issue.number | 1 | |
| dc.journal.title | Archive for Rational Mechanics and Analysis | |
| dc.language.iso | eng | |
| dc.page.final | 95 | |
| dc.page.initial | 53 | |
| dc.publisher | Springer | |
| dc.relation.projectID | 74/91 | |
| dc.rights.accessRights | restricted access | |
| dc.subject.cdu | 517.956.2 | |
| dc.subject.keyword | iterative procedure | |
| dc.subject.keyword | existence theorem | |
| dc.subject.keyword | averaging | |
| dc.subject.keyword | nonlinear elliptic partial differential equation | |
| dc.subject.keyword | inverse-type problem | |
| dc.subject.keyword | zero net current conditions | |
| dc.subject.keyword | relative arrangements | |
| dc.subject.keyword | non-local terms | |
| dc.subject.keyword | convergence | |
| dc.subject.ucm | Geometría diferencial | |
| dc.subject.ucm | Ecuaciones diferenciales | |
| dc.subject.unesco | 1204.04 Geometría Diferencial | |
| dc.subject.unesco | 1202.07 Ecuaciones en Diferencias | |
| dc.title | On a nonlocal stationary free-boundary problem arising in the confinement of a plasma in a stellarator geometry | |
| dc.type | journal article | |
| dc.volume.number | 134 | |
| dcterms.references | ALMGREN, F. & LIEB, E., Symmetric rearrangement is sometimes continuous, J. Amer. lvlath. Soc., 2,1989,683-772. BERETTA, E. & VOGELlUS, M., An inverse problem originating from magnetobydrodynamics Il: tbe case of tbe Grad-Shafranov Eqllation, Indiana University Mathematics Journal, 41,1992,1081-1117. BERESTYCKI, H. & BREZIS, R., On a free bOllndary problem arising in plasma physics, NOIl/ilJeal' Anal., 4, 1980, 415-436. BLUM, J., Numerical Simulation and Optimal Control in Plasma Physics, Wiley, Gallthier-Villars, 1989. BOOZER, A. H., Establishment of magnetic coordina tes for given magnetic field, Phys. Fluids, 25, 1982, 520-521. CHONG, K. M. & RICE, N. M., Equimeasurable rearrangements of fumctions, Queen's University, 1971. DÍAZ, J. I., Modelos bidimensionales de equilibrio magnetohiidrodilámico para Stellarators, Informe # 1 Formulación. CIEMAT Repports. Madrid, December 1991, 38 pp. DÍAZ, J. I., Modelos bidimensionales de equilibrio magnetohiidrodilámico para Stellarators, Informe # 2 Existencia de soluciones. CIEMAT Repports. Madrid, July 1992,30 pp. DÍAZ, J. I., Modelos bidimensionales de equilibrio magnetohiidrodilámico para Stellarators, Informe # 3. Multiplicidady dependencia de parámetros. CIEMA T Repports. Madrid, March 1993, 29 pp. DÍAZ, J. I., & MOSSINO, J., Isoperimetric inequalities in the parabolic obstacle problems,.J. Mah. Pures Appl., 71,1992, 233-266. DÍAZ, J. I., & RAKOTOSON, J. M., On a two-dimensional stationary free boundary problem arising in the confinement of a plasma in a Stellarator, C. R. Acad. Sci. Paris, 317, Série I. 1993, 353-358. FEDERER, H., Geometric Measure Theory, Springer-Verlag, 1969. FREIDBERG, J. P., Ideal Magnetohydrodynamics, Plenum Press, 1987. FRIEDMAN, A., Variatinal Principles and Free-Bounwldary Problems, Wiley, 1982. GREENE, J. M. & JOHNSON, J. L., Determination of hydromagnetic equilibria, Phys. Fluids, 27, 1984, 2101-2120. GRAD, H., Hu, P. N. & STEVENS, D. c., Adiabatic evolution of plasma equilibrium, Proc. Nat. Amd. Sci. USA, 72,1975, 3789-3793. GILBARG, D. & TRUDINGER, N. S., Elliptic Partial Differeltial Equations of Second Order, Springer-Verlag, 1977. HENDER, T. C. & CARRERAS, B. A., Equilibrium ca1culation for helical axis Stellarators, Phys. Fluids, 27, 1984, 2101-2120. MOSSINO, J., A priori estimates for a model of Grad-Mercier type in plasma confinement, Applicable Analysis, 13, 1982, 185-207. MOSSINO, J. & RAKOTOSON, J. M., Isoperimetric inequalities in parabolic equations, Annali Scuola Norm. Sup. Pisa, Série IV, 13, 1986,51-73. MOSSINO, J. & TEMAM, R., Directional derivative of the increasing rearrangement mapping and application to a queer differential equation in plasma physics, Duke Math . .J., 48, 1981, 475-495. RAKOTOSON, J. M., Some properties of the relative rearrangement, .J. Math. Anal. App/., 135, 1988,488-500. RAKOTOSON, J. M., A differentiability result for the relative rearrangement, Diff. Int. Eqns, 2, 1989, 363-377. RAKOTOSON, J. M., Relative rearrangement for highly nonlinear equations, Nonlin. Anal., Theory Meh¡. Appl. 24, 1995, 493-507. RAKOTOSON, J. M., & SIMON, B., Relative rearrangement on a measure space. Application to the regularity of weighted monotone rearrangement. Part. I Appl. Mah. Lett., 6, 1993,75-78; Part. II., Appl. Math. Lett. , 6, 1993,79-92. RAKOTOSON, J. M., & TEMAM, R., A co-area formula witb applications to monotone rearrangement and to regularity, Arch. Rational M ech. Anal., 109, 1991, 213-238. SIMON, B., Réarrangement relatir sur un espace measuré et applicationls, These, Univ. de Poitiers, 1994. SIMON, B., Réarrangement relatir sur un espace measuré et applicationls, These, Univ. de Poitiers, 1994. TEMAM, R., A nonlinear eigenvalue problem: equilibrium space of a confined plasma, Arch. Rational Mech. Anal., 60, 1975, 51~73. TEMAM, R., Remarks on a free boundary problem arising in plasma physics, Comm. in Partial Diff. Equations, 2, 1977, 563-585. ZIEMER, W., Weakly Differentiable Funtions, Springer-Verlag, 1989. | |
| dspace.entity.type | Publication | |
| relation.isAuthorOfPublication | 34ef57af-1f9d-4cf3-85a8-6a4171b23557 | |
| relation.isAuthorOfPublication.latestForDiscovery | 34ef57af-1f9d-4cf3-85a8-6a4171b23557 |
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