On nonlinear stability of linear pinch

References

• Grad , H , Kadish , A and Stevens , DC . 1974 . A free boundary Tokamak equilibrium . Comm. Pure Appl. Math. London , 27 : 39 – 57 .
   Google Scholar
• Agostinelli , C . 1966 . Magnetofluidodinamica , Cremonese, Roma : Monography CNR .
   Google Scholar
• Chandrasekar , S . 1961 . Hydrodynamic and Hydromagnetic Stability , Oxford : International Series of Monographs on Physics .
   Google Scholar
• Jeffrey , A . 1966 . Magnetohydrodynamics , London : Oliver and Boyd .
   Google Scholar
• (a) P. Roberts, An Introduction to Magnetohydrodinamics, Longmans, 1967; (b) Chandrasekhar, S. and Fermi, E. Problems of gravitational stability in the presence of a magnetic field, Astrophys. J. 118 (1953), pp. 116–141
   Google Scholar
• Bashtovoy , VG , Berkovsky , BM and Vislovich , AN . 1987 . Introduction to Thermomechanics of Magnetic Fluids, Hemisphere Pbl.Corp. , Berlin : Springer-Verlag .
   Google Scholar
• Grad , H . 1966 . Variational principle for a guiding-center plasma . Phys. Fluids , 9 : 225 – 251 .
   Google Scholar
• Agostinelli , C . 1954 . Soluzioni stazionarie delle equazioni della magneto-idrodinamica interessanti la Cosmogonia . Atti dell'Acc. Naz. Lincei Cl. Sci. Fis. Mat. Nat. , 17 : 216 – 221 .
   Google Scholar
• Agostinelli , C . 1957 . Figure di equilibrio ellissoidali per una massa fluida elettricamente conmduttrice uniformemente ruotante, con campi magnetici variabili col tempo . Atti dell'Acc. Naz. Lincei Cl. Sci. Fis. Mat. Nat. , 23 : 409 – 414 .
   Google Scholar
• Cowling , TG . 1957 . Magnetohydrodynamics , New York : Interescience .
   Google Scholar
• Goldberger , M . 1956 . One fluid hydro magnetics and the Boltzmann equation . Proc. Roy. Soc. London , A236 : 260 – 281 .
   Google Scholar
• Oppeneim , A . 1960 . Equilibrium configuration of a plasma in a guiding center limit . Inst. Math. Sci. rep.NYO-9353 , September
   Google Scholar
• Ferrari , I . 1960 . Su un teorema di unicita' per le equazioni dell'idromagnetismo . Atti Sem. Mat. Fis. Univ. Modena , 9 : 205 – 217 .
   Google Scholar
• Galdi , P and Padula , M . 1990 . A new approach to energy theory in the stability of fluid motion . Arch. Rational. Mech. Anal. , 110 : 187 – 186 .
   Web of Science ®Google Scholar
• Ladyzhensaja , OA and Solonnikov , VA . 1960 . The mathematical problems of hydrodynamics and magnetohydrodynamics for a viscous incompressible fluid . Tr. Mat. Inst. Steklov , 59 : 115 – 173 .
   Google Scholar
• Ladyzhensaja , OA and Solonnikov , VA . 1973 . The linearization principle and invariant manifold for problems of magnetohydrodynamics . Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Steklov AN SSSR , 38 : 46 – 93 .
   Google Scholar
• Mulone , G and Rionero , S . 2003 . Necessary and sufficient conditions for nonlinear stability in the magnetic Benard problem . Arch. Ratl. Mech. Anal. , 166 : 197 – 218 .
   Web of Science ®Google Scholar
• Rionero , S . 1968 . Sulla stabilita' magnetoidrodinamica in media con vari tipi di condizioni al contorno . Ricerche di Mat. Univ. Napoli , 17 : 64 – 78 .
   Google Scholar
• Rudraiah , N . 1981 . Nonlinear interaction of magnetic field and convection in three-dimensional motion . Publ. Astron. Soc. Japan , 33 : 721 – 737 .
   Google Scholar
• Rudraiah , N . 1986 . Double diffusive magnetoconvection . J. Phys. , 27 : 233 – 266 .
   Google Scholar
• Scarpellini , B . 2000 . Loss of stability in the magnetic Benard proble . Nonlinear Diff. Equ. Appl. , 7 : 157 – 185 .
   Google Scholar
• Rappaz , J and Topuzani , R . 1991 . On a two-dimensional magnetohydrodynamic problem I. Modelling and analysis . Math. Model. Numer. Anal. , 26 : 347 – 364 .
   Google Scholar
• Meir , AJ and Schmidt , PG . 1996 . Variational methods for a stationary MHD flow under natural interface conditions . Nonlin. Anal. Theory, Methods Appl. , 26 : 659 – 689 .
   Web of Science ®Google Scholar
• Fosdic , R and Tang , H . 2007 . Electrodynamics and thermomechanics of material bodies . J. Elasticity , 88 : 255 – 297 .
   Google Scholar
• Besson , O , Boulanger , J , Chevalier , P-A , Rappaz , J and Topuzani , R . 1991 . Numerical modelling of electromagnetic casting processes . J. Comp. Phys. , 92 ( 2 ) : 482 – 507 .
   Google Scholar
• Romerio , MV and Secretan , M-A . 1086 . Magnetohydrodynamic equilibrium in alluminium electrolitic cells . Comput. Phys. Rep. , 3 : 327 – 360 .
   Google Scholar
• Giga , Y and Yoshida , Z . 1990 . A dynamic free-boundary problem in plasma physics . SIAM J. Math. Anal. , 21 : 1118 – 1138 .
   Google Scholar
• Maillard , P and Romerio , MV . 1996 . A stability criterion for an infinitely long Hall–Heroult cell . J. Comput. Appl. Math. , 71 : 47 – 65 .
   Google Scholar
• Padula , M . 2009 . On nonlinear stability of MHD equilibrium figures . Adv. Math. Fluid Mech. , : 301 – 331 .
   Google Scholar
• M. Padula and V.A. Solonnikov, On the free boundary problem of magnetohydrodynamics, in a issue in honour of professor Seregin, Zapiski Nauchnykh Sem. Leningrad, to appear
   Google Scholar
• Guidoboni , G and Jin , BJ . 2005 . On the nonlinear stability of Marangoni–Benard problem with Free Surface in the Boussinesq approximation . Math. Models Methods Appl. Sci , 15 ( 1 ) : 1 – 22 .
   Google Scholar
• Babovski , H and Padula , M . 1992 . A new contribution in the nonlinear stability of a discrete velocity model . Comm. Math. Phys. , 144 : 87
   Google Scholar
• Landau , LD and Lifshitz , EM . 1960 . Electrodynamics of Continuous Media , Oxford, , UK : Pergamon Press .
   Google Scholar
• M.Padula, and V.A. Solonnikov, Existence of Non-steady Flows of an Incompressible Viscous Drop of Fluid in a Frame Rotating with Finite Angular Velocity, World Scientific, Singapore
   Google Scholar
• Solonnikov , VA . 2004 . On the stability of nonsymmetric equilibrium figures of rotating viscous incompressible liquid . Interfaces Free Boundaries , 6 : 461 – 492 .
   Google Scholar
• Rudraiah , N , Krishnamurthy , BS and Mathad , RD . 1996 . The effect of oblique magnetic field on the surface instability of a finite conducting fluid layer . Acta Mech. , 119 : 187 – 201 .
   Google Scholar
• Padula , M and Solonnikov , VA . 2000 . On Raileigh–Taylor stability . Ann. Univer. Ferrara , 45 : 307 – 336 .
   Google Scholar
• De Groot , SR and Mazur , P . 1969 . Non-equilibrium Thermodynamics , Amsterdam : North-Holland .
   Google Scholar
• Padula , M and Solonnikov , VA . 2008 . A simple proof of linear instability of rotating liquid drops . Ann. Univer. Ferrara , 54 : 97 – 111 .
   Google Scholar
• Padula , M and Solonnikov , VA . Initial boundary value problem for a mass of electroconducting fluid in the vacuum , (in preparation)
   Google Scholar