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Abstrakty
We prove the existence and uniqueness of weak solutions to the variational formulation of the Maxwell-Boussinesq approximation problem. Some further regularity in W1,2+δ, δ > 0, is obtained for the weak solutions. The shape sensitivity analysis by the boundary variations technique is performed for the weak solutions. As a result, the existence of the strong material derivatives for the weak solutions of the problem is shown. The result can be used to establish the shape differentiability for a broad class of shape functionals for the models of Fourier-Navier-Stokes flows under the electromagnetic field.
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Rocznik
Tom
Strony
1193--1215
Opis fizyczny
Bibliogr. 27 poz.
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autor
autor
autor
- Mathematics Dep/FCUL and CMAF, University of Lisbon 1749-016 Lisboa, Portugal
Bibliografia
- ALEKSEEV, G.V. (2006) Steady model of magnetohydrodynamic viscous fluid with heat transfer (in Russian). Uspekhi Mekhaniki, 66-116.
- BOISGÉRAULT, S. and ZOLÉSIO, J.P. (2000) Shape derivative of sharp functionals governed by Navier - Stokes flow. Partial Differential Equations (Praha, 1998), Res. Notes Math., Chapman & Hall/CRC 406, 49-63.
- BOISGÉRAULT, S. and ZOLÉSIO, J.P. (2001) Boundary variations in the Navier-Stokes equations and Lagrangian functional. Shape optimization and optimal design (Cambridge, 1999), Lecture Notes in Pure and Appl. Math. 216, Dekker, New York, 7-26.
- CONSIGLIERI, L. (2006) Steady-state flows of thermal viscous incompressible fluids with convective-radiation effects. Math. Mod. and Meth. in Appl. Sci. 16 (12), 2013-2027.
- CONSIGLIERI, L. (2008) A (p - q) coupled system in elliptic nonlinear problems with nonstandard boundary conditions. J. Math. Anal. Appl. 340 (1), 183-196.
- CONSIGLIERI, L. and SHILKIN, T. (2000) Regularity to stationary weak solutions for generalized Newtonian fluids with energy transfer. Zapiski Nauchnyh Seminarov POMI 271, 122-150.
- CONSIGLIERI, L., NEČASOVÁ, S. and SOKOLOWSKI, J. (2008) New approach to the incompressible Maxwell-Boussinesq approximation: Existence, uniqueness and shape sensitivity. Necas Center for Mathematical Modeling, Preprint no. 2008-017.
- DELFOUR, M. and ZOLÉSIO, J.P. (2001) Shapes and Geometries: Analysis, Differential Calculus, and Optimisation. SIAM Series on Advances in Design and Control, Philadelphia.
- DRUET, P.E. (2009) On Weak Solutions to the Stationary MHD-Equations Coupled to Heat Transfer with Nonlocal Radiation Boundary Conditions. Nonlinear Anal. Real World Appl. 10, 2914-2936.
- DUNFORD, N. and SCHWARTZ, J.T. (1958) Linear Operators, Part I. Interscience Publ., New York.
- DUVAUT, G. and LIONS, J.L. (1972) Inéquations en thermoélasticité et magétohydrodynamique. Arch. Rat. Mech. Anal. 46, 241-279.
- DZIRI, R. and ZOLÉSIO, J.P. (1996) Shape derivative for the heat-conduction equation with Lipschitz continuous coefficients. Boll. Un. Mat. ltd,. B 10 (3), 569-594.
- DZIRI, R. and ZOLÉSIO, J.P. (1997) Shape existence in Navier- Stokes flow with heat convection. Annali della Scuola Normale Superiore di Pisa 24 (1), 165-192.
- DZIRI, R., MOUBACHIR, M. and ZOLÉSIO, J.P. (2004) Dynamical shape gradient for the Navier-Stokes system. Comptes Rendus Mathematique 338 (2), 183-186.
- GEHRING, F.W. (1973) The Lp-integrability of the partial derivatives of a quasi conformal mappings. Acta Math. 130, 265-277.
- GERBEAU, J.F. and LE BRIS, C. (1997) Existence of solution for a density-dependent magnetohydrodynamic equation. Advances in Differential Equations 2 (3), 427-452.
- GERBEAU, J.F. and LE BRIS, C. (1999) A coupled system arising in magnetohydrodynamics. Applied Mathematics Letters 12 (3), 53-57.
- GROGER, K. (1989) A W1,p- estimate for solutions to mixed boundary value problems for second order elliptic differential equations. Mathematische Annalen 283, 679-687.
- HOMBERG, D. and SOKOLOWSKI, J. (2003) Optimal shape design of inductor coils for surface hardening. SIAM J. Control Optim. 42 (3), 1087-1117.
- MEIR, A.J. (1994) Thermally coupled magnetohydrodynamics flow. Appl. Math, and Comp. 65, 79-94.
- MEIR, A.J. and SCHMIDT, P.G. (2001) On electromagnetically and thermally driven liquid-metal flows. Nonlinear Analysis 47, 3281-3294.
- MURAT, F. and SIMON, J. (1976) Sur la Contrôle par un Domaine Géometrique. Publications du Laboratoire d’Analyse Numerique, Univ. de Paris VI.
- PIRONNEAU, O. (1984) Optimal Shape Design for Elliptic Systems. Springer series in Computational Physics, Springer-Verlag, New York.
- PLOTNIKOV, P.I. and SOKOLOWSKI, J. (2006) Domain dependence of solutions to compressible Navier-Stokes equations. SIAM J. Control Optim. 45 (4), 1147-1539.
- PLOTNIKOV, P.I., RUBAN, E.V. and SOKOLOWSKI, J. (2008) Inhomogeneous boundary value problems for compressible Navier-Stokes equations, well-posedness and sensitivity analysis. SIAM J. Math. Anal. 40 (3), 1152-1200.
- SERMANGE, M. and TEMAM, R. (1983) Some mathematical questions related to MHD equations. Comm. Pure Appl. Math. 36, 635-664.
- SOKOLOWSKI, J. and ZOLÉSIO, J.P. (1992) Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer Series in Computational Mathematics, 16. Springer Verlag.
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Bibliografia
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