LOCAL AND GLOBAL WELL-POSEDNESS TO MAGNETO-ELASTICITY SYSTEM

A BSTRACT . In this article we employ classical tricks to give local and global well-posedness to Magneto-Elasticity System. Different from many cases, we consider the equation which the magnetic field satisfies is Landau-Lifshitz system without viscidity, i.e. the Schr¨odinger flow. As is well known, people can not obtain global existence of Schr¨odinger flow at general cases. However, the reason why we do what others can not do is the Schr¨odinger flow with non-zero convection term.

[1]  A. Schlomerkemper,et al.  Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity , 2019, Discrete & Continuous Dynamical Systems - S.

[2]  Wenjing Zhao.  Local well-posedness and blow-up criteria of magneto-viscoelastic flows , 2018 .

[3]  T. Roubíček,et al.  A thermodynamically consistent model of magneto-elastic materials under diffusion at large strains and its analysis , 2017, Zeitschrift für angewandte Mathematik und Physik.

[4]  Ning Jiang,et al.  On Well-Posedness of Ericksen-Leslie's Hyperbolic Incompressible Liquid Crystal Model , 2016, SIAM J. Math. Anal..

[5]  Chun Liu,et al.  Existence of Weak Solutions to an Evolutionary Model for Magnetoelasticity , 2016, SIAM J. Math. Anal..

[6]  Tomáš Roubíček,et al.  Rate-Independent Systems: Theory and Application , 2015 .

[7]  J. Zeman,et al.  Existence results for incompressible magnetoelasticity , 2013, 1311.4097.

[8]  Gilles Carbou,et al.  Global weak solutions for the Landau–Lifschitz equation with magnetostriction , 2011 .

[9]  Christof Melcher,et al.  Thin-Film Limits for Landau-Lifshitz-Gilbert Equations , 2010, SIAM J. Math. Anal..

[10]  I. Shafrir,et al.  On a hyperbolic–parabolic system arising in magnetoelasticity , 2009 .

[11]  F. Lin,et al.  On the initial‐boundary value problem of the incompressible viscoelastic fluid system , 2008 .

[12]  C. Melcher.  A dual approach to regularity in thin film micromagnetics , 2007 .

[13]  Ping Zhang,et al.  On hydrodynamics of viscoelastic fluids , 2005 .

[14]  T. Gilbert.  A phenomenological theory of damping in ferromagnetic materials , 2004, IEEE Transactions on Magnetics.

[15]  Antonio DeSimone,et al.  A constrained theory of magnetoelasticity , 2002 .

[16]  N. Walkington,et al.  An Eulerian Description of Fluids Containing Visco-Elastic Particles , 2001 .

[17]  Gilles Carbou,et al.  Regular solutions for Landau-Lifschitz equation in a bounded domain , 2001, Differential and Integral Equations.

[18]  Antonio DeSimone,et al.  Existence of Minimizers for a Variational Problem in Two‐Dimensional Nonlinear Magnetoelasticity , 1998 .

[19]  François Alouges,et al.  On global weak solutions for Landau-Lifshitz equations: existence and nonuniqueness , 1992 .

[20]  H. Tiersten.  Variational Principle for Saturated Magnetoelastic Insulators , 1965 .

[21]  H. Tiersten.  Coupled Magnetomechanical Equations for Magnetically Saturated Insulators , 1964 .

[22]  F. Lin,et al.  Nonparabolic dissipative systems modeling the flow of liquid crystals , 1995 .

[23]  David Kinderlehrer,et al.  Theory of magnetostriction with applications to TbxDy1-xFe2 , 1993 .

[24]  Hantaek Bae.  Navier-Stokes equations , 1992 .

[25]  R. Temam,et al.  Navier-Stokes equations: theory and numerical analysis: R. Teman North-Holland, Amsterdam and New York. 1977. 454 pp. US $45.00 , 1978 .

[26]  T. Gilbert.  A Lagrangian Formulation of the Gyromagnetic Equation of the Magnetization Field , 1955 .

[27]  Lev Davidovich Landau,et al.  ON THE THEORY OF THE DISPERSION OF MAGNETIC PERMEABILITY IN FERROMAGNETIC BODIES , 1935 .