Existence of Weak Solutions to Some Vortex Density Models

We study the weak solutions to equations arising in the modeling of vortexmotions in superfluids such as superconductors. The global existence of measure-valued solutions is established with a bounded Radon measure as initial data. Moreover, we get a local space-time Lq estimate for the continuous part of the solution, and we prove the global existence of a distributional weak solution for a particular case. We also consider a modification to the model in order to physically account for the different signs of vortices, and we present, in one space dimension, the global existence of weak solutions with the initial data in BV for the modified model.

[1]  Qiang Du,et al.  Analysis and Approximation of the Ginzburg-Landau Model of Superconductivity , 1992, SIAM Rev..

[2]  E Weinan,et al.  Dynamics of vortex liquids in Ginzburg-Landau theories with applications to superconductivity. , 1994, Physical review. B, Condensed matter.

[3]  Andrew J. Majda,et al.  Vorticity and the mathematical theory of incompressible fluid flow , 1986 .

[4]  V. I. Yudovich,et al.  Non-stationary flow of an ideal incompressible liquid , 1963 .

[5]  S. Jonathan Chapman,et al.  A Hierarchy of Models for Type-II Superconductors , 2000, SIAM Rev..

[6]  Michael Taylor,et al.  Pseudo differential operators , 1974 .

[7]  A. Majda.  Compressible fluid flow and systems of conservation laws in several space variables , 1984 .

[8]  P. Lions,et al.  User’s guide to viscosity solutions of second order partial differential equations , 1992, math/9207212.

[9]  Ping Zhang,et al.  RAREFACTIVE SOLUTIONS TO A NONLINEAR VARIATIONAL WAVE EQUATION OF LIQUID CRYSTALS , 2001 .

[10]  P. Hartman.  Ordinary Differential Equations , 1965 .

[11]  Andrew J. Majda,et al.  Concentrations in the one-dimensional Vlasov-Poisson equations, I.: temporal development and non-unique weak solutions in the single component case , 1994 .

[12]  Ping Zhang,et al.  On the hydrodynamic limit of Ginzburg-Landau vortices , 1999 .

[13]  Radjesvarane Alexandre,et al.  On the Boltzmann equation for long‐range interactions , 2002 .

[14]  R. Schwarzenberger.  ORDINARY DIFFERENTIAL EQUATIONS , 1982 .

[15]  J. Delort.  Existence de nappes de tourbillon en dimension deux , 1991 .

[16]  P. Lions,et al.  Ordinary differential equations, transport theory and Sobolev spaces , 1989 .

[17]  Qiang Du.  Convergence Analysis of a Numerical Method for a Mean Field Model of Superconducting Vortices , 2000, SIAM J. Numer. Anal..

[18]  Qiang Du,et al.  Analysis and computation of a mean-field model for superconductivity , 1999, Numerische Mathematik.

[19]  C. M. Elliott,et al.  Numerical analysis of a mean field model of superconducting vortices , 2001 .

[20]  P. Lions.  Mathematical topics in fluid mechanics , 1996 .

[21]  B. Stoth,et al.  Ill-posedness of the mean-field model of superconducting vortices and a possible regularisation , 2000, European Journal of Applied Mathematics.

[22]  Andrew J. Majda,et al.  Concentrations in the one-dimensional Vlasov-Poisson equations. II: screening and the necessity for measure-valued solutions in the two component case , 1994 .

[23]  J. Cooper.  SINGULAR INTEGRALS AND DIFFERENTIABILITY PROPERTIES OF FUNCTIONS , 1973 .

[24]  Dorsey.  Vortex motion and the Hall effect in type-II superconductors: A time-dependent Ginzburg-Landau theory approach. , 1992, Physical review. B, Condensed matter.

[25]  S. Jonathan.  A MEAN-FIELD MODEL OF SUPERCONDUCTING VORTICES IN THREE DIMENSIONS* , 1995 .

[26]  A. Majda,et al.  Concentrations in regularizations for 2-D incompressible flow , 1987 .

[27]  V. Styles,et al.  Analysis of a mean field model of superconducting vortices , 1999, European journal of applied mathematics.

[28]  A. Majda,et al.  Oscillations and concentrations in weak solutions of the incompressible fluid equations , 1987 .

[29]  Lawrence C. Evans,et al.  Weak convergence methods for nonlinear partial differential equations , 1990 .

[30]  C. M. Elliott,et al.  Viscosity Solutions of a Degenerate Parabolic-Elliptic System Arising in the Mean-Field Theory of Superconductivity , 1998 .

[31]  Zhiming Chen,et al.  AN UPWINDING MIXED FINITE ELEMENT METHOD FOR A MEAN FIELD MODEL OF SUPERCONDUCTING VORTICES , 2000 .