ON THE DYNAMICAL PROCESS GENERATED BY A SUPERCONDUCTIVITY MODEL

We present a result on the existence and uniqueness of a weak solution to the time-dependent GINZBURG-LANDAU equations. These equations desc ribe a macroscopic model for a superconductor Omega near the critical temperature. Mathematically, they form a semilinear, weakly parabolic system for the unknowns (psi, A, Phi). Any two solutions related throu gh a gauge transformation describe the same physical state of the supe rconductor. We choose the gauge ''Phi = -div A'' to reduce this system to a strongly parabolic one. (I.e., we ''factor out'' the linear spac e of all Phi' = Phi + div A.) Assuming sufficient smoothness of the bo undary partial derivative Omega of the superconductor and of the exter nal magnetic field H as well, we thew show that the reduced system for (psi, A, -div A) possesses a unique weal solution whose spatial regul arity (differentiability in x) at any time t greater than or equal to 0 is at least as high as at t = 0. We prove that these solutions gener ate a dynamical process in a suitable Cartesian product of fractional Sobolev spaces. For the global existence in time t greater than or equ al to 0, we assume only integral(0)(t)(integral(Omega)\partial derivat ive H/partial derivative t\(2) dx)(1/2) dt < infinity for all t greate r than or equal to 0. If H is tau-periodic in time (tau > 0), then so is the dynamical process which becomes a dynamical system if H is time -independent).