Asymptotic global fast dynamics and attractor fractal dimension of the TDGL-equations of superconductivity

Abstract

In this paper, we study a model system of equations of the time dependent Ginzburg–Landau equations of superconductivity in a Lorentz gauge, in scale of Hilbert spaces \(E^{\alpha }\) with initial data in \(E^{\beta }\) satisfying \(3\alpha + \beta \ge \frac{N}{2}\), where \(N=2,3\) is such that the spatial domain of the equations Open image in new window. We show in the asymptotic dynamics of the equations, well-posedness of the dynamical system for a global exponential attractor \({\mathcal {U}}\subset E^{\alpha }\) compact in \(E^{\beta }\) if \(\alpha >\beta \), uniform differentiability of orbits on the attractor in \(E^{0}\cong L^{2}\), and the existence of an explicit finite bounding estimate on the fractal dimension of the attractor yielding that its Hausdorff dimension is as well finite. Uniform boundedness in \((0,\infty )\times \Omega \) of solutions in \(E^{\frac{1}{2}}\cong H^{1}(\Omega )\) is in addition investigated.