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 $$E^{\alpha }$$ E α with initial data in $$E^{\beta }$$ E β satisfying $$3\alpha + \beta \ge \frac{N}{2}$$ 3 α + β ≥ N 2 , where $$N=2,3$$ N = 2 , 3 is such that the spatial domain of the equations [InlineEquation not available: see fulltext.]. We show in the asymptotic dynamics of the equations, well-posedness of the dynamical system for a global exponential attractor $${\mathcal {U}}\subset E^{\alpha }$$ U ⊂ E α compact in $$E^{\beta }$$ E β if $$\alpha >\beta $$ α > β , uniform differentiability of orbits on the attractor in $$E^{0}\cong L^{2}$$ E 0 ≅ 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 $$(0,\infty )\times \Omega $$ ( 0 , ∞ ) × Ω of solutions in $$E^{\frac{1}{2}}\cong H^{1}(\Omega )$$ E 1 2 ≅ H 1 ( Ω ) is in addition investigated.
Analysis and Mathematical Physics – Springer Journals
Published: Oct 15, 2015
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