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

Asymptotic global fast dynamics and attractor fractal dimension of the TDGL-equations of... 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

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

, Volume 6 (3) – Oct 15, 2015
25 pages

/lp/springer-journals/asymptotic-global-fast-dynamics-and-attractor-fractal-dimension-of-the-Pf5HoZ4SjM
Publisher
Springer Journals
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-015-0114-4
Publisher site
See Article on Publisher Site

### 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 $$E^{\alpha }$$ 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.

### Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Oct 15, 2015