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Scaling limits and regularity results for a class of Ginzburg-Landau systems. (English) Zbl 0944.35006

The paper deals with the regularity and asymptotic behaviour of solutions of the Ginzburg–Landau system \[ u^\varepsilon_t-\Delta u^\varepsilon+{u^\varepsilon\over\varepsilon^2} (1-|u^\varepsilon|^2)=0 \] with \(u:{\mathbb R}^d\to {\mathbb R}^k\), which is the natural gradient flow of the energy \(e_\varepsilon(u)\) defined by \[ e_\varepsilon(u):={|\nabla u|^2\over 2}+{W(u)\over\varepsilon^2} \quad\text{with}\quad W(u):=\tfrac 14(1-|u|^2)^2. \] More generally the authors study the quasilinear system of PDE \[ u^\varepsilon_t-\Delta u^\varepsilon+{2-p\over p} {\nabla [e_\varepsilon(u_\varepsilon)]\over e_\varepsilon(u_\varepsilon)} +{u^\varepsilon\over\varepsilon^2}(1-|u^\varepsilon|^2)=0 \] for which \(\int [e_\varepsilon(u)]^{p/2}\) is a Lyapunov functional. In the case \(d>p=k\) the authors prove, under suitable regularity assumptions on the initial data, that the renormalized energy density \(e_\varepsilon(u^\varepsilon)/\ln(1/\varepsilon)\) concentrates for a short time on a codimension \(k\)-manifold flowing by mean curvature. This result confirms those previously obtained by other authors using matched asymptotic expansions. The larger part of the paper is however devoted to the proof of some \(\varepsilon\)-regularity theorems, saying that if a weighted integral of the energy density is “sufficiently small” in a region, then the energy density is bounded in a smaller region. As a byproduct this result leads to partial regularity theorems which are uniform in \(\varepsilon\). In the special case \(d=p=k=2\) a stronger result is obtained, replacing the smallness assumption by a boundedness assumption.
Reviewer: L.Ambrosio (Pisa)

MSC:

35B25 Singular perturbations in context of PDEs
35K55 Nonlinear parabolic equations
35B65 Smoothness and regularity of solutions to PDEs
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