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We address the global existence and uniqueness of solutions for the anisotropically reduced 2D Kuramoto-Sivashinsky equations in a periodic domain with initial data \begin{document}$$ u_{01} \in L^2 $$\end{document} and \begin{document}$$ u_{02} \in H^{-1 + \eta} $$\end{document} for \begin{document}$$ \eta > 0 $$\end{document}.
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This content will become publicly available on March 7, 2026
Solutions of the Ginzburg–Landau equations with vorticity concentrating near a nondegenerate geodesic
It is well-known that under suitable hypotheses, for a sequence of solutions of the (simplified) Ginzburg–Landau equations-\Delta u_{\varepsilon} +\varepsilon^{-2}(|u_{\varepsilon}|^{2}-1)u_{\varepsilon} = 0, the energy and vorticity concentrate as\varepsilon\to 0around a codimension2stationary varifold – a (measure-theoretic) minimal surface. Much less is known about the question of whether, given a codimension2minimal surface, there exists a sequence of solutions for which the given minimal surface is the limiting concentration set. The corresponding question is very well-understood for minimal hypersurfaces and the scalar Allen–Cahn equation, and for the Ginzburg–Landau equations when the minimal surface is locally area-minimizing, but otherwise quite open. We consider this question on a3-dimensional closed Riemannian manifold(M,g), and we prove that any embedded nondegenerate closed geodesic can be realized as the asymptotic energy/ vorticity concentration set of a sequence of solutions of the Ginzburg–Landau equations.
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- Award ID(s):
- 2106516
- PAR ID:
- 10615403
- Publisher / Repository:
- European Math Soceity
- Date Published:
- Journal Name:
- Journal of the European Mathematical Society
- Volume:
- 27
- Issue:
- 4
- ISSN:
- 1435-9855
- Page Range / eLocation ID:
- 1527 to 1561
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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