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Absence of splash singularities for surface quasi-geostrophic sharp fronts and the Muskat problem

 

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Opened Access Absence of splash singularities for surface quasi-geostrophic sharp fronts and the Muskat problem
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Author: Gancedo García, Francisco
Strain, Robert M.
Department: Universidad de Sevilla. Departamento de Análisis Matemático
Date: 2014
Published in: Proceedings of the National Academy of Sciences, 111 (2), 635-639.
Document type: Article
Abstract: In this paper, for both the sharp front surface quasi-geostrophic equation and the Muskat problem, we rule out the “splash singularity” blow-up scenario; in other words, we prove that the contours evolving from either of these systems cannot intersect at a single point while the free boundary remains smooth. Splash singularities have been shown to hold for the free boundary incompressible Euler equation in the form of the water waves contour evolution problem. Our result confirms the numerical simulations in earlier work, in which it was shown that the curvature blows up because the contours collapse at a point. Here, we prove that maintaining control of the curvature will remove the possibility of pointwise interphase collapse. Another conclusion that we provide is a better understanding of earlier work in which squirt singularities are ruled out; in this case, a positive volume of fluid between the contours cannot be ejected in finite time.
Cite: Gancedo García, F. y Strain, R.M. (2014). Absence of splash singularities for surface quasi-geostrophic sharp fronts and the Muskat problem. Proceedings of the National Academy of Sciences, 111 (2), 635-639.
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URI: http://hdl.handle.net/11441/44880

DOI: 10.1073/pnas.1320554111

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