On a singular incompressible porous media equation

Susan Friedlander; Francisco Gancedo; Weiran Sun; Vlad Vicol

doi:10.1063/1.4725532

On a singular incompressible porous media equation

Susan Friedlander, Francisco Gancedo, Weiran Sun, Vlad Vicol

Mathematics

Research output: Contribution to journal › Article › peer-review

14 Scopus citations

Abstract

This paper considers a family of active scalar equations with transport velocities which are more singular by a derivative of order β than the active scalar. We prove that the equations with 0 < β ≤ 2 are Lipschitz ill-posed for regular initial data. On the contrary, when 0 < β < 1 we show local well-posedness for patch-type weak solutions.

Original language	English (US)
Article number	115602
Journal	Journal of Mathematical Physics
Volume	53
Issue number	11
DOIs	https://doi.org/10.1063/1.4725532
State	Published - Nov 27 2012

All Science Journal Classification (ASJC) codes

Statistical and Nonlinear Physics
Mathematical Physics

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10.1063/1.4725532

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title = "On a singular incompressible porous media equation",

abstract = "This paper considers a family of active scalar equations with transport velocities which are more singular by a derivative of order β than the active scalar. We prove that the equations with 0 < β ≤ 2 are Lipschitz ill-posed for regular initial data. On the contrary, when 0 < β < 1 we show local well-posedness for patch-type weak solutions.",

author = "Susan Friedlander and Francisco Gancedo and Weiran Sun and Vlad Vicol",

note = "Funding Information: S.F. was supported in part by the National Science Foundation (NSF) Grant No. DMS-0803268. F.G. was supported in part by the Grant No. MTM2008-03754 of the Ministerio de Ciencia e Innovation (Spain), the Grant No. StG-203138CDSIF of the European Research Council, and the NSF Grant No. DMS-0901810. V.V. was supported in part by American Mathematical Society (AMS)-Simons travel award.",

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N2 - This paper considers a family of active scalar equations with transport velocities which are more singular by a derivative of order β than the active scalar. We prove that the equations with 0 < β ≤ 2 are Lipschitz ill-posed for regular initial data. On the contrary, when 0 < β < 1 we show local well-posedness for patch-type weak solutions.

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On a singular incompressible porous media equation

Abstract

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