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

16 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

Access to Document

10.1063/1.4725532

Cite this

@article{193c4d30fb9c45b699631162ec97aa05,

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.",

year = "2012",

month = nov,

day = "27",

doi = "10.1063/1.4725532",

language = "English (US)",

volume = "53",

journal = "Journal of Mathematical Physics",

issn = "0022-2488",

publisher = "American Institute of Physics",

number = "11",

}

TY - JOUR

T1 - On a singular incompressible porous media equation

AU - Friedlander, Susan

AU - Gancedo, Francisco

AU - Sun, Weiran

AU - Vicol, Vlad

N1 - 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.

PY - 2012/11/27

Y1 - 2012/11/27

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.

AB - 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.

UR - https://www.scopus.com/pages/publications/84870516502

UR - https://www.scopus.com/pages/publications/84870516502#tab=citedBy

U2 - 10.1063/1.4725532

DO - 10.1063/1.4725532

M3 - Article

AN - SCOPUS:84870516502

SN - 0022-2488

VL - 53

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

IS - 11

M1 - 115602

ER -

On a singular incompressible porous media equation

Abstract

All Science Journal Classification (ASJC) codes

Access to Document

Other files and links

Fingerprint

Cite this