Ill-posedness / Well-posedness Results for a Class of Active Scalar Equations.

Ill-posedness / Well-posedness Results for a Class of Active Scalar Equations.

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Susan Friedlander , USC
Fine Hall 322

Please note special day (Tuesday).    We discuss a class of active scalar equations where the transport velocities are more singular than the active scalar. There is a significant difference in the well-posedness properties of the problem depending on whether the Fourier multiplier symbol for the velocity is even or odd. The "even" symbol non-diffusive or weakly diffusive equations are ill-posed in Sobolev spaces. However the critically diffusive equations are globally well posed in both the odd and even cases. Examples of "even" equations are the magnetogeostrophic equation that is a model for the geodynamo and the modified porous media equation. This is joint work with Francisco Gancedo, Weiran Sun and Vlad Vicol.