Bounded Mean Oscillation and the Uniqueness of Active Scalar Equations

Details Bounded Mean Oscillation and the Uniqueness of Active Scalar Equations Bounded Mean Oscillation and the Uniqueness of Active Scalar Equations

2011-08-12

arxiv.org/abs/1108.2735v1

Mathematics

Analysis of PDEs

Scientific paper

We consider a number of uniqueness questions for several wide classes of active scalar equations, unifying and generalizing the techniques of several authors \cite{Yudovich63,Yudovich95,BertozziSlepcev10,BertozziBrandman10,BRB10,Rusin11}. As special cases of our results, we provide a significantly simplified proof to the known uniqueness result for the 2D Euler equations in $L^1 \cap BMO$ \cite{Vishik99} and provide a mild improvement to the recent results of Rusin \cite{Rusin11} for the 2D inviscid surface quasi-geostrophic (SQG) equations, which are now to our knowledge, the best results known for this model. We also obtain what are (to our knowledge) the strongest known uniqueness results for the Patlak-Keller-Segel models \cite{BlanchetEJDE06,Blanchet09,BRB10}. We obtain these results via technical refinements of energy methods which are well-known in the $L^2$ setting but are less well-known in the $\dot{H}^{-1}$ setting. The $\dot{H}^{-1}$ method can be considered a generalization of Yudovich's classical method \cite{Yudovich63,Yudovich95,MajdaBertozzi} and is naturally applied to equations such as the Patlak-Keller-Segel models with nonlinear diffusion, and other variants \cite{BertozziBrandman10,BertozziSlepcev10,BRB10}. An important tool in our analysis is a Sobolev embedding lemma which shows that velocity fields $v$ with $\grad v \in BMO$ are locally log-Lipschitz.

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