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Inviscid models generalizing the 2D Euler and the surface quasi-geostrophic equations

Any classical solution of the 2D incompressible Euler equation is global in time. However, it remains an outstanding open problem whether classical solutions of the surface quasi-geostrophic (SQG) equation preserve their regularity for all time. This paper studies solutions of a family of active scalar equations in which each component $u_j$ of the velocity field $u$ is determined by the scalar $θ$ through $u_j =\mathcal{R} Λ^{-1} P(Λ) θ$ where $\mathcal{R}$ is a Riesz transform and $Λ=(-Δ)^{1/2}$. The 2D Euler vorticity equation corresponds to the special case $P(Λ)=I$ while the SQG equation to the case $P(Λ) =Λ$. We develop tools to bound $\|\nabla u||_{L^\infty}$ for a general class of operators $P$ and establish the global regularity for the Loglog-Euler equation for which $P(Λ)= (\log(I+\log(I-Δ)))^γ$ with $0\le γ\le 1$. In addition, a regularity criterion for the model corresponding to $P(Λ)=Λ^β$ with $0\le β\le 1$ is also obtained.