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

Original · EN

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 uⱼ of the velocity field u is determined by the scalar θ through uⱼ =R Λ⁻¹ P(Λ) θ where 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 ∇ u||ₗ∞ for a general class of operators P and establish the global regularity for the Loglog-Euler equation for which P(Λ)= ((I+(I-Δ)))γ with 0≤ γ≤ 1. In addition, a regularity criterion for the model corresponding to P(Λ)=Λβ with 0≤ β≤ 1 is also obtained.