TY - JOUR

T1 - Dissipative models generalizing the 2D navier-Stokes and surface quasi-Geostrophic equations

AU - Chae, Dongho

AU - Constantin, Peter

AU - Wu, Jiahong

PY - 2012/12/1

Y1 - 2012/12/1

N2 - This paper is devoted to the global (in time) regularity problem for a family of active scalar equations with fractional dissipation. Each component of the velocity field u is determined by the active scalar θ through RΛ-1P(Λ)θ, where R denotes a Riesz transform, Λ = (-Δ)1/2, and P(Λ) represents a family of Fourier multiplier operators. The 2D Navier-Stokes vorticity equations correspond to the special case P(Λ) = I, while the surface quasi-geostrophic (SQG) equation corresponds to P(Λ) = Λ. We obtain the global regularity for a class of equations for which P(Λ) and the fractional power of the dissipative Laplacian are required to satisfy an explicit condition. In particular, the active scalar equations with any fractional dissipation and with P(Λ) = (log(I - Δ))γ for any γ > 0 are globally regular.

AB - This paper is devoted to the global (in time) regularity problem for a family of active scalar equations with fractional dissipation. Each component of the velocity field u is determined by the active scalar θ through RΛ-1P(Λ)θ, where R denotes a Riesz transform, Λ = (-Δ)1/2, and P(Λ) represents a family of Fourier multiplier operators. The 2D Navier-Stokes vorticity equations correspond to the special case P(Λ) = I, while the surface quasi-geostrophic (SQG) equation corresponds to P(Λ) = Λ. We obtain the global regularity for a class of equations for which P(Λ) and the fractional power of the dissipative Laplacian are required to satisfy an explicit condition. In particular, the active scalar equations with any fractional dissipation and with P(Λ) = (log(I - Δ))γ for any γ > 0 are globally regular.

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U2 - 10.1512/iumj.2012.61.4756

DO - 10.1512/iumj.2012.61.4756

M3 - Article

AN - SCOPUS:84887852378

VL - 61

SP - 1997

EP - 2018

JO - Indiana University Mathematics Journal

JF - Indiana University Mathematics Journal

SN - 0022-2518

IS - 5

ER -