We study solutions to the 2D quasi-geostrophic (QGS) equationand prove global existence and uniqueness of smooth solutions if α ∈ ( 1 2 , 1]; weak solutions also exist globally but are proven to be unique only in the class of strong solutions. Detailed aspects of large time approximation by the linear QGS equation are obtained.

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This paper establishes several existence and uniqueness results for two families of active scalar equations with velocity fields determined by the scalars through very singular integrals. The first family is a generalized surface quasi-geostrophic (SQG) equation with the velocity field u related to the scalar θ by u = ∇⊥Λ β−2 θ, where 1 < β ≤ 2 and Λ = (−∆)1/2 is the Zygmund operator. The borderline case β = 1 corresponds to the SQG equation and the situation is more singular for β > 1. We obtain the local existence and uniqueness of classical solutions, the global existence of weak solutions and the local existence of patch type solutions. The second family is a dissipative active scalar equation with u = ∇⊥(log(I − ∆))µθ for µ > 0, which is at least logarithmically more singular than the velocity in the first family. We prove that this family with any fractional dissipation possesses a unique local smooth solution for any given smooth data. This result for the second family constitutes a first step towards resolving the global regularity issue recently proposed by K. Ohkitani.National Research Foundation of KoreaNational Science FoundationMinisterio de Ciencia e InnovaciónEuropean Research Counci

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Solutions of the d-dimensional generalized MHD (GMHD) equations@ t u þ u Á ru ¼ ÀrP þ b Á rb À nðÀDÞ a u;( are studied in this paper. We pay special attention to the impact of the parameters n; Z; a and b on the regularity of solutions. Our investigation is divided into three major cases: (1) n40 and Z40; (2) n ¼ 0 and Z40; and (3) n ¼ 0 and Z ¼ 0: When n40 and Z40; the GMHD equations with any a40 and b40 possess a global weak solution corresponding to any L 2 initial data. Furthermore, weak solutions associated with aX 1 2 þ d 4 and bX 1 2 þ d 4 are actually global classical solutions when their initial data are sufficiently smooth. As a special consequence, smooth solutions of the 3D GMHD equations with aX 5 4 and bX 5 4 do not develop finite-time singularities. The study of the GMHD equations with n ¼ 0 and Z40 is motivated by their potential applications in magnetic reconnection. A local existence result of classical solutions and several global regularity conditions are established for this case. These conditions are imposed on either the vorticity o ¼ r  u or the current density j ¼ r  b (but not both) and are weaker than some of current existing ones. When n ¼ 0 and Z ¼ 0; the GMHD equations reduce to the ideal MHD equations. It is shown here that the ideal MHD equations admit a unique local solution when the prescribed initial data is in a Ho¨lder space C r with r41: r