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The nonlocal Fokker–Planck equations for a class of stochastic differential equations with non-Gaussian α -stable Lévy motion in Euclidean space are studied. The existence and uniqueness of weak solution are obtained with rough drift. The solution is shown to be smooth on spatial variable if all derivatives of the drift are bounded. Moreover, the solution is jointly smooth on spatial and time variable if we assume further that the drift grows like a power of logarithm function at infinity. [ABSTRACT FROM AUTHOR]

Abstract: The long time behavior of solutions of 2D quasi-geostrophic equations with damping on the whole space is studied. The existence of the global attractor in is proved. New difficulties come from nonlocal terms and the Banach space . We overcome them by using commutator estimates and positivity inequalities associated with fractional Laplacian. [Copyright &y& Elsevier]

Abstract: In this paper we consider the long time behavior of solutions of a version of gKdV equation on the real line. The existence of the global attractor in is proved when the force f belongs to . Moreover, we investigate the change of the attractor according to the change of f and the coefficient γ of damping term. Finally, several numerical experiments confirm all the theoretical conclusions. [Copyright &y& Elsevier]