Abstract. In this paper, we establish analyticity of the Navier-Stokes equations with small data in critical Besov spacesḂ 3 p −1 p,q . The main method is so-called Gevrey estimates, which is motivated by the work of Foias and Temam [19]. We show that mild solutions are Gevrey regular, i.e. the energy bound ep,q ), globally in time for p < ∞. We extend these results for the intricate limiting case p = ∞ in a suitably designed E∞ space. As a consequence of analyticity, we obtain decay estimates of weak solutions in Besov spaces. Finally, we provide a regularity criterion in Besov spaces.

Abstract. In this paper we consider the integral Volterra operator on the space L 2 ð0; 1Þ. We say that a complex number is an extended eigenvalue of V if there exists a nonzero operator X satisfying the equation XV ¼ VX. We show that the set of extended eigenvalues of V is precisely the interval ð0; 1Þ and the corresponding eigenvectors may be chosen to be integral operators as well.

We prove the existence and Gevrey regularity of local solutions of the three dimensional periodic Navier-Stokes equations in case the sequence of Fourier coefficients of the initial data lies in an appropriate weighted p space. Our work is motivated by that of Foias and Temam ([10]) and we obtain a generalization of their result. In particular, our analysis allows for initial data that are less smooth than in [10] and can also have infinite energy. Our main tool is a variant of the Young convolution inequality enabling us to estimate the nonlinear term in weighted p norm.

No abstract

In this paper, following the techniques of Foias and Temam, we establish Gevrey class regularity of solutions to a class of dissipative equations with a general quadratic nonlinearity and a general dissipation including fractional Laplacian. The initial data is taken to be in Besov type spaces defined via "caloric extension". We apply our result to the Navier-Stokes equations, the surface quasi-geostrophic equations, the Kuramoto-Sivashinsky equation and the barotropic quasi-geostrophic equation. Consideration of initial data in critical regularity spaces allow us to obtain generalizations of existing results on the higher order temporal decay of solutions to the Navier-Stokes equations. In the 3D case, we extend the class of initial data where such decay holds while in 2D we provide a new class for such decay. Similar decay result, and uniform analyticity band on the attractor, is also proven for the sub-critical 2D surface quasi-geostrophic equation.