No abstract

In this paper, we study an LES model for the approximation of large scales of the 3D Boussinesq equations. This model is obtained using the approach first described by Stolz and Adams, based on the Van Cittern approximate deconvolution operators, and applied to the filtered Boussinesq equations. Existence and uniqueness of a regular weak solution are provided. Our main objective is to prove that this solution converges towards a solution of the filtered Boussinesq equations, as the deconvolution parameter goes to zero.

Abstract. By using topological methods, mainly the degree of a tangent vector field, we establish multiplicity results for T -periodic solutions of parametrized T -periodic perturbations of autonomous ODEs on a differentiable manifold M . In order to provide insights into the key notion of T -resonance, we consider the elementary situations M = R and M = R 2 . So doing, we provide more comprehensive analysis of those cases and find improved conditions.

We consider the 3D simplified Bardina turbulence model with horizontal filtering, fractional dissipation, and the presence of a memory term incorporating hereditary effects. We analyze the regularity properties and the dissipative nature of the considered system and, in our main result, we show the existence of a global exponential attractor in a suitable phase space.

We consider the Cauchy-problem for a parabolic equation of the following type:where f = f (u, |x|) is supercritical. We supply this equation by the initial condition u(x, 0) = φ, and we allow φ to be either bounded or unbounded in the origin but smaller than stationary singular solutions. We discuss local existence and long time behaviour for the solutions u(t, x; φ) for a wide class of nonhomogeneous non-linearities f . We show that in the supercritical case, Ground States with slow decay lie on the threshold between blowing up initial data and the basin of attraction of the null solution. Our results extend previous ones allowing Matukuma-type potential and more generic dependence on u.Then, we further explore such a threshold in the subcritical case too. We find two families of initial data ζ(x) and ψ(x) which are respectively above and below the threshold, and have arbitrarily small distance in L ∞ norm, whose existence is new even for f (u, r) = u q−1 . Quite surprisingly both ζ(x) and ψ(x) have fast decay (i.e. ∼ |x| 2−n ), while the expected critical asymptotic behavior is slow decay (i.e. ∼ |x| 2/q−2 ).