Lorenzo Brandolese

Large time decay and growth for solutions of a viscous Boussinesq system

In this paper we analyze the decay and the growth for large time of weak and strong solutions to the three-dimensional viscous Boussinesq system. We show that generic solutions blow up as t→∞ in the sense that the energy and the L-norms of the velocity field grow to infinity for large time for 1 ≤ p < 3. In the case of strong solutions we provide sharp estimates both from above and from below and explicit asymptotic profiles. We also show that solutions arising from (u0, θ0) with zero-mean for the initial temperature θ0 have a special behavior as |x| or t tends to infinity: contrarily to the generic case, their energy dissipates to zero for large time.

ASYMPTOTIC BEHAVIOR OF THE ENERGY AND POINTWISE ESTIMATES FOR SOLUTIONS TO THE NAVIER-STOKES EQUATIONS

In this paper we deal with the asymptotic behavior, in the space-time variables, of weak and strong solutions to the Navier-Stokes system. For an incompressible viscous fluid which fills the whole space R n , in the absence of external forces, the @tu + r · (u u) = u r p u(x,0) = a(x) div(u) = 0. Here u : R n ◊ (0,1(! R n (n 2) denotes the velocity field and p(x,t) is the pressure. Starting with the pioneering work of Leray (21), a considerable number of papers is concerned with questions related to the large-time behavior of the L 2 -norm of u(t). The problem of finding optimal decay rates for the energy of generic weak solutions is now well understood. Indeed, Wiegner (41) showed that ||u(t)||2 C(1 +t) (0 < (n + 2)/4), if such decay holds for the solution e t a of the heat equation starting with the same data. This improved previous results by Kato (17), Schonbek (29) and Kaijkiya- Miyakawa (19). The bound on is now known to be optimal: optimality was first discussed in (30) and, more recently, in (28), (13), (14) with dierent methods. However, exceptional flows which decay much faster do exist. For example, it is known since a long time that, in dimension n = 2, there exists a very particular and explicit solution of the Navier-Stokes equations with radial vorticity. This condition on the vorticity implies that the nonlinearity has the potential form (i.e. r ·(u u) = r p), so that u is also a solution of the homogeneous heat equation. It was pointed out by Majda and Schonbek that for such flow ||u(t)||2 has an exponential decay at infinity (see e.g. (30), (10), (28)). In dimension 2, no other examples with such a property seem to be known. Similar flows with exponential decay exist in higher even dimension and a general method for their construction is described in (32). All these solutions, sometimes called generalized Beltrami flows, turn out to solve simultaneously (NS) and the heat equation. As discussed in (32), it seems impossible to adapt these examples to the n = 3 case or for general odd dimensions.

Fine Properties of Self-Similar Solutions of the Navier–Stokes Equations

We study the solutions of the nonstationary incompressible Navier–Stokes equations in

Application of the Realization of Homogeneous Sobolev Spaces to Navier--Stokes

{\mathbb{R}^d}, d\geqq2

Poisson kernels and sparse wavelet expansions

, of self-similar form

Large Time Behavior of the Navier–Stokes Flow

{u(x,t)=\frac{1}{\sqrt t}U\left(\frac{x}{\sqrt t}\right)}