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We establish new results on convergence, in strong topologies, of solutions of the parabolic-parabolic Keller–Segel system in the plane to the corresponding solutions of the parabolic-elliptic model, as a physical parameter goes to zero. Our main tools are suitable space-time estimates, implying the global existence of slowly decaying (in general,… (More)
We show that the solutions to the non-stationary Navier–Stokes equations in R d (d = 2, 3) which are left invariant under the action of discrete subgroups of the orthogonal group O(d) decay much faster as |x| → ∞ or t → ∞ than in generic case and we compute, for each subgroup, the precise decay rates in space-time of the velocity field.
The initial value problem for the conservation law ∂ t u + (−) α/2 u + ∇ · f (u) = 0 is studied for α ∈ (1, 2) and under natural polynomial growth conditions imposed on the nonlinearity. We find the asymptotic expansion as |x| → ∞ of solutions to this equation corresponding to initial conditions, decaying sufficiently fast at infinity.
We show that solutions u(x, t) of the non-stationnary incompressible Navier–Stokes system in R d (d ≥ 2) starting from mild decaying data a behave as |x| → ∞ as a potential field: u(x, t) = e t∆ a(x) + γ d ∇ x h,k δ h,k |x| 2 − dx h x k d|x| d+2 K h,k (t) + o 1 |x| d+1 (i) where γ d is a constant and K h,k = t 0 (u h |u k) L 2 is the energy matrix… (More)
We study the solutions of the nonstationary incompressible Navier–Stokes equations in R d , d ≥ 2 , of self-similar form u(x, t) = 1 √ t U x √ t , obtained from small and homogeneous initial data a(x). We construct an explicit asymptotic formula relating the self-similar profile U (x) of the velocity field to its corresponding initial datum a(x).
We show that the vorticity of a viscous flow in R 3 admits an atomic decomposition of the form ω(x, t) = ∞ k=1 ω k (x − x k , t), with localized and oscillating building blocks ω k , if such a property is satisfied at the beginning of the evolution. We also study the long time behavior of an isolated coherent structure and the special behavior of flows with… (More)
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 p-norms of the velocity field grow to infinity for large time for 1 ≤ p < 3. In the case of strong solutions we provide sharp… (More)
We exhibit a sufficient condition in terms of decay at infinity of the initial data for the finite time blowup of strong solutions to the Camassa–Holm equation: a wave breaking will occur as soon as the initial data decay faster at infinity than the solitons. In the case of data decaying slower than solitons we provide persistence results for the solution… (More)
We study incompressible Navier–Stokes flows in R d with small and well localized data and external force f. We establish pointwise estimates for large |x| of the form c t |x| −d ≤ |u(x, t)| ≤ c t |x| −d , where c t > 0 whenever t 0 f (x, s) dx ds = 0. This sharply contrasts with the case of the Navier–Stokes equations without force, studied in Brandolese… (More)