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We unify a few of the best known results on wave breaking for the Camassa– Holm equation (by R. Camassa, A. Constantin, J. Escher, L. Holm, J. Hyman and others) in a single theorem: a sufficient condition for the breakdown is that u′0 + |u0| is strictly negative in at least one point x0 ∈ R. Such blowup criterion looks more natural than the previous ones,… (More)

We study the solutions of the nonstationary incompressible Navier–Stokes equations in R , d ≥ 2, of self-similar form u(x, t) = 1 √

We establish new convergence results, in strong topologies, for 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)

Given a finite sequence of times 0 < t1 < · · · < tN , we construct an example of a smooth solution of the free nonstationnary Navier-Stokes equations in Rd, d = 2,3, such that: (i) The velocity field u(x, t) is spatially poorly localized at the beginning of the evolution but tends to concentrate until, as the time t approaches t1, it becomes… (More)

We study the global existence and space-time asymptotics of solutions for a class of nonlocal parabolic semilinear equations. Our models include the Nernst–Planck and the Debye–Hückel drift-diffusion systems as well as parabolic-elliptic systems of chemotaxis. In the case of a model of self-gravitating particles, we also give a result on the finite time… (More)

We show that the solutions to the non-stationary Navier–Stokes equations in R (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/2u + ∇ · 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 ≥ 2) starting from mild decaying data a behave as |x| → ∞ as a potential field: u(x, t) = ea(x) + γd∇x ∑ h,k δh,k |x| − dxhxk d|x|d+2 Kh,k(t) + o( 1 |x|d+1 ) (i) where γd is a constant and Kh,k = ∫ t 0 (uh|uk)L2 is the energy matrix of the flow. We deduce… (More)

We study the large time behavior of the energy of a class of Navier–<lb>Stokes flows in Rn (n ≥ 2) with special symmetries. Inside this class, we<lb>construct examples of solutions such that the energy norm decays faster than<lb>t−(n+2)/4. Institut Girard Desargues, Université Lyon 1, 21, avenue Clude Bernard, 69622<lb>Villeurbanne Cedex, FRANCE, e-mail:… (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)