We investigate the large-time behavior of the solutions of the two-dimensional Keller-Segel system in self-similar variables, when the total mass is subcritical, that is less than 8 π after a proper adimensionalization. It was known from previous works that all solutions converge to stationary solutions, with exponential rate when the mass is small. Here we remove this restriction and show that the rate of convergence measured in relative entropy is exponential for any mass in the subcritical range, and independent of the mass. The proof relies on symmetrization techniques, which are adapted from a paper of J.I. Diaz, T. Nagai, and J.-M. Rakotoson, and allow us to establish uniform estimates for L p norms of the solution. Exponential convergence is obtained by the mean of a linearization in a space which is defined consistently with relative entropy estimates and in which the linearized evolution operator is self-adjoint. The core of proof relies on several new spectral gap estimates which are of independent interest. R 2 n 0 dx < 8 π , then there exists a solution u, in the sense of distributions, that is global in time and such that M = R 2 u(t, x) dx is conserved along the evolution in the euclidean space R 2 . There is no non-trivial stationary solution of (1.1) and any solution converges to zero locally as time gets large. In order to study the asymptotic behavior of u, it is convenient to work in self-similar variables. We define R(t) := √ 1 + 2 t, τ (t) := log R(t), and the rescaled functions n and c by u(t, x) := R −2 (t) n τ (t), R −1 (t) x and v(t, x) := c τ (t), R −1 (t) x .
We study a three dimensional continuous model of gravitating matter rotating at constant angular velocity. In the rotating reference frame, by a finite dimensional reduction, we prove the existence of non radial stationary solutions whose supports are made of an arbitrarily large number of disjoint compact sets, in the low angular velocity and large scale limit. At first order, the solutions behave like point particles, thus making the link with the relative equilibria in N -body dynamics.
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