Refined asymptotics for the subcritical Keller-Segel system and related functional inequalities (Q2845868)

Some properties of solutions to the drift-diffusion equation with logarithmic potential NEWLINE\[NEWLINE \partial_t\rho = \text{div}\left[ \frac{1}{N} \nabla\rho + 2 \chi \rho \left( \nabla \log(|x|) \star \rho \right) \right]\;, \quad t>0\;, \;x\in\mathbb{R^N}\;, NEWLINE\]NEWLINE are established in one space dimension (\(N=1\)) and in two space dimensions (\(N=2\)), the latter being restricted to radially symmetric solutions. Recall that, when \(N=2\), the above equation coincides with the Smoluchowski-Poisson equation and the parabolic-elliptic Keller-Segel model for chemotaxis. Assuming that the initial condition satisfies NEWLINE\[NEWLINE \int_{\mathbb{R}^N} \rho(0, x)\;dx = 1\;, \quad \int_{\mathbb{R}^N} x \rho(0, x)\;dx = 0\;, NEWLINE\]NEWLINE these properties remain true for \(\rho(t)\) for all positive times and it is shown that, if \(\chi\in (0,1)\), a suitably rescaled version of \(\rho\) converges exponentially fast to a stationary state (in rescaled variables) for the \(2\)-Wasserstein metric. As already noticed in [\textit{J.~Dolbeault} and \textit{B.~Perthame}, C. R., Math., Acad. Sci. Paris 339, No. 9, 611--616 (2004; Zbl 1056.35076)], a key tool in the analysis is the logarithmic Hardy-Littlewood-Sobolev inequality, and one of the outcomes of the paper under review is an alternative proof of this inequality when \(N=1,2\). The equation is also shown to generate a contractive dynamical system for a specific Fourier distance when \(N=1\).