Stability with explicit constants of the critical points of the fractional Sobolev inequality and applications to fast diffusion
Nicola De Nitti, Tobias König
TL;DR
This work analyzes stability of critical points for the fractional Sobolev inequality with explicit constants. It establishes a quantitative, single-bubble stability inequality around Talenti bubbles, providing an explicit constant $c^1_{CP}(s)$ and showing it is strictly less than the natural spectral gap $\gamma_{N,s}^2$, with precise optimality conditions. The results extend to Palais-Smale-type sequences and enable a rigorous link between stability and dynamics, which the authors exploit to derive an explicit extinction/convergence rate for the fractional fast-diffusion equation: the solution converges to the extinction profile with rate dictated by $\kappa_{N,s}=\frac{\gamma_{N,s}^2}{(N+2-2s)(p-1)}$ where $p=2^*_s-1$. This yields actionable, explicit decay estimates in $L^{2^*_s}$ and, via interpolation, in $L^{\infty}$, connecting sharp nonlocal Sobolev stability to precise dynamical behavior in fractional diffusion problems.
Abstract
We study the quantitative stability of critical points of the fractional Sobolev inequality. We show that, for a non-negative function $u \in \dot H^s(\mathbb R^N)$ whose energy satisfies $$\tfrac{1}{2} S^\frac{N}{2s}_{N,s} \le \|u\|_{\dot H^s(\mathbb R^N)} \le \tfrac{3}{2}S_{N,s}^\frac{N}{2s},$$ where $S_{N,s}$ is the optimal Sobolev constant, the bound $$ \|u -U[z,λ]\|_{\dot{H}^s(\mathbb R^N)} \lesssim \|(-Δ)^s u - u^{2^*_s-1}\|_{\dot{H}^{-s}(\mathbb R^N)}, $$ holds for a suitable fractional Talenti bubble $U[z,λ]$. {For functions $u$ which are close to Talenti bubbles, we give the sharp asymptotic value of the implied constant in this inequality.} As an application {of this}, we derive an explicit polynomial extinction rate for positive solutions to a fractional fast diffusion equation.