Stability in Gagliardo–Nirenberg–Sobolev Inequalities Flows, Regularity, and the Entropy Method

We start by discussing stability results in Gagliardo–Nirenberg–Sobolev inequalities from a variational point of view. Using a non scale invariant form of the inequalities, which is equivalent to entropy-entropy production inequalities arising in the study of large time asymptotics of solutions to fast diffusion equations, we first establish non constructive estimates where the distance to the manifold of optimal functions is measured by a relative Fisher information. When the tails of the initial data have a certain decay, solutions to the fast diffusion equation converge to self-similar Barenblatt functions in the strong topology of uniform convergence in relative error after some finite time. This threshold time plays a fundamental role in obtaining a constructive stability result. Up to the threshold time, that is, in the initial time layer, the carré du champ method provides improved decay rates of the relative entropy. After the threshold time, in the asymptotic time layer, improved rates of decay can be deduced from improved spectral gap estimates in the linearized problem, under appropriate orthogonality conditions. In the subcritical regime, these orthogonality conditions follow from an appropriate choice of the coordinates which amount to fix the center of mass at the origin. In the critical case, that is for Sobolev’s inequality, scale invariance has to be taken into account. This can be rephrased as a strategy for computing the relative entropy with respect to a notion of best matching self-similar Barenblatt functions in place of the standard approach where entropy is defined relatively to a fixed family of self-similar solutions. Best matching is adapted to nonlinear evolution equations and degenerates in the asymptotic regime into more standard orthogonality conditions. With this method, we provide fully constructive stability estimates, to the price of a small restriction of the functional space which is inherent to the method.