Optimal stability of the critical two-dimensional Gagliardo-Nirenberg-Sobolev inequality in density metrics, and sharp convergence rates for the critical-mass Keller-Segel equation

Carlen and Figalli (Duke Math. J. 2013) proved that the deficit of the sharp critical two-dimensional Gagliardo-Nirenberg-Sobolev inequality $\pi\int u^6\,dx\le\big(\int|\nabla u|^2\,dx\big)\big(\int u^4\,dx\big)$ controls the $L^1$ distances of the power densities $u^6$ and $u^4$ from the family of optimizers: with exponent $1/2$ in the $u^6$ metric (for small deficit) and exponent $(p-1)/(4p)$ in the $u^4$ metric (under entropy and $p$-moment bounds), but with constants inherited from the compactness-based Bianchi-Egnell stability estimate for the Sobolev inequality and hence non-explicit. Through a duality with a fast-diffusion equation, the same stability theory yields stability for the logarithmic Hardy-Littlewood-Sobolev inequality and algebraic, not known to be sharp, convergence rates for properly dissipative critical-mass ($8\pi$) Keller-Segel solutions. Later literature resolved the log-HLS side (Carlen 2025 proved unconditional stability with an explicit constant and optimal quadratic order, improving the Keller-Segel $L^1$ rate to $t^{-1/16}$) and produced constructive stability constants for the Gagliardo-Nirenberg-Sobolev family in entropy-type metrics (Bonforte-Dolbeault-Nazaret-Simonov) and explicit sharp constants at the Sobolev level (Dolbeault-Esteban-Figalli-Frank-Loss), while Buseghin-Collot determined the complementary finite-second-moment regime. What remains open is the optimal stability theory in the metrics tied to the Keller-Segel application: the optimal exponent for $u^4$-density stability under the natural entropy and moment bounds (is $(p-1)/(4p)$ improvable?), explicit values of the stability constants, and sharp rates, upper bounds matched by lower-bound examples, for the convergence of properly dissipative critical-mass Keller-Segel solutions to the stationary family.

Background

The sharp two-dimensional critical Gagliardo-Nirenberg-Sobolev (GNS) inequality states that for every non-negative function $u$ in the Sobolev space $W^{1,2}(\mathbb{R}^2)$ (the space of square-integrable functions with square-integrable weak first derivatives),

$$\pi\int_{\mathbb{R}^2}u^6(x)\,dx \;\le\; \left(\int_{\mathbb{R}^2}|\nabla u(x)|^2\,dx\right)\left(\int_{\mathbb{R}^2}u^4(x)\,dx\right),$$

with equality exactly for the functions $v_{\lambda,x_0}(x)=(1+\lambda^2|x-x_0|^2)^{-1/2}$ and their multiples; the sharp constants for the whole GNS family were computed by del Pino and Dolbeault. Its deficit is

$$\delta_{GNS}[u]=\left(\int_{\mathbb{R}^2}|\nabla u|^2\,dx\right)^{1/2}\left(\int_{\mathbb{R}^2}u^4\,dx\right)^{1/2}-\left(\pi\int_{\mathbb{R}^2}u^6\,dx\right)^{1/2}.$$

Quantitative stability theory asks: when $\delta_{GNS}[u]$ is small, in what sense must $u$ be close to an optimizer? For the Sobolev inequality, Bianchi and Egnell proved that the deficit controls the $L^2$ distance of the gradient from the gradients of optimizers; their proof is a compactness-contradiction argument and yields no explicit value of the constant. Carlen and Figalli (2013) transported this estimate through a derivation of the critical two-dimensional GNS inequality from the four-dimensional Sobolev inequality (an argument due to D. Bakry, presented in Section 2 of their paper), measuring closeness not in norms of $u$ but in the $L^1$ distances of the power densities $u^6$ and $u^4$ from the extremal family: the density $u^6$ is the natural variable for optimal-transport arguments, while $u^4$ is the density that appears in the Keller-Segel application below. Their Theorem 1.2 states that for $\|u\|_6=\|v\|_6$ (where $v=v_{1,0}$) and $\delta_{GNS}[u]\le\delta_1$, one has $\inf_{\lambda>0,x_0\in\mathbb{R}^2}\|u^6-\lambda^2v_{\lambda,x_0}^6\|_{L^1}\le K_1\delta_{GNS}[u]^{1/2}$; their Theorem 1.4 states that if, besides $\|u\|_4=\|v\|_4$, the entropy $\int u^4\log(u^4)\,dx$ is at most $A$ and the $p$-th moment $\int|y|^pu^4(y)\,dy$ is at most $B$ for some $1<p<2$, and $u^4$ is centered, then $\inf_{\lambda>0}\|u^4-\lambda^2v_\lambda^4\|_{L^1}\le K_2\delta_{GNS}[u]^{(p-1)/(4p)}$ whenever $\delta_{GNS}[u]\le\delta_2$. The constants $K_1,\delta_1$ are universal but non-explicit; $K_2,\delta_2$ depend only on $p,A,B$ and are non-explicit; and nothing is known about the optimality of the exponents. In particular $(p-1)/(4p)$ degenerates to $0$ as $p\downarrow1$, reflecting that entropy and $p$-moment bounds only barely control the tails of near-extremal functions.

The application runs through the logarithmic Hardy-Littlewood-Sobolev (log-HLS) functional

$$F[\rho]=\int_{\mathbb{R}^2}\rho(x)\log\rho(x)\,dx+2\left(\int_{\mathbb{R}^2}\rho(x)\,dx\right)^{-1}\iint_{\mathbb{R}^2\times\mathbb{R}^2}\rho(x)\log|x-y|\,\rho(y)\,dx\,dy,$$

defined for densities $\rho\ge0$ with $\rho\log\rho$ and $\rho\log(e+|x|^2)$ integrable. The sharp log-HLS inequality, proved independently by Beckner and by Carlen and Loss, states that $F[\rho]\ge C(M)=M(\log(M/\pi)-1)$ for every density of mass $M$, with equality exactly at the Barenblatt-type profiles $\sigma_{\kappa,M}(x)=\frac{M}{\pi}\frac{\kappa}{(\kappa+|x|^2)^2}$ and their translates; write $\delta_{HLS}[\rho]=F[\rho]-C(M)$ for its deficit. Two further functionals connect this to the GNS inequality. The fast-diffusion entropy $H_{\kappa,M}[\sigma]=\int_{\mathbb{R}^2}\frac{|\sqrt{\sigma}-\sqrt{\sigma_{\kappa,M}}|^2}{\sqrt{\sigma_{\kappa,M}}}\,dy$ is minimized exactly at $\sigma_{\kappa,M}$ and is extremely sensitive to tails: $H_{\kappa,8\pi}[\sigma_{\mu,8\pi}]=\infty$ whenever $\mu\ne\kappa$. The dissipation functional $\mathcal{D}[\sigma]=\frac{1}{\pi}\big(\|\nabla u\|_2^2\|u\|_4^4-\pi\|u\|_6^6\big)$, where $u=\sigma^{1/4}$, is a restatement of the GNS inequality and is therefore non-negative. The bridge is the fast-diffusion equation $\partial_t\sigma=\Delta\sqrt{\sigma}+2\sqrt{\pi/(\kappa M)}\,{\rm div}(x\sigma)$: along its solutions $\frac{d}{dt}F[\sigma(\cdot,t)]=-\frac{8\pi}{M}\mathcal{D}[\sigma(\cdot,t)]$ (Carlen-Carrillo-Loss), so smallness of the GNS deficit along the flow forces decay of the log-HLS deficit.

The parabolic-elliptic Keller-Segel system for chemotaxis (the directed motion of cells along the gradient of a chemical attractant they themselves emit) is $\partial_t\rho={\rm div}(\nabla\rho-\rho\nabla c)$ with $c=-\frac{1}{2\pi}\log|\cdot|*\rho$ on $\mathbb{R}^2$; it is the gradient flow of $F$ for the 2-Wasserstein metric, the distance between probability measures induced by optimally transporting mass at quadratic cost. The total mass $M$ is decisive (Dolbeault-Perthame): below the critical mass $8\pi$ solutions spread and vanish, above it they collapse in finite time, and at exactly $8\pi$ they aggregate in infinite time (Blanchet-Carrillo-Masmoudi), the stationary states being the $\sigma_{\kappa,8\pi}$. Blanchet-Carlen-Carrillo constructed properly dissipative critical-mass solutions, i.e. solutions with $H_{\kappa,8\pi}[\rho(\cdot,0)]<\infty$ along which $H_{\kappa,8\pi}[\rho(\cdot,T)]+\int_0^T\mathcal{D}[\rho(\cdot,t)]\,dt$ stays bounded; this is a thick-tail class, since the stationary states have $|x|^{-4}$ tails and hence infinite second moment. For these solutions Carlen and Figalli proved (Theorem 1.9) that if $H_{\kappa,M}[\rho]$, $F[\rho]$ and $\mathcal{D}[\rho]$ are finite, then $\|\rho-\sigma_{\mu,M}\|_{L^1}\le C\,\delta_{HLS}[\rho]^{(1-\epsilon)/20}$ for some $\mu$ (for small deficit, any $\epsilon>0$); and (Theorem 3.5) that for centered initial data $F[\rho(\cdot,t)]-C(8\pi)\le C_1(1+t)^{-(1-\epsilon)/8}$, $\inf_{\mu>0}\|\rho(\cdot,t)-\sigma_{\mu,8\pi}\|_{L^1}\le C_2(1+t)^{-(1-\epsilon)/160}$, and the selected scale obeys $(\mu(t)-\kappa)^2\le C/\log(e+t)$, so that $\|\rho(\cdot,t)-\sigma_{\kappa,8\pi}\|_{L^1}\le\big(C/\log(e+t)\big)^{1/2}$. Convergence thus occurs on two time scales: algebraically toward the one-parameter family of stationary states, and then only logarithmically in the selection of the correct spatial scale.

Later work reshaped this landscape. Carlen (2025) proved an optimal log-HLS stability theorem: for every mass-one density $\rho$ in $L^1(\mathbb{R}^2,\ln(e+|x|^2)\,dx)$, the deficit dominates $\frac{1}{8}\inf_{g\in M}\|\rho-g\|_{L^1}^2$, where $M$ is the manifold of optimizers; this is unconditional, carries an explicit constant and the optimal quadratic order, and improves the Keller-Segel $L^1$ rate to $t^{-1/16}$. Bonforte-Dolbeault-Nazaret-Simonov (2025) developed the entropy method — deriving stability by flowing the function along the fast-diffusion equation associated with the inequality and tracking an entropy functional along the flow — for the whole subcritical GNS family, which contains the critical two-dimensional inequality, obtaining constructive stability constants in relative-entropy/Fisher metrics (distances measured by a relative entropy and its dissipation along the flow, which control $L^1$ distances of the power densities; under tail conditions they list as open to remove). Dolbeault-Esteban-Figalli-Frank-Loss (2025) computed explicit dimensionally sharp stability constants for the Sobolev and Gaussian log-Sobolev inequalities, the base of the Carlen-Figalli chain. In the complementary regime of finite second moment, Buseghin-Collot (2026) classified the critical-mass dynamics (radial data in their classification note, arbitrary data in their companion determination paper): such solutions concentrate around the stationary profile at the universal scale $\lambda(t)\sim\sqrt{\mu/(8\pi)}/\sqrt{\ln t}$, where $\mu$ is the second moment of the data, with $L^1$ remainder $O(t^{-1+\epsilon})$, and they conjecture the rate $t^{-1}$ to be optimal. What no later work addresses is the optimal form of the Carlen-Figalli stability theory in its own density metrics under its own hypotheses, and the sharpness of the algebraic rates in the properly dissipative class; stating precisely these two linked questions is the purpose of this problem.

Problem Statement

Throughout, $u$ is a non-negative function in the Sobolev space $W^{1,2}(\mathbb{R}^2)$ (square-integrable functions with square-integrable weak first derivatives), $v_{\lambda,x_0}(x)=(1+\lambda^2|x-x_0|^2)^{-1/2}$, $v=v_{1,0}$, and

$$\delta_{GNS}[u]=\left(\int_{\mathbb{R}^2}|\nabla u|^2\,dx\right)^{1/2}\left(\int_{\mathbb{R}^2}u^4\,dx\right)^{1/2}-\left(\pi\int_{\mathbb{R}^2}u^6\,dx\right)^{1/2}$$

is the deficit in the sharp inequality $\pi\int u^6\,dx\le(\int|\nabla u|^2\,dx)(\int u^4\,dx)$.

Part A: optimal GNS stability in the density metrics.

A.1 (optimal exponent in the $u^4$ metric). For $1<p<2$ and $A,B<\infty$, consider non-negative $u\in W^{1,2}(\mathbb{R}^2)$ with $\|u\|_4=\|v\|_4$, $\int u^4\log(u^4)\,dx\le A$, $\int|y|^pu^4(y)\,dy\le B$ and $\int y\,u^4(y)\,dy=0$. Determine the optimal exponent $\alpha(p)$ such that

$$\inf_{\lambda>0}\|u^4-\lambda^2v_\lambda^4\|_{L^1}\le K\,\delta_{GNS}[u]^{\alpha(p)}$$

holds for all such $u$ (with $K$ and a smallness threshold depending only on $p,A,B$). Carlen and Figalli give $\alpha(p)\ge(p-1)/(4p)$; the question is whether this can be improved, for instance toward the exponent $1/2$ available in the $u^6$ metric, or whether tail obstructions under mere entropy-and-$p$-moment bounds force a smaller value, with an explicit near-extremal sequence exhibiting the obstruction. The analogous questions for the $u^6$-metric theorem (optimal exponent and constant; Carlen and Figalli's Remark 1.3 already notes that the smallness restriction there is removable by enlarging $K_1$) belong to the same objective as the $u^4$-metric questions.

A.2 (explicit constants). Provide explicit numerical values for the constants $K_1,\delta_1$ of the $u^6$-metric theorem and for $K_2,\delta_2$ (as functions of $p,A,B$) of the $u^4$-metric theorem, equivalently a constructive proof of stability in these density metrics, or prove that specific values are optimal.

Part B: sharp convergence rates for critical-mass Keller-Segel in the properly dissipative class. Here $\rho(x,t)$ solves the parabolic-elliptic Keller-Segel system $\partial_t\rho={\rm div}(\nabla\rho-\rho\nabla c)$ with $c=-\frac{1}{2\pi}\log|\cdot|*\rho$ on $\mathbb{R}^2$. Its free energy is the logarithmic Hardy-Littlewood-Sobolev functional

$$F[\rho]=\int_{\mathbb{R}^2}\rho\log\rho\,dx+\frac{1}{4\pi}\iint_{\mathbb{R}^2\times\mathbb{R}^2}\rho(x)\log|x-y|\,\rho(y)\,dx\,dy,$$

which at mass $8\pi$ is bounded below by $C(8\pi)=8\pi(\log 8-1)$, with equality exactly at the stationary densities $\sigma_{\kappa,8\pi}(x)=\frac{8\kappa}{(\kappa+|x|^2)^2}$, $\kappa>0$, and their translates. A solution is properly dissipative (in the sense of Blanchet-Carlen-Carrillo) if $H_{\kappa,8\pi}[\rho(\cdot,0)]<\infty$ for some $\kappa>0$ and $H_{\kappa,8\pi}[\rho(\cdot,T)]+\int_0^T\mathcal{D}[\rho(\cdot,t)]\,dt$ stays bounded, where $H_{\kappa,M}[\sigma]=\int_{\mathbb{R}^2}\frac{|\sqrt{\sigma}-\sqrt{\sigma_{\kappa,M}}|^2}{\sqrt{\sigma_{\kappa,M}}}\,dy$ and $\mathcal{D}[\sigma]=\frac{1}{\pi}(\|\nabla u\|_2^2\|u\|_4^4-\pi\|u\|_6^6)$ with $u=\sigma^{1/4}$. Take centered initial data, $\int x\rho(x,0)\,dx=0$. For such solutions Carlen and Figalli proved $F[\rho(\cdot,t)]-C(8\pi)\le C_1(1+t)^{-(1-\epsilon)/8}$ and $\inf_{\mu>0}\|\rho(\cdot,t)-\sigma_{\mu,8\pi}\|_{L^1}\le C_2(1+t)^{-(1-\epsilon)/160}$ for every $\epsilon>0$, with the $L^1$ rate improved to $O(t^{-1/16})$ by Carlen's 2025 log-HLS stability theorem; the last infimum is attained at a scale $\mu(t)$ contained in a compact subinterval of $(0,\infty)$, and $(\mu(t)-\kappa)^2\le C/\log(e+t)$.

B.1 Is the free-energy decay rate sharp: does there exist such a solution with $F[\rho(\cdot,t)]-C(8\pi)\ge c\,t^{-1/8}$ for all large $t$ (up to the $\epsilon$ in the known upper bound), or can the exponent $1/8$ be improved?

B.2 Which of the resulting convergence rates are sharp: the $L^1$ distance $\inf_{\mu>0}\|\rho(\cdot,t)-\sigma_{\mu,8\pi}\|_{L^1}$ (known to be $O(t^{-1/16})$), and the scale-selection bounds $(\mu(t)-\kappa)^2\le C/\log(e+t)$ and $\|\rho(\cdot,t)-\sigma_{\kappa,8\pi}\|_{L^1}\le(C/\log(e+t))^{1/2}$? Produce solutions with matching lower bounds, or improve the upper bounds.

Parts A and B form one linked objective, exactly as in the source: quantitative GNS stability in the $u^4$ density metric is the engine that, through the duality chain from $\mathcal{D}$ to $\delta_{HLS}$ to $L^1$ distance, produces the Keller-Segel rates, so any improvement of the exponent in A.1 propagates to B.1 and B.2, and conversely lower-bound examples in Part B delimit what Part A can achieve.

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