Quantitative dynamics of multi-bubble regimes for the Sobolev-critical fast diffusion equation and Yamabe-type flows

The prescribed scalar curvature equation $-\Delta u=Ku^{(n+2)/(n-2)}$ on $\mathbb{R}^n$, $n\ge3$, is quantitatively rigid in the single-bubble energy range: Ciraolo, Figalli, and Maggi proved that if the $H^1$-energy of a positive solution is below the energy of two bubbles (concretely $\int|\nabla u|^2\le\tfrac32 S^n$), then a small curvature defect $\delta(u)$ forces $u$ to be a single Aubin-Talenti bubble plus a remainder controlled linearly by $\delta(u)$, and used this to give the first quantitative extinction rate for the Sobolev-critical fast diffusion flow on $\mathbb{R}^n$; the multi-bubble energy range was explicitly out of scope. Later literature has settled the static half of that exclusion: Figalli and Glaudo proved the linear multi-bubble estimate in dimensions $3\le n\le5$ and refuted it for two or more bubbles in $n\ge6$, and Deng, Sun, and Wei proved the sharp nonlinear replacements in $n\ge6$. The surviving open problem is the dynamical half: for the fast diffusion equation $u_t=\Delta(u^{(n-2)/(n+2)})$ (which encodes the conformal Yamabe flow) and for Yamabe-type gradient flows, in energy regimes supporting two or more bubbles -- regimes that arise on bounded domains with Dirichlet data and on compact manifolds, since on the whole space extinction is always single-bubble -- establish a quantitative theory: classification of extinction and singularity-formation profiles, explicit convergence rates for general data, sharp laws for interacting-bubble blow-up, and control of the remainder modulo the multi-bubble manifold via energy and entropy dissipation.

Background

Conformal metrics and bubbles. For $n\ge3$, a Riemannian metric on $\mathbb{R}^n$ conformal to the flat metric has the form $g=u^{4/(n-2)}\,\mathrm{d}x^2$ with $u>0$, and its scalar curvature is constant $\kappa>0$ exactly when $u$ solves the prescribed scalar curvature equation $-\Delta u=\kappa u^{p}$ with $p=\frac{n+2}{n-2}$ and finite energy $\int_{\mathbb{R}^n}|\nabla u|^2$. By the classical classification of Obata and of Gidas-Ni-Nirenberg, all positive finite-energy solutions for constant $\kappa$ are the Aubin-Talenti bubbles $v_\kappa[z,\lambda](x)=\lambda^{(n-2)/2}v_\kappa(\lambda(x-z))$, where $v_\kappa(x)=\left(\frac{n(n-2)}{\kappa}\right)^{(n-2)/4}\left(1+|x|^2\right)^{-(n-2)/2}$. Bubbles are precisely the extremals of the sharp Sobolev inequality $\|\nabla u\|_{L^2}\ge S\|u\|_{L^{2^*}}$ with $2^*=\frac{2n}{n-2}$ and $S$ the best constant; after the natural normalization a single bubble carries gradient energy $S^n$, so the energy counts (approximately) the number of bubbles present.

The defect functional and the bubbling obstruction. Writing the equation as $-\Delta u=Ku^p$ with $K$ a positive curvature function, Ciraolo-Figalli-Maggi measure how far $K$ is from a constant by $\delta(u)=\|Ku^p-K_0(u)\,u^p\|_{L^{2n/(n+2)}}$, where $K_0(u)=\int Ku^{2^*}/\int u^{2^*}$ is the $u^{2^*}$-weighted average of $K$ (the exponent $2n/(n+2)$ is the conjugate of $2^*$). If $\delta(u)$ is small, must $u$ be close to a single bubble? Not in general: a sum $u=\sum_{i=1}^m v_1[z_i,\lambda_i]$ of bubbles with centers $z_i$ very far apart has $\delta(u)$ arbitrarily small while staying far from any one bubble, a phenomenon called bubbling. Struwe's global compactness theorem shows qualitatively that this is the only obstruction: whenever $\delta(u)\to0$ (equivalently, the residual $\Gamma(u)=\|\Delta u+u^p\|_{H^{-1}}$ of the Euler-Lagrange equation tends to zero), $u$ approaches the manifold $\mathcal{T}$ of sums of weakly-interacting bubbles, but with no quantitative control.

The single-bubble quantitative theorem and its flow application. Ciraolo-Figalli-Maggi proved the first quantitative statement, in the regime where bubbling is excluded by an energy bound: assuming the energy is below that of two bubbles (for concreteness $\int_{\mathbb{R}^n}|\nabla u|^2\le\tfrac32 S^n$, the constant $\tfrac32$ being replaceable by any constant strictly less than $2$), small $\delta(u)$ implies $u=v_1[z,\lambda]+\rho$ with $\|\nabla\rho\|_{L^2}\le C_0\,\delta(u)$, and the linear dependence is optimal. As an application they studied the fast diffusion equation $u_t=\Delta(u^m)$ at the critical exponent $m=\frac{n-2}{n+2}$, for which the metric $g(t)=u(t)^{4/(n-2)}\mathrm{d}x^2$ evolves by a (non-volume-preserving) Yamabe flow $\partial_t g=-R_{g(t)}g(t)$, the gradient flow of the Yamabe functional (the total-scalar-curvature energy $\int R_g\,\mathrm{d}V_g$, up to a constant factor). Solutions vanish at a finite extinction time $T$; del Pino and Sáez proved that for data $u_0(x)=O(|x|^{-(n+2)})$ the extinction profile is a single bubble, while Daskalopoulos and Sesum showed that data trapped between two Barenblatt profiles (the self-similar separated solutions of fast diffusion) obey a different asymptotic. Ciraolo-Figalli-Maggi upgraded the del Pino-Sáez profile to a quantitative rate: the relative error $\|u(t,\cdot)/u_{T,z,\lambda}(t,\cdot)-1\|_{L^\infty}$, where $u_{T,z,\lambda}(t)$ is the explicit separated-variable solution of the critical flow carrying a single shrinking bubble, decays like $C^*(T-t)^{\kappa(n)}$, obtained by converting the energy dissipation of the rescaled flow into control of $\delta$ and then invoking their structure theorem.

What has been settled since. The static multi-bubble question has been completely resolved, with a dimension dichotomy that was unexpected: Figalli-Glaudo proved that for $3\le n\le5$ the linear estimate $\mathrm{dist}(u,\mathcal{T})\le C\,\Gamma(u)$ holds for any finite number of weakly-interacting bubbles, while for two or more bubbles in $n\ge6$ it fails, with implicit counterexamples built from the linearized equation near a sum of bubbles; Deng-Sun-Wei then proved the sharp replacements $\mathrm{dist}(u,\mathcal{T})\le C\,\Gamma(u)|\log\Gamma(u)|^{1/2}$ for $n=6$ and $\le C\,\Gamma(u)^{\frac{n+2}{2(n-2)}}$ for $n\ge7$, with matching counterexamples, together with interaction bounds between the bubbles. On the flow side, on the whole space Figalli-Glaudo show that every nonnegative solution with datum in $L^{(2^*)'}(\mathbb{R}^n)=L^{2n/(n+2)}(\mathbb{R}^n)$ extinguishes with a single-bubble profile at a quantitative rate, so genuinely multi-bubble dynamical regimes do not occur there; they arise instead for the Dirichlet problem on bounded domains -- where Jin-Xiong proved the one-bubble extinction dichotomy only under a two-bubble energy threshold, Bonforte-Figalli and Jin-Xiong obtained sharp extinction rates for subcritical exponents and for a Brezis-Nirenberg-type spectral perturbation of the critical equation (the equation with an added lower-order spectral term, as in the classical Brezis-Nirenberg problem), and Sire-Wei-Zheng constructed by gluing special solutions whose energy concentrates at finitely many points -- and for Yamabe-type flows on compact manifolds, where Chen-Kim-Musso constructed ancient (defined for all negative time) Yamabe flows on the sphere whose backward limits are wedge sums (one-point unions) of finitely many spheres. A Łojasiewicz–Simon inequality (a lower bound on the norm of the gradient of an analytic functional by a power of its energy gap, used to extract convergence rates for gradient flows) has very recently been established near finitely-many-bubble configurations of the Yamabe functional on bounded domains by Jin-Xiong-Zhou, as elliptic groundwork for exactly these parabolic questions.

Problem Statement

The source paper (Ciraolo-Figalli-Maggi) developed its quantitative structure theorem and fast-diffusion estimates only under the single-bubble energy restriction $\int_{\mathbb{R}^n}|\nabla u|^2\le\tfrac32 S^n$ (any constant below the two-bubble threshold $2S^n$ suffices) and left the complementary multi-bubble energy range without quantitative results. The static part of that exclusion has since been settled (Figalli-Glaudo 2020; Deng-Sun-Wei 2025): a sharp quantitative multi-bubble decomposition now exists in every dimension, linear in the defect for $3\le n\le5$ and of sharp nonlinear order $\Gamma|\log\Gamma|^{1/2}$ in $n=6$ and $\Gamma^{\frac{n+2}{2(n-2)}}$ in $n\ge7$ otherwise, where $\Gamma=\|\Delta u+u^p\|_{H^{-1}}$. The surviving problem is the dynamical part:

Notation. Throughout, $n\ge3$ is the dimension; $p=\frac{n+2}{n-2}$ is the critical Sobolev exponent; $S$ is the best constant in the Sobolev inequality $\|\nabla u\|_{L^2(\mathbb{R}^n)}\ge S\|u\|_{L^{2^*}(\mathbb{R}^n)}$ with $2^*=\frac{2n}{n-2}$; $v_\kappa[z,\lambda]$ is the Aubin-Talenti bubble defined in the Background, and a single bubble carries gradient energy $S^n$, so the "two-bubble energy threshold" is $2S^n$; $\delta(u)$ is the curvature defect and $\Gamma(u)=\|\Delta u+u^p\|_{H^{-1}}$ the equation residual, both defined in the Background; and $\mathcal{T}$ denotes the manifold of sums of finitely many weakly-interacting bubbles.

For the Sobolev-critical fast diffusion equation $u_t=\Delta\left(u^{(n-2)/(n+2)}\right),$ which encodes the conformal Yamabe flow via $g(t)=u(t)^{4/(n-2)}\mathrm{d}x^2$, and for Yamabe-type gradient flows, establish a quantitative theory of the energy regimes in which the solution can support two or more bubbles. These are the regimes on bounded domains with Dirichlet boundary conditions above the two-bubble energy threshold, and on compact manifolds such as the sphere; on the whole space $\mathbb{R}^n$ they do not occur at extinction, since extinction is always single-bubble for nonnegative data in $L^{(2^*)'}(\mathbb{R}^n)=L^{2n/(n+2)}(\mathbb{R}^n)$ (Figalli-Glaudo, Theorem 5.1). The theory should determine, with explicit constants and sharp rate laws:

1. the classification of long-time and extinction behavior: which profiles arise as limits -- finite sums of weakly-interacting bubbles, steady states where such exist (for instance on annuli or topologically nontrivial domains), or other singular profiles -- and in which relative configurations;

2. explicit convergence rates for general initial data above the two-bubble threshold, in relative error and natural norms, analogous to the single-bubble rates already available below it;

3. a sharp description of singularity formation and blow-up when multiple bubbles interact, including the optimal rate laws -- noting that already the static distance estimate is nonlinear in the defect for $n\ge6$, so linear-in-defect dynamical rates cannot be expected there;

4. whether, and how, the energy and entropy dissipation identities of the flow can be converted into effective control of the infinite-dimensional remainder modulo the multi-bubble manifold.

The four items above are independent sub-questions, each admitting a separate answer.

References

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