Papers

In this article we prove that the solution to the focusing $H^{1/2}$-critical nonlinear Schrödinger equation in dimension $d\geq 5$ scatters outside finitely many arbitrarily small spacetime cones. We will discuss the relationship between Theorem~\ref{thm:finite-bad-directions} and the resolution of solitons. There are two key ingredients in the proof: the localized below-ground-state scattering in Theorem~\ref{thm:cone-scattering}, and the characterization in Theorem~\ref{thm:measure-convergence} of the asymptotic behaviour of the residual part (i.e., after subtracting the scattering part). As a byproduct, we also establish an upgrading machinery: if the solution is below the ground state (resp. small) along a time sequence in a spacetime cone, then it is also below ground state (resp. small) uniformly for large time in a shrinked spacetime cone. We expect this to be useful in future works using concentration compactness and our spacetime cone approach.

We obtain Calderón–Zygmund estimates for weak solutions to nonlinear parabolic systems with polynomial $p$-growth and the right-hand side in non-divergence form. Our approach does not require differentiability of the vector field with respect to the gradient variable and provides a unified treatment of both the singular and degenerate regimes. In particular, we establish local higher-integrability estimates for the gradient of the solution under appropriate dependency on the integrability of the right-hand side.

We construct weak solutions of the three-dimensional incompressible Navier–Stokes equations on the torus. The convex-integration scheme is based on the moving-dipole construction of Bruè, Colombo, and Kumar~\cite{BrueColomboKumar2024}. For the explicit exponent $\bar p=\frac65+5\times10^{-5},$ and for any two mean-zero, divergence-free vector fields in $L^2(\mathbb T^3)$, we construct a weak solution whose traces at times $0$ and $1$ approximate the prescribed fields arbitrarily well and which satisfies \[ u\in C([0,1];L^2(\mathbb T^3)), \qquad \nabla u\in C([0,1];L^{\bar p}(\mathbb T^3)). \] Exploiting the time-locality of the iteration, we also obtain exact nonuniqueness for a dense set of initial data in \(L^2_\sigma(\mathbb T^3)\).The principal perturbations are localized, rescaled copies of Hill's spherical vortex. The Hill scaling preserves both the kinetic-energy scale and the \(L^{6/5}\)-scale of the velocity gradient. The construction uses localization of the potential exterior flow, long-orbit averaging of moving vortex cores, an auxiliary source correction, and a temporal corrector compatible with uniform-in-time Sobolev control.

In this paper, we investigate the dynamic stability of certain steady-state solutions to the periodic boundary value problem for compressible isentropic Navier-Stokes system under the van der Waals equation of state in one space dimension. These steady-state solutions correspond to the admissible solutions describing two-phase coexisting phase transitions, where the integral average of the specific volume belongs to the Maxwell region. We first construct a semi-discrete staggered grid difference scheme to prove the local existence of solutions to the periodic problem, without imposing the standard stability hypothesis \(p_v<0\). Then, by virtue of rigorous piecewise a priori estimates, we demonstrate that the periodic boundary value problem for van der Waals fluids possesses a global solution existing for all time, and this solution converges uniformly to the admissible steady state as time tends to infinity. This result firmly establishes the nonlinear stability of the admissible phase-transition solutions under general small initial disturbances.

We introduce a novel visco-elastic Mullins-Sekerka system with a prescribed constant contact angle at the boundary. The system is derived as an $H^{-1}$-$H^1$-type gradient flow of an energy consisting of the perimeter together with capillary, elastic, and second-gradient contributions. Building on the framework of Hensel and Stinson (Arch. Ration. Mech. Anal. 248, 2024), we introduce a measure-valued solution concept featuring a sharp De Giorgi-type energy-dissipation inequality. Moreover, we establish existence of solutions via an implicit time discretization scheme, and prove existence of $BV$ solutions under an energy-conservation hypothesis.

We prove optimal $(n-1)$-semiconvexity estimates for solutions to the classical obstacle problem, possibly with a smooth right-hand side. We deduce that the mean curvature of the free boundary is universally bounded above in all dimensions. Existing examples show that full curvature bounds fail in dimensions three and higher. Noticeably, these results are novel even for a constant right-hand side in the $n=2$ case.

We prove that weakly stable solutions of Serrin's problem are compact and therefore round balls.

We construct and classify all global solutions to the singular equation $$-\Delta u=f(X)\cdot u^{-\gamma}$$ supported in a general Lipschitz epigraphical cone, where $f(X)$ is a locally Dini continuous function with $0<\lambda\leq f(X)\leq\Lambda$. The existence and non-existence of a global solution is solely determined by the exponent $\gamma$ of the equation and the “frequency" of the cone. Moreover, in order to classify all global solutions, we introduce several new methods. First, we use the local data to estimate the global growth rate, which in turn establishes the boundedness of the “asymptotic slope" of the global solution. Second, by establishing a nonlinear variant of Kemper's boundary Harnack principle, we classify all global solutions through an “oscillation reduction" argument on the “asymptotic slope".

In two dimensions, we show that dissipation in one spatial direction is sufficient to enforce the energy equality for every weak solution at the natural energy level. In particular, neither anomalous energy loss nor creation can occur. The main difficulty is that the missing directional regularity prevents the usual self-testing argument. We overcome this obstruction through two observations: The pressure is square-integrable by a directional Riesz-transform estimate, and the less regular component is still a renormalized solution. As an application of energy rigidity, we derive a weak-strong uniqueness principle.

We consider the inhomogeneous doubly nonlinear parabolic systems of the form \begin{equation*}\partial_t (|u|^{q-1}u)-\operatorname{div}(|Du|^{p-2}Du)=\operatorname{div}(|F|^{p-2}F)\end{equation*} in a bounded space-time cylinder $\Omega_T=\Omega\times(0,T)\subset \mathbb{R}^{N+1}$. We study the local regularity properties for weak solutions in the subcritical range $p\leq\frac{N(q+1)}{N+q+1}$ and $0<q<\frac{N+2}{N-2}$. Under an extra integrability assumption $|u|\in L_{\loc}^{\rr}(\Omega_T)$, we establish a quantitative bound for $|u|$. Here, the exponent $\rr$ satisfies $\mathrm{\lambda}_{\rr}=N(p-q-1)+p\rr>0$. In addition, we prove local higher integrability of $|Du|$ in the range $p=\frac{N(q+1)}{N+q+1}$ and $\frac{N+p}{N-p}<q<\frac{N+2}{N-2}$, provided that $|u|\in L_{\loc}^{\rr}(\Omega_T)$ holds.

We study a shape optimization problem among convex bodies in $\mathbb{R}^n$ that minimize or maximize weighted perimeters of the form $\int_{\partial\Omega} \phi(|x|) \, \mathrm{d} H^{n-1}(x)$ under a standard perimeter constraint. We prove the existence of extremals for general weight functions in any dimension. In dimension two, we prove that the degenerate needle configuration $(-a,a)\times \{0\} \subset \mathbb{R}^2$ is the optimizer for a wide family of weights, including $|x|^p$ for $p \in (0,2]$ and $|x|^{-\alpha}$ for $\alpha \in (0,1)$, among convex curves satisfying a symmetry assumption.

We study a symmetric system of transport equations with solenoidal coefficients. This system reduces to an evolutionary equation with a skew-symmetric spatial operator in the real Hilbert space of square-integrable vector-functions, and by general results we claim that there always exists a generalized solution of the Cauchy problem. Uniqueness of this solution is equivalent to skew-adjointness of the spatial operator. We demonstrate that in the case of locally Lipschitz coefficients satisfying a linear growth condition the spatial transport operator is indeed skew-adjoint. For scalar transport equation this result remains true under the weaker DiPerna-Lions conditions.

We study small-noise large deviations for the stochastic heat equation (SHE) on the torus with unbounded, multiplicative space-time white noise. We establish uniform large deviation principles (ULDPs) over unbounded sets of initial data, alongside novel well-posedness and regularity results. The ULDPs are obtained for both continuous-in-space and $L^p$ initial data. For these two classes, the ULDP holds uniformly over $L^q$-bounded subsets, with $q<\infty$ and $q<p$ allowed respectively, so these sets are highly unbounded in the state space. The admissible range of $q$ is dictated by the growth of the noise coefficient in the SHE. Crucially, our methods allow us to reach down to $q=1$. This yields ULDPs over $L^1$-bounded subsets, enabling the study of exit times in physical systems with mass conservation.

We establish a quantitative stability estimate for the critical Hardy inequality on bounded domains containing the origin. Our result improves the existing quantitative stability estimate by reducing the exponent in the distance function from $N^{2}$ to $N$, replacing the Lorentz-Zygmund framework with the Luxemburg norm of the critical exponential Orlicz space $\operatorname{Exp}L^{\frac{N}{N-1}}(\Omega)$, and avoiding any cut-off modification of the virtual extremizers. As a consequence, we also obtain an improved quantitative stability estimate for the critical Hardy inequality with the logarithmic weight considered by Cianchi and Ferone, where the exponent is likewise reduced from $N^{2}$ to $N$ and the distance is measured directly from the virtual extremizer without truncation. The proof is completely rearrangement-free and relies on a critical Hardy inequality with a remainder term, scale-invariant Sobolev inequalities, and a refined dyadic summation argument. These ingredients yield stronger quantitative stability estimates for both forms of the critical Hardy inequality.

This study examines Cauchy problems governed by highly singular, time-measurable pseudo-differential operators (singular measurable families of Fourier multipliers). We show that the symbols of these operators can exhibit arbitrary blow-up behavior. In particular, we prove the existence and uniqueness of solutions even when the symbols grow super-exponentially in time and frequency. As a concrete application, we solve evolutionary equations driven by fractional Laplacians of any negative order. Additionally, we establish unique strong solutions under the sole condition that the symbol is locally integrable in frequency, even in the presence of severe blow-up at the initial time.

We study a class of $p$-Laplace equations $$\Delta_p u-\lambda u^{p-1}+ u^{q-1}=0$$ on a closed $n$-dimensional Riemannian manifold $(M,g)$ with $\operatorname{Ric}\geqslant(n-1)g$. For $1<p<2$, $p<q<p^*$, and $0<\lambda<S_{p,q}^{-1}$, where $$S_{p,q}=\frac{q-p}{2}(\frac p n)^{\frac p 2}\Big(\frac{(p^*-1)^2(2-p)}{(p_*-1)(q-1)(p^*-q)}\Big)^{\frac{2-p}{2}},$$ with $p_*=\frac{(n-1)p}{n-p}$ and $p^*=\frac{np}{n-p}$, we prove that the constant $\lambda^{\frac{1}{q-p}}$ is the unique positive solution of the equation. In contrast, for $p>2$ and $p<q<p^*$, the uniqueness fails for every $\lambda>0$; aside from the constant solution, the equation admits a positive nonconstant solution. This answers Véron's problem raised in \cite{Ver92}.

In this paper, we study the existence of nontrivial solutions for the following Gross-Pitaevskii system involving mass-critical exponent: \[ \left\{ \begin{array}{ll} -\Delta u_{1}+V_1(x)u_{1}=a_{1}u_{1}^3+\beta u_{1}u_{2}^2+\mu u_{1}& \hbox{ in }\Omega,\\ -\Delta u_{2}+V_2(x)u_{2}=a_{2}u_{2}^3+\beta u_{2}u_{1}^2+\mu u_{2}&\hbox{ in }\Omega, u_{1},u_{2}\ge 0 &\hbox{ in }\Omega, u_1=u_2=0 &\hbox{ on }\partial\Omega, \end{array}\right. \] with the constraint \[ \int_\Omega (u_1^2+u_2^2)=1, \] where $\Omega$ is an unbounded smooth domain in $\mathbb{R}^2$, $a_1, a_2, \beta$ are positive parameters, $V_i$ are trapping potentials, and $\mu\in\mathbb{R}$ is an unknown Lagrange multiplier. We derive the existence of solutions by computing the Leray-Schauder degree for the parameters $a_1,a_2, \beta$, which are away from some critical values. The system may have semi-trivial solutions of the form $(u_1, 0)$ or $(0, u_2)$. Our novelty is that we provide mechanisms ensuring that the solutions we find are nontrivial, i.e., $u_1>0$ and $u_2>0$.

In this paper, we study a class of eigenvalue problems involving both local and nonlocal operators, namely the classical Laplacian and the fractional Laplacian, under mixed boundary conditions. More precisely, we consider the problem \begin{equation}\label{1} \left\{ \begin{aligned} \mathcal{L}u &= \lambda f(u), \quad u>0 &&\text{in }\Omega,\\ u&=0 &&\text{in }U^c,\\ \mathcal{N}_s(u)&=0 &&\text{in }\mathcal{N},\\ \frac{\partial u}{\partial\nu}&=0 &&\text{on }\partial\Omega\cap\overline{\mathcal{N}}, \end{aligned} \right. \tag{$P_\lambda$} \end{equation} where \( U=\Omega\cup\mathcal{N}\cup \bigl(\partial\Omega\cap\overline{\mathcal{N}}\bigr), \) \(\Omega\subseteq\mathbb{R}^n\) is a bounded open set with smooth boundary, \(\lambda>0\) is a real parameter, \(f\) is continuous function with \(f(0)=0\), and \[ \mathcal{L}=-\Delta+(-\Delta)^s, \qquad s\in(0,1). \] We establish a characterization theorem for the existence of positive weak solutions to problem (P_{\lambda}). Motivated by the classical elliptic framework developed by Molica Bisci and Rădulescu \cite{MolicaBisciRadulescu2017}, we establish a corresponding characterization result for mixed local-nonlocal operators under mixed boundary conditions.

For every prescribed integer $N\ge2$, we construct uniformly rotating unit-vorticity vortex patches $\omega=\mathbf{1}_D$ for the planar Euler equation such that $D$ is connected and $\mathbb{R}^2\backslash D$ has exactly $N$ bounded connected components. These components are genuine zero-vorticity holes, rather than opposite-sign vortex patches or regions carrying a second nonzero vorticity level. The angular velocities lie in the rigidity-compatible interval $(0,1/2)$, and the domains converge in measure to the Rankine disk as the holes collapse. The construction starts from a fixed co-rotating polygonal configuration of $N$ unit-vorticity vortex patches and removes a shrinking spatial copy of that configuration from the Rankine disk. An exact complement identity solves all inner-boundary equations before the outer circle is perturbed. The remaining defect is generated by the $N$-th exterior multipole and has size $\varepsilon^{N+2}$. The resulting outer correction feeds back into the normalized inner problem at size $\varepsilon^{2N}$. Separate renormalization of the outer and inner equations produces a limiting affine system with a lower-triangular derivative. Its diagonal blocks are the nonresonant Rankine operator and the angular-velocity-augmented linearization of the fixed seed configuration. We also determine the first corrections to the outer boundary, the hole boundaries, and the angular velocity.

Let $D\subset\mathbb{R}^n$, $n\ge2$, be a bounded convex domain, and let $u_D$ be the torsion function for the restricted half-Laplacian. We prove that $D^2u_D$ is negative definite at every point of $D$. The argument is based on the reflected harmonic extension in a slit domain. Quantitative Schauder estimates in slit domains yield parameter-uniform estimates for the first and second derivatives of the edge remainder; a Schur-complement calculation then determines the inertia of the extended Hessian near the slit edge. Superharmonicity of the logarithmic Hessian determinant and the Gleason–Wolff zero-set theorem exclude interior degeneracy. A method of continuity starting from the unit ball proves the result for smooth uniformly convex domains, and an exhaustion argument treats arbitrary bounded convex domains.

We establish scale-invariant interior $C^{1,\alpha}$ estimates for bounded viscosity solutions of $(-\Delta)^su+b\cdot\nabla u=f$ for $s\in[1/2,1)$ with locally Hölder $b$ and $f$. The critical case uses Silvestre's parabolic theorem; the subcritical case uses Schauder estimates and interpolation. Applying this viscosity estimate to drifted Green sections, for finite Radon data above the critical order we obtain sharp solution and gradient potentials of orders $2s$ and $2s-1$, together with weak-*–to–strong local $W^{1,1}$ stability; hence the Green-potential SOLA is approximation-independent. For the normalized whole-space kernels, we identify the classical second-order limits as $s\uparrow1$, including the logarithmic kernel in dimension two; at the critical order, the whole-space gradient becomes a zero-order singular integral. For zero-exterior problems with compactly supported drift, we also prove $u/d^s\in C^{s-\varepsilon}(\overline\Omega)$ and identify the obstruction when the drift reaches the boundary.

We establish sharp bilinear decoupling under shell-type Fourier support restrictions and show that its optimal coefficient need not be optimal for the corresponding periodic bilinear Strichartz estimate. For $N_1\geq N_2$, the sharp decoupling coefficient is $N_2^{(2d-7)/4}(1+N_2^2/N_1)^{1/4}$, up to an $N_2^\varepsilon$ loss. This yields a shell-type bilinear Strichartz estimate with the same coefficient on every fixed rectangular torus, including irrational tori. On the standard square torus, we improve this Strichartz estimate for $d\geq6$, obtaining the optimal coefficient $(N_2^{d-4}+N_2^{d-2}/N_1)^{1/2}$, up to an $N_2^\varepsilon$ loss. The proof combines localization in spatial frequency and energy, a circle-method estimate for the localized high-frequency factor, and a linear estimate on arbitrarily oriented strips. As applications, we obtain estimates for toral eigenfunctions, mixed additive energy, and separated nonlinear smoothing for the periodic Zakharov system.

We consider the Bessel operator defined by \[ B_\lambda =-\frac{d^2}{dx^2}+\frac{\lambda^2-1/4}{x^2}, \] on $(0,\infty)$, with $\lambda>-1$. We study the fractional power $B_\lambda^s$, $s\in (-1,1)$, $s\neq 0$, and the logarithm $\log^kB_\lambda $, $k\in \mathbb N$, of $B_\lambda$. We obtain pointwise representations of these operators and asymptotic Taylor expansions of the operators $B_\lambda^s$ in terms of logarithmic operators $\log^kB_\lambda $. We also obtain $\log B_\lambda$ as the solution of an extension problem.

We study the dynamics of the kinetic wave equation associated to the three dimensional Schrödinger equation close to Rayleigh-Jeans equilibria. We first prove that the linearised operator generates a semigroup of contractions in $L^2((0,\infty);\sqrt \omega \dd\omega )$. Considering the family of nonsingular Rayleigh-Jeans spectra, we prove that the linearised operator possesses a spectral gap, despite the non-compactness of the integral collisional operator, and thus obtain an exponential relaxation for the linear semigroup. We then prove bilinear and trilinear estimates in the relevant norm for the nonlinear terms and deduce global well-posedness and exponential relaxation for sufficiently small relative perturbations of Rayleigh-Jeans equilibria. To our knowledge, this is the first global strong well-posedness and asymptotic stability result near a nonzero thermodynamic equilibrium for the full spatially homogeneous four-wave kinetic equation associated with the cubic nonlinear Schrödinger equation.

We study a one-dimensional free boundary problem arising in combustion theory, where the motion of the interface is governed by a prescribed Neumann boundary flux and a zero Dirichlet boundary condition. We treat both the half-line case and the bounded interval case. For both settings, we employ maximal $L^p$-$L^q$ regularity as our main analytical tool. In the half-line case, the solutions need not decay at infinity, even though the spatial derivatives belong to $L^q(\mathbb{R}_+)$. To handle the evolution law of the free boundary, we introduce a derivative formulation that avoids second-order boundary traces. By combining maximal $L^p$-$L^q$ regularity and Schauder estimates, we establish the local-in-time existence, uniqueness, and regularity of solutions, as well as the evolution law of the free boundary.

We study the transport of polynomial differential operators under the two-parameter commuting evolution \[ P_{t,s} = \exp\left( \frac{t}{2}D^2-\frac{s}{3}D^3 \right), \qquad D=\frac{d}{dx}. \] Conjugation of the position operator gives \[ P_{t,s}xP_{t,s}^{-1} = x+tD-sD^2, \] which leads to a recursive normal-ordering expansion and to a family of Heat–Airy polynomials arising as its derivative-free coefficients. Our main purpose is to study the interaction of this transport with moving absorbing boundaries. For functions satisfying \[ v_t=\frac12v_{xx}, \qquad v_s=-\frac13v_{xxx}, \qquad v(t,s,f(t,s))=0, \] the boundary condition generates a hierarchy of relations among the spatial jets of \(v\). Combined with the restrictions of transported differential equations and their spatial derivatives, these identities produce compatibility equations for \(f\). We prove a general restriction principle showing that, at every spatial-jet level, the compatibility equations of the classical Heat problem are obtained by restricting the corresponding Heat–Airy hierarchy to the slice \(s=0\). For second-order operators, this framework yields a finite characteristic system and nonlinear boundary equations, together with explicit solvable examples. For a third-order linear-potential operator, the additional Airy-flow identity permits a stronger sequential recovery: the first Heat–Airy compatibility equation determines \(f_s(t,0)\), and the next spatial-jet equation then reproduces the nonlinear compatibility condition obtained in the pure Heat theory from a \(4\times4\) characteristic determinant. Thus the additional commuting flow reorganizes the moving-boundary compatibility problem into a hierarchy while retaining the classical Heat theory as a distinguished restriction.

Let \(\Omega\Subset\mathbb H^N\), \(N\in\{3,4,5\}\), be a bounded connected \(C^2\) domain. We prove that the pure critical Dirichlet problem \[ -\Delta_{\mathbb H}u=u^{\frac{N+2}{N-2}} \quad\text{in }\Omega, \qquad u=0 \quad\text{on }\partial\Omega \] admits a positive solution whenever \(H_d(\Omega;\mathbb F_2)\neq0\) for some \(1\le d\le N-1\). This gives a hyperbolic Bahri-Coron theorem for \(3\le N\le5\) under \(C^2\) boundary regularity. Under conformal reduction, the hyperbolic geometry produces a positive potential and leads to dimension-dependent bubbling mechanisms. In dimension three, the required energy drop follows from the balance between diagonal corrections and pair interactions at fixed large multiplicity. In dimensions four and five, it is obtained at the matched scale through a normalized defect estimate and an all-pairs source-transfer bound. A unified Thom-barycenter construction converts these analytic estimates into the topological contradiction. Thus nontrivial domain topology forces existence despite critical loss of compactness and the additional geometric potential.

We study a long-range perturbation of the perimeter functional by a nonlocal repulsive term defined through the Yukawa kernel, minimized under a volume constraint. Because the kernel decays exponentially, the competition between surface tension and repulsion is governed by two independent quantities, the screening rate and the volume. Specifically we prove that (i) above an explicit critical screening rate, minimizers exist at every volume, whereas earlier work required the volume to be large; (ii) below an explicit volume threshold minimizers exist for every screening rate; (iii) at small volume, and uniformly in the screening rate, the ball is the unique minimizer up to translation; and (iv) there exists a sharp volume threshold, depending on the screening rate, for the ball to be a stable volume-constrained critical point. The threshold is given in closed form for every screening rate and reduces to the known unscreened value when $\alpha=0$. Our proofs overcome new technical changes due to the lack of homogeneity in the Yukawa kernel.

In this work we analyze the eigendecomposition of the Hessian matrix of the Robin function $\mathcal{R}(x)$ for the spectral fractional Laplacian in orthogonally invariant domains. We prove that, if $\Omega$ a smooth bounded convex domain invariant under the action of an orthogonal transformation $\mathcal{O}$ then, for $\overline{t}\in\{a\in\Omega:\mathcal{O}(a)=a\}$, the gradient vector $\nabla\mathcal{R}(\overline{t})$ is an eigenvector of the Jacobian $D\mathcal{O}$ associated to the eigenvalue $1$. Moreover, if $\Omega$ is a domain invariant under the reflection about a hyperplane $\pi_{v}=\{x\in\mathbb{R}^N:x\cdot v=0\}$, there exists $\eta>0$ such that $\mathbb{H}(\overline{t})v=\eta v$ where $\mathbb{H}$ denotes the Hessian matrix of $\mathcal{R}(x)$. Consequently, if $\Omega$ is invariant under the reflection about the hyperplanes $\pi_{v_i}$ for a linearly independent set $\{v_1,\ldots,v_N\}$, then the origin is a non degenerate critical point of $\mathcal{R}(x)$. A short proof of the Brezis-Peletier-like formulas for $\nabla \mathcal{R}$ is also provided which allows us to prove the former results for any $0<s<1$.

We describe certain finite-time singularities of the Kähler-Ricci flow arising in the analytic minimal model program. Assuming that convergence to an asymptotically conical Kähler-Ricci shrinker is realized by holomorphic maps, we prove that, in a fixed holomorphic gauge, the nearby flow is modeled on the shrinker at the level of Kähler potentials. Consequently, every noncollapsed Kähler-Ricci flow through singularities in complex dimension two is modeled on a shrinker-cone-expander transition, confirming a strong form of Song's conjectural picture. We also show analogous results in higher dimensions under the Calabi ansatz, and improve known results in the compact shrinker case. These give the first compact Ricci flows through conical singularities whose small-scale behavior is fully described.