Papers

We investigate the Morse index and spectral asymptotics of least-energy solutions to the planar Choquard equation \begin{equation*} \begin{cases} -\Delta u = \displaystyle\left(\int_{\Omega} \frac{u^{p+1}(y)}{|x - y|^\alpha}dy\right) u^p & \text{in } \Omega, \\ u > 0 & \text{in } \Omega, \\ u = 0 & \text{on } \partial \Omega, \end{cases} \end{equation*} where $\Omega\subset\mathbb R^2$ is a smooth bounded domain and $0<\alpha<1$. Under some geometric assumptions, we obtain sharp asymptotics for the first four eigenpairs of the linearized problem as $p\to+\infty$. The first eigenvalue is exactly $(2p+1)^{-1}$. The second and third eigenfunctions give the translation modes of the limiting bubble, and their eigenvalue expansions are determined by the Hessian of the Robin function at the concentration point. The fourth eigenfunction gives the dilation mode with eigenvalue $1+3(4-\alpha)/(2p)+o(p^{-1})$. As a consequence, we derive a comparison between the spectral indices of the solution and the critical-point indices of the Robin function.The proofs combine variational estimates, blow-up analysis and nonlocal Pohožaev identities.

Invariant manifold theory for partial differential equations (PDEs) is technically challenging as we often lack spectral gaps. Motivated by recent progress using renormalization in various areas of mathematics, we introduce a spectral renormalization method for semilinear parabolic equations that creates an artificial spectral gap at high frequencies while leaving the low modes unchanged. For the resulting family of renormalized equations, classical inertial-manifold theory yields finite-dimensional invariant manifolds whose dimension diverges as the renormalization parameter tends to zero. We show that the reduced dynamics on these manifolds approximate the original semiflow through an approximate semi-conjugacy, with an $O(\epsilon^2)$ error on bounded finite-time intervals. Under global incremental dissipativity, the semi-conjugacy is uniform in time and yields an $O(\epsilon^2)$ Hausdorff estimate for the global attractors, if they exist. We also establish further dynamical properties of the spectral renormalization such as persistence of hyperbolic equilibria and their Morse indices. We illustrate the construction in settings with and without a natural spectral gap and embed it into a two-parameter fast-slow PDE framework, where the artificial gap enables us to construct slow manifolds for systems on bounded domains with arbitrary dimension. Thus our new approach provides a finite-dimensional dynamical approximation even in regimes where an exact inertial manifold for the original equation is not available. In particular, this provides a practical balance between fully invariant manifolds and direct estimates.

In this paper, we study the hydrodynamics stability of the 3-D plane Couette flow $v_s=(y,0,0)$ with constant temperature state $\rho_s$ for the Boussinesq system in a finite channel $\mathbb{T}\times [-1,1]\times \mathbb{T}$ with non-slip boundary condition in $y$ direction. We prove that if the initial velocity field $v_{in}$ and initial temperature $\rho_{in}$ satisfy \begin{align*} \|v_{in}-(y,0,0)\|_{H^2} \leq c_0 \nu, \ \ \|\rho_{in}-\rho_s\|_{H^2} \leq c_0 \nu^2, \end{align*} for some $c_0>0$ independent of the viscosity and thermal diffusion $\nu$, then the solutions to 3D Boussinesq system do not transit from the Couette flow and constant temperature state. To our best knowledge, this is the first result about the nonlinear stability of plane Couette flow for 3-D Boussinesq system in the domain with non-slip boundary condition. This result, together with [18], implies that the strong boundary layer effect does not lead to strong instability.

We show that the map $u_*\colon B_1\subset\mathbb R^3\to\mathbb S^2$, $x\mapsto x/|x|$, minimizes the anisotropic energy \begin{align*} \int_{B_1} \Big( k_1(\mathrm{div}\, u)^2+k_2 (u\cdot \mathrm{curl}\, u)^2 +k_3 |u\times\mathrm{curl}\, u|^2 \Big) \, dx, \end{align*} among $\mathbb S^2$-valued maps agreeing with it on the boundary, for values of $k_1,k_2,k_3>0$ beyond the known regime $k_1\leq k_2$. Our main tool is a new stability estimate for $u_*$ as a minimizing sphere-valued harmonic map.

We show the existence of infinitely many singular positive forward self-similar solutions for the semilinear heat equation $\partial_t u = \Delta u + u^p$ for $x \in \R^N$, $t > 0$, where $N \geq 3$ and $N/(N-2) \leq p \leq (N+2)/(N-2)$. We also investigate the asymptotic behavior and the bound of the profiles of the solutions as $|x| \to 0$ and $|x| \to \infty$, respectively. Furthermore, we show nonuniqueness of positive soutions of the Cauchy problem for the equation with $p = N/(N-2)$ by using singular self-similar solutions.

The one-dimensional compressible Navier–Stokes–Fourier–Poisson system under the Boltzmann relation is studied for the existence of small-amplitude viscous shock profiles connecting the Rankine–Hugoniot states of the quasineutral Euler system with effective pressure $P_{\mathrm{eff}}=\rho(\theta+1)$. This profile is unique up to translation, of Lax type, and orbitally stable in time without a zero-mass assumption on the perturbation. The proof is based on a center-manifold reduction of the traveling-wave ODE and on the method of $a$-contraction with shifts, together with a relative-entropy functional augmented by electric energy and a sharp estimate of the viscous flux associated with $P_{\mathrm{eff}}$.

We show that for a horizontally stretched Bunimovich billiard, for almost every stretching parameter, there exists a sequence of eigenfunctions concentrating in the rectangular part. Thanks to the work of Markarian–Oliffson Kamphorst–Pinto de Carvalho, we know that for an open set of parameters such billiards are ergodic. The proof is a result of interaction with ChatGPT 6: we were curious if it could establish the existence of bouncing ball modes for Bunimovich or Sinai billiards. That was not successful until we suggested varying the wings. That produced an essentially complete argument for the existence of quasimodes. We then suggested using Hassell's parameter-variation argument which led to the result about eigenfunctions. The argument was then significantly simplified and clarified by the authors.

In this paper, we study a system of partial differential equations modeling the dynamics of nematic electrolytes on a two-dimensional torus $\mathbb{T}^2$. The model couples the Poisson–Nernst–Planck equations for the evolution of ion concentrations and electrostatic potential with the Ericksen-Leslie equations for the flow and orientation of nematic liquid crystals. We prove the global existence of weak solutions to this system. The proof relies on a Ginzburg-Landau approximation scheme. Key steps include deriving uniform energy estimates and establishing the strong convergence of the director field by utilizing a Pohozaev-type identity to handle the lack of compactness in the Ericksen stress tensor.

Let $(M^{2m},g,f,J)$ be a complete gradient Kähler–Ricci shrinker satisfying \[ \operatorname{Ric}+\nabla^2 f=\frac12 g. \] We prove that if the scalar curvature is constant, then the soliton is rigid. More precisely, there exists an integer $k\in\{0,\ldots,m\}$ such that \[ R\equiv k \] and \[ (M^{2m},g,J)\cong \bigl(N^{2k}\times\mathbb C^{m-k}, g_N+g_{\mathrm{Euc}},J_N\oplus J_0\bigr), \] where $(N^{2k},g_N,J_N)$ is Kähler–Einstein and \[ \operatorname{Ric}_{g_N}=\frac12 g_N. \] As part of the proof, we establish a Riemannian rigidity criterion: for a complete nonsteady gradient Ricci soliton with constant scalar curvature, the condition $\mathcal{L}_{\nabla f}\operatorname{Ric}=0$ implies radial flatness and hence rigidity.

In this paper, the detailed behavior of solutions of both the Stokes and the Navier–Stokes system in the half space is investigated when the initial data belongs to $L^1$. We first give a detailed asymptotic expansion of the solution to the Stokes system supplemented with $L^1$ initial data. With the aid of this expansion, we provide two sufficient conditions on the initial data so that the associated solutions of the Navier–Stokes system do not belong to $L^1$, corresponding to long time and short time behaviors, respectively. Moreover, for the $n$- dimensional Navier–Stokes system, a pointwise spatial lower bound of order $n$ is derived in a conic neighborhood of the $x_n$-axis, provided that the initial data decays faster than order $n$. One of the main difficulties to get precise form of leading order term is the non-commutativity of the Leray projection operator with the Laplacian. Our strategy is to employ the special structure of the Green tensor observed by Han and a cancellation property of the initial data. This also yields the sufficient conditions on the initial data to get precise asymptotic behavior via comparing the linear and nonlinear parts.

We prove that the Benjamin-Bona-Mahony equation is almost surely globally well-posed with respect to random Gaussian initial data of negative Sobolev regularity in $H^{s}(\mathbb{T})$ for any $s>-\frac 14$. This result is sharp in view of the mild probabilistic ill-posedness due to Oh-Tzvetkov (2026). This also improves on a previous result of the author which established this only in a logarithmically negative regularity. To break through this logarithmic regularity barrier, we combine the $I$-method with the low-high argument from Bona-Tzvetkov (2007), which motivates a refined first-order expansion for solutions.

We study a fractional logistic-type elliptic problem on a bounded domain with homogeneous exterior conditions and linear and nonlinear integral interactions. The interaction kernels may be nonsymmetric and sign-changing, so the problem need not admit a variational formulation. In the symmetric competitive regime, coercivity yields a weak solution by global minimization. For arbitrary fixed values of the interaction parameter, explicit smallness conditions allow us to apply Schauder's fixed-point theorem without symmetry or sign restrictions on the kernels. Additional sign assumptions give nonnegative solutions. When the nonlinear exponent satisfies $q\geq2$, a contraction condition yields uniqueness in an invariant ball and convergence of the Picard iteration. Finally, as the fractional order approaches one, we prove subsequential convergence to a solution of the local Dirichlet problem, together with convergence of the fractional energies to the Dirichlet energy. Uniqueness of the local solution in the limiting ball gives convergence of the entire family.

We prove unique continuation for the real periodic Benjamin–Bona–Mahony equation in the energy space: a solution that equals a constant on a nonempty space–time strip is that constant everywhere. In particular, Rosier's unique continuation conjecture [23] holds without any smallness or mean assumption. It even suffices to observe, at one point $x_0$ and for $t$ in an interval, the solution and the potential $\psi=(1-\partial_x^2)^{-1}(u+u^2/2)$ of its flux: $u(x_0,t)=C$ and $\psi(x_0,t)=C+C^2/2$ imply $u\equiv C$, and the second condition cannot be dropped when $C=-1$. The proof uses an exponentially weighted Hamiltonian of indefinite sign with a coercive dissipation identity, and it applies to every nonconstant real polynomial flux. Consequently, Rosier's stabilization results for two localized feedback laws and for a boundary feedback hold unconditionally, for arbitrary data. Stationary step profiles show that the theorem fails in $H^s$ for $0\leq s<1/2$. We also give a generating-series proof of single-point unique continuation for the linearized equation and discuss related control results.

When a free energy admits several local minimisers, the associated mean-field particle system exhibits a metastable behavior, undergoing random transitions between these minimisers. However, when the number of particles is large, these transitions are rare events and typically occur at a time-scale which is out of reach for simulations. In this work, we introduce a system of particles coupled with a slow macroscopic external variable which approximately follows a Langevin dynamics associated to a coarse-grained free energy. The empirical distribution of the particle system thus samples all local minimisers of the initial free energy, with transition times which are now independent from the number of particles. The construction is inspired by a theoretical tool: the two-scale approach for proving log-Sobolev inequalities for Gibbs measures.

The familiar wave kinetic theory revolves around turbulent energy cascades characterized by random phases of the individual normal modes. By contrast, recent years have revealed turbulent cascades where the transfer of energy is mediated by phase-coherent dynamics. Such coherent cascades have been previously established in deterministic systems with a tightly controlled algebraic structure of the mode interactions, such as nonlinear Schrödinger equations. Here, we show that the presence of phase-coherent cascades is robust with respect to substantial alterations of the mode couplings. We prove that, in infinite-dimensional resonant Hamiltonian systems with fully resonant normal mode spectra and random nonlinear couplings, a structured subset of couplings — an infinitesimal fraction of the total — suffices to support coherent cascades characterized by phase alignment, the formation of power-law spectra, and either unbounded growth of Sobolev norms or a finite-time blow-up. Numerical simulations further demonstrate the emergence of coherent cascades from random initial data through a process of phase locking, suggesting a mechanism for their emergence in broader classes of initial data and systems. We thus show that disordered mode interactions can sustain, rather than block, coherent energy transport across arbitrarily distant scales.

In this work, we establish a sharp logarithmic Choquard inequality by combining Beckner's entropy inequality with the sharp Pitt-type inequality. Motivated by this estimate, we classify the positive classical solutions, in a suitable integrability class, of the critical logarithmic Choquard problem \[ \mathcal{L}_\Delta u = \sigma u + \frac{1}{\|u\|_2^2} \left(G_{\ln}(u)-\frac{4}{N}\int_{\mathbb{R}^N}u^2\ln u\,dx\right)u \qquad\text{in }\mathbb{R}^N, \] where $\sigma\in\mathbb{R}$, $N\geq 1$, $\mathcal{L}_\Delta$ is the logarithmic Laplacian, and $G_{\ln}(u)=\ln(1/|x|^4)*u^2$. We construct explicit solutions as limits of critical fractional Choquard bubbles and prove that every positive classical solution in the prescribed class has the form \[ u_{\sigma,t}(x) = e^{\frac{N}{4}(\sigma-B_{N,\mathcal{L}})} B_{N,0}\left(\frac{t}{t^2+|x-x_0|^2}\right)^{N/2}, \qquad t>0,\quad x_0\in\mathbb{R}^N, \] where \[ B_{N,0} =\left(\frac{\Gamma(N)}{\pi^{N/2}\Gamma(N/2)}\right)^{1/2}, \qquad B_{N,\mathcal{L}} =2\ln 2+4\psi(N/2)-2\psi(N)+\frac{4}{N}\ln B_{N,0}. \]

We study the sharp-interface limit of the fractional Allen–Cahn equation in $\R^n$, $n\geq 2$, in the strongly nonlocal regime $s\in(0,\frac12)$, for initial data consisting of finitely many nested transition layers. We identify a coupled geometric law for the motion of the resulting fronts: each velocity contains fractional mean curvature and a nonlocal interaction potential generated by the other fronts. These interactions persist while the fronts remain separated, in contrast to the independent motion in the critical regime $s=\frac12$. Assuming that the coupled law admits a smooth evolution with strictly nested sets on a given time interval, we prove that the Allen–Cahn solutions converge locally uniformly away from the moving fronts to the corresponding integer-valued phases.

For probability measures on $\mathbb{R}^d$ supported in fixed compact sets, we prove that quadratic optimal transport plans are quantitatively stable in Wasserstein distance under perturbation of both marginal measures, assuming that one of the initial measures satisfies an upper Ahlfors regularity condition with exponent strictly greater than $d-1$. Under the same assumption, we also prove novel quantitative stability results for optimal transport maps. Furthermore, we prove a coercivity theorem which states that any probability measure on the product space must be quantitatively close to the set of optimal plans, if it has similar marginals and a similar transport cost to the optimal value. Finally, we prove a quantitative uniqueness theorem which acts as a quantitative counterpart to Brenier's theorem. The Wasserstein diameter of the set of optimal plans is controlled by the distance of one marginal measure to a regular measure for which uniqueness holds. In this way, "almost uniqueness" of optimal plans is quantified by the source measure being "almost regular". Examples are provided which prove that the exponents of all estimates are sharp.

In this paper we study the numerical approximation and convergence analysis of a steady-state heat conduction problem with mixed boundary conditions. The physical model is governed by a hemivariational inequality depending on a heat transfer parameter $\alpha > 0$. We consider the finite element approximation of this penalized problem, as well as its corresponding limit problem with a prescribed constant temperature on a part of the boundary. The main theoretical result establishes the strong convergence of the discrete solutions. Specifically, we prove the double-limit convergence of the finite element approximations to the limit solution as the mesh size $h$ tends to zero and the penalization parameter $\alpha$ tends to infinity, independently and simultaneously. We further complement the convergence analysis with an error estimate of optimal order for the discrete limit problem, and we explain why an estimate uniform in the penalization parameter cannot be expected. The theoretical results are illustrated by numerical simulations on four examples verified by the method of manufactured solutions.

We construct the first convex planar counterexample to the Schiffer and Pompeiu conjectures: a bounded strictly convex non-disc domain with real-analytic boundary carrying a nonconstant solution of $\Delta u+u=0$ with $u=1$ and $\partial_\nu u=0$ on the boundary. Together with the higher-dimensional constructions, this gives convex non-ball counterexamples in every dimension from two to eighteen, and in dimensions twenty and twenty-one, with real-analytic boundaries diffeomorphic to spheres and indicator Fourier transforms vanishing on the unit sphere. In dimensions $4$, $6$, $8$, $10$ and $14$, planar reductions through compact Lie algebras of rank two use Harish-Chandra's radial part formula and Kostant's convexity theorem. In dimensions $3$, $5$ and $7$, an axisymmetric formulation on the unit ball gives a quartic equation in weighted coefficient spaces. The exact two-sided inverse of the residual-corrected linearisation combines a Dirichlet Helmholtz inverse, a holomorphic boundary real-part problem and the material-derivative identity; it gives the planar Schiffer domain and, by one dimension-parametrised argument, the domains in dimensions $5$, $9$, $11$ to $13$, $15$ to $18$, $20$ and $21$. We also construct the first convex planar counterexample to Berenstein's conjecture: a strictly convex non-disc domain with real-analytic boundary carrying a sign-changing solution of $\Delta u+u=0$ with $u=0$ and $\partial_\nu u=1$ on the boundary. Planar existence for this problem is due to Colbrook, Sadeghi and Stepaniants. All existence proofs in this paper are computer-assisted.

This paper gives a readable and accessible version of the profile-construction part of OpenAI's manuscript "Finite Time Blowup for Navier-Stokes". We regard OpenAI's work as a major advance on the Navier-Stokes Millennium Prize Problem. The profiles are smooth and axi-symmetric. Inserting them into the Navier-Stokes equations gives a residual consisting of a divergence-form term and a remainder vanishing to infinite order in $1-t$ on fixed similarity sectors. The associated stress and radial shear satisfy the admissible cone condition. The stress need not be small. We introduce a new linear model that provides a clearer explanation of the inner core construction. The cancellation by oscillatory pulses will be treated in the companion paper "Finite-Time Blowup for Navier-Stokes with Smooth Forcing, Part II: Residual Correction via Oscillatory Pulses".

We consider the problem of minimizing the Ginzburg-Landau functional among $\mathbb{R}^n$-valued maps from the unit ball $B \subset \mathbb{R}^n$ with vortex boundary condition $u(x) = x$ on $\partial B$. We show that, for every Ginzburg-Landau parameter $\varepsilon > 0$, there exists a unique minimizer given exactly by the rotationally symmetric vortex solution in dimension $n\geq 2$.

We investigate the local stability of direct Mach configurations for the two-dimensional steady compressible Euler equations. Given a piecewise constant Mach configuration that consists of an incident shock, a reflected shock, a Mach stem and a slip line, with subsonic downstream states, we prove the following: under a transversality condition on the shock polar loops and a suitable condition on the truncation segment, every sufficiently small perturbation of the incoming flow gives rise to a Mach configuration that is a small perturbation of the background one, in the sense that the reflected shock, the Mach stem and the slip line, together with the downstream subsonic flow, stay close to their background counterparts; moreover, the solution is unique in the class of solutions satisfying the a priori estimate (for incoming flows that are small in a slightly stronger norm). Lagrangian coordinates are employed to transform the unknown contact discontinuity curve into a fixed boundary and to reduce the full Euler system to a first-order elliptic system for the flow direction and the pressure. The key idea in dealing with the contact discontinuity is to solve a mixed boundary value problem, in divergence form with discontinuous coefficients, for a single elliptic equation, so that the contact discontinuity conditions are naturally preserved as compatibility conditions for the solutions of the elliptic problem.

Let $\Omega\subset\mathbb C$ be a bounded simply connected Lipschitz domain with conformal metric $g=\omega|dz|^2$, where $\omega\in C^2(\Omega)\cap L^\infty(\Omega)$ is positive in the interior and may tend to zero at the boundary. If the Gaussian curvature satisfies $K_g\leq K$, $K>0$, and $KM<4\pi$, where $M=\int_\Omega\omega\,dA$, then the Neumann eigenvalues, numbered by $0=\mu_1(\Omega;\omega)<\mu_2(\Omega;\omega)\leq\mu_3(\Omega;\omega)\leq\cdots$, satisfy $\frac{1}{\mu_2(\Omega;\omega)}+\frac{1}{\mu_3(\Omega;\omega)}\geq\frac{2}{\mu_2(D_K(M))}$. Here $D_K(M)$ is the geodesic disk of area $M$ in the surface of constant curvature $K$. We prove that equality holds exactly when the interiors are isometric. This completes Conjecture 1.2 of Langford–Laugesen; in particular, the model disk maximizes the first positive Neumann eigenvalue in this class.

We study the uniform null-controllability, in the vanishing viscosity limit, of the transport–diffusion equation $u_t + M u_x = \varepsilon u_{xx}$ on $[0,L]$ with a Dirichlet boundary control at $x=0$. Our results concern the uniform null-controllability time $T_{\mathrm{unif}}$, defined as the infimum of the times for which the null-controllability cost remains bounded as $\varepsilon \to 0$. We determine this time exactly for both signs of the transport velocity: $T_{\mathrm{unif}} = 2 L / M$ in the positive-speed case $M>0$, and $T_{\mathrm{unif}} = (2 + 2 \sqrt{2}) L / |M|$ in the negative-speed case $M<0$. The positive-speed result disproves the conjecture $T_{\mathrm{unif}}=L/M$ suggested by Coron and Guerrero. For negative speed, the conjectured threshold had already been disproved by Lissy. Our result determines the exact threshold. We establish the lower bounds by constructing adjoint solutions that violate uniform observability below the respective thresholds. In the positive-speed case, the construction exploits the asymptotic structure of the Blaschke products and model spaces associated with the exponential family generated by the adjoint spectrum. The same model-space structure is used to reduce the upper-bound problems for both signs of $M$ to infinite-time observability inequalities, which are proved through a representation of the boundary-to-interior map and estimates of its Hilbert–Schmidt norm.

Let k be an integer from 1 to n, let f be a positive smooth function on the unit sphere, and consider the normalized flow X_t = -f({nu})|X|^{alpha} sigma_k({kappa}) {nu} + beta X, beta = C_n^k. In the supercritical range alpha > k+1, we prove the flow exists for all time and converges exponentially in C^{infty}, from every smooth strictly convex initial hypersurface enclosing the origin, to the unique smooth strictly convex solution of f(x)r^{alpha} sigma_k({kappa}) = beta u. The convergence is driven by the relative residual p = (log u)_t = beta - f(x)r^{alpha} {sigma}_k({kappa})/u, whose L^{infty} norm is nonincreasing and decays with the explicit exponent beta(alpha-k-1); the residual estimates are proved before, and independently of, the curvature estimates. Existence of a strictly convex solution of the stationary equation in this range, for arbitrary positive angular data, was previously obtained by Bryan-Ivaki-Scheuer through an expanding-type flow started from a barrier; the contribution here is the convergence of the normalized contracting flow from arbitrary initial data, with an explicit exponential rate, together with uniqueness of the limit.

In 2005, Tuval et al. introduced a continuum model for the interaction of aerobic bacteria, oxygen, and an incompressible fluid. For the associated three-dimensional chemotaxis-consumption systems, global classical regularity for large data has remained a longstanding question, even after omitting the nonlinear convection term from the fluid equation. We answer this question and prove global classical solvability of the chemotaxis–Stokes Cauchy problem in \(\mathbb{R}^3\), with linear cell diffusion and bilinear signal consumption, under the following data assumptions. The initial density and signal are smooth, bounded, and nonnegative, with finite density mass and second moment and finite signal Fisher energy; the divergence-free initial velocity belongs to \(H^4\). No smallness assumption is imposed, and a positive constant signal background is allowed. The solution remains bounded on every finite time interval. The main new ingredient is a localized entropy estimate uniform under density normalization, even when the resulting consumption coefficient becomes arbitrarily large. A terminal representation of the signal separates two complementary mechanisms: consumption excludes normalized density concentration at positive-signal points, while a localized weighted estimate improves density integrability at zero-signal points. Together, these mechanisms yield decay of a scale-invariant density quantity and allow a local $\varepsilon$-regularity criterion to be applied without a separate smallness assumption on the signal gradient.

We study the nonlinear stability of the quadratic plane Poiseuille flow $U(y)=(y^2,0)$ for the two-dimensional incompressible Navier–Stokes equations on $\mathbb T\times\mathbb R$. We prove that for initial perturbations of size $O(\nu^{2/3})$ in a $\nu$-dependent weighted Sobolev space, the flow is globally nonlinearly stable. In particular, the nonzero streamwise modes experience enhanced dissipation and decay exponentially on the characteristic time scale $O(\nu^{-1/2})$, while the zero mode remains uniformly controlled. Thus, in this weighted Sobolev topology, \(O(\nu^{2/3})\) is a sufficient stability scale without logarithmic loss.

In this paper we study a class of 1-homogeneous solutions of the one-phase problem coming from the Cartan isoparametric cubic polynomials, which exist only in dimensions $d=5,8,14,26$. Our main result shows that they are unstable in dimensions $d=5,8,14$, and minimizing in dimension $d=26$. This provides the first example of a singular minimizing cone in the Alt-Caffarelli problem that is neither axially symmetric nor of Lawson-type.

We prove uniqueness of the blow-up at every singular point of the one-phase free boundary problem for which one blow-up has an isolated singularity. The result applies to Lipschitz stationary solutions and does not require any integrability assumption on the cone. In this sense, our result completes the picture for uniqueness of tangent cones at isolated singularities in the one-phase problem. Inspired by Simon's work in the minimal surface setting, we prove an infinite dimensional Łojasiewicz inequality for the spherical Weiss' energy. This yields an epiperimetric inequality for non-minimizing solutions, which leads to the uniqueness result.