Papers

In this paper, we prove that a smooth, properly embedded ancient mean curvature flow that is asymptotic to an $O(n)\times O(m)$ symmetric minimal quadratic cone for $n +m \geq 10$, and lies on one side of the cone has to have unique asymptotics in the parabolic region. When additionally assuming mean convexity, we upgrade unique asymptotics to full uniqueness, and show that such flow has to be a stationary flow given by one of the leaves of the Hardt-Simon foliation. This is the first rigidity/unique asymptotics theory for ancient mean curvature flow with a $\textbf{singular minimal cone as the asymptotic model}$.

In this paper, we study the Couette-Taylor instability of a viscous fluid between two rotating cylinders in the small-gap, slow rescaled rotation rate, high Reynolds number regime, focusing on the very counter-rotating case $\mu < \mu_c \approx -0.8$ where the primary instability is non-axisymmetric. Starting from the Navier-Stokes equations, we derive a limit system that captures the leading-order dynamics and compute the critical Taylor number $T_c(\mu)$ together with the critical axial and azimuthal wavenumbers. Near criticality, the weakly nonlinear behaviour is governed by a system of two coupled complex Ginzburg-Landau equations. All coefficients of this amplitude system including the cubic nonlinear terms are evaluated numerically from the linearised eigenfunctions and the associated adjoint problem. The reduced equations admit helicoidal waves (travelling in both the axial and azimuthal directions) and ribbon waves (standing axially, travelling azimuthally), and their existence and stability criteria are discussed. We also examine more exotic spatially modulated solutions that satisfy a third-order dynamical system, whose complete classification remains an open challenge.

We study the exterior Dirichlet problem for the mixed Hessian quotient equation \[ \frac{\sigma_k(\eta(D^2 u))}{\sigma_l(\eta(D^2 u))} = 1, \] where $\eta(M) = (\operatorname{tr} M)I - M$. We establish existence and uniqueness of smooth admissible solutions with prescribed quadratic asymptotics at infinity, and obtain full derivative decay of the remainder. The proof relies on a three-stage subsolution construction.

We study spatial decay properties of solutions to the dispersion generalized Benjamin-Ono equation. We first establish persistence properties in weighted Sobolev spaces under a sharp condition between decay and regularity, improving previous results. Additionally, extending recent works on the Benjamin-Ono equation by Linares and Ponce [27], we also show that decay at two distinct times implies additional regularity of solutions. The proof relies on weighted energy estimates and pseudo-differential methods, which build on an iterative mechanism that trades decay for regularity.

We introduce fully nonlinear curvature flows of horo-convex hypersurfaces in the sphere that preserve an arbitrarily prescribed spherical quermassintegral. The non-local normalization is defined relative to a fixed ambient origin, and we prove that the evolution is governed by a single smooth fixed-origin equation for all time. For monotone, homogeneous curvature functions satisfying concavity and inverse concavity, we establish preservation of horo-convexity, uniform curvature pinching, a direct estimate for the non-local coefficient. A Tso-type argument then yields global curvature bounds and long-time existence. We further prove exponential decay of the traceless second fundamental form and exponential $C^\infty$ convergence to the geodesic sphere centered at the fixed origin whose radius is determined by the preserved quermassintegral.

We study self-similar solutions of a multi-phase Stefan problem for a heat equation on the half-line $x>0$ with a constant initial data and with Dirichlet, Neumann or Robin boundary condition at the fixed boundary $x=0$. In the case of Dirichlet boundary condition we prove that a nonlinear algebraic system for determination of the free boundaries is gradient one and the corresponding potential is an explicitly written strictly convex and coercive function. Therefore, there exists a unique minimum point of the potential, coordinates of this point determine free boundaries and provide the desired solution. In the case of Neumann boundary condition the study is complicated by the fact that number of phase transitions undergone by a solution (called its type) cannot be directly determined from the boundary data. For each fixed type $n$ the system for determination of the free boundaries is again gradient and the corresponding potential is proved to be strictly convex and coercive, but in some wider non-physical domain. On the base of these properties it is proved that the Dirichlet-to-Neumann map is a strictly increasing continuous function. This allows to establish existence and uniqueness of a solution to Stefan-Neumann problem, and to specify the type of this solution. The same technique is further applied to Stefan-Robin problem with a positive connection coefficient. In the last section we also study some particular ill-posed Stefan-Robin problem with a negative connection coefficient, using again the variational approach developed in the previous sections.

This paper studies doubly nonlinear parabolic obstacle problems. We introduce a renormalized entropy formulation in which the reaction measure is encoded by evaluating the entropy multiplier at the obstacle. For diffusion depending only on the gradient and continuous time-independent obstacles, we prove a global $L^1$ comparison estimate, uniqueness of the solution, and a minimality principle among a measure-free entropy supersolution class. For diffusion depending on both the solution and its gradient, we prove existence for continuous time-dependent obstacles with a boundary-controlled decomposition. Using two ordered penalization procedures, we obtain convergence of the approximate solutions and of the corresponding reaction terms. The limiting reaction consists of an absolutely continuous part with bounded density and a finite Radon measure concentrated on a prescribed compact spatial region. A one-sided time regularization yields strong convergence of the gradients and identifies the nonlinear flux. Together, the results yield existence, uniqueness and $L^1$ stability under their common assumptions.

We prove that, for every $n\ge 2$, every positive entire solution of the critical CR Yamabe equation $4\Delta_b u = n^2 u^{(Q+2)/(Q-2)}$, $Q=2n+2$, on the Heisenberg group $\mathbb H^n$ is a Jerison-Lee bubble. No integrability, decay, boundedness, or symmetry is assumed. Together with the theorem of Catino, Li, Monticelli, and Roncoroni in $\mathbb H^1$, this classifies the positive entire solutions in every dimension. Both Euclidean routes to such a statement lose their starting configuration here. Hyperplane reflections are not CR automorphisms, and a CR inversion preserves its Koranyi sphere only setwise, so the difference between a solution and its Kelvin transform need not vanish on the sphere one inverts in. Nor is there a substitute a priori bound to fall back on: the only scale-invariant estimate available for every positive solution is a critical Morrey bound, which concentration saturates and which yields neither decay nor a small-mass regularity principle. We proceed instead from two exact consequences of the Green representation, which every positive solution is shown to satisfy. Reciprocity with a bubble $U$ converts the distance from $U$ into a nonnegative convex deficit, so that no information about the sign of a linearized quadratic form is required; differentiating the same reciprocity along the conformal orbit of $U$ gives a nonlinear barycentre identity, which forbids that deficit from concentrating at a single point of the CR sphere. Together these isolate the bubble manifold in a class carrying no energy bound. Quantizing the Jerison-Lee tensor defect against this isolation, and alternating the resulting budget with the Morrey bound, drives the energy-growth exponent into the range where the defect must vanish. Both identities use only conformal covariance and an exact positive Green representation.

This paper is a modern review of C. B. Morrey's 1955 paper on the derivation of the equations of hydrodynamics from statistical mechanics. It has two main parts. First, assuming the existence of certain invariant N -particle phase distributions, Morrey derives formal balance laws for mass, momentum, and energy. Second, he attempts to construct a concrete sequence of such distributions by imposing cell-wise constraints that match the prescribed macroscopic fields (h̊o}, u, e). From this construction, he obtains a Gibbs-type limiting form and finally the Euler equations. Compared to classical results, this regime employs the scaling law of N\epsilon^3 = O(1) and leads to a more flexible pressure Peq . Our goal is to explain what Morrey is trying to do, why it's natural and where its most delicate points lie. We will rewrite the argument in a more transparent way, distinguishing carefully between formal derivations, physical input, and statements that still need justification.

We prove asymptotic stability of the degree-one vortex in the $(1+2)$-dimensional abelian Yang-Mills-Higgs model at the self-dual coupling, for small equivariant perturbations in weighted Sobolev spaces. The abelian Yang-Mills-Higgs model is a classical relativistic field theory on $(1+2)$-dimensional Minkowski space, describing a complex-valued field coupled to an electromagnetic potential and admitting topological solitons known as vortices. This paper is the final and main part of a three-paper series. Under the orthogonal gauge condition used here, perturbations of the vortex are governed by a system of nonlinear Klein-Gordon equations for the dynamical variables, coupled to an elliptic equation for the temporal component of the electromagnetic potential. The linearized operator has continuous spectrum $[1,\infty)$ and a single positive gap eigenvalue (internal mode) of multiplicity two, whose spectral properties, associated distorted Fourier theory, and linear decay estimates are developed in the two companion papers. The main difficulty is the long-time analysis of the coupled radiation–internal-mode dynamics. In two space dimensions the dispersive decay of the Klein-Gordon radiation is relatively weak, while the internal mode decays only on the long time scale dictated by nonlinear radiation damping. At the same time, the Klein-Gordon equations for the radiation contain non-spatially localized variable coefficient quadratic interactions, which cannot be treated perturbatively and require a normal form analysis. We prove decay of the radiation by combining a good-bad decomposition, a flat-sharp decomposition, and a space-time resonance analysis carried out relative to the flat Klein-Gordon flow. The passage between the flat analysis and the Klein-Gordon flow with potential is achieved through ILED and transference estimates derived from the distorted Fourier theory.

We study the linearized dynamics near the degree-one vortex of the $(1+2)$-dimensional abelian Yang-Mills-Higgs model at the self-dual coupling, restricted to equivariant perturbations in the orthogonal gauge. The linearized operator is a selfadjoint matrix Schrödinger operator $\mathbf{M}$ on radial $L^2_{\mathrm{rad}}(\mathbb{R}^2;\mathbb{R}^4)$ with continuous spectrum $[1,\infty)$ and a two-dimensional internal mode at a unique gap eigenvalue $\lambda^2 \in (0,1)$, as established in Part I of our three-paper series on asymptotic stability of the ground state vortex. In this second part of the series, we prove linear estimates for $\mathbf{M}$ for applications in Part III. Specifically, we prove dispersive and local-energy decay estimates, as well as a transference relation which allows us to implement the space-time resonance method with respect to the flat Klein-Gordon operator in the nonlinear analysis in Part III. The engine for proving linear estimates for $\mathbf{M}$ in our approach is the distorted Fourier transform associated with $\mathbf{M}$. The construction of the distorted Fourier transform together with a detailed analysis of the underlying generalized eigenfunctions occupy the first half of this paper. For this we exploit the super-symmetric factorization of $\mathbf{M}$, and the diagonal structure of the super-symmetric partner operator, to relate the problem to the Weyl-Titchmarsh theory of two strongly singular scalar half-line Schrödinger operators.

This is the first of three papers proving asymptotic stability of the degree-one vortex under equivariant perturbations in the $(1+2)$-dimensional abelian Yang-Mills-Higgs model at self-dual coupling. In the orthogonal gauge, the linearized dynamics are governed by a selfadjoint matrix Schrödinger operator $\mathbf{M}$. The super-symmetric partner operator is a diagonal matrix whose diagonal entries are strongly singular radial Schrödinger operators on $\mathbb{R}^2$. After a conjugation, this reduces the spectral problem to the analysis of two strongly singular scalar half-line operators. Combining analysis with rigorous interval arithmetic, we prove absence of threshold resonances and show that the discrete spectrum of $\mathbf{M}$ consists of exactly one positive gap eigenvalue (internal mode) with a two-dimensional eigenspace. We also certify that the relevant nonlinear Fermi Golden Rule coefficients form a definite quadratic form, yielding effective nonlinear damping of the internal mode. These spectral inputs form the basis of the stability analysis in the subsequent papers.

Previous works of Gu-Huang-Meng-Zhou~\cite{Gu-Huang-Meng-Zhou} and Huang-Lei-Zhou~\cite{Huang-Lei-Zhou} established global strong solutions away from vacuum for arbitrarily large initial data when $\alpha$ lies in a suitable range. In contrast, we show that, for a class of small positive exponents $\alpha$ $(\alpha<\frac{1}{2}$), there exist smooth initial data with density uniformly separated from vacuum whose corresponding solutions develop finite-time implosion singularities. Our construction is based on smooth self-similar imploding profiles of the compressible Euler equations. After reformulating the system in self-similar coordinates, the viscous and capillary effects appear as exponentially decaying perturbations. We control the resulting non-autonomous system through weighted high-order energy estimates, a stable-unstable decomposition of the linearized operator, and a finite-dimensional selection of the unstable components. The constructed solutions remain smooth before the singular time and converge, after rescaling, to the prescribed imploding profile. In particular, at the blowup time $T$, the density becomes infinite at the origin, while the effective velocity $u + d \alpha \rho^{\alpha-2} \nabla \rho$ is unbounded in every neighborhood of the origin. These results complement the aforementioned global existence theory and exhibit a distinct finite-time blowup mechanism for the small-$\alpha$ regime, where the effective bulk-viscosity structure may no longer be positive.

The space of $W^{2,2}$-isometric immersions of a surface into $\mathbb{R}^3$ arises naturally in the variational theory of thin elastic sheets: it is precisely the finite-bending class, where the bending energy — the $L^2$-norm of the second fundamental form — is finite. For sheets with negative Gaussian curvature, previous work has identified branch points, where "too many" asymptotic directions meet — or, equivalently, where the index of the Gauss map is not $-1$ — as a potentially important mechanism in shape selection and pattern formation. Such branch points are precluded for $C^2$-isometric immersion. We show that this index-based notion of branch points extends to the full finite-bending class: Namely, for every $W^{2,2}$ isometric immersion of a negatively-curved surface, the index of the Gauss map is well-defined at every point, and the set of branch points is discrete. We further show that the index is stable under $W^{2,2}$-convergence, and thus, an isometric immersion with branch points cannot be approximated by $C^2$-isometric immersions. Conversely, we show that every negatively-curved metric locally admits $W^{2,2}$-isometric immersions (in fact, $C^{1,1}$) with branch points of arbitrary order. Consequently, $C^2$-isometric immersions are, in general, not dense among finite-bending ones, in stark contrast with the flat and positively curved cases.

In this paper, we establish the global existence and large-time behavior of strong solutions for the two- and three-dimensional periodic compressible Navier-Stokes-Korteweg system with arbitrarily large initial data $(\rho_0,u_0)\in H^3\times H^2$. The viscosity coefficients satisfy the BD relation $\mu(\rho)=\nu\rho^\alpha$ and $\lambda(\rho)=2\nu(\alpha-1)\rho^\alpha$, while the capillarity coefficient is given by $\kappa(\rho)=\varepsilon^2\alpha^2\rho^{2\alpha-3}$. For the case $\alpha<1$, we first enlarge the admissible parameter range for the global existence of strong solutions established in Gu-Huang-Meng-Zhou [arXiv:2603.11762 (2026)] by exploiting the doubly parabolic structure of the density-effective velocity system. We then develop a time-discretization strategy to establish uniform integrability estimates for the effective velocity, yielding a uniform upper bound for the density. Furthermore, we introduce a novel bootstrap argument to successively improve these integrability estimates, which leads to a uniform positive lower bound for the density. Finally, we derive global-in-time higher-order estimates and prove the large-time behavior \[ \left\|\rho(t)-\frac{1}{|\mathbb{T}^N|}\int_{\mathbb T^N}\rho_0 dx\right\|_{H^3} +\|\nabla u(t)\|_{H^1} \longrightarrow0, \qquad t\to\infty, \] without imposing any smallness assumption on the initial data. For the critical case $\alpha=1$, we improve the admissible parameter range established in Huang-Meng-Zhang [arXiv:2602.00455 (2026)] and establish a uniform upper bound for the density.

We study the problem of minimising the $L^\infty$ norm of a function of the $k$-th derivative over a class of maps, subject to a constraint involving the $L^\infty$ norm of a function of the map and its lower-order derivatives, for any integer $k\geq 2$. We impose boundary conditions corresponding to the $k$-th order analogues of the classical “clamped” and “hinged” cases. By employing the method of $L^p$ approximations, we establish the existence of a special $L^\infty$ minimiser, which solves a divergence PDE system with measure coefficients as parameters. This system constitutes the counterpart of the Aronsson–Euler equations for the constrained variational problem under consideration. Furthermore, we establish a lower bound for the eigenvalue. The present work extends the second-order vectorial results of Clark and Katzourakis [Generalised second order vectorial $\infty$-eigenvalue problems, PRSE A, 1-21, 2024] to the general higher-order setting.

In this article, we investigate the global topology of a complete non-compact Riemannian $2$-manifold $\Sigma$ admitting a Schrödinger operator with non-negative potential and finite Morse index. While classical results of Fischer-Colbrie classify such manifolds under the assumption of vanishing index or geometric stability as immersed minimal surfaces in a Riemannian $3$-manifold, we show that the curvature at infinity $\lambda_\infty^*(\Sigma)$ of the Fischer-Colbrie metric $g^*$—a complete conformal metric determined by a positive function furnished by Fischer-Colbrie's theorem—governs the global topology and geometric rigidity of $\Sigma$ without imposing either assumption. More precisely, we derive a fundamental identity relating $\lambda_\infty^*(\Sigma)$ to the area growth of $(\Sigma,g^*)$, show that all critical points of the distance function $d_p^*$ from a fixed base point $p$ are confined to a bounded region, and, as a corollary, obtain a quantitative bound for the number of ends. We further distinguish two complementary geometric viewpoints. On the one hand, a quantitative condition on $\lambda_\infty^*(\Sigma)$ forces $\Sigma$ to be diffeomorphic to the Euclidean plane $\mathbb{R}^2$. On the other hand, when $\Sigma$ has exactly one end, another condition on $\lambda_\infty^*(\Sigma)$ guarantees that every Busemann function on $(\Sigma,g^*)$ is an exhaustion. By clarifying the relationship between these two regimes—the critical-point structure of distance functions relative to a base point and the global behavior of Busemann functions at infinity—we exhibit two complementary manifestations of how $\lambda_\infty^*(\Sigma)$ controls the global geometry and topology of $\Sigma$.

We study positive stable solutions of the supercritical Lane–Emden equation in the first Joseph–Lundgren interval. For a family of dimensions, we construct nonradial stable entire solutions with exponent close to the upper endpoint of this interval. This disproves a radiality conjecture of Chan and Wei. The construction begins with a smooth positive nonconstant solution on the sphere, which is obtained by matching a polar cap to an inner neck. A sharp expansion of the lowest shifted eigenvalue proves that the resulting singular cone is strictly stable. Finally, a minimal-solution and rescaling argument replaces the cone by a smooth stable entire solution while preserving its sphere variation. To the best of our knowledge, this is the first nontrivial example of nonradial stable solutions for the Emden-Fowler equation.

Let $\lambda_k(\Omega)$ denote the $k$th Dirichlet Laplacian eigenvalue of a bounded planar domain $\Omega$, with eigenvalues counted with multiplicity. We disprove a conjectured higher-index extension of the Payne–Pólya–Weinberger inequality that was recorded as an open problem by Ashbaugh. For every integer $m\geq3$, we construct a bounded planar domain $\Omega_m$ with $C^\infty$ boundary such that $$ \frac{\lambda_{2m}(\Omega_m)}{\lambda_m(\Omega_m)} >\frac{13}{5} >\frac{j_{1,1}^2}{j_{0,1}^2}, $$ where $j_{\nu,1}$ is the first positive zero of $J_\nu$, the Bessel function of the first kind of order $\nu$. For $m=3$, the annulus $\{x\in\mathbb{R}^2:1/10<|x|<1\}$ already provides such a counterexample.

We prove a Kolmogorov–Riesz compactness theorem for Banach lattices over spaces of homogeneous type. Especially, we consider the one for $L^1$. Our characterization is formulated in terms of boundedness, tightness, and approximation by averaging operators, which replace translations in the absence of a group structure. The result requires neither the Vitali covering property nor continuity assumptions on the measure of balls, lower volume bounds, or finiteness of the underlying measure space, thereby extending several recent compactness criteria in the literature. Our approach begins with a new proof of the Kolmogorov–Riesz theorem in $L^1$, which avoids reflexivity and yields a unified treatment of $L^p$-spaces for $1\le p<\infty$. We then establish an approximation theorem based on Christ's dyadic cubes and use it to characterize the closure of compactly supported continuous functions in general ball Banach function spaces. Applications include compactness criteria for rearrangement-invariant Banach function spaces and Morrey spaces. In the Euclidean setting, we also identify the resulting approximation space with the classical heat-semigroup (equivalently, mollifier) closure.

We develop a diffeological framework for the geometry of Euler–Reynolds subsolutions of the incompressible Euler equations. Passing to a lifted formulation in which the velocity, quadratic flux, and trace-free Reynolds stress are treated as independent variables, we construct a diffeological limit space obtained as the weak closure of smooth strict subsolutions. Its internal tangent spaces provide an intrinsic notion of infinitesimal deformation despite the absence of any underlying manifold structure. We prove that ambient realizations of internal tangent vectors satisfy the linearized Euler–Reynolds equations in the sense of distributions, thereby establishing a natural first-order theory for lifted weak limit spaces. We further describe the tangent directions compatible with the Euler locus and identify the kernels of the natural velocity, flux, and full-state observables. These results distinguish observable perturbations from hidden stress-gauge directions that encode infinitesimal variations of the Reynolds stress. Finally, we introduce a finite-mixture model for lifted Euler–Reynolds states whose differential realizes explicit tangent directions and relates Reynolds stress to infinitesimal phase splitting. Under a genericity assumption, every deviatoric stress tensor is realized by such a first-order mixture defect, yielding phase-counting bounds and a minimality result for the observable hierarchy.

We prove the existence of stability optimizers for the Yamabe inequality on closed Riemannian manifolds of dimension at least three with positive Yamabe invariant that satisfy two threshold conditions. Remarkably, the compactness threshold we uncover is different from the special case of the round sphere treated previously by the second author. More precisely, it is given by sequences blowing up in one instead of two bubbles, reflecting the compactness of Yamabe minimizers in the non-spherical case. Using the classical asymptotic analysis of Aubin–Schoen test functions, we prove that the stability constant is strictly below the one-bubble threshold in dimension at least six and when the manifold is not locally conformally flat. In the complementary case, namely in dimensions three through five or when the manifold is locally conformally flat, we find a new positive-mass-type condition which is sufficient for the strict inequality.

For several natural phenomena, the use of the surface area functional is a first approximation. In order to capture microstructures, numerous models in applied sciences employ directionally dependent functionals, known as anisotropic energies. Since anisotropic energies are not invariant under rigid motions, their critical points do not enjoy the same conservation laws as isotropic minimal surfaces. For instance, the monotonicity formula for the density ratio is not known to hold for minimizers of general anisotropic energies. Consequently, the study of anisotropic minimal surfaces is more challenging than the study of their isotropic counterparts. In this mini-course, we give an overview of the state of the art in the existence and regularity theory of anisotropic minimal surfaces. In particular, we first focus on solutions of the anisotropic Plateau problem and subsequently move to the investigation of the rectifiability and regularity theory for critical points of anisotropic energies. To conclude, we provide applications to the min-max theory for the construction of closed optimally regular anisotropic minimal hypersurfaces in closed Riemannian manifolds.

In this article, we study the Lebesgue space inequalities for extension operators associated with hyperbolic surfaces over rectangular regions. We characterize the corresponding operator norms in terms of the side-lengths. As an application, we present new restriction estimates for a class of hypersurfaces with additive structure.

We prove that solutions arising from smooth, sufficiently small, compactly supported perturbations of non-vacuum constant states in three-dimensional irrotational compressible flow must blow up in finite time, without any symmetry assumptions or other restrictions on the initial data. Moreover, we prove that shock inevitably forms at the boundary of the maximal Cauchy development, and that its formation time agrees, in the small-data asymptotic regime, with the lifespan predicted by the radiation-field analysis.

In this article, we establish the nonlinear time-asymptotic stability of viscous shock wave with arbitrary large amplitude to the Cauchy problem for the compressible magnetohydrodynamics system. In contrast to prior related work, which relied on various smallness assumptions on both the amplitude and the speed of the viscous shock wave, here we show that any viscous shock profile is time-asymptotically stable under small initial disturbance with zero mass condition without making any assumptions on the amplitude and speed of the viscous shock wave. Moreover, while most of the stability results are obtained for constant viscosity in the momentum equation, the present results hold even for the general bulk viscosity as a smooth function of specific volume. The key part of the proof is introducing a new variable which provides an effective diffusion in the specific volume.

We investigate heteroclinical solutions of a vector-valued Bose-Einstein condensation system involving the $p$-Laplacian. The main difficulty comes from the degeneracy of the $p$-Laplacian and the possible nonsmooth behavior of the potential wells. Under suitable assumptions on the double-well potential, we establish the existence and detailed asymptotic behavior of heteroclinical solutions. In particular, we prove the strict monotonicity of every component, classify the blow-up profiles near the potential wells through Weiss-type monotonicity formulas, and obtain an almost homogeneity property of the solution. Furthermore, after re-parametrizing the heteroclinical solution by its $l_p$-arc length, we derive a curvature estimate for the trajectory near the potential wells. These results provide the geometric control needed for the construction of radial barrier functions for the corresponding Bose-Einstein condensation system.

We establish a trapped surface formation theorem for the spherically symmetric Einstein Yang Mills equations in a double-null gauge. The theorem concerns characteristic initial data posed on a pair of transversely intersecting null hypersurfaces and allows nontrivial incoming data. The proof extends the singular characteristic method of An and Lim for the Einstein Maxwell Charged Scalar Field System to the non-abelian Yang Mills setting, where the curvature coupling and gauge field nonlinearities introduce new structural difficulties. This paper constitutes the first part of a program toward weak cosmic censorship for the Einstein Yang Mills system.

We provide a general ansatz for the paracontrolled approach introduced by Gubinelli, Imkeller, Perkowski for treating singular SPDEs. The ansatz proposed covers a large class of equations. It is described via decorated trees that encode its coefficients and its stochastic data. The main novelty is the recursive definition of the paracontrolled stochastic iterated integrals that involve a well-chosen combinatorial set of words. The paralinearisation is performed via a lift of iterated paraproducts to modelled distributions and the reconstruction theorem coming from Regularity Structures. We also provide a fixed point argument and the renormalised equation with this ansatz when one uses the preparation map formalism that encompasses the BPHZ renormsalition. The last main result is the convergence of the renormalised stochastic data via the BPHZ renormalisation via an adaptation of the spectral gap approach. One obtains in the end a general solution theory for parabolic singular SPDEs via the paracontrolled approach.

We prove exact boundary controllability for a class of one-dimensional singular/degenerate wave equations of Sturm–Liouville type, \[ u_{tt}-(x^\alpha u_x)_x-\beta x^{\alpha-1}u_x-\mu x^{\alpha-2}u=0, \qquad x\in(0,1), \] with a Dirichlet condition at the regular endpoint and a boundary control acting at the singular endpoint \(x=0\). The analysis is carried out in the fractional energy space % \[ X=\mathcal H^{\nu+1/2}\times \mathcal H^{\nu-1/2}, \] associated with the corresponding singular Sturm–Liouville operator. Using the spectral decomposition induced by Bessel functions, we establish admissibility of the boundary observation operator and derive precise lower estimates for the observation coefficients. Exact observability is obtained through Ingham-type inequalities together with an abstract observability result for unitary groups. The proof treats simultaneously the subcritical, critical logarithmic, and limit-point regimes by means of singular Sturm–Liouville theory and boundary traces identified through the Lagrange bracket of the singular Sturm–Liouville expression.