Papers

An investigation of local aspects of the space of axisymmetric traveling-wave solutions to the three-dimensional Euler equations near Hill's zero-flux spherical vortex and Norbury's small-flux vortex rings is the endeavor undertaken herein. The aforementioned family consists of idealized, swirl-free, and nonsmooth objects, their azimuthal relative vorticity being prescribed by a scaled characteristic function of the compact vortex core. We confront the desingularization question, both within the swirl-free solution subspace and upon admitting swirl, and prove that each member of the Hill–Norbury family of sufficiently small flux is the limit of traveling-wave solutions to the Euler equations that are smooth to infinite order and have compact toroidal vorticity support. In the latter swirling regime, however, a curious flexibility is revealed: total vorticity may diverge along sequences converging in a natural desingularization topology. Our construction, founded upon a five-dimensional reformulation of the Stokes stream function, is an amalgamation of delicate thin-domain analysis, singular-parameter Newton-type nonlinear inversion, and topological fixed-point arguments. To the best of the authors' knowledge, this constitutes the first verification that the small-flux Hill–Norbury family is nowhere isolated from the collection of smooth vortex rings.

We study the critical behavior of the inhomogeneous semilinear heat equation \[ u_t-\Delta u=|u|^p+w(x) \qquad\text{in }(0,\infty)\times\mathbb{R}^N, \] where \(N\geq3\), \(p>1\), and \(w\in L^1(\mathbb{R}^N)\) is nontrivial and has zero total mass: \[ \int_{\mathbb{R}^N}w(x)\,dx=0. \] This zero-mass case was posed as an open problem by Bandle, Levine, and Zhang in 2000 and has remained unresolved since then. We provide a complete answer to this problem by proving nonexistence of global weak solutions for \[ 1<p\leq\frac{N}{N-2}. \] Combined with the known supercritical existence result for sufficiently small data, this shows that the critical exponent separating the nonexistence and existence regimes remains \[ p_c=\frac{N}{N-2}, \] the same as in the positive-mass case.

We prove a Michael-Simon inequality for a class of varifolds in Euclidean space that includes all integral varifolds with locally bounded first variation of arbitrary dimension and codimension. In codimension $2$, this inequality is sharp and implies the sharp isoperimetric inequality for varifolds, which attains equality precisely for the round disk. Our proof uses the optimal transportation approach of Brendle-Eichmair.

Let $\Omega\subset\mathbb R^2$ be a smooth bounded domain containing the origin and invariant under reflection across the coordinate axes, and let $0<\lambda<\lambda_1(\Omega)$, where $\Lambda_1 (\Omega)$ is the first eigenvalue for $-\Delta$ on $\Omega$ under Dirichlet boundary conditions. For every fixed integer $k\ge1$ and all sufficiently small $\varepsilon>0$, we construct a positive solution of \[ -\Delta u=\lambda u e^{u^{2+\varepsilon}} \quad\hbox{in }\Omega, \qquad u=0\quad\hbox{on }\partial\Omega, \] which blows up at the origin as a tower of $k$ Liouville bubbles. The concentration scales are strongly separated. The proof is based on a Lyapunov–Schmidt reduction adapted to these different scales.

Weighted Hardy inequalities on a domain $\Omega\subset\R^n$ can be deduced from lower Hausdorff-content bounds on suitable visible parts of the boundary, and it is natural to look for visible sets described by straight segments, which are easier to verify than the curve-based visual boundary of Koskela and Lehrbäck. We show that this approach is subject to an intrinsic obstruction. If a boundary point $\xi$ is seen from $x\in\Omega$ along a segment satisfying a uniform thickness condition, then $\Omega$ contains a truncated cone with vertex $\xi$; consequently the quantitatively straight visible boundary $\partial^{\mathrm{str}}_{x,\alpha}\Omega\cap B(x,C_0d_\Omega(x))$ is a finite union of Lipschitz graphs, is $(n-1)$-rectifiable, and carries no $\lambda$-Hausdorff content for any $\lambda>n-1$. Since content conditions with $\lambda\le n-1$ already imply the $(p,\beta)$-Hardy inequality for $\beta<p-n+\lambda$ without any visibility assumption, segment-based criteria are either unsatisfiable or redundant: they never reach the regime $\beta\ge p-1$ in which visibility is genuinely needed. We show that the exponent $n-1$ is attained, so that the rectifiability theorem is sharp; we exhibit a boundary point lying in the visual boundary of Koskela and Lehrbäck but in no $\partial^{\mathrm{str}}_{x,\alpha}\Omega$; and we observe that raw straight visibility, without the thickness condition, escapes the obstruction, is sufficient for $\lambda\le n-1$, and remains open for $\lambda>n-1$.

We consider a discrete coagulation-fragmentation-diffusion system on a bounded domain of $\mathbb{R}^d$. In dimension $d\leq 2$, we prove mass conservation (that is absence of gelation) when the coagulation coefficient satisfies the critical additive bound $a_{i, j} \leq C(i + j)$ for some constant $C > 0$. We thereby close the remaining gap between the spatially homogeneous setting and the spatially inhomogeneous diffusive setting. In dimension $d\leq 4$, we impose a higher decay rate on the coagulation and fragmentation coefficients, but we nonetheless prove regularity and absence of gelation under mild assumptions on the diffusion rates.

We consider a 3D rotating Navier-Stokes equation with transport-stretching noise, posed between two horizontal plates, and study the joint limit of vanishing vertical viscosity and rapid rotation. Our driving noise depends only on the horizontal coordinates and its vertical component vanishes with the viscosity. Provided that the initial data is purely horizontal, we construct martingale weak solutions which converge in $L^2_{\omega}L^\infty_tL^2_x$ to the layered strong solution of a 2D stochastic Navier-Stokes equation with damping. The damping coefficient is dependent on the limit of the ratio between vertical viscosity and inverse rotation rate.

We study the quantitative codimension-two estimate for the singular set in the boundary neighborhoods of the solutions of \begin{equation*} \Delta u+V(x)u=0\qquad\text{in }\Omega, \qquad u=0\qquad\text{on }\partial\Omega, \end{equation*} where $\Omega\subset\mathbb R^n$ is a bounded $C^{1,\mathrm{Dini}}$ domain and $V\in L^p(\Omega)$ for some $p>n$. We first prove the explicit upper bound for the doubling index is given by $C(n,p,\Omega)(1+\|V\|_{L^p(\Omega)}^{\frac{2p}{3p-2n}})$.The analytic input is an interior volume estimate for solutions of second order elliptic equations with uniformly elliptic Dini leading coefficients and $V\in L^p$. Using the quantitative doubling index bound, boundary flattening, we show an explicit upper bound for singular sets in the neighborhood of the boundary of the $C^{1,\mathrm{Dini}}$ domain.

In this paper, we study the Cauchy problem for a coupled system of damped wave equations with mixed product-type nonlinearities. We have succeeded in determining the critical curve, formulated in terms of the power exponents of nonlinearities, for the global (in time) existence of small data mild solutions and the nonexistence of global mild solutions. Moreover, the global existence result is established by means of Schauder's fixed point theorem and Banach's contraction principle, whereas the nonexistence analysis relies on a new iteration scheme adapted to the product-type structure of nonlinearities. To the best of our knowledge, this work provides the first critical threshold results for such systems.

We consider the non-linear cubic defocusing Schrödinger equation on the 2D Torus ($T^2$) with a potential $V(x)\in H^{2^+}(T^2)$. The initial data, $u_0$,is given by a centered gaussian variable with covariance $(-\Delta+V)^{-1}$, and thus is almost surely in $H^{0^-}(T^2)$. We show that it is almost surely well-possed for all time in the weak sense, and that $u-e^{it(\Delta -V)}u_0$ is in $H^{s}(T^2)$ for some $s>0$, and that the solution is the limit of the solutions to the truncated equation. This result is a generalization of the results first given by Bourgain in Invariant Measures for the {2D}-Defocusing Nonlinear {Schrödinger} Equation.

In this paper, we prove a boundedness result for parabolic Riesz transforms in the range $p \le 2$ associated with degenerate parabolic operators whose elliptic part is in divergence form with complex coefficients depending measurably on space and time. The degeneracy is determined by a spatial weight in the Muckenhoupt class $A_2(\mathbb{R}^n)$. The argument relies on off-diagonal estimates for the parabolic gradient of the resolvent family, together with weighted embeddings. We obtain boundedness in a range of exponents strictly below $2$, with explicit ranges quantified by the reverse Hölder class of the weight. For real coefficients, this quantification is sharp and extends down to $1$.

We study the simultaneous asymptotic behaviour of homogenization and the Bourgain–Brezis–Mironescu limit for quadratic fractional energies with periodic coefficients. The functionals depend on two small parameters: the period of the coefficients and the fractional exponent approaching the local limit. We prove that the $\Gamma$-limit is completely determined by the relative scaling of these parameters. In the two extreme regimes, one recovers the expected separation of scales, yielding either the homogenized Dirichlet energy or the energy with averaged coefficients. At the critical logarithmic scaling, however, the two mechanisms coexist, and the limit is a convex combination of the homogenized and averaged energies, with weights depending explicitly on the relative scaling of the two asymptotic processes. The proof combines the homogenization analysis of the short-range interactions with a blow-up argument showing that long-range interactions are simply averaged using a Riemann–Lebesgue-type argument.

We prove that, on closed three-manifolds, the limit interface of a sequence of stable abelian Higgs critical points with uniformly bounded energy is a finite union of smooth closed immersed geodesics, possibly with multiplicity and self-intersections. This follows from an integral estimate for the diffuse curvature, derived from an integral bound on the discrepancy.

We discuss the application of $L^2$-based velocity averaging to the Wigner kinetic equation governing the quantum evolution of the Wigner transform of a density operator of a particle subjected to an external electromagnetic field. This extends the $L^2$-based averaging lemma from [F. Golse, J. Möller: Commun. Math. Sci. (2026)] to electromagnetic fields. We make use of the magnetic or gauge-invariant Wigner transform due to Stratonovich [Sov. Phys. D., (1)414-418, 1956] which differs from the usual definition by a phase factor involving the circulation of the magnetic vector potential along a line segment. The magnetic Wigner transform obeys the Wigner equation in the generalized phase space variables $(x,v=\xi-A(t,x))$, which is a more natural setting for the application of velocity averaging. Since the magnetic Wigner equation contains a first order derivative in the spatial variable, we obtain compactness in $L^2$ but no gain in regularity.

In this paper we prove a quantization property for Palais-Smale sequences of functionals related to Liouville equations on compact surfaces. While these equations have been extensively studied over the past decades, such a quantization property had not been established so far, and has been often bypassed by the Struwe monotonicity trick. Our result, which relies on careful $L^p$ estimates to obtain gradient bounds and to perform neck analysis, allows to directly apply variational methods to obtain existence of solutions in non-resonant regimes. We also give an example of clustering of blow-up points and a stronger assumption that, on the contrary, guarantees that the blow-up points are isolated.

We study the minimisation of the first solenoidal Maxwell eigenvalue $\lambda_1^{\mathrm{Max}}(\Omega)$ in perfectly conducting cavities through the scale-invariant functional $J_{\mathrm{Per}}(\Omega):=\mathrm{Per}(\Omega)\lambda_1^{\mathrm{Max}}(\Omega)$ in $\mathbb R^3$. While the corresponding unconstrained problem is known to be ill-posed, we focus on bounded convex cavities and prove that $J_{\mathrm{Per}}$ has a positive infimum in this class. The proof relies on the study of the three main possible behaviours for sequences of convex domains with bounded perimeter: convergence to a bounded convex domain with non-empty interior, collapse to a planar domain or collapse to a one-dimensional domain. The case of a planar collapse is handled through projection arguments, yielding a lower bound, while that of a one-dimensional collapse is controlled by a different argument, showing that in that case $J_{\mathrm{Per}}$ must diverge to $+\infty$. Although the question of existence of an optimal shape (that is, ruling out planar collapse) remains open, we derive several negative results and obstructions to existence; typically, no minimiser with a $\mathscr C^{3,+}$ boundary exists. This relies on a fine analysis of first-order optimality conditions.

In this paper, we prove that symmetric, doubly-periodic, three-dimensional hydroelastic waves of small amplitude, bifurcating from a non-uniform shear flow, are necessarily two-dimensional to leading order. A detailed analysis of the solvability condition at quadratic order reveals that all Fourier coefficients with transverse dependence vanish identically. The dimensional reduction is controlled entirely by the vertical structure of the shear flow and is robust under changes to the dynamic boundary condition.

We consider the stabilization of two-dimensional periodic hydroelastic waves in deep water beneath a massless nonlinear Cosserat sheet by means of a localized pres- sure feedback. For sufficiently regular solutions, under an explicit condition relating gravity, flexural rigidity and the localization profile, we prove a uniform spacetime estimate and a localized dissipation observability inequality. As a consequence, the hydroelastic Hamiltonian decays quantitatively as long as the solution remains in a small geometric regime. The proof combines an energy dissipation identity with a nonlinear multiplier method adapted to the infinite-depth geometry, together with a coercivity analysis of the Cosserat bending energy.

Let $m\geq1$, $k\geq0$, $n=m+k$, and write $x=(v,z)\in\mathbb R^m\times \mathbb R^k$, where $v$ is the active velocity variable and $z$ is passive with respect to $Y=v\cdot\nabla_y+\partial_t$. We consider \[ \operatorname{div}_x(A\nabla_xu)-Y(wu)=0, \] where $w=w(x)\in A_2(\mathbb R^n)$ and the measurable matrix $A$ has ellipticity and size controlled by $w$. For arbitrary such weights, including weights depending on the active variables, we prove a weighted hypoelliptic transfer estimate, a kinetic Sobolev inequality, local boundedness, the weak Harnack and Harnack inequalities, local Hölder continuity, and the strong minimum principle. The Harnack argument uses a full-variable weighted measure-data compactness theorem. An $\mathrm L^p$ velocity-averaging estimate gives compactness of smooth moments of $wu$, and weighted Poincaré inequalities in all diffusive variables reconstruct $u$. At a two-phase limit, testing with non-negative profiles produces transport directions given by weighted mean velocities. A $\mathrm{BV}$ argument applied to the limiting subsolution inequality yields the no-jump principle needed for expansion of positivity and a weighted kinetic ink-spots lemma. The theory includes the weight $|\lambda|^{1-2s}$ in the extension of Garofalo and Tralli for fractional powers of the Kolmogorov operator.

We provide kernel estimates for pseudo-differential operators in the $S(m,g)$ Weyl-Hörmander calculus classes. These new results lead to boundedness theorems like the weak (1,1) type for these classes and the boundedness of these operators from Hardy spaces to Lebesgue spaces $L^p$. We prove that under a suitable hypoelliticity condition on the weigh $m,$ one can get the weak (1,1) inequality for the Fefferman order established previously by the second author in analogy to the classical $L^p$-boundedness theorem due to Fefferman. Our approach makes an extension to $S(m,g)$ classes of previous estimates proved by Álvarez and Hounie, Nagase, Kumano-Go, etc, for the Hörmander classes including new regularity properties in this framework from the Hardy space $H^p$ to $L^p$ and also on Lorentz spaces. We have identified that the Uniform Hörmander Condition plays a fundamental role in some of our boundedness theorems.

We prove that every bounded stable entire solution $v$ of the Allen–Cahn equation in $\R^3$ is one-dimensional. As a consequence, the full De Giorgi conjecture in $\mathbb{R}^4$ is true. We also obtain local curvature estimates for stable solutions. The proof strategy is inspired by the recent breakthrough work of Chan, Fernández-Real, Figalli and Serra [J. Amer. Math. Soc. 2026], by reducing the stabilty condition for the Allen-Cahn equation to a weak stability condition on a surface (the zero set) and then utilizing Gauss-Bonnet formula. For this purpose, we first use the stability condition to get a sublinear bound for a weighted integral that controls the zeros where the solution is far from planar. If such zeros exist, we isolate a bounded set of them and join $1-v^2$ near this set to derivatives of one-dimensional transitions farther away. By controlling the interaction between these transitions, we derive the weak stability condition on the zero set, which is then used to bound a weighted integral of the squared curvature on the regular part of the zero set by a cutoff gradient integral and a controlled error. We use this inequality to bound the intrinsic area and construct logarithmic cutoffs. The resulting compactly supported test function has a negative contribution near this set that exceeds all joining and cutoff errors, contradicting stability.

In this paper, we develop a polyhedral version of Milman's centro-affine spectral framework for full-dimensional origin-symmetric polytopes. A weighted graph Laplacian on the facets plays the role of the smooth centro-affine Laplacian. For $n\geq 3$, we prove that its first nonconstant even eigenvalue satisfies \[ \lambda_{1,e}(P)\geq n-1+\frac{1}{n-1} \] whenever $P$ is a local minimizer of $\lambda_{1,e}$ among origin-symmetric polytopes with the same facet normals, but not necessarily the same normal fan.

We study local convergence to the dynamical $\Phi^4_3$ model under a vanishing quintic perturbation. More precisely, on the three-dimensional torus we consider $$\partial_tu_\varepsilon=\Delta u_\varepsilon-\varepsilon^\alpha u_\varepsilon^5+\xi_\varepsilon+C_\varepsilon u_\varepsilon+\widetilde C_\varepsilon u_\varepsilon^3$$ for $\alpha\in(\frac56,1)$, where $\xi_\varepsilon$ is a spatial mollification of space-time white noise. Although the quintic coefficient vanishes, its contractions generate divergent linear and cubic contributions. We identify suitable mass and cubic counterterms that compensate these divergences. Using paracontrolled calculus, we construct the required enhanced stochastic data and prove that their renormalized higher-order components vanish, while the remaining coordinates converge to the enhanced data of the dynamical $\Phi^4_3(\lambda)$ model. We further establish local well-posedness and stability of the associated deterministic solution map. Consequently, for well-prepared initial conditions, $u_\varepsilon$ converges in probability, as a random local solution germ, to the renormalized dynamical $\Phi^4_3(\lambda)$ solution. In particular, the cubic counterterm allows convergence strictly below the threshold $\alpha=1$ arising when only linear renormalization is allowed.

Let $L_v=-\Delta-2v\cdot\nabla$ on $L^2(\mathbb R^n,e^{2v\cdot x}\,dx)$, where $v\in\mathbb R^n\setminus\{0\}$ is a constant drift vector. We prove that the full vector Riesz transform $\nabla L_v^{-1/2}$ satisfies a weak-type $(1,1)$ estimate for real-valued functions with constant at most $2$, uniformly in both the dimension and the drift vector. Our proof extends the fractional obstacle method of Ouyang, Spector and Stockdale (https://arxiv.org/abs/2608.18068) to the exponentially weighted setting. Interpolation then yields strong $L^p$ bounds for $1<p\le 2$, with constants depending only on $p$.

We study the large-time behavior of solutions to the aggregation–diffusion equation $$ u_t=\Delta u+\nabla\cdot\bigl(u\,\nabla K*u\bigr) \qquad \text{in } \mathbb{R}^d. $$ Our main results identify a transition, governed by the regularity and singularity of the interaction kernel, between Gaussian and nonlinear self-similar asymptotics. We prove that for regular interaction kernels, as well as for sufficiently mild singular repulsive kernels, the nonlinear drift is asymptotically negligible and solutions exhibit the same Gaussian large-time behavior as solutions of the linear heat equation. In contrast, for the critical logarithmic kernel $ K(x)=-\log |x|, $ the interaction persists at the diffusive scale and leads to genuinely nonlinear asymptotics. More precisely, for every mass \(M>0\), we prove the existence and uniqueness of a self-similar solution of mass \(M\) and show that every solution with mass $M$ and with finite second moment converges in all $L^p$-norms to this self-similar solution as \(t\to\infty\).

A central conjecture in the regularity theory of harmonic maps asserts that the singular set of a stationary harmonic map has codimension at least three. In this paper, we construct counterexamples to this folklore conjecture. We construct stationary harmonic maps whose singular sets $S$ satisfies $\dim S=n-2$ and $\mathcal{H}^{n-2}(S)=0$ . The target is a smooth compact manifold diffeomorphic to $\mathbb{S}^2 \times \mathbb{T}^3$, with metric arbitrarily close in $C^\infty$ to the standard product metric.

We establish stability principles under compact gluing for the reverse square-root inequality and the Riesz transform. Below a common Sobolev dimension, the reverse estimate passes from finitely many model manifolds to their connected sum. A boundary-compatible Poisson parametrix also proves the Neumann reverse inequality for every $1<p<\infty$ for real symmetric uniformly elliptic divergence-form operators on exterior Lipschitz domains. For the forward transform, we glue a bounded-core model and an independent whole-space end model that agree with the coefficients on overlapping regions. A homogeneous Fredholm construction then resolves the form-valued error. The resulting range is controlled by the two model exponents, with the additional restriction $p<n$ for Dirichlet data. More broadly, our method illuminates the interplay between the bounded interior and exterior problems, and provides a flexible framework for studying other boundary value problems.

We prove that every solution of the classical rational Calogero-Moser system can be approximated, after subtracting a common linear drift from the particle positions, by rational multisoliton solutions of the focusing continuum Calogero-Moser equation, uniformly for all time. The construction identifies the multisoliton inverse spectral matrix as a rank-one dissipative perturbation of Moser's time-dependent Hermitian position matrix. A uniform lower bound on the particle separation then yields quantitative control of the multisoliton poles. Their real parts and real velocities approximate the shifted particle positions and velocities with errors of order $\varepsilon^2$, while the pole heights are positive and sum exactly to $\varepsilon$. Using the exact Poisson-kernel representation of the multisoliton density, we obtain convergence to the atomic particle measure in bounded-Lipschitz distance at rate $O(\varepsilon(1+|\log\varepsilon|))$, uniformly in time. We also derive the large-time distribution of pole height at fixed $\varepsilon$, with one distinguished branch retaining the limiting height and the remaining heights decaying quadratically in time. Explicit two-particle formulas illustrate the approximation and show that the real-position error estimate is sharp.

We study the unforced viscous Katz–Pavlović dyadic model with non-negative smooth initial data for two shell settings. For super-exponential scales $N_n=N_0^{b^n}$, with $N_0>1$ and $1<b<2$, we prove that $\alpha=1/(b+2)$ is the sharp dissipation threshold: every such datum produces a globally smooth solution at and above this value, whereas suitable compactly supported data lose regularity in finite time below it. For geometric scales $N_n=N_0\Lambda^n$, global regularity holds when $\alpha\geq1/3$ and $4\alpha-1>(\log2)/(2\log\Lambda)$. In particular, for $\Lambda>2^{3/2}$, the result reaches $\alpha=1/3$ and, together with the known classical blow-up theorem, identifies the sharp threshold in classical shell ratios.

We derive a thermodynamically-consistent model of fluid flow through a poroelastic medium. Starting from elastic and fluid free-energy densities, an energy-dissipation rate, and a kinematic constraint, the force-balance equations are derived using variational principles, with the pressure–density constitutive relation emerging as a direct consequence of the variational structure; the same kinematic constraint also supplies the total-flux transport structure. In the ideal-gas limit, the model linearization recovers the classical linear Biot equations. For power-law fluid energies, it yields isentropic pressure–density relations. A key advantage of the variational formulation is that extensions to richer physics, such as thermal effects, chemical reactions, or multi-component fluids, can be incorporated systematically by augmenting the energy and dissipation functionals without redesigning the force-balance or transport closure. We support the model with an energy-compatible two-field discretization and study consolidation under a surface load with three lateral-boundary treatments and three fluid-compressibility exponents.