Papers

We compute the limit of the eigenvalue of the $k$-Hessian operator on the unit ball in $\mathbb{R}^n$ when $k\rightarrow \infty$, assuming that the ratio $\frac{n}{k}$ remains fixed. When $n< 2ke$, we moreover identify the limit of the corresponding eigenfunctions. We also derive monotonicity results and consider when the ratio varies in $n$.

We show that the Navier-Stokes equations on $\mathbb R^2\times\mathbb T$ are globally well posed for initial vorticities that are both helically symmetric and integrable. This class of data typically generates velocity fields of infinite kinetic energy, so this result is not covered by the classical finite-energy well-posedness theory. This result is supercritical from the perspective of the three-dimensional Navier-Stokes scaling, and it is made possible by the special structure of helical flows using time-weighted Kato-type spaces.

This paper establishes the exponential $L^2$-stability of the unique solutions to initial-boundary value problems for linear parabolic partial differential equations with general drift and zero-order coefficients in bounded domains. The key idea lies in constructing a suitable Dirichlet form with respect to a weighted measure $\mu = \rho\,dx$ and identifying the corresponding sub-Markovian $C_0$-semigroup of contractions on $L^2(U, \mu)$ with the unique weak solution. Remarkably, the exponential $L^2$-stability remains valid even when the zero-order term vanishes, and it holds robustly for all drift coefficients $\mathbf{H} \in L^p(U, \mathbb{R}^d)$ with $p \in (d, \infty)$.

Let $P$ denote normalized nearest-neighbor averaging on $\mathbb{Z}^d$ and set $A=I-P$. We consider $\partial_tu+A^su=f(u),t>0, x\in\mathbb{Z}^d, 0<s<1.$ For a Fisher-KPP reaction with $a=f'(0)>0$, compactly supported data have logarithmic propagation rate $a/(d+2s)$. In the logistic case, each fixed level set lies between two constant multiples of $\exp(at/(d+2s))$; this localization remains valid for data with the critical algebraic tail. Nondecreasing front-like data in one dimension instead propagate at rate $a/(2s)$, and the resulting acceleration rules out planar traveling fronts of finite speed in every dimension. We also analyze the singular limit $s\uparrow1$. If $\tau_s=-\log(1-s)$, then throughout $at<\tau_s$ compactly supported solutions follow the Wulff shape of the local nearest-neighbor equation. After a constant multiple of $\tau_s$, their propagation is exponential. Half-space data exhibit the corresponding transition from the directional local speed to the exponent $a/(2s)$. The analysis rests on sharp bounds for the subordinated kernel, algebraic barriers, a compactness–Liouville argument, and a decomposition of the fractional walk into its nearest-neighbor and rare long-jump parts.

Inspired by the monotonicity formula of Alt, Caffarelli, and Friedman, we consider a natural variant of the ACF functional in positively curved 2-dimensional settings where we integrate over sublevel sets of Green's function rather than disks. We prove sharp almost-monotonicity formulas for our new functional in the case of convex planar domains (both with the pole on the boundary and in the interior) and in the setting of 2-dimensional complete manifolds with Euclidean volume growth and nonnegative Gaussian curvature. The tools include the Schwarz-Christoffel formula and Riesz decomposition using conformal coordinates. As a consequence, using quasiconformal estimates, we give a new Lipschitz bound in the manifold setting for minimizers of the two-phase free boundary problem of Alt, Caffarelli, and Friedman, with the constant depending on only the total integral curvature, in contrast to work of Teixeira and Zhang, which gives constants depending on pointwise bounds for the Riemann curvature tensor and its derivatives. Our methods also recover the Lipschitz bound up to a Neumann boundary of Gemmer, Moon, and Raynor in the planar convex domain case.

We prove that a class of nonlocal nonlinearities on periodic Sobolev spaces can be expressed as a trace of the unique mild solution to a local dispersive problem consisting of a system of forced Schrödinger equations. The proof is based on a reformulation of the nonlocal nonlinearities as the resonant orbit average of a real-analytic function under the unitary group generated by the free Schrödinger operator $\mathrm{i} \partial_x^2$. We illustrate our result by providing an equivalent local reformulation for a recently obtained nonlocal amplitude equation that formally captures the dynamics of parabolic systems close to a conserved Hopf instability.

We introduce a nonlocal regularization of optimal transport that bridges Kantorovich and Monge formulations. The regularization penalizes oscillations through an interaction kernel singular near the diagonal. We show that, for sufficiently strong singularities, every finite-energy transport plan is induced by a map, yielding equivalence between the regularized Kantorovich and Monge problems. For fractional kernels, the regularizer reduces to a fractional Sobolev seminorm that can allow jump discontinuities, leading to existence results for broad classes of source and target measures. We derive first-order optimality conditions and analyze the vanishing-regularization limit. The regularized functionals $\Gamma$-converge to the classical Kantorovich problem, while the first-order $\Gamma$-limit selects, whenever they exist, finite-energy Monge minimizers with minimal nonlocal energy. When no such minimizer exists, we identify a different asymptotic regime and establish a sharp scaling law in a prototypical mass-splitting example. Finally, after lifting the problem to the product of transport plans, we derive a dual formulation using the cone of copositive functions and propose a fixed-point numerical scheme illustrating the regularization.

We prove that exact quadratic energy cancellation alone does not force growth of the spatial analyticity radius. On $\mathbb{T}^3$, we construct an explicit symmetric, translation-invariant, first-order bilinear operator $Q$ that preserves the divergence-free class and, for every smooth real-valued divergence-free vector field $v$, satisfies $ \int_{\mathbb{T}^3} Q(v,v)\cdot v\,dx=0.$ For every prescribed sufficiently small time $T>0$, the equation $\partial_t u-\Delta u=Q(u,u)$ admits a global smooth, real-valued, mean-zero, divergence-free solution $u$ such that $\operatorname{rad}(u(0))=\operatorname{rad}(u(T))=1$. The construction reduces the dynamics on an invariant cyclic-shear class to viscous Burgers and tunes a Cole-Hopf heat profile so that its nearest complex zero returns to its initial distance from the real torus. For every $1<\alpha<2$ and every prescribed sufficiently small $T>0$, we also construct a symmetric sparse frequency set, its associated Fourier projection, and trigonometric-polynomial initial data for the projected dissipative surface quasi-geostrophic equation. The resulting unique global smooth solution $\theta$ has infinite analyticity radius initially but satisfies $0<\operatorname{rad}(\theta(T))\leq1$. An additively separated Fourier cascade yields coefficientwise exponential lower bounds, while uniform comparison estimates control the feedback interactions. Thus entire analyticity need not persist even from trigonometric-polynomial data. Together, the two constructions show that an exact energy identity alone does not determine the frequency geometry governing analyticity-radius growth.

We analyze the numerical approximation of time-harmonic scattering by highly heterogeneous penetrable obstacles. These problems are especially challenging in the high-frequency regime, where the size of the scatterer $L$ is much larger than the wavelength, i.e., the wavenumber $k$ is such that $kL \gg 1$. Here, we further consider the situation where the scatterer contains different materials, with a characteristic size $\varepsilon$ such that $k\varepsilon \ll 1$. We propose a high-order multiscale finite element method, and provide an error analysis that is explicit in both $k$ and $\varepsilon$. Crucially, our error estimates suggest that using a high-order method should reduce the computational cost for large frequencies, which is corroborated by numerical examples.

We derive a family of degenerate vortices with non trivial swirls and fat tails which can be used to bifurcate both stationary and self similar solutions to the three dimensional axi-symmetric incompressible Euler equations.

We study the Liouville property for nonnegative weak solutions of the $(p,q)$-Laplacian elliptic differential inequality $$\Delta^{m}_{p}u(x)+\Delta^{m}_{q}u(x)+V(x)u^{s}(x)\leq 0$$ and the associated parabolic differential inequality $$\partial_t u(x,t) \geq \Delta^{m}_{p}u(x,t)+\Delta^{m}_{q}u(x,t)+V(x,t)u^{s}(x,t)$$ on a forward geodesically complete noncompact Finsler measure space $(M,F,m)$ with finite reversibility. Under several sets of integral growth conditions on the positive potential function over certain annular domains, we prove that any nonnegative weak solution vanishes almost everywhere. The proofs of our results are essentially based on the nonlinear capacity method.

This paper provides the instability counterpart to our recent construction of nonspherically symmetric approximating $\kappa$-self-similar naked-singularity solutions for the Einstein–scalar field system. These singular solutions contain pervasive nonspherical borderline terms, and to prove instability the delicate renormalization procedure developed in [2] does not extend to the more singular setting considered here. To overcome these difficulties, we introduce a new iteration scheme adapted to singular backgrounds whose leading-order geometry depends on the angular variables. At each step, the nonlinear coefficients are frozen using the preceding double-null geometry, and the resulting equations are solved in a triangular order. In this way, the nonspherical borderline terms are incorporated into the approximate geometry rather than treated as perturbative errors, yielding successively sharper estimates. After sufficiently many iterations, the scheme controls the singular angular structure and produces an approximate spacetime whose difference from the exact solution satisfies the required bounds. In particular, these bounds provide an existence region large enough to carry out the instability argument. We further prove that anisotropic perturbations of the outgoing data, arbitrarily small in a scale-critical norm, lead to the formation of a trapped surface. We also formulate and verify a matter-focusing condition under which a parabolic flow argument guarantees the existence of a corresponding marginally outer trapped surface (MOTS). Together, these results establish the nonlinear instability of $\kappa$-self-similar naked singularities beyond spherical symmetry in the Einstein–scalar field system and introduce a framework for applications across Einstein systems.

We study entire graphical translating solutions of the mean curvature flow, \[ \operatorname{div}\!\left(\frac{\nabla G}{\sqrt{1+|\nabla G|^2}}\right) =\frac{1}{\sqrt{1+|\nabla G|^2}} \qquad\text{in }\mathbb R^N. \] Every such graph is mean-convex, since its mean curvature is the vertical component of its unit normal. In dimension two, mean-convex translating solitons are convex, and an entire graphical translator is therefore the rotationally symmetric bowl soliton. In higher dimensions Wang constructed non-rotational entire convex translating graphs. We prove that a further loss of rigidity occurs at the Bernstein dimension: for every $N\ge 8$ there exists a one-parameter family of entire graphical translators that are mean-convex but not convex. The construction starts from the Bombieri–De Giorgi–Giusti (BDG) entire minimal graph in $\mathbb R^8$ and develops a singular perturbation theory for the translator equation around it. The main new feature is a transition layer near Simons' cone: the translating term breaks the odd symmetry of the minimal graph, and after a suitable recentering the matching problem is governed by a parabolic inner equation. A detailed analysis of this layer, together with weighted Jacobi theory on the BDG graph and global barriers, yields the desired entire solutions.

We study three asymptotic regimes for a parabolic–elliptic Keller–Segel system with porous-medium diffusion and the volume-filling sensitivity function $u(1-u)$. First, when $D=\epsilon^2\to0$, $\delta=1$, $1\le m<2$, and the chemotactic sensitivity coefficient is below an explicit $m$-dependent parabolicity threshold, we prove strong convergence to a scalar nonlinear diffusion equation. Second, as $\delta=\epsilon\to0$ with $D=1$, we obtain a hyperbolic–elliptic Keller–Segel limit by means of a kinetic reduction argument. Third, under the scaling $\chi=\epsilon^{-1}$, $D=\epsilon^2$, $\delta=1$, and for characteristic initial data of finite perimeter, we prove convergence on fixed time intervals to the time-independent initial patch in $BV(\Omega;\{0,1\})$. Because the sensitivity function is nonlinear in the density, weak convergence alone is insufficient to identify the chemotactic flux. Strong compactness of the density is therefore required in all three regimes and is obtained, respectively, through energy–entropy estimates, kinetic reduction, and $BV$ compactness.

We study the two-dimensional incompressible magnetohydrodynamic system near the Couette equilibrium $\bigl((y,0)^{\rm T},(\beta,0)^{\rm T}\bigr)$ on $\mathbb T\times\mathbb R$, in the anisotropic regime where both viscosity and magnetic diffusivity act only in the vertical direction. For the inviscid linearized problem with $|\beta|>1/2$, we prove sharp linear-in-time growth of the vorticity and current density at the level of the time rate. In contrast, the horizontal components of the velocity and magnetic perturbations remain uniformly bounded, while the vertical components exhibit quantitative inviscid damping at the rate $\langle t\rangle^{-1}$. For the nonlinear problem, we introduce shear-adapted Fourier multipliers that simultaneously capture enhanced dissipation, critical-time effects, and echo-type resonant interactions. Under a suitable horizontal background magnetic field and a quantitative compatibility condition between the magnetic field strength, viscosity, and magnetic diffusivity, we establish global nonlinear stability for divergence-free perturbations satisfying $ \left\| \bigl( \mathbf v_{\rm in}-(y,0)^{\rm T}, \mathbf H_{\rm in}-(\beta,0)^{\rm T} \bigr) \right\|_{H^N} \leq \varepsilon_0\min\{\mu,\nu\}^{1/2}, N\geq4$. Moreover, the nonzero horizontal Fourier modes decay in a shear-adapted $H^N$ norm at the enhanced-dissipation rate $e^{-c\min\{\mu,\nu\}^{1/3}\,t},$ and the vertical velocity and magnetic components gain an additional inviscid-damping factor $\langle t\rangle^{-1}$.

Let $A\subseteq\mathbb{R}$ support a probability measure whose Fourier–Stieltjes transform tends to zero at infinity. We prove that, for every $\varepsilon\in(0,1)$, there is a closed, $1$-periodic, nowhere dense set $E\subseteq\mathbb{R}$ such that \[ m(E\cap I)\ge1-\varepsilon \] for every interval $I$ of length $1$, while $E$ contains no affine copy of $A$. Thus every set supporting a Rajchman measure satisfies the Erdős similarity conjecture in a uniform large-set form. The proof combines equidistribution modulo one for large dilates of the measure with a multiscale family of low-density periodic blockers; no quantitative rate of Fourier decay is used. We also refine a classical theorem of Ivašev-Musatov, showing that for every Hausdorff gauge $h$ there is an $h$-null compact Rajchman support $K$ satisfying \[ \overline{\dim}_{\mathrm B}^{\log} K=\dim_{\mathrm P}^{\log} K=1. \] The value $1$ is sharp for both dimensions.

This paper considers the energy landscape of the t-SNE algorithm. While this algorithm has enjoyed broad adoption, the non-convexity of the associated energy has made it difficult to rigorously understand what the algorithm captures in many settings. In particular, a number of well-known numerical examples, several of which are reproduced in this article, suggest a complicated energy landscape with many local minimizers that do not respect the topology or clustering structure of the underlying data. This work seeks to provide first steps towards a rigorous explanation of these phenomena. Specifically, for a general family of energies, which include both the original t-SNE algorithm and recently identified large data limits, and for densities in feature space which obey a continuous symmetry, we construct infinite families of distinct critical points. These critical points are based upon identifying pairs of discrete symmetries, one in the original feature space and the other in the target embedding space, which are preserved under gradient dynamics. These critical configurations exhibit many characteristics, such as topology breaking and spurious clustering, which are often observed empirically. Finally, numerical and analytical examples are given throughout as a means of illustrating the approach.

We prove that harmonic morphisms between sub-Riemannian Lie groups are smooth, are symmetries of the sub-Riemannian Laplacian and are conformal submersions that satisfy a particular PDE. Moreover, we give some partial results for harmonic morphisms beyond Lie groups. We show the existence of harmonic coordinates for harmonically homogeneous spaces, and that smooth harmonic morphisms are symmetries of the Laplacian in all sub-Riemannian manifolds. Finally, we compute the first variation of the horizontal energy along a harmonic morphism of sub-Riemannian Lie groups: it is given by a linear form on the Lie algebra of the target, the modular mismatch, which vanishes for Riemannian and Carnot targets, but surprisingly not in general. Therefore, unlike in the Riemannian case, harmonic morphisms of sub-Riemannian Lie groups need not be harmonic maps.

We construct a nonempty $C^k$-open family of smooth, autonomous, divergence-free velocity fields on $\mathbb{T}^3$ that generate fast dynamos. The proof proceeds by first establishing fast dynamo action for a large class of smooth, time-periodic velocity fields exhibiting the classical stretch–fold–shear mechanism. We then realize the associated period map within a smooth autonomous flow using carefully tuned return dynamics to a global Poincaré section. In both settings, the proof first isolates unstable distributional eigenmodes for the ideal dynamo operator on an anisotropic Banach space of distributions, and then shows that this spectral instability persists under the singular perturbation $\varepsilon \Delta$. The resulting estimates are uniform as $\varepsilon\to0$ and stable under $C^k$ perturbations of the velocity field, yielding an open set of smooth fast dynamo vector fields. This result resolves the Fast Dynamo Conjecture of Zeldovich and Sakharov, as recorded in Arnold's book of problems.

We study dimension-free differential estimates for Brenier maps under first-order perturbations of the two marginals. Caffarelli's contraction theorem yields such estimates when the source and target potentials satisfy a pointwise Hessian comparison. We show that the same conclusion remains true under arbitrary globally Lipschitz perturbations of both marginals, without additional assumptions. In particular, this answers the conjecture due to Fathi, Mikulincer, and Shenfeld. More precisely, let $d\ge1$, $L\ge0$, and let $B:\mathbb{R}^d\to\mathbb{R}$ be globally $L$-Lipschitz. We prove that the Brenier map from the standard Gaussian measure $\gamma_d$ to the probability measure proportional to $\mathrm{e}^{-B}\gamma_d$ has a globally Lipschitz representative whose bound depends only on $L$. The Brenier potential also belongs to $C^{1,1}(\mathbb{R}^d)$ and satisfies dimension-free two-sided Hessian bounds. Let $\mathfrak{C}(0,L)$ denote the upper-bound constant, then \[ \log\mathfrak{C}(0,L)=4L^2+\log L+\mathcal{O}(1) \qquad(L\to\infty), \] and a one-dimensional example with Brenier map $T_L$ satisfies $\log\operatorname{Lip}(T_L)\ge L^2/2$. We deduce this Gaussian estimate from an anisotropic two-marginal result. Let $V,W:\mathbb{R}^d\to\mathbb{R}$ be potentials, and let $Q,P$ be positive-definite matrices. Using the distributional curvature bounds $D^2V\preceq Q$ and $D^2W\succeq P$, we establish matrix Hessian bounds for Brenier maps between arbitrary globally Lipschitz perturbations of the reference marginals. Most importantly, the constants keep the directional geometry of the essential gradient ranges. For affine perturbations, our estimates recover the sharp noncommuting Caffarelli tensor. We further obtain pointwise displacement bounds for Gaussian perturbations and spectral Hessian estimates for compact Gaussian mixtures.

We study heat content and spectral properties of subordinated sub-Laplacians on arbitrary Carnot groups. We consider restrictions of such operators to bounded open sets with zero Dirichlet boundary condition and study their spectral properties. In particular, for a large class of subordinators we give explicit eigenvalue estimates in terms of the subordinator and the eigenvalues of the sub-Laplacian. We also provide large-time asymptotics of the heat content for the subordinated sub-Laplacian in terms of its spectral gap. For fractional sub-Laplacians we prove short-time asymptotics for the corresponding heat content and relative heat content. Our approach combines semigroup methods, probabilistic techniques, geometric measure theory, and heat kernel estimates.

We study of the long-term behavior of a coupled wave-heat system. The system consists of a wave equation and a heat equation on two adjacent Lipschitz domains coupled by a common interface, with the heat equation being allowed to incorporate spatially dependent coefficients. We first establish strong asymptotic stability independent of the domains and coefficients. To this end, we employ the framework of closure relations, which reduces the spectral analysis of the coupled system to that of a wave equation. Secondly, we analyze non-uniform decay rates for classical solutions to the coupled system. Using a non-orthogonal decomposition of the state space, we reduce the problem to a resolvent-type estimate which is independent of the heat domain and heat coefficients. With this, we extend the known non-uniform decay rates to spatially dependent heat coefficients, yielding logarithmic decay under no assumptions and polynomial decay under the Geometric Control Condition.

We construct exact traveling-wave solutions for a semilinear parabolic equation with a specific nonlinearity. This nonlinearity arises, in particular, in voting models based on branching Brownian motion. Previously known results are special cases of those presented here.

A classical theory of Amrouche, Girault and Giroire (1994) resolves the Laplace equation on $\mathbb{R}^n$ through an isomorphism between weighted Sobolev spaces. Inspired by their framework, we develop the corresponding $L^p$ theory for the fractional Laplacian: we introduce weighted fractional Sobolev spaces $\Lambda^{s,p}(\mathbb{R}^n)$, gauged at top order by $\|(-\triangle)^{\frac{s}{2}}u\|_{L^p}$, and prove that $(-\triangle)^{\frac{s}{2}}:\Lambda^{s,p}(\mathbb{R}^n)/\mathcal P_{[s-n/p]}\to L^p(\mathbb{R}^n)$ is an isomorphism for all $s\in(0,2)$ and $p\in(1,\infty)$; thus $(-\triangle)^{\frac{s}{2}}u=f$ is solvable for every $f\in L^p(\mathbb{R}^n)$, uniquely modulo an explicit finite-dimensional space of polynomials. More generally, extending the scale by duality to negative orders, the fractional Laplacians of the appropriate orders map its spaces isomorphically onto one another and compose exactly. A key ingredient is a family of weighted Hardy and Poincaré inequalities adapted to $\|(-\triangle)^{\frac{s}{2}}u\|_{L^p}$, established here for $1\le s<2$ by a reduction to the gradient. Since these inequalities hold a priori only on a dense class, while the density of $C_{\mathrm{c}}^\infty(\mathbb{R}^n)$ in $\Lambda^{s,p}(\mathbb{R}^n)$ is itself nontrivial, we first prove the isomorphism on the closure of the Schwartz class, which coincides with $\Lambda^{s,p}(\mathbb{R}^n)$ since the difference of two solutions is an entire $s$-harmonic function, hence a polynomial; the density follows as a by-product. As an application, we obtain existence and uniqueness, with explicit kernels and compatibility conditions, for the fractional Stokes system in $\mathbb{R}^n$.

In this paper, let $u\in C^4(B_{10})$ with $D^2u\geq -KI$ define a $2$-admissible graph $M=\{(x,u(x)):x\in B_{10}\}\subset\R^{n+1}$ satisfying \[ \sigma_2(\kappa[u])=f(x). \] We prove an interior curvature estimate depending on the Lipschitz norm of the right-hand sides. The proof combines a shifted Jacobi inequality for \(b=\log(H+J_0)\), a parallel hypersurface transformation that makes the Newton tensor uniformly elliptic and local boundedness estimate then reduces the pointwise bound to a weighted $L^1$ estimate, which is completed using the Jacobi energy inequality and integration by parts.

We prove an arbitrary-dimensional halfspace theorem for global minimizers of a smooth uniformly elliptic anisotropic perimeter: if the nonempty boundary of a minimizing set is contained in a halfspace, then the set itself is a halfspace. No evenness of the integrand or regularity of the minimizing boundary is assumed. The proof combines wall contact with a plane-peeling argument for the complementary phase defect and a simultaneous second blow-down. We also give a direct, self-contained exterior-barrier proof of the two-dimensional stationary anisotropic statement. Its rigidity conclusion is already covered by Bergner's earlier halfspace theorem; the proof is retained because it works directly with the anisotropic Euler-Lagrange operator.

We prove a scaling-critical regularity criterion involving only one velocity component for finite-energy suitable weak solutions of the three-dimensional incompressible Navier–Stokes equations. Let $2<p<\infty$ and $m=3p/(p-2)$, so that $2/p+3/m=1$. We show that a singularity cannot occur provided \[ \sum_{j\in\mathbb Z} \|\dot\Delta_j u^3\|_{L^p(0,T;L^m(\mathbb R^3))}<\infty, \] that is, $u^3\in\widetilde L^p(0,T;\dot B^0_{m,1}(\mathbb R^3))$. The assumption is a spatial-frequency $\ell^1$ refinement of the still unresolved critical condition $u^3\in L^p_tL^m_x$ and is complementary to the Lorentz-in-time refinement $L^{p,1}_tL^m_x$ obtained by Wang, Wu, and Zhang. By Bernstein embedding, the result extends to $u^3\in\widetilde L^p_t\dot B^s_{q,1}$ on the nonnegative-smoothness critical line $s=-1+2/p+3/q\ge0$. The principal innovation is a frequency–scale matching scheme embedded in the local energy inequality. Each dyadic block of $u^3$ is retained until it is paired with the vertical scale selected by a one-dimensional backward heat kernel. Low frequencies gain from the slab thickness, high frequencies from transferring the projection to the localized flux and applying an inverse Bernstein estimate, and spatially separated pressure sources from harmonic decay. These mechanisms generate a two-sided $\ell^1$ kernel, converting spatial-frequency summability into summability of physical-scale energy increments. Consequently, we obtain a uniform Type-I local energy bound without a Lorentz refinement in time; compactness and one-component rigidity then exclude singular blow-up limits whose third velocity component vanishes.

This paper is concerned with the precise asymptotics, as time goes to infinity, of a transport problem in a half plane coupled with a nonlinear nonlocal boundary condition. This system arises from a class of models for the spatial spread of epdemics, its space independent version being the classical Kermack-McKendrick model. Using ideas pertaining to the study of nonlocal equations of the Fisher-KPP type, and exploiting the particular structure of the model, we prove that any initially localized solution will lag behind the minimal traveling wave, with a delay that grows logarithmically in time.

The Schrödinger-Lohe (SL) model is a coupled system of nonlinear Schrödinger equations describing the temporal-spatial evolution of the component wave functions, and it corresponds to the infinite-dimensional counterpart of the Lohe matrix model for quantum synchronization. In this paper, we study a rigorous mean-field limit of the SL model and provide a quantitative estimate on the fluctuations between one-marginal distribution and one particle distribution in 2-Wasserstein distance. For the derived kinetic SL equation, we present sufficient conditions leading to the complete and practical synchronizations.

We disprove McKean's conjecture, which asserts that the entropy dissipation is monotone nonincreasing along solutions, or equivalently that the entropy is convex in time, for the spatially homogeneous Landau-Coulomb equation. We provide an explicit counterexample which consists of a Maxwellian equilibrium under radially symmetric, smooth perturbation on an annulus of scale $R\gg 1$. Developing the derivative of the entropy dissipation in powers of $R$ reveals that the leading order terms can be positive for a suitably chosen perturbation. Remarkably, the counterexamples are close to equilibrium in relative entropy and standard Sobolev norms. Since the Landau equation arises from the Boltzmann equation in the grazing collisions limit, our counterexample also shows that McKean's conjecture is not true in general for the Boltzmann equation.