Papers

This paper establishes an asymptotic formula for negative Sobolev norms as the fractional order tends to zero. In the Euclidean setting, under a mild boundedness condition, the product of the order and the norm raised to the power $p$ converges to a dimension-independent constant multiple of the corresponding $L^p$ norm. The proof relies on heat-kernel regularization, weak compactness, and an Abelian–Tauberian argument. The result is further extended to a general measure space equipped with a family of bounded and continuous operators that covering nonlinear and non-semigroup settings. We also present several applications in analysis and probability, including an absolute-continuity criterion for measures, a random-distribution regularity result, a construction of square-integrable local times for fractional Brownian motion, and limiting formulas for truncated maximal operators and martingales.

We establish quantitative stability estimates for weighted Hardy inequalities in the local and fractional settings, showing that the Hardy deficit controls the distance of a normalized admissible function to the family of virtual extremizers. In the local case, for $p>1$, $\alpha\in[0,1)$, $0<p-\alpha p<N$, and $u\in C_c^1(\mathbb{R}^N)$ normalized by $\int_{\mathbb{R}^N}|u|^p|x|^{\alpha p-p}\,dx=1$, we prove \begin{equation*} \int_{\mathbb{R}^N}|\nabla u|^p|x|^{\alpha p}\,dx-\Big(\frac{N-p+\alpha p}{p}\Big)^p\int_{\mathbb{R}^N}\frac{|u|^p}{|x|^{p-\alpha p}}\,dx \ge C\,\operatorname{dist}_{\alpha}(u,\mathcal{M}_\alpha)^{\max\{4,2p\}}, \end{equation*} where $\mathcal{M}_\alpha$ is generated by the virtual extremizer with scale-invariant distance. We extend this to the weighted fractional Hardy inequality: for $s\in(0,1)$, $\alpha=\alpha_1+\alpha_2\ge0$, $\alpha_1p,\alpha_2p\in(-N,sp)$, and $0<sp-\alpha p<N$, the fractional Hardy deficit satisfies \begin{equation*} \int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}|x|^{\alpha_1p}|y|^{\alpha_2p}\,dx\,dy-\mathcal{C}\int_{\mathbb{R}^N}\frac{|u|^p}{|x|^{sp-\alpha p}}\,dx \ge C\,\operatorname{dist}_{s,\alpha}(u,\mathcal{M}_{s,\alpha})^{\max\{4,2p\}}. \end{equation*} For $p\ge2$ the distance is measured on $u$, while for $1<p<2$ it is measured on a power-type transformation of $u$. Both recover the unweighted results when the weights vanish. We also prove weighted Sobolev and fractional Sobolev extension and embedding inequalities on bounded smooth domains avoiding the origin. The proof is rearrangement-free, combining these inequalities with scale-invariant Poincaré–Sobolev estimates on annuli, telescoping oscillation decompositions, and weighted Lorentz-space embeddings. The distinction between $p\ge2$ and $1<p<2$ stems from the convexity estimate used to obtain a remainder term from the deficit.

We propose a thresholding scheme for the half-harmonic map heat flow from the flat torus $\mathbb{T}^d$ into the unit sphere $\mathbb{S}^m$ in $\mathbb{R}^{m+1}$, whose iterates are given by convolution with the Poisson kernel, replacing the Gaussian kernel of the classical Merriman-Bence-Osher algorithm, and pointwise normalization. We prove that the piecewise constant interpolants subconverge to a weak solution of the half-harmonic maps heat flow that attains the initial map strongly in the energy space and satisfies the energy-dissipation inequality. For a spectrally truncated variant, convergence holds under a condition on the step size. A key ingredient is the derivation of the precise energy-dissipation identity directly from a refined within-step estimate, without resorting to De Giorgi interpolation typically used in the minimizing movement framework. This provides a direct way to recover the appropriate energy-dissipation property in the limit and, in turn, enables weak-strong uniqueness: every energy-dissipating weak solution coincides with the strong solution whenever the latter exists.

We study SPDEs associated with $2$D active scalars driven by incompressible transport noise of Kraichnan type, with regularity exponent $\alpha\in (0,1)$; our examples include the Euler, SQG and IPM systems. We investigate whether a number of “turbulent” phenomenologies, which are well understood in the linear Kraichnan model, persist in this nonlinear setting, uniformly in vanishing viscosity approximations. First, for suitable values of $\alpha$ and initial data in $L^p_x$, we establish anomalous regularization estimates, measured in appropriate endpoint Besov-type spaces of regularity $\beta=\beta(\alpha,p)>0$. Remarkably, these results allow for some scaling supercritical regimes of the parameters $\alpha,p$; on the other hand, for (sub)critical parameters, we recover the same regularity exponent $\beta=1-\alpha$ as in the linear case. Second, in the (sub)critical case, we further prove anomalous integrability, namely solutions becoming instantaneously $L^\infty_x$-valued at positive times, uniformly in the viscosity; moreover, in this case we establish strong existence and pathwise uniqueness of solutions to the inviscid SPDE, which are recovered as the unique vanishing viscosity limit. Finally, in the $2$D Euler case, for $\alpha\in (0,1/2)$, we establish anomalous dissipation of enstrophy and sharpness of anomalous regularization.

In this paper, we systematically exhibit the roles of logistic source on global boundedness and convergence of classical solutions, and global existence and eventual smoothness as well as convergence of generalized solutions for a singular chemotaxis urban crime model. Roughly speaking, this work shows that suitably (can be precise) strong logistic damping ensures boundedness and convergence of classical solutions, and weak logistic damping ensures global existence of generalized solution and its eventual smoothness as well as convergence in two and three dimensions. The former in particular implies that strong logistic damping can prevent the formation of crime hotspots (spatial patterns). Previous known results in this respect are thus significantly improved and extended.

We establish a structure theorem for measures satisfying left-invariant PDE constraints in Carnot groups. Our generalization, that takes inspiration from the paper arxiv:1601.06543 by the first named author and F. Rindler, requires the introduction of the hypoelliptic wave cone, a sub-Riemannian version of the wave cone from the theory of Compensated Compactness, and it employs classical results in Harmonic Analysis on homogeneous spaces. Then, we utilize this result to extend the Rank-one Theorem for functions of bounded horizontal variation ($BV_H$) to all Carnot groups. This is achieved by considering a suitable curl-type left-invariant operator, that provides necessary differential constraints for horizontal gradients, and by studying the hypoellipticity of said constraints.

We introduce a Green–potential heat flow for maps from a closed Riemannian surface. Inspired by Parker–Wolfson renormalization~\cite{ParkerWolfson1993,Parker1996}, the conformal factor is reconstructed at each time from an energy-based source determined by the current map. The resulting parabolic–elliptic system incorporates renormalization into the evolution while leaving the Dirichlet energy functional and its harmonic critical points unchanged. We establish local well-posedness for $H^3$ initial data and a continuation criterion in terms of energy concentration. For sufficiently large coupling, we exclude finite-time bubbling and obtain global solutions. At infinite time, the energy identity provides only weighted $L^2$ control of the tension field. Localizing to regular regions of concentration charts yields harmonic profiles along subsequences of prescribed times. We represent these profiles in the flow's packet charts, organize them into a finite bubble tree, and decompose the limiting energy into harmonic component energies and neck energies. A critically normalized radial example illustrates how the modified flow can prevent finite-time blow-up and produce harmonic bubbles with no-neck property.

In this paper, we investigate the elastic inverse scattering problem in a two-dimensional cavity with an impedance boundary condition. We prove that the scattered fields inside the cavity satisfy a reciprocity relation, and prove that both the shape of cavity and the impedance parameter on the boundary can be uniquely determined from the scattered fields measured on an interior curve. Numerically, we first employ the linear sampling method to reconstruct the shape, and then recover the impedance parameter by combining potential theory with the least squares method. Several numerical experiments are presented to demonstrate the feasibility and effectiveness of our methods.

We study the strong-shear limit of bistable traveling fronts in infinite cylinders with periodic transverse boundary conditions. We prove that, for every sufficiently regular periodic shear profile, the front speed normalized by the flow amplitude converges as the amplitude tends to infinity. After suitable longitudinal rescaling and translations, the corresponding profiles converge along subsequences to full fronts of the limiting degenerate equation. Furthermore, under a Hörmander-type non-degeneracy condition on the shear, the limiting front is proved to be regular and unique up to translation, and the whole normalized family converges uniformly. A main difficulty is that the sign-changing bistable reaction allows the limiting transition to split through intermediate transverse equilibria. We rule out this possibility by exploiting the instability of such equilibria together with suitable regularization and comparison arguments. Finally, in contrast with the combustion case, where every nonconstant mean-zero shear yields a positive limiting speed, we construct an example showing that, for a fixed bistable reaction, smooth mean-zero shears can produce negative, zero, or positive limiting speeds. In particular, the example shows that a shear which accelerates propagation at small amplitudes may reverse the propagation direction when its amplitude becomes large.

We analyse Schrödinger operators with accretive potentials in weighted spaces. We find conditions on potentials and weights for which the Dirichlet realisation, introduced by generalised form methods, has non-empty resolvent set. We establish a domain and graph norm separation property, as well as sufficient conditions for the compactness and Schatten class of the resolvent. Moreover, we investigate the relation between discrete spectra and eigenfunctions of operators in standard and weighted spaces. As applications we extend results on the completeness of eigensystems of operators with accretive potentials from standard to weighted spaces and analyse operator matrices exhibiting a Schur dominance property, in particular, related to a wave equation with strong accretive damping.

We study harmonic mappings between the source domain in the $\alpha$-Grushin plane and the target domain in the $\beta$-Grushin plane for $\alpha$ not necessarily equal to $\beta$ and $\alpha\in [0,1)$. Our harmonic mappings arise as a Euler–Lagrange system of equations related to the Dirichlet energy and are studied in weak and strong forms. The key results enclose the second order regularity of harmonic mappings, the Bochner identity, the weak Harnack estimates for coordinate functions of a harmonic mapping, as well as the Caccioppoli estimates and the Liouville type theorem. Furthermore, we discuss several examples of (Grushin) harmonic mappings and, in particular, relate them to holomorphic mappings in the plane.

Let $1 \leq q < p$ and let $\lambda_1$ be the first eigenvalue of the $p$-Laplacian in a bounded domain $\Omega$. We study the energy functional $$ E_\lambda(u)=\frac{1}{p}\left(\int_\Omega|\nabla u|^p\,dx -\lambda\int_\Omega|u|^p\,dx\right)-\mathcal{F}(u), \quad u\in W_0^{1,p}(\Omega), $$ where $\mathcal{F}$ is positively $q$-homogeneous and vanishes along the first eigenspace. Assuming a suitable relation between $\mathcal{F}$ and the $\kappa$-th power of the principal part of $E_{\lambda_1}$ near this eigenspace, we describe the behavior of $E_{\lambda_1}$ according to the relations $p\kappa<q$, $p\kappa=q$, or $p\kappa>q$. In particular, the functional is unbounded from below in the first case, and has a negative infimum in the last case. We then study how sufficiently small $q$-homogeneous perturbations of $\mathcal{F}$ influence the geometry of $E_\lambda$. In this way, we describe assumptions guaranteeing the existence of three critical points in a left neighborhood of $\lambda_1$ and two critical points in a right neighborhood of $\lambda_1$, which indicates an $S$-shaped structure of the solution set. The results are applied to double-phase functionals and the nonlinear Fredholm alternative.

We extend the global existence results for critical parameters of [Bianchini and Leccese, J. Math. Anal. Appl. (2024)] to the multidimensional problem \begin{equation*} \partial_t u + \text{div } (b(t,x) u^{1+k}) = \Delta u \end{equation*} for $b:(0,\infty)\times\mathbb{R}^d\to\mathbb{R}^d$ non-autonomous fields, in the case \begin{equation*} b\in L^\infty_{\mathrm{loc}} \bigl((0,\infty),L^{p,\infty}(\mathbb{R}^d)\bigr), \qquad p>d\ge 2. \end{equation*} For the critical case, we study the difference between two ordered solutions whose conserved small mass absorbs the drift. Finitely many increments then reach an arbitrary solution.

We prove global well-posedness of the periodic unidirectional Euler-alignment system in every dimension for supercritical dissipation of order $0<\alpha<1$ and arbitrary non-vacuum initial data in the Sobolev class of the local theory. The key idea is to propagate simultaneously two moduli of continuity at critical scales for the density $\rho$ and the potential gradient $\Gamma=\nabla\Lambda^{-\alpha}u$, both of which have scaling-invariant amplitudes. We also establish exponential alignment of the velocity and exponential convergence of the density to a traveling profile.

Despite its physical importance, the stability of Prandtl boundary layers under wall suction remains relatively unexplored in rigorous PDE analysis. In this paper, we establish the global-in-$x$ asymptotic stability of a suction profile. For any prescribed algebraic decay rate in $x$, we prove that sufficiently small inflow perturbations with sufficiently rapid spatial decay yield downstream decay at that rate. The von Mises transformation reduces the Prandtl system to a degenerate parabolic equation with a favorably directed convection term. We first identify the downstream decay mechanism in an associated uniformly parabolic problem, then control the boundary degeneracy using a sharp Hardy-type estimate. Together, these arguments yield stability and decay for the original Prandtl system.

This paper is devoted to the global dynamics of a nonlocal Lagrangian-averaged Navier-Stokes-$\alpha$ (LANS-$\alpha$)-type system (also known as the viscous Euler-Poincaré system) in one, two, and three spatial dimensions. In the unforced periodic setting with zero mean, we establish global well-posedness for the initial-value problem and prove that the corresponding solution semigroup is dissipative and possesses a compact global attractor consisting of the singleton $\{0\}$. Furthermore, we obtain the following global-in-time algebraic asymptotic stability estimate: \[ \|u(t,\cdot)\|_{H^3}\lesssim t^{-1/2},\qquad \forall\, t>0, \] which is derived from higher-order energy inequalities combined with a uniform Gronwall-type argument. Our results provide a complete description of the long-time behavior of the unforced periodic dynamics for this nonlocal LANS-$\alpha$-type system.

We study the semiclassical dynamics of solitary waves for the three-dimensional cubic–quintic nonlinear Schrödinger equation with an external potential. For initial data given by a modulated free ground state $P_\omega$, we prove that the wave-packet center follows the associated classical Hamiltonian flow. For $\omega\in\mathcal I$, we obtain qualitative persistence near the soliton orbit. Under the additional positive-slope condition $\omega\in\mathcal I_+$, we establish an $O(\eps)$ approximation in the scaled $H^1$ norm on every fixed time interval. We also derive $O(\eps^2)$ concentration estimates for the mass, momentum, and center of mass.

Let $0<\kappa<1$ and set $p_+=2/(1-\kappa)$ and $p_-=2/(1+\kappa)$. We construct a coefficient $\kappa\leq a\leq\kappa^{-1}$, smooth outside a compact set of Lebesgue measure zero, for which the $W^{1,2}_{p_+}$ estimate fails for the one-space dimensional nondivergence form parabolic equation. The corresponding solution has second spatial derivative in the weak $L_{p_+}$ space, but not in $L_{p_+}$. By duality, the $W^{1,2}_{p_-}$ a priori estimate also fails. Conversely, we demonstrate that the $W^{1,2}_p$ estimate holds and the equation is uniquely solvable for $|p-2|<c\kappa$, showing that the size of the solvability interval around $2$ has optimal order $\kappa$ as $\kappa\downarrow0$, which addresses a question raised in [17]. These results imply that the Alexandrov maximum principle holds for $p>2-c\kappa$ and this order is sharp as $\kappa\to 0$. The corresponding results for divergence form equations are also obtained.

We study Keller–Segel systems on $ \mathbb{T}\times\mathbb{R}$ subject to a stochastic Couette transport flow. We prove that sufficiently strong mixing suppresses chemotactic finite-time blow-up with high probability; thus yielding a unique global-in-time mild solution for initial data of arbitrary mass. Moreover, we obtain a quantitative estimate for the probability of blow-up, showing that it decays exponentially with increasing strength of the stochastic shear. Our analysis is based on the enhanced dissipation generated by the stochastic shear flow. To prove our main result, we first construct a maximal local mild solution through pathwise estimates for the random solution operator of passive scalar transport equations and a fixed-point argument. To obtain global control, we decompose the evolution into time blocks and exploit mixing-induced decay of the nonzero spatial Fourier modes. A nonlinear bootstrap argument then yields global existence above an almost-sure finite random threshold for the shear strength which is obtained by an application of the Borel-Cantelli theorem.

In this paper, we study local minimizers of the energy functional$J(\mathbf{u}) = \int_D (|\nabla \mathbf{u}|^2 + Q^2(x)\chi_{\Omega_{\mathbf{u}}})\,dx$ which give rise to a singular cooperative system. Here \(\mathbf{u}=(u_1,\dots,u_m): D\to\R_+^m\) is a vector-valued unknown function, \(\Omega_{\mathbf{u}}:=\{|\mathbf{u}|>0\}\) is the positive set, \(\chi_{\Omega_{\mathbf{u}}}\) is the characteristic function of the positive set \(\Omega_{\mathbf{u}}\), and \(Q(x)\) is the Bernoulli weight function. This problem was first introduced by Caffarelli, Shahgholian, and Yeressian in the poineer work (Duke Math. J. 167(10), 2018), where the regularity theory for minimizers and for the free boundary \(\partial\Omega_{\mathbf{u}}\) was established for nondegenerate Bernoulli weights, i.e., \(Q(x)\ge Q_{\rm min}>0\). The present paper studies the same free boundary problem but for a degenerate Bernoulli weight. The main difficulty lies in the coupling between the nondegeneracy of \(Q(x)\) and the vector-valued nature of the problem. We show that the the free boundary set \(\partial\Omega_{\mathbf{u}}\cap\{Q(x)=0\}\) decomposes into a nondegenerate part and a degenerate part. Nondegenerate points admit nontrivial blow-up limits; degenerate points have zero weighted density. We classify nondegenerate points into single-phase, multi-phase non-branching, and multi-phase branching points, analyzing their geometric structure. The branching-point singular structure for multi-phase points is purely vectorial and new in this setting. Finally, following Naber–Valtorta (Ann. Math. 185, 2017) and Edelen–Engelstein (Trans. Amer. Math. Soc. 371, 2019), we prove the nondegenerate set has locally finite $\mathcal{H}^{n-2}$-measure and is countably $(n-2)$-rectifiable.

Following a program outlined by Bartnik, and also employing nonlinear potential techniques we developed in the last years, we study lightlike singularities of variational solutions to the prescribed Lorentzian mean curvature equation with possibly unbounded (rough) curvature. This corresponds to allowing for unbounded charges in the Born-Infeld model.

Nirenberg's theorem gives interior $C^{2,\alpha}$ regularity for continuous viscosity solutions of fully nonlinear uniformly elliptic equations $F(D^2u)=0$ in dimension two. In dimensions five and higher, the work of Nadirashvili–Tkachev–Vlăduţ shows that such solutions need not be $C^2$. We construct a family of $C^{1,1}\setminus C^2$ functions $\{w_m\}_{m\geq3}$ and uniformly elliptic operators $F_m$ such that $F_m(D^2 w_m)=0$ in $\mathbb{R}^{d_m}$, where $d_m=m(m+1)/2-1$. These solutions are given by $\operatorname{tr}(X^3)/\sqrt{\operatorname{tr}(X^2)}$ on the space of traceless real symmetric $m\times m$ matrices. When $m=3,4$, this family recovers the known singular solutions associated with the five-dimensional Cartan isoparametric cubic and the nine-dimensional Hsiang minimal cubic, respectively. Our construction treats all $m\geq3$ in a unified way by direct analysis on traceless symmetric matrices. We also prove a geometric criterion for pointwise $C^{2,\beta}$ regularity of continuous viscosity solutions $u$ of uniformly elliptic equations $F(D^2u)=0$ in every dimension $n\geq 3$. Suppose that $n-2$ smooth hypersurfaces pass through a point $p$, each carrying a smooth vector field along which the directional derivative of $u$ is constant on that hypersurface. If these vector fields are linearly independent at $p$, then $u$ is twice differentiable at $p$ and admits a pointwise $C^{2,\beta}$ expansion for some $\beta\in(0,1)$. Our criterion is quantitative and yields interior $C^{2,\beta}$ regularity under uniform geometric hypotheses. As an application, we extend Nirenberg's theorem to a general class of solutions in dimensions three and higher. As a consequence, we establish interior $C^{2,\beta}$ regularity for solutions of Dirichlet problems with certain group symmetries. This extends a result of Nadirashvili–Vlăduţ for axially symmetric problems.

In his 1957 paper on the Dirichlet problem for higher-order elliptic equations in the plane, Agmon conjectured that there should exist higher-order equations in dimensions greater than two for which a potential-theoretic approach based on multi-layer integral operators can be successfully implemented. We show that this is indeed the case for the polyharmonic operator $\Delta^m$ of arbitrary order $m\in\mathbb{N}$ in $\mathbb{R}^n$, $n\geq 2$, even in geometrically rough domains permitting singularities beyond the scope of the existing theory of higher-order elliptic boundary value problems. Central to our approach is a new genre of multi-layer singular integral operators, which play for $\Delta^m$ the role assumed by the classical harmonic double layer potential in the treatment of the Dirichlet problem for harmonic functions.

In this paper, we study the deformed Hermitian-Yang-Mills (dHYM) equation on the non-compact Calabi-Yau manifold $Z = {\rm{Tot}}({\bf{K}}_{X})$, where $X$ is a rational homogeneous variety. Since $X$ is a Fano variety, $Z$ is a resolution of the singularity at the vertex of the affine cone ${\rm{Aff}}(X)$, provided the cone is built using the anticanonical polarization ${\bf{L}} = {\bf{K}}_{X}^{-1}$. Using the cohomogeneity-one symmetry of the Ricci-flat Kähler metric obtained via the Calabi ansatz on $Z$, we reduce the fully nonlinear PDE underlying the dHYM equation to a scalar, asymptotically autonomous ordinary differential equation (ODE). From this, we determine the exact condition on the topological phase that guarantees global existence of solutions. As an application, we show that every holomorphic line bundle over $Z$ admits a smooth, globally defined Hermitian connection solving the dHYM equation, provided the total phase lies in an explicit open interval determined by the Lie-theoretic data. Also, we prove a rigidity result classifying the exact geometric conditions under which the dHYM solution collapses into a classical Hermitian-Yang-Mills (HYM) connection. The results established generalize previous constructions and provide a substantial new class of examples. Furthermore, the approach presented allows one to study the behavior of the dHYM solutions through ODE methods. Using this approach, we construct the first explicit non-trivial example of a Hermitian-Einstein connection on a line bundle over a non-toric Calabi-Yau manifold which is not dHYM.

We study nonnegative entire spacelike graphs in Lorentz–Minkowski space satisfying $\sigma_k(A[u])=u^p$. For $2\le k<n$ and $p\ge k$, pointwise strict spacelikeness and closed $k$-admissibility yield universal bounds for the height and Lorentz factor. Their proof combines block coercivity in the full Gårding cone, a Lorentzian cutoff, and hyperbolic-cap comparison. For $2\le k<n/2$ and $k\le p\le k(n+2)/(n-2k)$, every such solution vanishes, without symmetry, decay, finite-energy, curvature-pinching, or global Hessian assumptions. In the subcritical range we use a trace-free Newton tensor, a weighted divergence identity, and a finite descent. At the critical endpoint, the divergence identity retains an exact nonnegative defect and a positive quartic term. Direct estimates cover $2k<n\le4k+2$. For $n\ge4k+3$, we use compactness, concentration of the $k$-Hessian measure at the first crossing, a single-pole Pohozaev argument, core counting, a recursive defect estimate, and Souplet-type radius selection. A geometric corollary gives a rigidity result for complete spacelike immersions under the corresponding curvature equation.

We establish a spectral weak-to-strong stability criterion and a sharp Sobolev dichotomy for Maxwell–Ohm evolution with prescribed velocity on $\T^d$, $d=2,3$. Uniform spatial spectral tightness, together with weak convergence of each fixed spatial Fourier mode and strong convergence of the electromagnetic initial states, yields strong convergence of the electromagnetic fields, weak convergence of the currents, and distributional convergence of the Lorentz forces. In particular, weak convergence of the velocity coefficients in $L^2_tH^s_x$ suffices when $s>d/2$. At the critical index $s=d/2$, this conclusion fails sharply. We construct smooth divergence-free velocities converging strongly to zero in the critical Sobolev space, together with electromagnetic initial states converging strongly to zero, while the corresponding terminal fields remain of order one and the currents stay bounded. In two dimensions the construction is based on a moving logarithmic core, whereas in three dimensions it uses a Fourier-capacity core built from Leray-projected lattice modes and a polarized six-component Maxwell packet with exact magnetic divergence constraint. In both dimensions, the Lorentz forces converge to an explicit nonzero measure supported on a single ray with fixed unit direction.

We construct global finite-energy weak solutions for a 1D barotropic Navier–Stokes–Maxwell system with displacement current and the algebraic Ohm law. The result holds for every \(\gamma>1\) and arbitrary finite-energy initial data, allowing vacuum and requiring only \(L^2\) initial electromagnetic fields. A key ingredient is a weak-to-strong compactness principle for the Maxwell–Ohm subsystem: weak convergence of the velocity coefficients in \(L^2_tH^1_x\), together with strong convergence of the initial fields, yields strong electromagnetic trajectories in \(C_tL^2_x\). This permits identification of the weak current and the distributional Lorentz force without strong convergence of the velocity. The electromagnetic closure is combined with the classical artificial-viscosity and artificial-pressure construction for compressible flow. A density-primitive effective-flux argument identifies the physical pressure and eliminates the physical and artificial pressure-concentration defects.

Let $\mathbf{C}=\partial E\subset\mathbb{R}^{n+1}$ be an area-minimizing hypercone. The cone may have nonisolated singularities. We prove that there is a constant $c_{\mathbf{C}}>0$ such that \[ \frac{\operatorname{Per}(F;B_R)-\operatorname{Per}(E;B_R)}{R^n} \geq c_{\mathbf{C}} \left(\frac{|F\triangle E|}{R^{n+1}}\right)^2 \] for every $R>0$ and every set $F$ of locally finite perimeter with $F\triangle E\Subset B_R$. This extends the unweighted quantitative inequality from regular area-minimizing hypercones to general area-minimizing hypercones. We follow the calibration argument in the author's earlier work on regular cones. The main change is the construction of the vector fields. The pointwise asymptotic estimates for positive Jacobi fields used in the regular case do not directly apply here. We use the minimal foliations constructed by Zhihan Wang on the two sides of the cone. On each leaf, we integrate projection kernels with a positive Jacobi field as the weight. The kernel construction gives the required divergence bound. Wang's weak Harnack inequality and growth estimates give the integral bounds needed to prove linear growth of the vector fields.

We construct global solutions of the periodic defocusing cubic half-wave equation whose $H^s$ norms grow exponentially as $t\to+\infty$ for every $1/2<s<3/2$. Such solutions exist at every positive mass. The construction starts from explicit concentrating trajectories of a linearly perturbed Szego equation and solves an infinite-time final-value problem for the full half-wave flow, including its negative-frequency component. We obtain the exact growth rate and conserved quantities. Finite Blaschke products allow us to prescribe finitely many concentration points and their relative widths, and we compute the limiting critical kinetic-energy measures. We also identify the endpoint Fourier decay of the constructed data and prove the sharp logarithmic regularity threshold on the resonant mass slice and for generic scalar parameters. In particular, to the best of our knowledge, this provides the first example of a genuine infinite cascade for a defocusing fractional nonlinear Schrödinger equation on the torus.

We establish Dirichlet–Neumann bracketing for the Dirichlet eigenvalues of $\psi(-\Delta)$ on bounded Lipschitz domains, where $\psi$ is an arbitrary complete Bernstein function. The eigenvalues lie between $\psi$ applied to the corresponding Neumann and Dirichlet eigenvalues of the Laplacian. Both inequalities are strict whenever $\psi$ admits no meromorphic continuation to $\mathbb C \setminus \{0\}$. The proof uses quadratic forms, operator monotonicity, and an analysis of equality in resolvent comparisons. Applying the bracketing to intervals and balls gives a unified proof of simplicity of interval eigenvalues and antisymmetry of second eigenfunctions in balls under the same condition on $\psi$. For fractional powers, these recover results of Fall, Ghimenti, Micheletti and Pistoia for the interval, and of Fall, Feulefack, Temgoua and Weth and, independently, Benedikt, Bobkov, Dhara and Girg for the ball. The argument extends these conclusions to a broader class of nonlocal operators.