Papers

In this paper, we revisit the singular asymptotics of the free surface near stagnation points for two-dimensional traveling gravity water waves with vorticity. We prove the nonexistence of full-density degenerate stagnation points beyond the strict two-sided linear growth regime. Our main tools are a modified Weiss-type monotonicity formula and a modified Almgren-type frequency formula. Together, they provide a new approach that completely avoids the use of a Bessel-type differential inequality, which is an essential tool used in the previous literature to prove the nonexistence of full-density degenerate stagnation points (Ann. I. H. Poincaré-AN, 29, 861–885, 2012). As consequences, we obtain uniform bounds for frequency-normalized blow-ups in arbitrary dimension and strong convergence in dimension two. As an application, we extend the Stokes conjecture for rotational waves to a broader class of vorticity distributions.

We show that the dual approach to Bourgain–Brezis estimates for Hodge systems is substantially more flexible than previously understood. For $1\leq l\leq n-1$, we introduce the trace-free Beurling–Ahlfors transform $S=\frac{n-l}{n}P-\frac lnP^\perp$, a canonical normalization of the generalized Beurling–Ahlfors transform on $l$-forms in $\mathbb{R}^n$. Its matrix symbol decomposes into scalar multipliers that are odd under suitable orthogonal reflections, yielding an endpoint cancellation estimate from finite measures to $L^\infty$ for $|D|^{-n}S$. This cancellation allows us to complete the Hilbertian case of the Bourgain–Brezis conjecture in every dimension and for every form degree. We then develop multilinear reflection estimates and obtain new critical Triebel–Lizorkin and Besov Bourgain–Brezis estimates. In particular, for every dimension and form degree, the Sobolev Bourgain–Brezis conjecture in $\dot W^{\frac np,p}$ holds for $p=\frac{2k}{2k-1}$, $k\geq1$, and hence for exponents arbitrarily close to $1$. We also derive endpoint Hodge decompositions and Hodge–Sobolev inequalities. Finally, except in the endpoint Besov case where the critical space already embeds into $L^\infty$, we prove that the associated bounded selections cannot be linear.

Analytical solutions to Maxwell's equations are essential for understanding the causal and instantaneous behavior of electromagnetic fields, yet they are challenging to obtain in open-space settings with general initial values and source terms. This work uses an infinite-order estimation scheme that yields closed-form analytical solutions to Maxwell's equations. The scheme is expressed as general function-to-function transformations that directly map initial values and source terms to the fields. Several case studies demonstrate the method, producing exact solutions for different initial and source configurations. The results provide both theoretical insight and practical benchmarks for computational electromagnetics.

We develop a nonlinear continuum transport model for localized traveling structures in a coupled two-phase medium. The model is derived from an Einstein-type material-balance formulation combining diffusion with directed transport induced by the relative gradient of an evolving background phase. The resulting system couples diffusion, nonlinear gradient-driven transport, and irreversible background depletion. The framework provides a mathematical model for chemotactic transport and a possible modeling perspective for processes associated with the formation of oil and gas deposits. We analyze traveling-wave solutions and identify beta = chi/D = 1 as the critical threshold for globally positive localized traveling bands under prescribed asymptotic conditions as the traveling coordinate s tends to plus or minus infinity. Such bands exist for beta greater than or equal to 1, whereas for 0 < beta < 1 the positive branch reaches zero background concentration at a finite traveling coordinate. For beta greater than or equal to 1, explicit profiles are obtained and, for fixed wave speed c > 0 and right-end state V_plus > 0, are unique up to spatial translation in the traveling coordinate s. We further derive the variable-coefficient linearized perturbation operator and establish a conditional finite-time a priori estimate for the mobile-phase perturbation under bounded background forcing using a maximum-principle argument.

Let a smooth, unforced, three-dimensional Navier–Stokes flow on the flat torus approach a finite terminal time. Its kinetic-energy densities converge, along the full time variable, to a unique endpoint measure. We prove that each point atom forces a same-parent, full-tail dynamical rigidity. A preassigned level-crossing catalogue and nested local Hodge projections produce orthogonal packets. From the entire packet tail, we extract a single backward adjoint whose terminal energy concentrates at the atom. Cauchy saturation then locks this adjoint to every sufficiently late packet and yields uniform two-parameter saturation of both the constrained Oseen propagator and its adjoint, together with vanishing first-order dissipation. Consequently, every sufficiently late fixed-root descendant has infinite delayed second-order action and non-integrable positive enstrophy production. Equivalently, a delayed second-order operator budget determined solely by the Navier–Stokes parent must fail arbitrarily close to the endpoint.

We obtain the first derivation of the 2D Vlasov-Poisson equation for classical particles with Coulomb interactions without any microscopic cutoff. The proof relies on our recent dual hierarchical approach to mean-field limits, together with a refined analysis of dual BBGKY hierarchies on linearized correlations, based on kinetic regularization effects. The result holds more generally for arbitrary singular interaction forces $K\in L^{2-\eta}_{loc}$ with $\eta>0$ small enough, in any dimension, and it extends to Brownian particles. It holds globally in time as long as the mean-field solution is regular enough.

We study reduced-order modeling for inverse problems in layered media, focusing on the recovery of impedance profiles from time-domain measurements. Using the Goupillaud structure, we formulate the forward problem as a discrete dynamical system and introduce a ROM-based objective defined at the level of reduced operators. Through numerical experiments on a 5-layer medium under various structured perturbations, we compare the ROM-based objective with a classical data misfit. The results show that the ROM-based inversion provides a more favorable reconstruction in the clean setting and in certain structured perturbation regimes, while remaining consistently competitive with the classical approach across all cases considered.

We study the time-dependent Maxwell system, formulated in the sense of distributions, for electromagnetic waves propagating through media with temporal and spatial material interfaces. Under explicit trace and regularity assumptions on the permittivity and permeability, we derive the jump conditions produced by temporal discontinuities and by subsequent spatial interfaces. These conditions are used to obtain generalized Snell laws for the wave vectors generated by temporal splitting, reflection, and transmission. We also derive the associated amplitude equations and organize them as a finite-dimensional linear system for a space-time slab geometry. Finally, we provide an explicit oblique-incidence example. Our approach does not impose a smooth-field ansatz and allows material parameters that need not be constant away from the interfaces.

In this paper, we study the Brunn-Minkowski inequality for $q$-th dual quermassintegrals with $q>n$. This problem was recently posed by Sadovsky and Zhang. First, by a second variation argument and a dimension reduction construction, we show that the inequality fails for arbitrary convex bodies when $q>n$, and fails even in the origin-symmetric class when $q>n+2$. Secondly, we prove the endpoint case $q=n+2$ for origin-symmetric convex bodies via Hadwiger's inequality for the polar moment of inertia. Finally, for unconditional convex bodies, we establish the inequality in the full range $0<q\le n+1$ by using a singular weighted Reilly formula and a coordinate-slice Hardy inequality. As applications, we derive several uniqueness results for the corresponding dual curvature measures.

We construct approximate traveling wave solutions for the $3d$ focusing cubic NLS perturbed by a rapidly decaying external potential. As an application, we demonstrate that there exists a forward-global, $H^1$ bounded, nonscattering solution at any mass-energy higher than the ground state threshold for the unperturbed equation.

We study the fully parabolic Keller–Segel system \[ u_t=\Delta u-\chi\nabla\!\cdot(u\nabla v)+\lambda u-\mu u^2, \qquad \tau v_t=\Delta v-v+u \] in a bounded smooth convex domain. For every fixed $\tau>0$, we prove that \[ \mu>\frac{N\chi}{4} \] guarantees global existence and uniform-in-time boundedness. This coefficient-explicit sufficient condition is independent of $\tau$ and involves no embedding or maximal-regularity constants. To the best of our knowledge, it is the first coefficient-explicit boundedness criterion that remains unchanged for all $\tau>0$ in arbitrary space dimension. The proof is built on a new auxiliary comparison function \[ Y_\tau =u+\frac{\chi\tau}{2}|\nabla v|^2-(\tau-1)\Delta v, \] which satisfies a closed scalar parabolic inequality for every $\tau>0$. When $\tau\ge1$, this inequality yields a direct pointwise comparison and an explicit bound for $u$. When $0<\tau<1$, it instead provides a uniform upper bound for $v$. Applying a parabolic squeezing argument to the transform $z=e^{-\chi v/2}$ then yields a uniform Hölder bound for $v$. Hölder–Sobolev interpolation and weighted maximal $L^p$-regularity subsequently give an $L^p$-bound for $u$ with sufficiently large $p$, and standard parabolic smoothing closes the argument.

We consider the magnetic Neumann Laplacian on rectangles, subject to non-homogeneous fields. We prove that the square is a local minimiser of the lowest eigenvalue among all rectangles of a fixed area for weak magnetic fields . We conjecture that it is a global minimiser both under the area or perimeter constraints and partially prove the conjecture by establishing lower and upper bounds to the principal eigenvalue.

We consider polyconvex functionals defined on vector valued functions, the model case being the graph area functional, and we analyze the relaxed energy with respect to the strict convergence in $BV$. In several cases where the relaxed area is known, by exploiting a continuity result by Reshetnyak we are able to find an explicit formula for wide classes of integrands with linear growth in the minors of the gradient. We preliminarily extend to the $BV$ setting a continuity property observed by Acerbi–Dal Maso in the Sobolev case. Finally, partial results concerning the relaxation of the vortex map with respect to the $L^1$ convergence are obtained.

We consider semilinear wave equations with time-dependent damping and mass and a derivative-type nonlinearity. By applying a Liouville transformation and solving a Volterra integral equation posed from infinity, we construct a positive exact solution to the adjoint equation without using an explicit representation of the linear propagator. This solution is used to derive a blow-up criterion for energy solutions with finite propagation, together with an upper bound for the lifespan. If the primitive of the damping coefficient grows at most logarithmically, we obtain the full shifted Glassey range, including the critical exponent, and the corresponding polynomial and exponential lifespan estimates. The result applies independently of the sign of the scale-invariant discriminant and therefore includes the mass-dominant regime. It also covers non-integrable oscillatory perturbations of the damping coefficient when their contributions are canceled by the corresponding mass terms.

The steady motion of a viscous incompressible fluid in a distorted pipe, containing several small particles of diameter $\eps^3$ and mutual distance $\eps$, is modeled through the Navier-Stokes equations with mixed boundary conditions. Apart from inhomogeneous Dirichlet boundary conditions on the particles, these involve the Bernoulli pressure and the tangential velocity on the inlet and outlet of the tube, while either the transversal flux rate or the pressure drop is prescribed along the pipe. Applying the energy method in homogenization theory, we study the asymptotic behavior of the solutions to these systems as $\eps \to 0$, without any restriction on the magnitude of the data, and show that the effective equations display an additional Brinkman term. An important feature of the present work concerns the required uniform bounds, which are achieved (in the case of the prescribed flux problem) by a contradiction argument based on Bernoulli's law for solutions of the stationary Euler equations.

We construct new families of uniformly rotating vortex-patch solutions of the two-dimensional incompressible Euler equations consisting of a simply connected outer patch and multiple interior interfaces, which can be interpreted geometrically as holes. More precisely, each solution consists of a single outer vortex patch enclosing $\mathbf m\geq2$ identical, highly concentrated inner components arranged at the vertices of a regular $\mathbf m$-gon; the entire configuration rotates rigidly in the clockwise direction. As the concentration parameter tends to zero, the inner components shrink and collapse simultaneously toward the origin, while the outer boundary converges to the unit circle. The corresponding vorticities converge, in the sense of measures, to a Rankine vortex supplemented by a point vortex of circulation $-\mathbf m$ at its center. The proof is based on a contour-dynamics formulation, a symmetry reduction to two nonlinear boundary equations, and a suitable singular rescaling. We then apply an implicit function theorem with a continuous parameter in symmetry-adapted Hölder spaces. To the best of our knowledge, this is the first analytical construction of a desingularization regime in which several concentrated inner components are contained in a common outer patch and collapse simultaneously toward its center.

This work introduces a robust deterministic framework for approximating solutions of the Boltzmann equation with binary collisions by discretizing their dependence on time, position, and velocity using Galerkin methods. By employing a family of parametric Galerkin closures based on $\varphi$-divergences in velocity space, we derive rigorous hierarchies of moment equations that govern fluid dynamic variables. Addressing the limitation that these closures alone do not guarantee dissipation of a $\varphi$-divergence entropy for the true binary collision operator, we restore this property by formulating a compatible approximate collision operator tailored to each closure. This constructed operator intrinsically retains fundamental physical properties essential for high-fidelity flow simulations, including Galilean invariance, exact conservation of mass, momentum, and energy, and strict dissipation of a $\varphi$-divergence entropy. Furthermore, we show that the resulting closed moment systems are symmetric-dissipative, yielding Cauchy problems that are well-posed locally in time. To translate this mathematical foundation into an efficient computational tool, we discretize the position and time variables with an entropy-stable discontinuous Galerkin (DG) finite element method. The fully implicit, entropy-stable space-time approach enables time steps far beyond typical CFL-limited step sizes and the direct computation of steady states. The robustness and accuracy of the methodology are verified and validated through numerical simulations on the supersonic nozzle flow of argon, mass flow through a channel, and heat transfer between parallel walls, demonstrating agreement with analytical benchmarks, experimental measurements, and stochastic particle simulations.

We construct, for any exponent $p\in(1,\infty)$, $p$-homogeneous rank-one convex integrands $F\colon \mathbb{R}^{2\times m}\to \mathbb{R}$ that are nowhere quasiconvex when $m$ is large. When $p$ is sufficiently close to $4$, such examples can be constructed on $\mathbb{R}^{2\times 3}$: for $p=4$, our example is an explicit quartic polynomial. Related constructions give, for every $p$, conjugation- and transposition-invariant examples on $\mathbb{R}^{d\times d}$, and examples on $\mathbb{R}^{4\times 2}$ for every $p\neq 2$.

We present several observations on the antimaximum principle (AMP) for the model problem $-\Delta_p u = \lambda |u|^{p-2} u + f$ in a bounded smooth domain $\Omega$, subject to the zero Dirichlet boundary conditions, and where the source function $f$ is nontrivial, nonnegative, and sufficiently regular. Denote by $\lambda_f$ the endpoint of validity of the AMP, so that every solution of the problem is negative in $\Omega$ for any $\lambda \in (\lambda_1,\lambda_f)$. Our discussion covers the following aspects: identification of a class of sources over which the AMP is uniform, lower semicontinuity of the map $f \mapsto \lambda_f$, bounds on $\lambda_f$, the nonexistence of negative solutions for sufficiently large $\lambda$ (extended AMP), the anticomparison principle, and the weakening of the source regularity from the Lebesgue to Morrey spaces. Some of the results are stated only in the linear case $p=2$. As a part of the discussion, we provide a few related open problems.

We prove positive-time regularity for harmonic map heat flows from finite-dimensional $\mathrm{RCD}(K,N)$ spaces into complete $\mathrm{CAT}(0)$ spaces, without assuming that either the source or the target is smooth. For bounded-image initial data, the $\mathrm{EVI}$ gradient flow of the Dirichlet energy admits a representative that is locally Lipschitz jointly in space and time. Moreover, its spatial pointwise Lipschitz constant satisfies an Eells–Sampson-type parabolic Bochner inequality. The main difficulty is that the smooth parabolic perturbations used to select contact points are unavailable on an $\mathrm{RCD}$ source. We overcome it by a sliced contact-selection principle that converts the elliptic ABP estimate on $\mathrm{RCD}$ spaces into the space–time contact selection required by the Hamilton–Jacobi argument.

We extend several characterizations of the $L^p$-solvability of the homogeneous Dirichlet problem to (possibly degenerate) elliptic operators $L=-\textrm{div}(wA\nabla)$ defined on a large class of open sets $\Omega$ in $\mathbb{R}^n$. This framework encompasses not only uniformly elliptic operators in Lipschitz domains, but also Caffarelli-Sylvestre-type operators, and boundaries $\partial\Omega$ that are not $(n-1)$-dimensional, for instance. We prove that, for $p\in (1,\infty)$, the $L^p$-solvability of the homogeneous Dirichlet problem is equivalent to the solvability of the Poisson-Dirichlet problem $Lu=wf-\textrm{div}(wF)$, i.e. to the existence of a solution $u$ satisfying a $L^p$ non-tangential estimate whenever $f$ and $F$ belong to suitable weighted $L^p$ tent spaces. Furthermore, it is also equivalent to the solvability of the Poisson-Regularity problem for data in appropriate weighted $L^{p'}$ tent spaces, where estimates are obtained for $\nabla u$ rather than $u$.

We prove stable self-similar blowup for the harmonic map heat flow in dimensions 3 to 6 for target manifolds which are slightly perturbed versions of the round sphere. Starting from the known blowup solution for the sphere, we construct self-similar blowup solutions for this class of target manifolds and prove that these solutions are asymptotically nonlinear stable. Although the blowup solution for the sphere is not explicitly known in this case, we can still use perturbative methods similar to those developed by Donninger, Schörkhuber, and the author in \cite{DonSchWit26} who consider a related problem for the wave maps equation.

Let $\Omega\subset\mathbb R^4$ be a bounded domain of class $C^6$, let $0<\alpha<4$, and let $K\in C^4(\overline\Omega)$ be positive. We study the Navier problem \[ \Delta^2u=\varepsilon^{8-\alpha}K(x)e^{u(x)} \left(\int_\Omega\frac{K(y)e^{u(y)}}{|x-y|^\alpha}\,dy\right), \qquad u=\Delta u=0\quad\text{on }\partial\Omega. \] Let $G$ be the Navier Green function, let $H$ be its regular part, and put $M_\alpha=8\pi^2(8-\alpha)$. The concentration points are governed by \[ \mathcal F_m(\boldsymbol\xi) =\sum_{i=1}^m \left[\log K(\xi_i)+\frac{M_\alpha}{2}H(\xi_i,\xi_i)\right] +M_\alpha\sum_{i<j}G(\xi_i,\xi_j). \] Every $C^1$-stable critical point of $\mathcal F_m$ produces a positive $m$-bubble solution whose scales are of order $\varepsilon^{-1}$ and whose nonlinear source converges to $M_\alpha\sum_i\delta_{\xi_i^*}$. If the critical point is nondegenerate, the corresponding $m$-bubble solution is locally unique, modulo permutations, in a fixed scaled modulation neighborhood. The linearized operator is nondegenerate on $H^2(\Omega)\cap H_0^1(\Omega)$, and \[ \operatorname{ind}(u_\varepsilon) =m+\operatorname{ind}\!\left(-D^2\mathcal F_m(\boldsymbol\xi^*)\right). \] A critical four-dimensional capacity controls the scale directions. The dilation block of the reduced Hessian is positive and equals $8\pi^2b_\alpha^2|\log\varepsilon|^{-1}I_m+o(|\log\varepsilon|^{-1})$, where $b_\alpha=(8-\alpha)/2$.

A Heisenberg Bieberbach manifold is a compact quotient of the Heisenberg group by a discrete torsion-free subgroup of its pseudo-Hermitian automorphism group. In this paper, we study the joint spectrum of the sub-Laplacian and the Reeb vector field on three-dimensional Heisenberg Bieberbach manifolds. In particular, we explicitly compute the multiplicities of the joint spectrum using theta functions and the character theory of finite groups.

We study long-time Strichartz estimates for the one-dimensional Schrödinger equation with quasi-periodic initial data. For two-frequency data with an algebraic frequency ratio, we observe that the behavior of the linear Schrödinger evolution changes depending on the algebraic degree of the ratio. Making use of this observation, we improve the Strichartz estimates on long time intervals. We also prove an endpoint $L^4$ Strichartz estimate. Our proofs use Roth-type Diophantine inequalities and Vinogradov-type mean value estimates for the Parsell–Vinogradov systems.

In this note, motivated by \cite{DW26}, we construct counterexamples to momentum-only Schauder estimates for linear kinetic equations with relativistic transport. The construction exploits a structural compatibility between the nonlinear momentum-to-velocity map and a single fixed pair of smooth diffusion and drift coefficients: after conjugation, the diffusion becomes the Laplacian and all induced first-order terms cancel. Thus the obstruction is realized for one fixed operator, uniformly elliptic on every bounded momentum set, rather than through a frequency-dependent family of coefficients. We further show that the failure persists with zero external forcing by constructing uniformly positive stationary solutions for which the hidden spatial oscillation is encoded in a bounded zeroth-order coefficient whose momentum Hölder seminorm tends to zero. Consequently, the full Lorentzian Hölder control used in \cite{HSTT} cannot in general be replaced by momentum-only Hölder control.

In this work, we establish new conditional Lipschitz stability results for electrical impedance tomography (EIT) with anisotropies, of recovering the conductivity in a conformal class of a known anisotropic conductivity in both two- and multi-dimensional cases. Then we employ the stability theory to understand the property of the deep Calderón method, one deep learning-based technique for image reconstruction in EIT that has shown promising empirical results, but still lacks theoretical underpinnings. Specifically, we relate the stability theory to the robustness of the method with the proper choice of the training data, and present numerical results in two-dimension to complement the theoretical analysis.

In this paper, we introduce a hybrid metric geometry on the space of absolutely continuous probability densities that combines optimal transport (Wasserstein geometry) and log-ratio composition (Aitchison geometry). The hybrid distance $D_\alpha$ is defined through a Benamou–Brenier-type dynamical formulation that couples spatial transport with a centered reaction term preserving total mass.We prove that $D_\alpha$ is a genuine metric and establish comparison estimates with the Wasserstein and Aitchison distances. In particular, we show that the topology induced by $D_\alpha$ is stronger than the narrow topology and weaker than the supremum topology generated by the Wasserstein and Aitchison metrics. We further prove that the metric space is geodesic. Within this framework, we develop the foundations of a gradient flow theory in the sense of Ambrosio–Gigli–Savaré, including the characterization of absolutely continuous curves, metric derivatives, metric slopes, and formal Jordan–Kinderlehrer–Otto schemes. We also investigate hybrid barycenters and their connections with Wasserstein barycenters and Aitchison barycenters. Finally, we discuss several partial differential equations, including logistic diffusion, Allen–Cahn equations with log-ratio constraints, and chemotaxis models with logarithmic growth, as formal gradient flows associated with the hybrid geometry, and compare the proposed framework with the Wasserstein–Fisher–Rao metric.

We study the inverse problem of recovering the constitutive tensors of a homogeneous anisotropic electromagnetic medium without magnetoelectric coupling (non-chiral) from its Fresnel surface, the characteristic variety of Maxwell's equations governing electromagnetic wave propagation. For known isotropic permeability, normalized to $\mu = I$, we prove that the Fresnel surface uniquely determines the permittivity tensor $\varepsilon$, and that the associated Fresnel polynomial is reducible precisely when $\varepsilon$ has a repeated eigenvalue. For general, positive-definite symmetric tensors $(\varepsilon,\mu)$, we prove that the Fresnel polynomial is generically irreducible over $\mathbb{C}$ and we identify the natural gauge symmetry under which it is invariant. Using geometric invariant theory, a powerful tool of modern algebraic geometry, we construct an affine quotient of the parameter space by this gauge action and prove that the induced Fresnel-polynomial map is birational onto its image. We deduce that, outside a proper real algebraic exceptional set, the real Fresnel surface determines $(\varepsilon,\mu)$ up to gauge. This establishes generic uniqueness for the inverse problem and, to our knowledge, provides a new application of affine geometric invariant theory to gauge freedom in a PDE inverse problem.

Let a bounded and smooth domain in the n dimensional Euclidean space be given and consider a boundary value problem of the half order Dirac operator and its power, with f be the value of the half derivative of the solution in the domain and, g be the trace value of the solution over the boundary. We will show that the solution will be a generalized function u in the half order Hilbert space which is the orthogonal sum of the solution that emerged as a domain integral from the domain value of the fractional half Dirac operator and the one that is the boundary integrals that emerges from the boundary or trace value of the solution. We use two different symbols to designate orthogonal sums for subspaces and functions in orthogonal subspaces of the Sobolev space.