Papers

Influenza A persists by continually changing the antigens that its host population recognizes. How fast it evolves and how many hosts it infects are not independent, because the immunity that selects new variants is produced by the infections the pathogen has already caused. We analyze a model of Andreasen et.\ al.\ that captures this eco-evolutionary feedback, and we extend it to the case where infection may kill the host. In the model infected and recovered hosts are distributions over a one-dimensional antigenic space coupled by cross-immunity. The endemic state is a traveling pulse: the infected distribution keeps a fixed shape and advances through antigenic space at constant speed. In the limit of weak antigenic mutation we construct the pulse asymptotically and find that: (i) the speed of antigenic evolution decreases as prevalence rises, (ii) less standing antigenic variation accompanies either faster or slower antigenic evolution, depending on whether pathogens differ in immune evasion or host lifespan (iii) the total number of infected hosts is largest at an intermediate virulence when the host birth rate is sufficiently low.

Consider the one-dimensional Schrödinger evolution $$ i\partial_tw+\partial_x^2w=Vw,\hspace{5mm}(t,x)\in [0,T]\times\mathbb{R},$$ where $V$ is a real-valued time-dependent potential belonging to $L^1_tL^\infty_x+L^2_{t,x}$. We prove that an $L^2$ solution to the above equation is determined by its mass density $|w|^2$ modulo a global unimodular phase factor. This answers a question recently posed by Jaming for a large class of potentials, and recovers his earlier result for the free Schrödinger evolution. Our proof is simple and robust. It combines the basic local smoothing properties of the above Schrödinger evolutions, Strichartz estimates, and a stripwise version of a unique continuation theorem of Ionescu and Kenig to show that the interaction function $$ F(t,x,y)=u(t,x)v(t,y)-u(t,y)v(t,x) $$ vanishes identically on $[0,T]\times\mathbb{R}^2$ whenever $u$ and $v$ are $L^2$ solutions to a Schröinder euqation with potential $V$ with the same mass density. The method immediately extends to several important nonlinear models such as the cubic NLS equation, which is globally well-posed in $L^2(\mathbb{R})$ with $V=|u|^2$. Interestingly, we also show that the above phase recovery property fails in dimensions two and higher, even for the free equation with Schwartz initial data.

In this study, we investigate continuous data assimilation for a class of partly diffusive FitzHugh–Nagumo neural networks with memristive feedback and linear synaptic coupling. The nudging algorithm implemented here only adds a feedback term to the membrane-potential equations, while the recovery and memductance variables remain unobserved. We obtain weighted energy estimates that give sufficient conditions for exponential convergence of the data-assimilated solution to the reference solution under full or coarse membrane potential observations. Moreover, we prove energy-class uniqueness in dimensions one and two. In dimension three, the uniqueness criterion is an \(L^4\) membrane-potential initial data. We also establish an observational-noise estimate for locally square-integrable forcing, yielding an asymptotic error bound when the noise energy is eventually bounded.

We study the hydrodynamic limit of kinetic flocking models with nonlinear velocity alignment and strong local Fokker–Planck forcing. In the limit of small Knudsen number the kinetic densities converge to a local Maxwellian centered around macroscopic quantities satisfying the compressible Euler system with isothermal pressure $p = \s \rho$, while macroscopic alignment is obtained by convolution of the microscopic velocity law with the standard Gaussian, resulting in a different nonlinearity. Such discrepancy has been observed already in \cite{black2025hydrodynamic}. In this note we rigorously justify the limit for admissible weak kinetic solutions and non-vacuous limiting macroscopic solution. Our analysis provides a quantitative rate of convergence in terms of relative entropy: $\cH(f_\e | \mu) \lesssim \e\bigl(1+|\log\e|^{p-2}\bigr)$ provided initially $\cH(f_\e(0) | \mu(0)) \leq \e$, where $p$ is the order of non-linearity in the alignment force. In the linear case $p=2$ we recover the known result \cite{karper2015hydrodynamic}.

We study critical Strichartz estimates for the resonant hyperbolic Schrödinger flow on the waveguide $\mathbb{R} \times \mathbb{T}^2$. For initial data localized to frequencies at most $N$, we prove that the unit-time $L^2\to L^{10/3}$ operator norm has sharp order $N^{1/5}$. We further track its dependence on the observation length, obtaining sharp estimates in the short- and long-time regimes and characterizing the frequency-dependent intervals on which the critical estimate remains lossless. The precise dependence on $T$ in the intermediate regime $N^{-1}<T<1$ remains open. The upper bounds are obtained by lifting the global endpoint estimate on the elliptic cylinder $\mathbb{R} \times \mathbb{T}$ to Hilbert-valued data, preserving its time summability, and combining this with periodic Bernstein and interpolation. The lower bounds exploit an exact integer null direction in the periodic frequency lattice together with a suitably scaled Euclidean wave packet. Finally, using a Bloch–Floquet transference argument, we recover the known elliptic unit-time estimate from compact torus theory and obtain a lossless elliptic estimate on frequency-dependent short time intervals.

We introduce a novel phase-field model approximating the Verigin problem with phase transition. For fixed interface thickness $\varepsilon>0$, we construct weak solutions through a time-discrete variational scheme combining a Wasserstein minimizing-movement for the density with Schätzle's selection procedure for the phase field, which yields strong compactness and an almost-minimizing property. We then investigate the sharp-interface limit $\varepsilon \downarrow0$. For well-prepared initial data, we establish strong compactness of the densities and phase fields, and we prove convergence of the time-integrated diffuse interfacial energy to the perimeter. The limiting evolution is identified as a weak solution of the Verigin problem with phase transition, satisfying an optimal energy dissipation inequality.

We consider the rigorous derivation of membrane theories from three-dimensional nonlinear elasticity. A salient feature of our framework is the inclusion of an orientation-preservation constraint together with energy growth that penalizes vanishing local volume. We embed the 3D variational problem into a space of parametrized measures, and obtain a membrane $\Gamma$-limit defined on a class of nondegenerate gradient Young measures. This formulation offers a twofold advantage: first, it uniquely identifies a non-relaxed membrane energy density capable of capturing the fine oscillations (microstructure) of minimizing sequences; second, the resulting membrane density preserves the unbounded energy growth near degenerate configurations. We further show that the classical relaxed membrane energy of Le Dret and Raoult is recovered as the minimum of our functional over all measures with a prescribed barycenter. The main difficulty lies in generating such measures by maps whose gradients have rank two almost everywhere, which we address through piecewise isometric (origami-like) constructions.

We study the three-dimensional (3D) incompressible Euler equation linearized about a smooth compactly supported stationary flow of Gavrilov type. Baldi's volume-preserving action–angle chart allows a reduction of the high-frequency amplitude equation on invariant tori. Using Baldi's chart, we identify a critical torus ensured by the compact cutoff. We show that, for a fixed axisymmetric covector with sufficiently large normal component, the reduced trace-free periodic cocycle on this torus is hyperbolic and has a positive Floquet exponent. A localized geometric-optics construction then lifts this Floquet instability to the linearized Euler equation, yielding exponential essential-norm growth in space $H^s$, $s\geq 0$.

We show that any convex co-compact hyperbolic surface with exponent of convergence of Poincaré series $\delta \in (\frac25,\frac{14}{27})$ has an essential spectral gap of size $\beta=\tfrac78(\tfrac12-\delta)+\tfrac{1}{32}\delta-\epsilon$ for any $\epsilon>0$. In particular, for $\delta=\frac12$ this becomes $\beta=\tfrac{1}{64}-\epsilon$. We show existence of the gap by proving a new Fractal Uncertainty Principle, using a two-ends Furstenberg theorem of O'Regan-Wu-Yi [arXiv:2607.08461].

We establish the acoustic and Stokes-Fourier hydrodynamic limits for the Boltzmann-Fermi-Dirac (BFD) equation on the torus of dimension 2 or more. For each limit, we consider fluctuations of appropriately scaled solutions of the BFD equation around a constant background Fermi-Dirac distribution. By adapting the weak solution framework developed by Bardos, Golse and Levermore for the classical Boltzmann equation to handle the cubic nonlinearity of the BFD collision operator, we show that these fluctuations converge weakly in time to a unique limit governed by the respective hydrodynamic equation.

We prove global well-posedness and scattering for the three-dimensional focusing energy-critical nonlinear Schrödinger equation with initial data below the ground-state energy and gradient thresholds. This extends the seminal result of Kenig and Merle (Invent. Math., 2006) to general data in $\dot H^1(\mathbb R^3)$. Classical concentration compactness and the exclusion of rapid frequency cascades reduce the problem to the quasi-soliton scenario. The main ingredient is a frequency-localized interaction Morawetz estimate based on an explicit convex weight whose time-dependent radius is determined implicitly by the mass density of the high-frequency component. The ground-state gap controls the focusing contribution with constants independent of the mass of this component. The radius is chosen to retain quartic spacetime control, while the continuity equation expresses the moving-radius contribution as a nonnegative square, up to frequency-truncation errors. We absorb these errors by closing the Morawetz estimate together with an auxiliary spacetime bound for the nonlinearity. The resulting quartic bound excludes the quasi-soliton scenario and completes the proof.

The Anderson tight-binding model is a fundamental model of quantum transport and localization in disordered media. Completing a problem left open by Bourgain and Kenig, this paper proves Anderson localization near the bottom of the spectrum for the lattice Anderson model with Bernoulli potential, in any dimension $d\ge 2$. The proof uses the multiscale framework of Fröhlich-Spencer and Bourgain-Kenig, and the main new ingredient is a probabilistic discrete unique continuation principle (PDUC) for the discrete Schrödinger equation. This PDUC is established via a bootstrap argument and a key probabilistic lemma proved by adaptively revealing the random potential.

This paper studies the three-dimensional Muskat problem for two immiscible fluids in a porous medium subject to gravity and surface tension. We consider the setting in which one fluid forms a bounded bubble surrounded by the other. For any value of the physical parameters, we prove the global-in-time nonlinear asymptotic stability of a translating spherical bubble for small initial perturbations in the critical Lipschitz class. This enlarges the class of critical spaces considered in previous two-dimensional bubble results. The proof combines spectral analysis of the linearized operator on $\mathbb{S}^2$ with multilinear singular integral estimates that exploit a crucial null structure in the nonlinear terms.

We prove that every bounded stable solution of the Allen–Cahn equation in $\mathbb R^3$ is one-dimensional. As a consequence, this shows the validity of De Giorgi's conjecture for monotone solutions in dimension four.

We address the inverse problem of reconstructing the initial temperature distribution in a one-dimensional non-homogeneous heat equation with Dirichlet boundary conditions from a finite number of pointwise-in-time measurements at a single spatial location. We propose an explicit approximation scheme that incorporates the source term and utilizes a refined sequence of measurement times. Unlike previous approaches that often rely on exponentially growing observation times, our chosen time sequence allows for efficient recovery within a fixed time horizon. We establish algebraic convergence rates under suitable regularity assumptions on the initial data and the forcing, and we validate the theoretical sharpness with numerical experiments for varying numbers of measurements.

This paper investigates the Schrödinger-Poisson system involving electromagnetic fields. Our approach is mainly based on the variational method. By developing the global compactness result, we establish the existence of higher-energy bound state and ground state solutions under suitable assumptions. In addition, under symmetry conditions, the existence and multiplicity of intertwining solutions are obtained for both finite and infinite symmetry group.

We establish global $W^{2,p}$ solvability for the Poisson equation with mixed Dirichlet–Neumann boundary conditions in two classes of domains in all dimensions $n\geq 2$. For $C^{1,\alpha}$ domains with a Reifenbeg flat interface, we obtain the optimal range $1<p<4/3$, provided that $\alpha>1-1/p$. For Lipschitz polyhedra with facewise boundary decompositions, we obtain solvability for $p$ close to $1$. In both settings, we establish endpoint $W^{2,1}$ solvability for data in an adapted atomic Hardy space. Applications of the methods to Lamé systems are also discussed.

We analyze solutions to a one-dimensional periodic linear model for peridynamic waves, focusing on the setting where the natural energy space allows for spatial discontinuities. Exploiting the high-frequency asymptotics of the peridynamic dispersion relation, we construct an explicit solution whose initial displacement has a single jump discontinuity but becomes continuous at every positive time. By time reversibility, this yields a continuous initial displacement whose evolution develops a jump at any prescribed time and is continuous at every other time. We further quantify the formation of this singularity through a sharp Hölder-norm blow-up estimate obtained via a Littlewood-Paley characterization of Hölder spaces. Finally, by superposing suitably weighted solutions, we construct an evolution exhibiting jump discontinuities on a dense set of times.

In this paper, we consider a three-dimensional non-homogeneous incompressible fluid in a bounded periodically perforated domain at the critical Stokes-capacity scale. The density obeys a transport equation. The viscosity depends on a quasi-static temperature perturbation, which solves a uniformly elliptic transmission problem with insulated outer boundary conditions and zero spatial mean. The limiting momentum equation contains a Brinkman resistance term generated by the critical perforation regime. For each fixed perforated domain, we construct global finite-energy weak solutions and derive a quantitative relative-energy stability estimate with respect to regular solutions of the homogenized system. As long as a regular effective solution exists, the squared relative error between any microscopic weak solution and the effective solution is bounded by the initial mismatch plus $O(\eps^{2})$. For well-prepared data, the density and uncorrected velocity converge at order $O(\eps)$ in $L^\infty(0,T;L^2)$, the temperature at the same order in both $L^\infty((0,T)\times\Omega)$ and $L^2(0,T;H^1)$, and the corrected velocity in $L^2(0,T;H^1)$. The key estimates are an $O(\eps)$ cell-capacity residual in the dual energy norm, an $O(\eps)$ corrector-gradient bound in $L^{6/5}$, and fixed-domain thermal stability. The critical boundary layers retain order-one viscous energy, represented in the limit by the Brinkman dissipation, so the uncorrected velocity gradient does not converge strongly to the effective gradient whenever the effective velocity is nonzero. Finally, we prove local existence of regular effective solutions for smooth data satisfying a finite set of boundary compatibility conditions, which closes the stability argument.

Let $1<m<n$, $q>0$, and $p\in\mathbb R$. We study positive $C^1_{\mathrm{loc}}$ weak solutions of \[-\Delta_m u=u^p|Du|^q\qquad\text{in }\mathbb R^n.\] If $0<q<m-1$, we prove that every such solution is constant provided \[p+q-m+1<\frac{(m-1)(m-q)^2}{(n-m)(m-1-q)}.\] If $q\geqslant m-1$, the same conclusion holds for every $p\in\mathbb R$. For $0<q<m$, the proof uses an auxiliary function and reduces the main differential estimate to a lower bound for a function of one variable. An explicit construction and a local maximum-principle argument then prove constancy without differentiating across $\{|Du|=0\}$. When $m=2$, our result answers the natural Liouville problem left open by Bidaut-Véron, García-Huidobro and Véron for the whole range $0<q<1$. The case $q=m$ is handled by an increasing change of the dependent variable, in the spirit of the related work of Bidaut-Véron, while the case $q>m$ follows from the theorem of Lu and Zhu.

In this paper, we reconstruct a wedge and its post-shock flow from a prescribed leading shock and a uniform supersonic incoming state. Full Euler Rankine–Hugoniot relations supply the shock data, while a reduced interior model combines homogeneous acoustic characteristic equations, entropy transport, and the Bernoulli relation. A hodograph transformation yields a linear hyperbolic equation for the physical coordinate. Comparing a shock-to-wedge mass bound with a hodograph non-degeneracy threshold gives a sufficient, data-dependent reconstruction criterion. Under the stated assumptions, this criterion ensures a unique \(C^1\) post-shock flow and a \(C^2\) wedge throughout the characteristic domain determined by the prescribed shock segment. Numerical reconstructions examine empirical grid convergence, Mach-number dependence, and discrepancies between the reduced-model and full-Euler inverse reconstructions.

We study the behavior of isolated singularities of stationary harmonic maps with generic boundary data. For round sphere targets, we prove that, for $4\leqslant n\leqslant7$, every stable stationary harmonic map from a bounded smooth $n$-dimensional domain to round $(n-1)$-sphere with generic smooth boundary data has only radial projections composed with orthogonal transformations as tangent maps at its singularities; for 7-dimensional domains and round $k$-sphere targets with $k\geqslant7$, every stable stationary harmonic map with generic smooth boundary data is smooth. These results follow from a general minimum-index principle for closed real-analytic target manifolds. Under suitable target hypotheses ensuring strong compactness and excluding lower-dimensional singularity models, we show that, in the first domain dimension in which singularities can occur, generic boundary data force the link of every singular tangent map to attain the smallest possible Morse index. This principle applies both to stable stationary harmonic maps and, under the corresponding stronger target hypotheses, to all stationary harmonic maps. If no non-constant tangent map in the relevant class has a link attaining this minimum, generic smoothness follows. As an application, we establish a rigidity at infinity theorem. For $3\leqslant n\leqslant7$, we show that any energy-minimizing map from $n$-Euclidean space to round $(n-1)$-sphere admitting the radial projection as a blow-down limit must itself be a translate of the radial projection.

We study mean field equations with multiple positive singularities on flat tori. We establish a uniform uniqueness criterion in terms of the total singular mass and the scale-invariant spectral quantity $\lambda_1(\mathbb T)|\mathbb T|$, where $\lambda_1(\mathbb T)$ denotes the first positive eigenvalue of the Laplacian. Combining nodal-set analysis for Jacobi fields with a cylindrical Alexandrov–Bol type inequality and sharp small-capacity asymptotics, we obtain the optimal leading coefficient $8/\pi^2$ in the total-mass uniqueness bound as $\lambda_1(\mathbb T)|\mathbb T|\to0$. Sharpness is demonstrated by examples with two symmetric singularities on degenerating rectangular tori that admit at least three distinct solutions. For the case of two singularities with total mass $2\rho$, we further derive uniqueness and multiplicity results near $\rho=4\pi$ from the critical-point structure of the associated symmetrized Green function. In particular, when its only critical points are the four nondegenerate two-torsion points, the equation admits a unique solution for $\rho$ just below $4\pi$ and exactly three solutions for $\rho$ just above $4\pi$.

We establish a stability threshold for the 3D Boussinesq equations around Couette flow $(y,0,0)$ with linearly stratified temperature profile $1+y$ on $\mathbb{T}\times\mathbb{R}^2$, with distinct viscosity $\kappa$ and thermal diffusivity $\mu$ and arbitrary Brunt-Väisälä frequency $\beta>0$. Assume that $\mu$ and $\kappa$ are comparable, and set $\nu=\min\{\kappa,\mu\}$. For sufficiently small \(a_0>0\) and \(s\geq4\), we prove that if the initial perturbation satisfies $\|(u_{in},\theta_{in})\|_{H^s}+\|(u_{0,in},\theta_{0,in})\|_{W^{s,1}}\lesssim\nu^{\frac{2}{3}} |\ln{\nu}|^{-2-2a_0}$, then the corresponding solution exists globally in time. We also establish Strichartz-type estimates for zero mode. Moreover, our estimates further allow for a nonlinear amplification of order $\nu^{-1/6}$ in the high-regularity norm of the zero mode of the first component $u^1_0$.

The classical Becker-Döring equations describe the formation of clusters by aggregation and fragmentation of monomers. If the total amount of mass is supercritical, larger and larger clusters are formed, leading to an asymptotic loss of mass in the long time limit. The two-component Becker-Döring system arises when clusters are built from two types of monomers. Here, we study the natural extension of the one-component system, where no energy or entropy is leaving or entering the system - the so called detailed balance assumption. We rigorously prove that all initial conditions admit a solution minimising the relative entropy as time approaches infinity. This shows, that the long time limit in the weak* topology is selected through the initial Type I and Type II masses. Furthermore, the proportion of lost Type I mass is determined by the limit point via the mixing ratio of Type I and Type II monomers that maximises the binding energy. The main difficulty of the two-component system is that no relative entropy is weak* continuous, which is crucial for the classical argument [2]. Instead, our proof is based on a discrete two-dimensional logarithmic Sobolev inequality, that bounds the entropy dissipation on the correct timescale. Our approach also improves current assumptions to determine the long time behaviour for the one-component system.

This paper is concerned with a no-flux initial-boundary value problem for the degenerate volume-filling chemotaxis system with source terms, \begin{align*} u_t = \nabla \cdot (D(u,v) \nabla u - h(u,v) \nabla v) + f(x, u, v), \quad v_t = \Delta v + g(u,v), \quad x\in \Omega, \ t>0 \end{align*} in a smoothly bounded domain $\Omega \subset \mathbb{R}^N$ $(N \in \mathbb{N})$. It is shown that when $D$, $h$, $f$ and $g$ satisfy suitable assumptions involving $D(1,\cdot)=h(0,\cdot)=h(1,\cdot)=f(\cdot,0,\cdot) = f(\cdot,1,\cdot) = 0$, for nonnegative initial data $u_0$ and $v_0$ with $u_0\le 1$, there exists a global weak solution $(u, v)$ with $u\le 1$. In addition, uniqueness of global weak solutions is established when $D(r,s) = D(r)$ for all $r\in[0,1]$ and $s\in[0,\infty)$, and $D$, $h$, $f$, $g$ and $v_0$ are supposed to satisfy additional conditions.

The two-component Becker-Döring model is a two-dimensional coagulation-fragmentation equation that describes the dynamics of clusters built from two different types of monomers growing and shrinking through monomer interaction. We show the existence and mass conservation of solutions and give a partial result regarding uniqueness. Furthermore, we prove under the assumption of detailed balance the entropy equation, followed by a detailed study of the equilibrium states and relative entropies. We can determine the minimum of all relative entropies under sequences of fixed Type I and Type II masses and fully describe the minimising sequences in the weak* topology.

In this paper, we prove global well-posedness and scattering below the energy space for the defocusing nonlinear Schrödinger equation with intercritical power nonlinearities in spatial dimensions $d\geq3$, improving the regularity threshold of Vişan and Zhang [Differ. Integr. Equations, 22(2009), 99-124].

We establish that the ellipticity of the minimal surface integrand is the sharp threshold for $\mathrm{W}^{1,1}$-regularity of solutions to the Dirichlet problem on $\mathrm{BV}$. Here, we particularly deal with the autonomous case of convex linear growth functionals.

We study the stability of the log–log blow-up regime for the focusing mass-critical nonlinear Schrödinger equation under small $H^s$ perturbations. Previously, stability was established for every $0<s<1$ in dimension two by Colliander and Raphaël [Math. Ann. (2009)] and subsequently extended to dimensions $d\geq3$ under the restriction $s>1/(1+\min\{1,4/d\})$ by Sun and the fourth author [J. Math. Pures Appl. (2020)]. We remove this restriction and establish stability throughout the full subcritical range $0<s<1$.