Papers

This paper is concerned with the Cauchy problem for a semilinear wave equation with arbitrary quadratic nonlinearities, posed on a cosmological supersymmetric spacetime of the product form (Open Milne) $\times K$, where $K$ is a compact, Ricci-flat $n$-dimensional Riemannian manifold. We establish the global future existence of solutions for small initial data. Our proof, based on a novel modified energy, avoids spectral decompositions of the Laplace operator on the internal manifold and does not require any special nonlinear coupling structure. This robust framework is expected to be readily extendable to tensorial wave equations, including the Einstein equations.

Let $M$ be a connected smooth $n$-manifold without boundary, where $n\geq3$, and let $\kappa\in\mathbb{R}$, with $\kappa\leq0$ if $M$ is open. We prove that every smooth Riemannian metric $g_0$ with $\mathrm{Scal}_{g_0}\geq\kappa$ is a locally uniform limit of smooth Riemannian metrics $g_i$ with $\mathrm{Scal}_{g_i}=\kappa$ that are locally uniformly bounded in $W^{1,\infty}$. As a corollary, combining this with Gromov's $C^0$-stability theorem, we obtain the perhaps surprising identity \[ \overline{\{g:\mathrm{Scal}_g=\kappa\}}^{\,C^{0,\alpha}_{\mathrm{loc}}}=\{g:\mathrm{Scal}_g\geq\kappa\}, \quad \forall \alpha \in(0,1). \] The restriction $\alpha<1$ is sharp. At $\kappa=0$, this proves and strengthens the Riemannian reverse-Burnett conjecture of Huneau and Luk.

We prove the existence of optimizers for a class of homogeneous Gagliardo-Nirenberg inequalities of the form $$ \|D^{s}\phi\|_{L^q} \leq C \|D^{s_1}\phi\|_{L^p}^{1-\theta} \|D^{s_2}\phi\|_{L^2}^{\theta}, \quad s_1<s_2, $$ under the scaling relation $$ \frac{1}{q}-\frac{s}{d} = (1-\theta)\left(\frac{1}{p}-\frac{s_1}{d}\right) + \theta\left(\frac{1}{2}-\frac{s_2}{d}\right), $$ and the strict interpolation condition $$ s<(1-\theta)s_1+\theta s_2. $$ When $s>s_1$, the problem reduces, after shifting the derivative orders, to the result of Bellazzini, Frank and Visciglia, whereas the case $s\leq s_1$ follows from the present paper. As an application, we study normalized solutions of the associated Euler-Lagrange equations under the constraint $$ \|D^{s_1}\psi\|_{L^p}^p=\lambda>0. $$ In particular, the case $p=2$ includes the stationary equation arising from the Majda-McLaughlin-Tabak (MMT) model.

We present an asymptotic analysis of two diffusive capture problems in general smooth closed three-dimensional geometries with multiple small reactive boundary patches of arbitrary shapes. (i) The narrow escape problem seeks to determine the escape rate of Brownian particles from an enclosed region through small boundary windows. (ii) The related Berg-Purcell (or narrow entrance) problem seeks to resolve the capture rate for signaling molecules diffusing outside the cell and entering through localized reactions at membrane-bound receptors. We obtain matched asymptotic solutions of these two problems and thus address the long-standing challenge of describing the role that curvature and local reactivities play in modulating diffusive capture rates. Our explicit expansions quantify local effects on diffusive capture through the sizes, shapes, and reactivities of the patches together with the principal curvatures of the manifold at each patch. In turn, we examine global effects on diffusive capture such as the spatial configuration of patches on the manifold, as encloded by the associated surface Neumann Green's function and its regular part. The accuracy of our asymptotic formulas is validated against a full numerical solution for an ellipsoidal domain. Overall, our results yield new insights on how geometry and stochasticity combine to shape the dynamics of various biological processes.

In this paper, we study isolated singularities of the following semilinear elliptic equation $-\Delta u+\frac12 x\cdot \nabla u+\frac{1}{q-1}u-u^q=0$ in $\Omega \setminus \{0\}$, where $n\ge 3$, $\Omega \subset \mathbb{R}^n$ is a domain, $0 \in \Omega$ and $q>1$. This equation arises in the study of blow-up profiles of semilinear heat equations. For $\frac{n}{n-2}< q < \frac{n+2}{n-2}$, we establish a complete classification of isolated singularities for nonnegative solutions and characterize the precise asymptotic behavior of singular solutions. Our results improve those of Guedda and Kirane (Trans. Amer. Math. Soc., 1995: 3595-3603), where analogous results were obtained only for radially symmetric positive solutions. In addition, we also derive the asymptotic behavior of solutions in the Serrin critical case $q=\frac{n}{n-2}$ and the supercritical case $q>\frac{n+2}{n-2}$.

Two identical scalar dynamical systems coupled through a scalar diffusion equation are studied herein, with respect to bifurcations from a symmetric steady-state to symmetric and asymmetric steady-states and to in-phase and anti-phase oscillations. Numerical continuations based on the developed theory show the shape of the bifurcation branches and the attracting nonlinear states far from bifurcation onset and confirm their, for the most part, derived stability. This study extends the work on the dynamical properties of a single scalar dynamical system coupled to its own delay through an adjacent diffusion field and quantifies further the delay of diffusive information transmission between two such Wentzell boundaries. The mathematical system is motivated by modeling biological membranes with yet unknown effective local fluxes that are coupled to bulk diffusion.

We prove that the Feldman–Ilmanen–Knopf (FIK) blowdown singularity has a nonempty open nonlinear formation basin in the full space of smooth Riemannian metrics and, on a smaller neighborhood, local marked first-order asymptotic moduli. Given a closed connected oriented Riemannian four-manifold, an implantation point, and a positive time bound, its oriented blow-up admits a relatively $C^{2,\alpha}$-open set of smooth metrics whose Ricci flows form, before that time, a localized FIK singularity with global Type-I curvature control. No symmetry, Kähler condition, or finite-dimensional tuning is imposed. Curvature stays uniformly bounded outside the implantation region; fixed-convention marked rescalings converge along the full singular-time sequence to the ancient FIK flow; and the exceptional sphere has sharp FIK asymptotics for area, intrinsic diameter, and curvature. To our knowledge, this is the first open full-metric formation theorem for a finite-time singularity modeled on a noncompact, noncylindrical shrinker. A self-contained spectral certification identifies the weighted FIK operator's nine-dimensional nonnegative space with the scaling and diffeomorphism directions, and exact modulation removes this block. On a smaller little-Hölder $h^{2,\alpha}$ neighborhood, a fixed positive-time restart yields the marked first-profile coordinate $\mathfrak{A}_1=\lambda_\infty^{-\gamma_1}V_\infty\in E_1$. This amplitude is a split $C^1$ submersion and locally the projection onto $E_1$. Equality of amplitudes characterizes marked first-order agreement; a transverse disk uniquely realizes each sufficiently small amplitude; normalized two-flow differences converge to the corresponding Jacobi field; and the amplitude determines the first quadratic scale and phase response. This gives a complete local marked first-order profile coordinate for FIK blowdown.

In this article, we establish elliptic gradient estimates for positive solutions of the Lane-Emden system on metric measure spaces satisfying the synthetic Ricci curvature-dimension condition $\mathrm{RCD}^*(-K,N)$. Our approach combines the weak differential calculus and the Bochner inequality available in the $\mathrm{RCD}^*(-K,N)$ setting with suitable auxiliary function arguments, extending classical gradient estimate techniques to nonsmooth spaces. As an application, we prove a Liouville-type theorem for positive solutions under appropriate geometric assumptions. This result helps us to identify constraints for which constant solutions exist. We also mention some cases where nonconstant solutions may exist in sequel. These results generalize corresponding results from the smooth Riemannian setting and contribute to the study of nonlinear elliptic systems on spaces with synthetic Ricci curvature lower bounds.

We present a new proof of the uniqueness of ground states for the pseudo-relativistic Hartree equation in the non-relativistic regime. The proof treats the Euler-Lagrange equation and the mass constraint as a single augmented system, with the Lagrange multiplier as an additional unknown. The key point is the invertibility of the augmented linearization at the non-relativistic ground state.

In this paper, we analyze global minimizers of the Landau–de Gennes functional in 3-dimensional domains in the Lyuksyutov regime. We prove smooth convergence to an $\mathbb{S}^4$-valued limit and obtain an $O(\mu^{-1})$ estimate for the distance to $\mathbb{S}^4$ as $\mu \to +\infty$, thereby extending the convergence results of Dipasquale et al.[Arch. Ration. Mech. Anal. 239: 599–678, 2021].

The present work extends a fixed-point framework previously developed for the standard planar four-velocity Broadwell model to the general four-velocity configuration with an arbitrary orientation of the discrete velocities. This provides a non-axis-aligned test of the framework and leads to a global-in-time existence and uniqueness result for classical solutions of the associated initial-boundary value problem in a rectangular domain. Under suitable assumptions on the initial and boundary data, we establish the existence and uniqueness of a global-in-time classical solution. The analysis identifies the geometric and analytic properties of the fixed-point approach that remain valid independently of the orientation of the discrete velocities. The results demonstrate the applicability of the framework to the general planar four-velocity Broadwell model beyond the standard axis-aligned configuration.

Let $\mathcal H=-\Delta+|x|^2$ be the Hermite operator on $\mathbb R^2$, and let $\Pi_\lambda$ denote the spectral projection corresponding to $\lambda=2N+2$. We prove the sharp log-free endpoint estimate $||\Pi_\lambda||_{L^2(\mathbb R^2)\to L^{10/3}(\mathbb R^2)}\lesssim\lambda^{-1/10}$. The proof uses a spectral decomposition in polar coordinates and combines Koch-Tataru localized spectral projection bounds with a Liouville-Green representation, van der Corput estimates for exponential sums, and a weighted $TT^*$ argument across radial scales.

Spatial transport can create apparent biological heterogeneity even when organoids are intrinsically identical. We develop a transport-to-phenotype null model for diffusively dosed liver-cancer organoid arrays. The model couples bulk diffusion and clearance to partially accessible adsorption, reversible surface residence, productive internalization, and intracellular state dynamics. Matched asymptotics reduce the perforated-domain problem to a Green-function system, while renewal resolvents describe desorption, re-adsorption, and residence-time effects. Across $2000$ random ten-organoid arrays with localized dosing, the predicted transport-only maturation coefficient of variation has median $0.623$; one-factor design changes move this median between $0.27$ and $0.86$. After matching array-mean exposure, distributed dosing reduces the baseline spread approximately fivefold. The analysis also shows that, in a conservative reflecting chamber, desorption changes uptake timing and allocation but not total eventual uptake; reductions in total uptake require a competing loss channel. Residence laws with equal means can nevertheless produce different transient phenotypes. The spatial reduction is verified against finite-element solutions of the full PDE, and the time reconstruction against numerical Laplace inversion. Finally, a large-batch theorem shows that increasing batch size averages independent process variation but not shared line or batch effects. The framework provides a geometry-specific null against which measured organoid heterogeneity can be assessed.

Let $L =-\Delta_{\mathbb{H}^n} +V$ be the Schödinger operator on the Heisenberg group $\mathbb{H}^n$, where $\Delta_{\mathbb{H}^n}$ is the sub-Laplacian, and $V$ is a nonnegative potential belonging to the reverse Hölder class $RH_q(\mathbb{H}^n)$ for some $q > Q/2$, where $Q:=2n+2$ is the homogeneous dimension of $\mathbb{H}^n$. In this paper, motivated by the work of De León-Contreras and Torrea \cite{DT}, we introduce the Lipschitz spaces $\Lambda_L^\alpha (\mathbb{H}^n)$, $0< \alpha <2$, adapted to $L$ via a pointwise second-order difference condition involving the critical radius function $\rho$ related to $V$, and also introduce another type of Lipschitz spaces $\Gamma^{\alpha/2}_L(\mathbb{H}^n)$, $0< \alpha <\infty$, adapted to $L$ in terms of the heat semigroup $e^{-tL}$. We show that for $0< \alpha <2-(Q/q)$, $\Lambda_{L}^\alpha (\mathbb{H}^n) =\Gamma_L^{\alpha/2} (\mathbb{H}^n)$ with equivalent norms. Applications of $\Gamma^{\alpha/2}_L(\mathbb{H}^n)$ to the regularity of the fractional powers of the operator $L$ are also given.

We study the spatial regularity of regular Lagrangian flows associated with vector fields in the DiPerna–Lions class \(\boldsymbol b\in L^1((0,T);W^{1,1}_{\loc}(\mathbb R^d)), \) under the standard growth and compressibility assumptions. For vector fields in \(L^1_tW^{1,p}_{\loc,x}\), with \(p>1\), the flow \(\boldsymbol X(t,\cdot)\) is known to satisfy a quantitative local Lipschitz estimate, which implies that it is Lipschitz continuous in the Lusin sense. We prove that, at the endpoint \(p=1\), this estimate admits an Osgood-type counterpart. More precisely, we construct an increasing function \(G\), with \(G(0+)=-\infty\), determined by the integrability of \(D\boldsymbol b\), such that \[ G\bigl(|\boldsymbol X(t,x)-\boldsymbol X(t,y)|\bigr) \leq G\bigl(|\boldsymbol X(s,x)-\boldsymbol X(s,y)|\bigr) + \int_s^t \bigl(k(\tau,x)+k(\tau,y)\bigr)\,\dd\tau, \] where \(k\) is locally integrable. As a consequence, the flow \(\boldsymbol X(t,\cdot)\) is uniformly continuous outside a set of arbitrarily small measure, with an explicit modulus of continuity determined by the integrability properties of \(D\boldsymbol b\). The resulting moduli include Hölder and log-Lipschitz regimes, as well as substantially weaker Osgood moduli. Our approach is based on a new family of weighted maximal operators associated with slowly varying functions in the sense of Karamata. We also provide examples showing that the resulting estimates are sharp in several respects and that the classical Lipschitz-type estimate may fail at the endpoint \(p=1\). Finally, we apply the flow estimates to transport equations, obtaining weighted logarithmic Sobolev regularity for transported scalars and corresponding lower bounds on functional and geometric mixing scales in the \(W^{1,1}\) setting.

We study radial nonlinear instability of compactly supported nonrotating equilibria of the three-dimensional Euler–Poisson system with a physical-vacuum boundary. Let $n^u(\mu)$ denote the radial instability index furnished by the turning-point theory of Lin and Zeng. Under general structural assumptions on the pressure law, suppose that $n^u(\mu)>0$ and that the equilibrium is not a mass extremum, $M'(\mu)\neq0$. Conditional on the existence of a sufficiently regular radial solution on the relevant time interval, we establish two nonlinear escape criteria. First, every perturbation with Hamiltonian strictly below that of the equilibrium exits a fixed neighborhood on a logarithmic time scale controlled by the least unstable linear growth rate. For the subclass of data for which the associated Lyapunov functional is initially nonnegative, we also obtain an explicit exponential lower bound in the weighted displacement norm. Second, sufficiently small perturbations whose Riesz projection onto the finite-dimensional unstable subspace is not too small escape on a logarithmic time scale. The second argument uses the exponential trichotomy of the linearized Hamiltonian flow and an invariant-cone estimate. In the polytropic class, the conditional estimates combine with the radial physical-vacuum local theory to yield unconditional nonlinear instability in the corresponding classical-solution topology. The results complement Jang's nonlinear instability theorem for Lane–Emden stars by treating mechanisms that do not require initial alignment with a leading growing eigenmode and by applying, conditionally, to unstable branches for general equations of state.

We study the global boundedness of weak solutions to a class of nonlinear elliptic equations in bounded Lipschitz domains, subject to conormal boundary conditions. Using techniques from Morrey space theory, Adams–Mazya trace inequalities, and higher integrability properties of the gradient, we derive uniform a priori bounds for the solutions. A crucial step in our analysis is the construction of appropriate measures associated with the solution and the application of the Hartman–Stampacchia lemma, which enables us to control the solution up to the boundary of the domain.

For each phase $\Theta\in(\pi/2,\pi)$, we construct a smooth, bounded, uniformly convex domain in $\mathbb R^3$ for which the zero-Dirichlet solution of the special Lagrangian equation has a nonconvex negative sublevel set.

We propose and analyze a frequency-stable boundary integral equation method for scalar wave scattering by periodic diffraction gratings in two dimensions. The standard quasi-periodic Green function is singular as a function of the frequency at Wood anomaly frequencies, which prevents a direct frequency-uniform error analysis of conventional integral equation discretizations. Our method is based on a modified quasi-periodic Green function obtained by subtracting the singular Rayleigh modes and introducing a finite-dimensional correction that restores the radiation condition. The resulting combined-field boundary integral equation is shown to be uniquely solvable whenever the underlying diffraction problem is uniquely solvable, and the corresponding scattered field depends continuously on the frequency, including at Wood anomalies. We then construct a fully discrete Nyström scheme using an Ewald representation of the Green function and a quadrature rule adapted to the logarithmic singularity of the kernel. For compact frequency intervals that may contain Wood anomaly frequencies, we prove unique solvability of the discrete system for sufficiently large discretization and truncation parameters and establish a frequency-uniform error estimate and bound on the condition number. Numerical examples illustrate the convergence of the scheme and its stability across Wood anomaly frequencies.

We present a neural network-based method for numerical analytic continuation of solutions of nonlinear partial differential equations (PDEs) and detection of their complex singularities. The proposed framework employs "unsafe" Padé activation units (PAUs) as activation functions, together with a novel backpropagation-free method for computing the weights and biases of the hidden layers. The training procedure is designed specifically for learning meromorphic functions with pole-type singularities. Unlike existing methods with fixed hidden-layer parameters, the proposed approach treats the weights and biases as time-dependent functions, allowing the network to adapt to the evolution of the complex singularities. Using this method, we can efficiently locate the complex singularities of the extended solution, track their trajectories over time, and, based on this dynamics, infer the formation of characteristic phenomena in the solutions of nonlinear PDEs. To demonstrate the performance of our method, we analyze three well-known cases: (1) the nonlinear heat equation, which exhibits finite-time blow up; (2) the nonlinear Burgers equation, which develops a shock; and (3) the nonlinear Schrödinger equation, in which rogue waves form.

This paper is concerned with the parabolic-elliptic chemotaxis system with weakly singular sensitivity and logistic source:~$ u_t=\Delta u-\chi\nabla\cdot(\frac{u}{v^\alpha}\nabla v) +ru-\mu u^2$, $0=\Delta v-v+u,$ under the homogeneous Neumann boundary in a smooth bounded convex domain $\Omega\subset\mathbb{R}^n$ for $n\ge 2$. where $\alpha\in(0,1)$ and $\chi,r,\mu>0$. If $\alpha\in(0,\frac{n+2}{2n})$ and $\mu>\mu_0$ with $\mu_0>0$ suitably large, we give the explicit expression of the upper bound for $u$ with respect to the coefficient $\mu$ after some time, without establishing the uniformly positive bound for $v$ from below. Furthermore, by dealing with the corresponding non-singular chemotaxis system via the transformation $z=v^{1-\alpha}$, it is proved that the solution $(u,v)$ converges to $(\frac{r}{\mu},\frac{r}{\mu})$ in $L^\infty$-norm as $t\rightarrow\infty$ if $\alpha\in(0,\frac{1}{2})$ and $\mu>\mu_\star$ sufficiently large, which is moreover enjoying exponential convergence when $\alpha\in(0,\frac{n+2}{n^2+4})$.

We consider the heat equation on $(0,L)$ with homogeneous Dirichlet condition at one endpoint and a Dirichlet boundary control at the other. If \(C_{\mathrm H}(T,L)\) denotes the optimal \(L^2\) null-control cost for initial data in \(H^{-1}(0,L)\), we prove that \[ C_{\mathrm H}(T,L) = \exp\left(\frac{\kappa_*L^2+o(1)}{T}\right), \qquad \kappa_* = \frac{\Gamma(\frac14)^4}{8\pi^3} \simeq 0.696601964842838, \qquad T\to0^+. \] The constant \(\kappa_*\) coincides with the upper-bound constant obtained by Dardé and Ervedoza (2019, ANPDE), which was expressed there through a convergent series. This closes the gap between the lower bound obtained in by Lissy (2015, JDE) and the upper bound established by Dardé and Ervedoza.

We establish quantitative $L^1$-stability estimates for the Bakry–Émery log-Sobolev and Talagrand inequalities with a universal exponent of 1/19 governing the corresponding deficits. Our approach relies on a Maurey-type argument combined with stability estimates for the Prékopa–Leindler inequality. In the radial setting, the exponent of both deficits can be improved to $1/2,$ which turns out to be optimal. As an application, we establish an estimate for the hypercontractivity deficit of the Hopf–Lax semigroup in the Bakry–Émery setting. In particular, these stability results provide elementary characterizations for the equality cases in the previously mentioned inequalities.

This paper provides a comprehensive treatment of inverse boundary value problems for (nonlinear) Kirchhoff plate equations under diverse general settings. We begin by establishing the global well-posedness of the nonlinear forward equations, which not only underpins the subsequent inverse analysis but also holds independent theoretical significance. The inverse problems are then examined for both passive and active measurement regimes. With a single passive boundary measurement, we establish the stable recovery of the unknown initial data. In the active regime with infinitely many boundary measurements, our results are twofold. For linear equations featuring generic time-dependent potentials-allowing for spatial unboundedness, we demonstrate the simultaneous recovery of both initial data and coefficients. For nonlinear equations, where both the nonlinearity and initial data are unknown, we develop a novel Runge approximation approach, together with carefully constructed geometric optics solutions and higher-order linearization around nonzero solutions, to prove their simultaneous determination. Furthermore, we introduce a delicate cut-off technique that provides an alternative means of addressing the scenario of vanishing initial data. Notably, the methodologies and results developed herein are readily generalizable to other boundary conditions and plate models, including the classical Euler-Bernoulli equation.

We study the clamped plate eigenvalue problem. We show that clamped plate eigenfunctions can be approximated in the $L^2$-sense by their oscillatory components, which solve the Helmholtz equation, with an error that decays exponentially fast in the frequency on any compact subdomain. Additionally, we establish an upper bound on the boundary localization of clamped plate eigenfunctions analogous to the one observed for membrane eigenfunctions.

We construct by convex integration a nonzero singular weak solution of the one-dimensional periodic cubic nonlinear Schrödinger equation that is compactly supported in time, has zero initial datum, and satisfies \[ u\in\bigcap_{\alpha<1/6} C_t^0 H_x^\alpha \cap \bigcap_{1\leq p<3} C_t^0 L_x^p. \] The key new ingredient is a two-carrier perturbation: its zero-output interaction cancels the previous error, without introducing any zero Fourier mode, which is crucial in the cubic NLS. Whenever a singular solution belongs to \(C_t^0L_x^3\), this nonlinearity agrees with the ordinary product \(|u|^2u\). Together with unconditional uniqueness at \(H^{1/6}\), this makes the threshold sharp within the singular solution class considered here.

Let $n\geq 2$, $1\leq \ell\leq n-1$, and $1<p<\infty$. We prove that every $v\in \dot W^{n/p,p}(\mathbb{R}^n;\Lambda^\ell)$ has a representative $u\in \dot W^{n/p,p}\cap L^\infty$ with $du=dv$ and $\max{|u|{\dot W^{n/p,p}},|u|{L^\infty}}\lesssim |v|{\dot W^{n/p,p}}$. Equivalently, $d[\dot W^{n/p,p}\Lambda^\ell]=d[(\dot W^{n/p,p}\cap L^\infty)\Lambda^\ell]$ with equivalent quotient norms. The proof reduces the selection problem to an endpoint graph estimate for a Riesz potential and the exact Hodge projection. Its main analytic input is a finite-dimensional-input Maz'ya–$\Phi$ inequality for operator-valued homogeneous kernels. For the Riesz/Hodge pair, a nonlinear spherical profile built from the projected kernel has exact atomic cancellation and is coercive by the identity $\int{S^{n-1}}P(\theta),d\bar{\sigma}=(\ell/n)\operatorname{Id}$. Frequency-localized graph closure and Hahn–Banach duality then return a bounded representative.

We give a new proof by flow decomposition of the known Grigor'yan type integral criterion for stochastic completeness of weighted graphs. The main ingredient is a finite-domain estimate relating the mass of a killed $1$-resolvent and the current reaching the boundary to relative capacities of intrinsic balls. This estimate yields logarithmic criteria in terms of capacity to infinity and relative capacity, and recovers the volume criterion by an intrinsic cutoff, without passing to a metric graph or refining the graph.

This paper develops a Hodge-theoretic feedback framework for flow-valued states, motivated by scalar parabolic thermostat problems with localized sensors and dual sources. On a connected graph, the edge space decomposes into a cut space and a harmonic cycle space. The ideal topological thermostat regulates the cut component toward a prescribed cycle-free transfer target and damps the harmonic component. For realizations through selected physical edge channels, the first Betti number gives the minimal sensing and actuation ranks required for exact compression of the full harmonic sector. Minimal rank does not imply local realization of the nonlocal Hodge projector; the remaining cut-harmonic blocks quantify spillover and forcing. For transmission networks, we formulate ideal and finite-channel nonlinear line-actuated swing closures and linearize the complete closed systems at an angle-cohesive synchronous equilibrium. With the actuator fixed, passive tangent swing motion preserves any pre-existing harmonic line-flow component. The ideal closure replaces this conservation law by exponential decay and appends a stable harmonic block without changing the reduced passive nodal-cut spectrum. Under the channel-rank conditions, finite realizations reproduce the harmonic compression, whereas autonomous harmonic decay and full reduced-state recovery require additional decoupling and Hurwitz conditions. A target-centred Bregman balance yields nonlinear dissipation for the ideal metric-compatible controller and for canonical colocated finite-channel feedback. Explicit theta-network, microgrid, ring, and synthetic IEEE 14-bus examples illustrate transfer regulation, cycle-flow damping, and localization-induced transients. Realistic device dynamics, constraints, and large-scale validation remain open.

This work investigates recurrent pattern formation in a two–species chemotaxis system with competitive dynamics under the assumption that the carrying capacities of both species are periodic in time. By considering a globally attracting periodic solution to the associated nonautonomous ODE system as a spatially homogeneous base state, we derive sufficient conditions for the existence of a critical chemotactic sensitivity of the first species, denoted by $\chi_{\text{crit}}$. Whenever this threshold is exceeded, the periodic state becomes linearly unstable over a nonempty set of times within each environmental cycle. This gives rise to a recurrent alternation of unstable and stable intervals, during which spatial perturbations are respectively amplified and damped. The analytical derivation is complemented with numerical simulations, in both periodic coexistence and competitive exclusion regimes. Finally, we discuss the temporal location of the instability sets and the possibility of negative instability thresholds.