Papers

We extend the concept of microlocal defect functionals to test functions belonging to the space $\mathrm{L}^\infty\cap\mathrm{VMO}_c$. Following L.~Tartar's remark that an extension of such concepts to $\mathrm{VMO}$ spaces should be possible, we establish a functional-analytic framework for this extension within the $\mathrm{L}^p-\mathrm{L}^q$ setting. Because the topological dual of $\mathrm{VMO}$ is the Hardy space $\mathcal{H}^1$, the resulting object takes the form of an H-distribution rather than a non-negative Radon measure. By assuming strict Hölder conjugate inequalities, we use the John-Nirenberg inequality over localized domains to construct these functionals. We show that the functional acts as a distribution on the inductive limit topology of the test spaces, giving geometric localisation principles for equations with rough $\mathrm{VMO}$ coefficients, that is, coefficients admitting sharp transitions of vanishing mean oscillation. We illustrate the framework in three settings: stratified transport, zero-order non-local cross-phase energies, and sub-critical acoustic scattering in high-contrast media. These differ in the nonlinearity generating the companion sequence but share a common geometric conclusion, since in each case the macroscopic energy defect is confined to the characteristic variety of the underlying flow, being supported where $\sum_j a_j({\bf x})\xi_j=0$.

The extension of microlocal defect measures to the $L^p$-$L^q$ framework necessitates the use of distributions which possess strictly positive finite order. A topological $\eps$-bounding definition of orthogonality for such distributions was recently introduced by Antonić, Mitrović, and Perić. Building on their framework, we systematically map the obstructions that arise when this natural approach is transported to strictly positive order, and we provide a complementary algebraic resolution. We first establish that while smooth spatial multipliers provide a robust framework for separating order-zero Radon measures, the partition-of-unity route to phase-space geometric separation breaks down for higher-order distributions: the separating frequency multiplier suffers an $\mathcal{O}(\delta^{-\kappa})$ derivative divergence when frequency supports touch, which we prove defeats the a priori bound at every order $\kappa \ge 1$ and produces an explicit non-orthogonality already at order one. Transitioning to algebraic phase-space projectors reveals a second severe obstruction: it is mathematically impossible to separate scalar distributions algebraically, as one-dimensional smooth projectors enforce topological triviality. We demonstrate that the resolution is achieved exclusively within vector-valued PDE systems. Rather than relying on arbitrarily constructed, non-physical geometric scalar cut-offs $\chi(x,\xi)$ to separate distinct modes, we redefine orthogonality canonically via zero-order algebraic matrix projectors. This replaces artificial geometric separation with the intrinsic spectral polarization of the governing system, yielding a coordinate-free microlocal orthogonality for systems of higher-order distributions.

In this paper, we follow Pan-Wu's idea [PW26] to prove local existence of the capillary wave kinetic equation. To be more specific, we will linearize the nonlinear operator and decompose it into dissipative part and bounded part. The main difficulty comes from the fact that the collision kernel has less symmetry compared with the one in gravity water wave. To overcome it, we are required to choose and pair each term delicately.

The below-threshold scattering problem is considered for the three-dimensional focusing energy-critical nonlinear Schrödinger equation. A main non-radial obstruction is the possible drift of the concentration center of a soliton-like critical element, which prevents a fixed-center localized virial estimate from closing directly. It is shown that such a compact critical solution must vanish if it has an $L_t^\infty L_x^q$ bound and its center drifts sufficiently slowly. The result covers the pure-energy, endpoint $L^4$, and finite-mass regimes and gives a corresponding conditional scattering criterion.

We prove a priori exponential growth and decay estimates for locally square integrable eigenfunctions with complex eigenvalues of the non-self-adjoint Schrödinger operator. There appear two natural critical exponents, given by the imaginary parts of square roots of twice an eigenvalue. Any associated eigenfunction has the same exponential growth or decay rate as one of these, or greater than the upper one. The results may be seen as extensions of the Phragmén–Lindelöf theorem, the Agmon estimates and Rellich's theorem to complex eigenvalues. For the proofs we adapt a commutator scheme of Ito–Skibsted~(2020) to the non-self-adjoint framework, presenting a unified approach to all of them. In order to control contributions from the imaginary parts of a potential and an eigenvalue, an additional commutator estimate is employed.

This second part of the series develops the statistical and applied layer of the theory built in Part I [1]. Unlike current Hilbert diffusion models [2-5], we focus not only on well-posedness of the infinite-dimensional generative dynamics, but on the noncommutative $A$-geometry, explicit entropy constants, and checkable lower bounds. The analytic estimate of Part I reduces stability of the backward evolution to control of a validation error $\mathcal{E}_{val}$. We prove three results. First, we construct a cylindrically weak denoising label for the infinite-dimensional score field, consistent with the exact logarithmic gradient. Second, the local parametric closure is replaced by a global nonparametric score closure, its complexity controlled via the Dudley entropy integral and uniform empirical bounds in $L^2(\nu_*;A)$. Third, for the trace-smoothed nonlinear class we construct minimax bounds by the Le Cam-Assouad method, showing the root statistical rate cannot be improved without additional quadratic structure. The final section applies the theory to anisotropic diffusion in a nanosystem model and checks the constants $c_A,C_A,C_{LSI}^A$ independently of any smallness condition, establishing explicit accuracy orders: the parametric trace-smoothed minimax score risk is $p/M$, while the uniform error of the risk functional is $\sqrt{p/M}$, showing the statistical plateau of order $p/M$ is not a proof artifact. The applied layer closes with an independent analytic benchmark for the isotropic case, a comparison with classical cell homogenization, and an approximation theorem for smooth logarithmic gradients by $A$-adapted spectral networks.

We derive a quasi-incompressible Cahn-Hilliard-Darcy phase-field model (qCHD) with the logarithmic Flory-Huggins free energy density function for two-phase flows in porous media. The model satisfies an energy-dissipation law. In the formal sharp interface limit, the qCHD model gives rise to the classical Muskat's problem. By exploiting estimates of the pressure from Darcy's equations, we establish global existence of weak solutions in both 2D and 3D to the qCHD model.

We consider minimisers in the homotopy class of a homeomorphism $f_0:R\to S$ between analytically finite Riemann surfaces with minimal $L^p$- conformal energy \[ \mathsf{E}_p(f:R,S)=\int_R \IK^p(z,f)\; d\sigma_R(z). \] The problem was first raised by Ahlfors in his celebrated proof of Teichmüller's theorem-the case $p=\infty$, but the existence, topological regularity and analytic regularity of these $L^p$ minimisers remained unknown for all $1<p<\infty$. Ahlfors established weak existence for $p\geq 2$. Here we prove that for all p, $1\leq p<\infty$, such minimisers exist, are unique and are diffeomorphisms. They are quasiconformal but not diffeomorphic at $p=\infty$.

We propose an alternative existence proof of global weak solutions to mean curvature flow and volume preserving mean curvature flow. We prove for the first time for a minimizing movements scheme the unconditional convergence towards a varifold solution, here a De Giorgi solution. The argument is purely variational and does not rely on comparison principles. The key novelty is an alternative proxy for the completely degenerate $L^2$ distance that is more robust than the one of Almgren-Taylor-Wang and Luckhaus-Sturzenhecker.

This paper is devoted to the transverse stability problem for small-amplitude periodic traveling waves of the $b$-Novikov equation, which arises as a two-dimensional extension of the $b$-family hierarchy. The perturbations under consideration fall into two groups. One group matches the fundamental period of the underlying wave along the longitudinal direction, while the other consists of profiles that are bounded or localized in the transverse direction. We perform spectral analysis of the associated linearized operator and derive precise stability and instability thresholds. The resulting conditions exhibit intricate dependence on two key parameters including the cubic coupling strength $b$ and the wavenumber $k$.

In this study, we introduce a general energy-based modelling approach for viscous poroelastic materials that feature diffusive transport in both Lagrangian and Eulerian frames. Our research produces refined weak formulations by using the reference map concept within the Eulerian configuration. We propose and implement a novel structure-preserving discretisation strategy, utilising mixed finite element methods. This paper highlights the spatial and temporal numerical convergence of our methods through a comparative analysis of Lagrangian and Eulerian schemes, thereby proving the robustness and usability of our approach. Furthermore, in the context of Eulerian multiphase flow, specifically of the quasi-static Euler-Euler type, our study demonstrates the existence of solitary fluid waves within poroviscoelastic media. This energy-based approach forms a basis for a deeper understanding of thermodynamical modelling and corresponding discretisation schemes for coupled poroelasticity, flow, and diffusion.

We develop a reaction–diffusion–chemotaxis model for spatial tumour–immune–chemokine dynamics that couples logistic tumour growth, immune-mediated killing, chemokine-dependent immune recruitment, chemotactic migration, and signal production. For the nondimensional system, we establish local classical solvability, nonnegativity, a uniform tumour-density bound, and global mass estimates for the immune and chemokine components. The tumour-free equilibrium is stable precisely when the baseline immune-control index satisfies \(\sigma_0/\delta>1\), whereas positive homogeneous coexistence is characterized by a scalar nonlinear equation. Linearization in the Neumann Laplacian eigenbasis yields a mode-dependent cubic dispersion relation, showing that chemotaxis does not alter the tumour-invasion threshold but can destabilize homogeneous coexistence through a finite-wavelength oscillatory instability above a critical sensitivity \(\xi_c\). A conservative finite-volume discretization with upwind chemotactic fluxes and implicit backward differentiation formula time integration is used to test these predictions. Numerical experiments recover the analytical equilibria and growth rates, identify the dominant unstable mode, reproduce the transition to spatial heterogeneity, and quantify the effects of immune recruitment, decay, and diffusion on the stability boundary. Grid-refinement, mass-balance, residual, and nonnegativity diagnostics support the computational reliability of the results.

We study a nonlinear size-structured transport model with distributed harvesting and prescribed recruitment flux, where environmental feedback is determined by a scalar population functional. After establishing global well-posedness on $L^1$, we reduce stationary equilibria to a scalar closure equation. This reduction reveals that loss of equilibrium uniqueness occurs through a generic fold, mathematically characterizing critical population transitions. On uniformly nonresonant equilibrium branches, we prove the existence of optimal stationary harvesting policies via the direct method of the calculus of variations. We then derive a boundary-corrected adjoint equation and establish an identity equating equilibrium sensitivity with the adjoint loop gain. This relation yields explicit criteria for the persistence and creation of optimal harvesting thresholds. Collectively, these results provide a unified analytical framework connecting environmental feedback, equilibrium structure, and optimal harvesting.

We study the structure of the initial layer arising in the relaxation-time limit of the collisional quantum hydrodynamic (QHD) system. When the initial data are not well prepared, a fast transient regime appears near the initial time, which prevents the uniform-in-time convergence of the momentum density to its limiting value. Using the method of matched asymptotic expansions, we derive a systematic asymptotic description of the solution with respect to the relaxation-time parameter $\tau$. In particular, we identify the fast time scale $t/\tau^{2}$ governing the initial layer and explicitly construct the corresponding inner expansion for the momentum density together with the outer expansion describing the slow dynamics. The leading-order outer dynamics are shown to coincide with the quantum drift-diffusion equation. The asymptotic expansion is rigorously justified by establishing uniform in $\tau$ energy estimates for the remainder terms under suitable regularity assumptions on the solutions. As a consequence, we prove the strong convergence of the momentum density in $L^\infty$ in time after subtracting the leading initial-layer correction. The analysis further shows that the convergence rate of order $\tau$ is optimal for general initial data and explains the improved rate in the well-prepared case.

This paper is devoted to the study of solutions for a class of operator equations governed by strongly monotone operators on real Riesz spaces continuously embedded into Banach lattices. By exploiting the intrinsic lattice structure of Banach lattices, we establish refined growth assumptions on nonlinear terms to guarantee that suitable neighbourhoods of positive and negative cones are invariant under the descending flow. Combining the descending flow invariant set technique with the theory of strongly monotone operators, we derive abstract existence theorems: the operator equation possesses at least one positive solution, one negative solution and one sign-changing solution. These abstract results are further applied to \((p,q)\)-Laplacian boundary value problems, yielding corresponding multiplicity conclusions on positive, negative and sign-changing solutions.

This paper investigates the existence of critical points for functionals defined on a Hilbert space $X$ which is continuously embedded into a Banach lattice $E$. A lattice decomposition of $E$ is constructed, which possesses both order disjointness and inner-product orthogonality. Accordingly, a corresponding decomposition of the Hilbert space $X$ is obtained. Under this decomposition, the associated functional satisfies the energy collapse condition and order-preserving property on certain subspaces, while exhibiting coerciveness on others. By combining the descending flow invariant set method with Morse theory, we establish the existence of multiple critical points for the abstract indefinite problems. Finally, applications to elliptic boundary value problems are provided.

Convection-dominated convection-diffusion problems often develop thin layers, where the solution has sharp transition profiles and its derivatives are highly localized. This creates a structural mismatch for standard physics-informed neural networks (PINNs), whose trial spaces are not designed to match the value–derivative structure of such layers. We propose a Layer-Resolving XNet Physics-Informed Neural Network (LRX-PINN) based on integrated Cauchy activations. The proposed basis is transition-type at the solution level, while its derivative recovers a localized Cauchy kernel. We show that this structure matches the scaling of convection-dominated layers, inherits the Cauchy approximation mechanism at the derivative-profile level, and identifies \(d/\|w\|\) as the effective physical width of a ridge neuron. For analytic layer profiles, this yields derivative-stable exponential approximation in the stretched coordinate and a layer-scaled estimate for the strong residual of the singularly perturbed operator. Numerical experiments on several convection-dominated benchmarks show that LRX-PINN achieves higher accuracy than PIKAN and Fourier-feature PINNs while using less than \(30\%\) of their trainable parameters. On more challenging benchmarks, embedding the proposed representation into hp-VPINN-based frameworks further improves the best results obtained by existing hp-VPINN-based baselines without changing their original loss functionals or stabilization strategies. These results show that neural representations aligned with layer structure provide a compact and effective approach for convection-dominated problems.

Bacterial chemotactic movement and collective aggregation have long attracted substantial interest in mathematical biology and applied modeling. Classical Keller–Segel-type systems, however, are typically formulated under idealized laboratory assumptions, such as smooth agar substrates, and thus cannot adequately capture the gliding dynamics of myxobacteria in naturally rough environments like soil. In this paper, we propose a unified framework that integrates stochastic modeling, rigorous analysis, and deep neural network-based simulation of chemotactic gliding–diffusion and aggregation processes on bounded domains. Starting from a lattice-based discrete agent description and a subordinated Langevin equation driven by an inverse stable subordinator at the microscopic level, we characterize anomalous gliding dynamics on rough surfaces and derive a macroscopic time-nonlocal Keller–Segel-type chemotaxis model with logarithmic sensitivity. We then establish a comprehensive solution theory for the resulting model, covering mass conservation, novel regularity results, local well-posedness in any spatial dimension, and global well-posedness in two and three. The analysis relies on several newly developed ingredients, including a fractional Lyapunov functional, a variational inequality adapted to the time-nonlocal structure, logarithmic Sobolev-type estimates, Bregman distance techniques, and a weighted bootstrap mechanism adapted to the singular sensitivity and time-nonlocal memory. Finally, we design a mesh-free, positivity-preserving, multi-objective, time-marching physics-informed neural network method with separate architectures and tailored variable transformations. Numerical experiments on complex geometries, including a butterfly-shaped domain, demonstrate the robustness, accuracy, and flexibility of the proposed computational framework across a range of Keller–Segel-type systems.

Given a one form $\lambda$ in $\mathbb{R}^N$ and $f: \mathbb{S}^{n} \to \mathbb{R}^N$ with $f^\ast \lambda = 0$ we discuss the maximal Hölder regularity of extensions $F: \mathbb{B}^{n+1} \to \mathbb{R}^N$ such that $F^\ast\lambda = 0$ in distributional sense. Our analysis applies to the Heisenberg groups $\mathbb{H}_n$. It implies in particular that for all $n \geq 1$ any smooth horizontal map $f: \mathbb{S}^{n} \to \mathbb{H}_n$ can be extended to a $C^\alpha$-map $F: \mathbb{B}^{n+1} \to \mathbb{H}_n$ for some $\alpha > 1/2$. Moreover, if $n \geq 3$ we find $C^\alpha$-embeddings from $\mathbb{B}^{n+1}$ into $\mathbb{H}_n$ for some $\alpha > \frac{1}{2}$. In the appendix we discuss an (as of now unverified) approach to extend these arguments to find $C^\alpha$-embeddings from $\mathbb{B}^{2}$ into $\mathbb{H}_1$ for some $\alpha > \frac{1}{2}$, assisted by GPT-6 Astra.

We introduce the notion of very weak suitable solutions for the Navier–Stokes equations. Here, the velocity and the pressure satisfy minimal conditions that make sense of the local energy inequality in the distributional setting. A well-known but still relevant question is to find sufficient conditions ensuring that very-weak solutions are in fact Leray solutions. Exploiting the local energy inequality within the general framework of local Morrey spaces, we establish such conditions. Local Morrey spaces provide a general framework that contains other useful functional settings in the theoretical analysis of the Navier–Stokes equations, such as Lebesgue, Lorentz, homogeneous Morrey, and parabolic Morrey spaces. As a by-product, we also derive some sufficient conditions to study the uniqueness and regularity of the resulting Leray solutions.

In this paper, we study Anderson localization near the spectral edge for the Anderson-Bernoulli model on $\mathbb{Z}$ with long-range hopping. When the hopping has a rational Laurent symbol, a quantitative version of the unique continuation principle can be proved, and localization occurs. For the unique continuation in the general case, we give some counterexamples and prove a weaker result for hopping that decays faster than exponential rate. To the best of our knowledge, this is the first localization result for the long-range Anderson model with pure Bernoulli potentials.

In this article, we study the weighted homogeneous Sobolev–Morrey spaces on domains in $\mathbb{R}^n$ with higher co-dimensional boundaries. Precisely, we systematically establish a real-variable theory of these spaces, including completeness, embedding theorems, Riesz potential characterizations, continuity, trace and extension theorems, and complex interpolation. Applying the boundedness of the trace and the extension operators, we obtain sharp weighted a priori estimates for solutions to the Dirichlet problem of divergence-form degenerate second-order elliptic equations on such domains in weighted Lebesgue spaces. The absence of a boundary manifold structure of these domains poses some essential difficulties, which are overcome by using some tools, such as the intrinsic properties of distance weights and the geometric structure of domains, different from those available in Lipschitz domains.

We consider the poisson problem $\wt{\mc{L}} u = E$ for the operator $\wt{\mc{L}} = \Delta_{\mb{R}^2} + 2\cosh^{-2}(s)$ on domains of the form $\Lambda : = \{ (s, z): |s| \leq \ell(z) \}$, where the width $\ell(z)$ of the domain $\Lambda$ varies with $z$. We prove the existence of solutions satisfying weighted estimates when the source term satisfies certain natural orthogonality conditions, and when $e^{-\ell}$ is small and slowly varying.

We investigate the existence, uniqueness, and regularity of the gradient flow of the Ambrosio-Tortorelli functional, viewed as an initial-boundary value problem, in arbitrary dimension. For any initial data, using a time-discrete Euler scheme, we establish the existence of a weak gradient flow satisfying a maximum principle. We also identify a functional space in which uniqueness holds. We further show that such gradient flow is smooth in the interior of the space-time domain. Under additional assumptions on the initial data, the regularity of the boundary of the domain, we prove optimal regularity for the solution, up to the space-time boundary.

In this paper we consider the Stefan problem endowed with a doubly nonlinear phase relaxation. More precisely we assume that the rate of convergence of the phase function depends on the temperature via an increasing continuous function, and it depends on the phase via a maximal antimonotone graph. We prove the existence and uniqueness of the solution of the relaxed problem, and we perform the asymptotic analysis toward the Stefan problem as the relaxation parameter approaches zero.

We prove a coarea-type result for scalar functions $f$ in fractional Sobolev spaces $W^{s, p} (\Omega)$ with $\Omega\subset \mathbb R^n$, $0<s<1$, and $1\leq p < \infty$. Our theorem shows that a.e. level set has zero Hausdorff $\mathcal{H}^{n-s}$ measure, where the level set $f^{-1} (y)$ is defined as the set all points at which $y$ is between the $\liminf$ and the $\limsup$ (as $r\downarrow 0$) of the averages of $f$ over the balls $B_r (y)$. A quite general construction of random series of wavelets shows also that with probability $1$ (many) level sets have indeed dimension $n-s$.

This paper studies finite-horizon optimal stopping problems for semilinear stochastic evolution equations in real, separable Hilbert spaces, together with their associated parabolic variational inequalities. We prove continuous differentiability of the value function in infinite dimensions, thereby obtaining a smooth-fit principle for the corresponding optimal stopping problem. The analysis has two parts. First, we prove existence and uniqueness, in a suitable weighted class, of a mild solution to the variational inequality and identify it with the optimal stopping value function. We also establish local spatial Lipschitz continuity of the value function by probabilistic methods, without requiring any smoothing property of the underlying transition semigroup. Second, under a global regularizing assumption on this semigroup, we prove higher-order spatial regularity: the value function is continuously Fréchet differentiable and its gradient is locally Hölder continuous. The abstract results are then applied to a stochastic heat equation with additive noise.

Since its introduction, the Keller–Segel model has become a cornerstone in the mathematical theory of chemotaxis and it has generated extensive analytical activity. Most studies consider homogeneous Neumann boundary conditions, which ensure mass conservation and simplify the qualitative analysis of solutions. To the best of the authors' knowledge, at present chemotaxis models incorporating boundary conditions that generate inward fluxes have only been studied in two recent papers, and we believe that this topic deserves and it may attract further mathematical attention. In this sense, in the present paper we investigate a two-species chemotaxis system with positive total flux. The model consists of two interacting populations, $u$ and $w$, coupled through elliptic/parabolic chemical signals $v$ and $z$, and subject to Robin-type boundary conditions allowing inward fluxes that depend on the product of the cellular and chemical densities. Unlike the classical conservative setting, the total mass is not preserved and it exhibits quadratic growth in time, exactly in line with one of the investigations above mentioned and dealing with a single-species taxis model. We show that, within the considered framework, standard logistic damping is not sufficient to compensate for the mass increase induced by the positive boundary flux. To restore control of the dynamics, stronger dissipative mechanisms involving gradient-dependent damping terms are required. Under suitable assumptions, we establish the global existence and boundedness of classical solutions in the presence of logistic-gradient damping.

We prove the existence of a global family of steady, doubly periodic gravity-capillary waves on Beltrami flows. This is the first rigorous existence result for genuinely three-dimensional inviscid surface waves, with or without vorticity, beyond the perturbative regime close to simple explicit solutions. The proof is based on reformulating the steady water wave problem as a bifurcation problem of the form `identity plus compact' and applying a global bifurcation argument in Hölder spaces. The main challenge is that the kernel of the linearisation at the bifurcation point is two-dimensional, and that both kernel elements are necessary to obtain genuinely three-dimensional solutions. Since this prevents the use of classical global bifurcation theory, we introduce a novel reformulation of the bifurcation problem using the parameterisation of a local family of solutions bifurcating from laminar flow. In this new parameter space, we apply a variation of analytic global bifurcation theory. Along the branch, we then prove a sharper blow-up alternative, namely blow-up of the surface gradient in $C^{0,\gamma}$.

We establish refined spectral asymptotics for the operator curl acting on a connected oriented closed Riemannian 3-manifold. We treat positive and negative eigenvalues separately and obtain explicit formulae for the first six global Weyl coefficients. With local Weyl coefficients we compute the first four coefficients as well as the sixth one, and we determine the fifth up to a universal constant. As a consequence, we prove that the eta function of curl (both in its local and global versions) is holomorphic in the complex half-plane $\operatorname{Re}s>-2$. Finally, under appropriate assumptions on the geodesic flow, we improve Bär's asymptotic formulae for the positive and negative counting functions, refining the remainder to $o(\lambda^2)$.