Papers

We develop an AI-assisted search-and-certification workflow for constructing Carleman weights for wave equations on semi-infinite domains. Starting from a weighted cross-term identity used to derive the Carleman estimate, we derive explicit analytical screening conditions and use an AI system to propose symbolic candidate weights subject to those conditions. The search identifies a logarithmic weight structure with concrete parameter values; the subsequent human verification by the authors then retains this structure, derives the admissible parameter range, and rigorously certifies the selected weight by verifying the required quantitative conditions. Based on the identified and certified Carleman weight, the subsequent proofs, analysis, and applications are carried out entirely by the authors: we establish a global Carleman estimate for the wave operator on the half-line and derive a weighted conditional lateral Cauchy stability result. The proposed weight is further used in a finite-depth one-sided reconstruction problem for semilinear wave equations on the half-line, where the initial-time flux induces an endpoint graph stabilizer and a contractive frozen nonlinear reconstruction map. Numerical experiments support the predicted contraction behavior and show improved robustness to noisy data from the Carleman weighting, with the induced endpoint graph term providing further stabilization at higher noise levels.

We study the dependence of parabolic solution operators on a finite signed measure describing the delay law. For a positive self-adjoint generator with compact inverse, a weighted dyadic spectral sum characterizes the norm of the semigroup-integrated memory perturbation. An exact realization identity transfers the matching lower estimate to two positive point delays in a fixed linear equation, using a common smooth, finite-spectral history. For the Dirichlet Laplacian on a nonempty bounded open set, the sharp worst-case modulus on a bounded ball of continuous $L^2$-valued histories is $d\sqrt{\log(e/d)}$, where $d$ is the bounded-Lipschitz distance between the delay measures. This is a rough-history endpoint result: logarithmic spatial regularity of order $\gamma>1/2$ restores resolution-uniform Lipschitz stability in the linear model. The critical order $\gamma=1/2$ retains a square-root double-logarithmic loss. More generally, a reciprocal-square summability criterion over occupied spectral bands gives the exact weighted threshold, including sparse spectra. Consequences include sharp finite-resolution Lipschitz constants, worst-case errors for prescribed midpoint quadrature of the memory measure, and semilinear upper estimates under Hilbert-space-valued local Lipschitz assumptions. The results distinguish sensitivity to the delay law from spatial approximation error and do not preclude uniform approximation of positive-time states on rough history balls.

We propose a method for closing moment chains for the Vlasov-Poisson kinetic system (the Landau fluid model), which yields a hyperbolic system of equations at each subsequent step. This system is interpreted as a multicomponent medium, where each component has its own density, velocity, and pressure. The coupling between the components is mediated by an electric field. The principle of closing moment chains is that the final component of the fluid is assumed to be pressureless.

In this paper, we study the existence, uniqueness, and quantitative estimates for weak solutions to linear elliptic Dirichlet problems of the form \[ -\operatorname{div}(\gamma \nabla u)+\langle \nabla\phi+\mathbf{H},\nabla u\rangle+(c+\alpha)u=f \quad\text{ in }U, \quad\; u=0 \quad\text{on }\partial U, \] where $U\subset \mathbb{R}^d$ is bounded, $\gamma\in[1,\infty)$ is a constant, $\phi\in H^{1,2}(U)\cap L^\infty(U)$, $\mathbf{H}\in L^p(U,\mathbb{R}^d)$ for some $p \in (d, \infty)$, and $c\in L^1(U)$ with $c\ge0$. A key feature of this setting is that the drift contains the low-regularity term $\nabla\phi$, which is only assumed to belong to $L^2(U,\mathbb{R}^d)$, while the zero-order coefficient is merely integrable. We prove that, even under these rough assumptions, well-posedness and quantitative energy and $L^2$ estimates remain valid. In addition, by using a previously established interpolation result, we characterize a trade-off between the integrability of the source term $f$ and that of the zero-order coefficient $c$, and show that well-posedness together with the corresponding quantitative estimates hold under these interpolated assumptions.

We consider the wave equation with variable wave speed in Euclidean space, with point sources modeled by Dirac delta initial displacement data. There are three special subsets of the whole space: (1) the source set where the delta initial conditions are supported, (2) the unknown set where the wave speed is not known a priori, and (3) the receiver set where the waves are measured. The inverse problem is to reconstruct the wave speed uniquely in the unknown set from an unlabeled collection of waves generated by point sources and measured in the receiver set. We use propagation of singularities and sharp finite speed of propagation to reduce this data to geometric travel-time data, whose form depends on how the three sets lie in relation to each other. We give three scenarios where this procedure leads to unique determination of the wave speed.

We prove that, on a bounded uniformly convex domain of class $C^{3,1}$, the first Robin eigenfunction is strictly log-concave and the Robin torsion function is strictly $1/2$-concave for all sufficiently large Robin parameters. This means that Conjecture 1.1 of Andrews, Clutterbuck and Hauer holds for uniformly convex $C^{3,1}$ domains in any dimension, and settles an open problem posed by Crasta and Fragalà when the domain has $C^{3,1}$ regularity. Our proof derives suitable uniform $C^2$-decay estimates by maximum principle arguments, thereby removing the imposed higher-order boundary regularity assumption required previously.

We study a family of functionals involving a product of logarithmic capacity and torsional rigidity. Using several geometric inequalities, we identify the discs as the only maximizers among planar convex sets for certain values of the parameters. Finally, we prove regularity of maximizers in the non-critical case. This is done by establishing a shape-differentiability theorem for the logarithmic capacity in the class of planar convex bodies.

We prove that the focusing and defocusing continuum Calogero–Moser equations are globally well-posed in $H^s_+(\mathbb R)$ for every integer $s\geq 3$. In the focusing case, this requires the initial data to satisfy the mass condition $$\|q_0\|_{L^2}^2<2\pi.$$ Our proof is based on a parabolic regularization of the equation, uniform a–priori estimates for the regularized solutions, and a Bona–Smith type approximation argument.

The Vlasov-Hartree system models a mixture of bosons and fermions interacting via Coulomb forces in which the bosons are described quantum-mechanically, while the fermions are described classically. For small initial data, we prove that the system exhibits modified scattering, i.e. the bosonic and fermionic subsystems each converge to solutions of the free Schrödinger and Vlasov equations respectively, but with coupled logarithmic corrections owing to the long-range nature of Coulomb interactions. We use a mixed Lagrangian and Fourier-based approach to derive explicit descriptions of the asymptotic dynamics of the solution and associated fields and Lagrangian trajectories. Notably, due to our purely Lagrangian analysis on the Vlasov side, we require no derivatives on the initial fermion density, the asymptotic fermionic profile is as regular as the initial data, and convergence occurs in almost the same space as the initial data.

We investigate positive radial solutions with an isolated nonremovable singularity for the semilinear elliptic equation \begin{align*} \Delta_{\mathbb{H}^N} u+\lambda u+u^p=0 \qquad\text{in }\mathbb{H}^N\setminus\{Q\}, \end{align*} where $N\geq 3$, $p>1$, $\lambda\le \frac{(N-1)^2}{4}$, and $Q\in\mathbb{H}^N$ is a prescribed pole. Our purpose is to describe how the locally Euclidean singular behavior near ${\{Q}\}$ interacts with the genuinely hyperbolic dynamics at infinity, and how this interaction changes across the Serrin and Sobolev critical exponents. For $\frac{N}{N-2}\le p<\frac{N+2}{N-2}$, we construct a family of positive radial singular solutions selecting the fast exponential mode at infinity. At the pole, these solutions exhibit the logarithmically corrected fundamental-solution profile when $p=\frac{N}{N-2}$, and the standard power-law profile when $\frac{N}{N-2}<p<\frac{N+2}{N-2}.$ At the conformally invariant pair $(p,\lambda)=(\frac{N+2}{N-2},\frac{N(N-2)}{4})$, we obtain an explicit singular solution and nonconstant Fowler-type solutions, and we classify all positive radial solutions with a nonremovable singularity via the positive periodic orbits of an associated autonomous equation. In the supercritical regime $p>\frac{N+2}{N-2}$ with $\lambda\le\frac{ N(N-2)}{4}$, we prove existence and uniqueness of the global positive radial singular solution, derive its two-term local asymptotic expansion near the pole, and establish a sharp trichotomy for its behavior at infinity.

In this paper, we investigate a class of magnetizable piezoelectric beam systems with two fractional infinite memory terms, whose fractional orders are denoted by \(\theta_1, \theta_2 \in [0,1]\). By introducing the Dafermos history variable, we construct an augmented state space and reformulate the system as an abstract evolution equation. The well-posedness of the system is established via semigroup theory. Through a frequency-domain analysis, we characterize the asymptotic behavior of the solutions. We show that the system is exponentially stable if \(\theta_1 = \theta_2 = 1\), and is polynomially stable in all other cases, with the explicit decay rate \(t^{-\frac{1}{2-2\theta_0}}\), where \(\theta_0 = \min\{\theta_1, \theta_2\}\). Furthermore, the optimality of the obtained decay rate is verified.

We construct a compressible isotropic energy on $\mathrm{GL}^+(3)$ combining polyconvexity and global true-stress-true-strain monotonicity (TSTS-M++), answering a question of Wollner, Holzapfel and Neff. For every $\mu>0$ and $k>0$, the energy $W_k(F)=\frac{\mu}{2k}[\exp(k(\lVert F\rVert^2+3J^{-1}+J-7))-1]$, $J=\det F$, is polyconvex and strictly rank-one convex. Its Cauchy-stress response satisfies TSTS-M++ globally if and only if $k\ge1/(8\sqrt3)$. The reference state is stress free with positive infinitesimal shear and bulk moduli. Stress bijectivity has the smaller sharp threshold $k_{\rm B}\approx0.0083$: at equality the stress map is a global homeomorphism with a nondifferentiable inverse, and above it a global smooth diffeomorphism. Thus, for $k_{\rm B}\le k<1/(8\sqrt3)$, the Cauchy-stress map remains globally bijective while TSTS-M++ fails at finite strain. In the TSTS-M++ regime, each prescribed inner radius of a finite plane-strain annulus with a traction-free outer wall has a unique radial equilibrium, and the inner pressure increases smoothly and strictly from zero to infinity. Comparisons with other energies distinguish the roles of polyconvexity, TSTS-M++, reduced convexity and endpoint growth. Under incompressible spherical inflation, global pressure inversion holds for the Demiray family with $b\ge b_*$, including $W_k|_{J=1}$ for $k\ge k_*$ and the stress-free $I_1$-exponential energy, and for the exponentiated Hencky energy when $k_{\mathrm{eH}}>3/8$. Incompressible neo-Hooke instead has a limit point. A further compressible $I_1$ energy retains the global constitutive properties and reference moduli of $W_k$ but exhibits two pressure turning points under incompressible spherical inflation, showing that these properties allow a descending pressure branch.

In this note, we revisit a Liouville-type theorem of Chae and Wolf for stationary Navier-Stokes equations in $\mathbb{R}^3$ [J. Differential Equations 261 (2016) 5541-5560]. We show that their logarithmic improvement of the classical $L^{9/2}$ condition is part of a substantially broader weighted framework. More precisely, we prove that a solution $u\in\dot H^1(\mathbb{R}^3)$ is necessarily trivial whenever $$ \int_{\mathbb{R}^3}|u(x)|^{9/2}\,\omega(|u(x)|)\,dx< +\infty, $$ for every positive, nondecreasing and bounded weight $\omega$ satisfying a mild growth condition near the origin. This structural condition encompasses the logarithmic weight due to Chae and Wolf, as well as a hierarchy of iterated-logarithmic weights and (genuinely) non-logarithmic examples, including a dyadic weight. Our result identifies a broader class of weighted integrability conditions under which the triviality of stationary Navier-Stokes solutions follows, and shows that the mechanism underlying the Chae-Wolf improvement is not intrinsically tied to a specific logarithmic weight or to a single logarithmic scale.

In this paper, we are concerned with the fine properties of entropy solutions of the Cauchy problem for the one-dimensional pressureless Euler system, wherein the initial density $\rho_0$ is a locally finite Radon measure and the initial velocity $u_0\in L^\infty_{\rho_0}$. We employ the solution formula introduced by [F.M. Huang and Z. Wang, Comm. Math. Phys. 222(1) (2001), 117–146.] for this Cauchy problem to analyze the entropy solutions and obtain various new fine properties of entropy solutions; these can be summarized in four aspects: (i) Characteristics and initial waves for the Cauchy problem; (ii) Fine local structures of entropy solutions; (iii) Divides and global structures of entropy solutions; (iv) Invariants and asymptotic behaviors of entropy solutions including the asymptotic profile and the corresponding decay rates. Through these results (i)-(iv), we establish the global dynamic patterns of entropy solutions of the Cauchy problem for the $1$-D pressureless Euler system with general initial data $\rho_0$ being locally finite Radon measures and $u_0\in L^\infty_{\rho_0}$.

The quasi-geostrophic equation is well known as a model for predicting potential temperature around low-pressure systems in high-latitude regions. When diffusion effects are added to this equation, it serves as a model for potential temperature at the sea surface. In either case, the nonlinear term represents the effect of the Coriolis force. This paper yields nonlinear profile in solutions to the surface quasi-geostrophic equation. This profile is determined uniquely from the perspective of large-time behavior of solutions. Components of solutions are arranged sequentially from slow to fast decay based on the parabolic scale. Then such an expansion is determined uniquely. Since the equation is subcritical in the context of large-time effects, the main components exhibit linear features. Nonlinear characteristics appear in components that have a smaller effect on solutions. From the perspective of the correlation between spacetime variables, nonlinear profiles exhibit characteristics that are distinctly different from those of linear profiles. Moreover, the nonlinear effects of this equation are expected to be weaker than those of other equations with the same scale due to their spatial structure. The main theorem publishes minute, but unique rotational flow arising from the Coriolis force.

In this article, we define classical and non-classical Besov and Triebel–Lizorkin spaces associated with the Grushin operator with or without drift. We establish various characterisations of these spaces, and study their complex interpolation, embedding properties, and fractional Leibniz rules.

Schultz [Comm.~Pure~Appl.~Math., 1998] first established dispersive estimates for the fundamental solution of the discrete wave equation on lattice graphs $\Z^d$ for $d=2,3$, which can be seen as a discrete analogue of the classical dispersive estimate for the Euclidean wave equation. His result was extended by Bi, Cheng and Hua to $d=4$ and $5$. In this paper, we give complete answers to all the remaining dimensions, including $d=5$. Our estimate is sharp for every $d\geq 5$. Moreover, the proof remains valid for the upper bound of the case for $d=4$.

Let $\Omega\subset\mathbb R^2$ be a bounded connected Lipschitz domain, and let $\{\mu_j(\Omega)\}_{j\geq1}$ and $\{\lambda_j(\Omega)\}_{j\geq1}$ denote the Neumann and Dirichlet Laplacian eigenvalues, respectively, counted with multiplicity. We prove that $$ \mu_3(\Omega)<\lambda_1(\Omega), $$ thereby removing the simple-connectivity assumption from the previously known planar result at the first Dirichlet threshold. We also establish a three-spectrum counting inequality relating the Dirichlet, Neumann, and conductivity spectra.

In this paper, we study the long-time asymptotics for nonlinear Fokker-Planck equations with spatial inhomogeneous diffusion, based on the law of energy dissipation. First, we derive a priori positive lower and upper bounds for solutions of the equations. Next, we give a sufficient condition for deriving the exponential decay of the dissipation function. Finally, we investigate whether the given sufficient condition is essential or not by numerical computation.

We study a coupled strongly damped wave–Moore–Gibson–Thompson (MGT) system with logarithmic source $f(u)=|u|^{\gamma-2}u\ln|u|$ in the full Sobolev-subcritical range $2<\gamma<2^*:=2n/(n-2)$. The main difficulty is that, in this full range, the logarithmic nonlinearity does not admit the standard compactness and difference estimates available in the lower subcritical regime. Using the augmented variable $w=v+\tau v_t$, we derive the exact energy-dissipation identity for the strongly damped system and exploit the associated coupled potential-well structure. For the local theory, a Faedo–Galerkin scheme combined with a closed nonlinear differential inequality and a spatial domain-splitting argument yields local existence. The strong damping provides the additional regularity needed to prove uniqueness throughout the full subcritical range and to establish a continuation principle. Below the well depth $d_\alpha$, we then obtain a sharp dynamical dichotomy: solutions with initial data in the stable set exist globally and decay exponentially, whereas solutions with initial data in the unstable set blow up in finite time. The blow-up argument reveals a structural cancellation specific to the augmented formulation: once the logarithmic contribution is reconstructed through the exact energy identity, the unfavorable MGT residual in Levine's concavity functional is absorbed algebraically. This allows the concavity method to close without imposing any additional sign condition on the initial velocities.

We define a class of infinite dimensional linear port-Hamiltonian delay systems, give a characterization for such systems, and provide an easily verifiable sufficient condition to belong to this class, thereby extending the study by Breiten, Hinsen and Unger in [IEEE Trans. Autom. Control vol. 69, no. 12 (2024), pp. 8924-8930, doi: 10.1109/TAC.2024.3464332]. The results rely on the operator semigroup theoretic treatment of linear delay PDEs.

We isolate and analyze the geometric PDE governing the evolution of the vorticity direction in the 3D incompressible (unforced) Navier-Stokes equations (NSE), restricted to the case of a critical spatial point singularity where the vorticity magnitude concentrates as $O(|x|^{-2})$, inhabiting the critical Lorentz space $L^{3/2, \infty}$. The PDE consists of the Harmonic Map Heat Flow (HMHF) into the sphere supplemented with the fluid transport, cross-diffusion and tangential strain. The question is whether the NSE mechanics can propagate logarithmic decay of the local mean oscillations of the direction – the condition $\xi \in \mathrm{bmo}_{1/|\log r|}$ – which (in this setting) was shown in the companion paper to prevent finite time blow-up. A key observation is that the $O(|x|^{-2})$ concentration, factored out of the viscous cross-diffusion, generates an outward radial drift $4\nu\, x/|x|^2$ at the core. Under a suitable condition we prove a barrier lemma for the total drift-diffusion operator and show that the HMHF nonlinearity is harmless for the quantity $\frac12|\xi - e|^2$ which is a subsolution on any hemisphere. This yields a transfer theorem: the $\mathrm{bmo}_{1/|\log r|}$ regularity of the direction at the core, uniformly up to the singular time, is controlled by a logarithmic modulus in time of the direction at the inner scale, together with the physical strain. Crucially, the strain enters the direction equation only through its tangential component which vanishes precisely when the direction is an eigenvector of the strain tensor. Since this includes the eigenvector carrying the maximal stretching, the result is in contrast to the classical geometric regularity theory which is built on depleting the full vortex-stretching term and thus confined to configurations of weak stretching.

Let $(M,g)$ be a closed Riemannian manifold of positive Ricci curvature. Let $f$ be a proper isoparametric function on $M$, and $\Gamma$ the unique level set of $f$ which is a minimal hypersurface. For any positive integer $k$, and $\lambda >0$ large enough, we construct a solution of the Allen-Cahn equation $ \Delta u + \lambda (u-u^3 ) =0$ which is constant along the level sets of $f$ and has exactly $k$ nodal components. We prove that the nodal components approach the minimal isoparametric hypersurface $\Gamma$ as $\lambda \rightarrow \infty$ and the energy is uniformly bounded for all such solutions.

We study front-type solutions of nonlinear dispersive-dissipative PDEs modeling the propagation of undular bores in a channel in two dimensions. The system extends the Korteweg-de Vries-Burgers (KdVB) equation by incorporating weak transverse motion, and admits the KdVB fronts as one-dimensional solutions. We investigate their stability under general two-dimensional perturbations. We prove that a one-dimensional front is a global asymptotic attractor in the weakly two-dimensional setting when the channel is sufficiently narrow in the transverse direction and the relative dispersion parameter lies in a range for stability to one-dimensional perturbations. Particularly, the front is unique up to spatial translations. The proof extends the energy method for temporally-modulated perturbed solutions, developed previously in the one-dimensional setting, to accommodate the transverse dynamics.

We prove that every global classical stable solution to the one-phase Bernoulli problem in $\mathbb R^4$ is one-dimensional. In particular, if nonempty, its free boundary consists of one or two parallel hyperplanes. This settles the stability conjecture for the Bernoulli problem in dimension four. As consequences, we obtain universal local Hessian and free-boundary curvature estimates for classical stable solutions in $\mathbb R^4$, as well as axial symmetry of global classical solutions with finite Morse index in $\mathbb R^4$.

We prove a weak-type $(1,1)$ estimate for the centered Hardy–Littlewood maximal function with respect to Euclidean balls with dimensional dependence $O(\sqrt{n} \log n)$. This improves the order of growth in the classical $O(n)$ estimate of Stein and Strömberg. The proof goes through a pointwise bound of the Hardy–Littlewood maximal operator by the heat maximal operator with $\sqrt{n}$ loss. The key technical aspect of our result is an improvement of the weak-type bound for the heat maximal operator from $O(\sqrt{n})$ to $O(\log n)$.

Data-Driven Computational Mechanics (DDCM) replaces traditional phenomenological constitutive models by directly reformulating boundary-value problems in terms of local material state data obtained from experiments or fine-scale simulations. Standard DDCM formulations, only enforcing equilibrium and compatibility, do not inherently guarantee compliance with the second law of thermodynamics. This breakdown occurs particularly when input material data sets are subject to noise or local physical non-admissibility. In this work, we present a variational DDCM framework specifically tailored to diffusion–reaction problems. Taking advantage of the simplicity of the thermodynamic constraint in gradient-flux systems, we propose an augmented formulation that explicitly enforces the second law of thermodynamics as a hard constraint within the energy-minimization problem. Although the set of thermodynamically admissible states is non-convex and fails to be weakly closed in the ambient phase space, we establish existence of minimizers by proving that the intersection of the admissible set with the subspace of fields that are compatible and in equilibrium is weakly sequentially closed via a compensated compactness argument. To enable practical computations, we analyze both a Lagrange multiplier formulation and a penalization scheme. We prove the $\Gamma$-convergence of the penalized functionals to the exact constrained problem and establish a fully discrete convergence framework incorporating spatial finite-element discretization and empirical data-set approximations. Numerical experiments confirm that the proposed penalty scheme effectively restores thermodynamic consistency even in the presence of severely corrupted material data.

We construct a framework for higher order unbounded rough drivers defined by rough paths over vector fields. This allows us to consider a purely Eulerian perspective for rough transport equations where the driving rough path vector field is allowed to have temporal $\mathfrak{p}$-variation for any $\mathfrak{p} \in (1,\infty)$. We use this framework to show the well-posedness of an equation with two transport terms; one driven by a rough path vector field with aforementioned $\mathfrak{p}$-variation in time, and the other driven by a DiPerna–Lions-type drift vector field.

We apply the variational method of Weighted Energy Dissipation (WED) to obtain a global-in-time approximation of solutions to fourth order degenerate parabolic equations of Cahn-Hilliard type. Differently from the standard approach to the existence theory, WED induces an elliptic regularization in time, not in space. The confinement on the phase field is guaranteed already for the approximation, a priori estimates follow without modification of Lyapunov functionals, and the approximations are weakly differentiable in time — even twice in the case of linear mobility. While WED has been widely used for the construction of gradient flows in Hilbert spaces, this appears to be the first application of WED to a metric gradient flow beyond second order PDEs of Wasserstein type.

Inspired by questions related to inverse problems and shape optimization, we derive an asymptotic expansion of the voltage potential, solution to a model elliptic second-order partial differential equation, under small perturbations of its boundary conditions. More precisely, the homogeneous Dirichlet or homogeneous Neumann boundary condition in a fixed, reference configuration of the problem is replaced by a Robin boundary condition with admittance $k_\varepsilon > 0$ on a “small” subset $\omega_\varepsilon$ of the boundary of the ambient domain, vanishing at the limit $\varepsilon \to 0$. In each of these two situations, a general asymptotic formula is established for the voltage potential, which rests on minimal assumptions about the shape of the vanishing subset $\omega_\varepsilon$ and the parameter $k_\varepsilon$. The scalings of these expansions, capturing the intensity of the perturbation, are measured by new quantities called “Robin-Dirichlet” capacity or a “Robin-Neumann” capacity, which depend on the geometry of $\omega_\varepsilon$ and on the value of the parameter $k_\varepsilon$. We analyze how these quantities compare to more classical measures of the “smallness” of $\omega_\varepsilon$, such as the capacity or the Neumann capacity of $\omega_\varepsilon$, according to the behavior of $k_\varepsilon$ as $\varepsilon \to 0$.