Papers

For the Selberg zeta function of geometrically finite infinite-area hyperbolic orbisurfaces with twists by finite-dimensional unitary representations, we establish a factorization formula in terms of a Weierstrass product of the Laplace resonances of the considered hyperbolic orbisurface, Barnes G-functions, gamma functions, and the singularity degrees of the representation. We thereby provide an interpretation of the zeros and poles of the Selberg zeta function by spectral and geometric entities of the orbisurface and the representation. This formula generalizes the factorization result by Borthwick, Judge and Perry to hyperbolic orbisurfaces with orbifold singularities as well as to unitary twists. Also in the untwisted case, the presence of orbifold singularities yields a separate, previously unobserved contribution to the factorization formula.

The integrability of the Kudryashov-Sinelshchikov equation in (1+3) dimension is detected using its point symmetries and the Singularity analysis method. The symmetries mostly points to the existence of infinite-dimensional Lie algebra with the presence of mostly arbitrary functions. The reduction process is initiated by considering these arbitrary functions in the general symmetry vectors. For certain functions, trivial solutions for the general parent equation is obtained whereas for some of the functions, the reductions does not yield an unmitigated result. Finally, a positive result is listed for one of the arbitrary function, for which the singularity analysis method is employed to study the Painleve property. The resultant PDE happens to be a fourth-order equation and its approval of the Painleve property hints at the integrability of the Kudryashov- Sinelshchikov equation in the general case.

We investigate finite-dimensional feedback stabilization for nonlinear nonautonomous stochastic parabolic equations driven by $Q$-Wiener, covering both additive and multiplicative perturbations. The control is given by a finite linear combination of localized indicator-type actuators whose supports are selected as part of the construction and may have arbitrarily small total measure. The feedback law is constructed by means of oblique projections onto suitable finite-dimensional subspaces. Within the variational Gelfand triple framework, we prove well-posedness of the closed-loop system under standard coercivity, growth, and global Lipschitz assumptions. By appropriately choosing the actuator configuration and feedback strength, we establish exponential mean-square stabilization of the stochastic dynamics and, for pure multiplicative noise, almost-sure stabilization. A fully discrete three-layer implementation complements the theoretical results. Numerical experiments illustrate the influence of number of actuators, noise intensity, and nonlinear effects on the closed-loop stabilization behavior.

Given $s\in (0,1)$ and a bounded Lipschitz domain $\Omega\subset \mathbb{R}^{n}$, we establish a quantitative Dahlberg theory for the $s$-harmonic measure of $\Omega$, $\omega_s^x$. In the nonlocal setting, the natural reference measure is an integral weight $\sigma_{s}$ in $\Omega^{c}$ that behaves like $(1-s)\text{dist}(\cdot, \partial \Omega)^{-s}$ close to the boundary. Our main result is a scale-invariant reverse-Hölder estimate for the density $d\omega_{s}^{x}/d\sigma_{s}$ on boundary-centered balls. As a consequence, we obtain $L^2(\Omega^c,\sigma_s)$-solvability of the exterior Dirichlet problem, with estimates for a nonlocal non-tangential maximal function and uniqueness in the natural distributional class. A weighted Gehring argument improves the reverse-Hölder exponent beyond $2$ and consequently yields $L^{q}$-solvability for a range of exponents extending strictly below $2$. Our results apply to general symmetric stable operators comparable to the fractional Laplacian. Moreover, the proofs are compatible with the limit $s\to 1^-$ and thus yield the corresponding results for the Laplacian in the nonlocal-to-local limit. The main new step is to convert a fractional Pohozaev identity for the Green function into uniform square estimates on distance level sets of a Lipschitz boundary. As applications, we derive optimal Sobolev regularity estimates for the homogeneous weighted Dirichlet problem and for the inhomogeneous Poisson problem with zero exterior data.

On $d$-dimensional cylinders $\mathcal{C}= \mathbb{R}^k\times N$, with a closed manifold $N$ as base and large scale dimension $k\in[1,d)$, we prove quadratic estimates in weighted $L^2$ space for Dirac operators perturbed by bounded, measurable and accretive coefficients. This gives in particular homogeneous Kato square root estimates on $\mathcal{C}$ for Riesz transforms associated with second order divergence form elliptic operators, having measurable coefficients with degeneracy governed by a Muckenhoupt $A_2$ weight. By localisation and scaling, it also yields local quadratic estimates for perturbed Dirac operators on general manifolds with locally thin cylindrical geometry, and possibly with zero injectivity radius.

We study four-wave kinetic equations in space dimension three with power-law dispersion $\omega(p)=|p|^a$ and collision kernels with high-frequency growth measured by $\beta$. In weighted $L^\infty$ spaces, we identify the sharp decay threshold $$ s_c=4\beta+3-\frac a2. $$ For $s>s_c$, we prove local well-posedness by establishing trilinear bounds for the full gain-loss collision operator. For $s<s_c$, we prove ill-posedness by constructing data concentrated near a high-low-low-high resonant configuration. This threshold captures the balance between the high-frequency strength of the kernel and the geometry of the resonant manifold. The proof also shows that gain-loss cancellations are essential in the most delicate regimes.

We present a general framework for producing families of harmonic maps from the unit ball into the Euclidean sphere starting from a fixed harmonic map. In particular, our analysis provides a theoretical background for a generalized radial projection which was recently introduced by Nakauchi. Our approach is based on work of Toth who established a machinery that generates eigenmaps between spheres from a given eigenmap. Finally, we point out how these families of harmonic maps can be used to construct solutions to associated variational problems such as intrinsic biharmonic and triharmonic maps, $p$-harmonic maps as well as extrinsic polyharmonic maps.

We study the existence of standing wave solutions for the following quasilinear Schrödinger equations with critical growth on the Heisenberg group $$ -\Delta_{\mathbb{H}} u +V(\xi)u-\Delta_{\mathbb{H}} (\left|u\right|^{2\alpha})\left|u\right|^{2\alpha-2} u= \lambda \left|u\right|^{q-2}u + \left|u\right|^{p-2}u \text{ in }\mathbb{H}^N $$ where $\mathbb{H}^N$ is Heisenberg group, $\Delta_{\mathbb{H}}$ is Kohn Laplacian operator, $4\alpha <q<p \leq 2\alpha Q^{*}$, \(Q^{*}= \frac{2Q}{Q-2}\) is the critical Folland–Stein exponent, $\lambda$ and $\alpha $ are positive parameters, $\alpha > \frac{1}{2}.$ By a suitable nonlinear change of variables, the quasilinear equation is transformed into a semilinear one, allowing the use of variational methods in the Folland–Stein Sobolev space $S^{1,2}(\mathbb{H}^N)$. Applying the mountain pass theorem together with a concentration–compactness argument adapted to the sub-Riemannian framework, we establish the existence of a nontrivial solution.

In this article, we investigate the following modified quasilinear equation with parameter driven by the $Q$-subLaplacian: \begin{align*} \begin{cases} -\Delta_Q u - \Delta_Q\bigl(|u|^{2\alpha}\bigr)\,|u|^{2\alpha-2} u = \lambda f(\xi,u) & \text{in } \Omega, \\[2mm] u = 0 & \text{on } \partial\Omega, \end{cases} \end{align*} where $\Delta_Q(\cdot):= \mathrm{div}_{\mathbb{H}}\bigl(|\nabla_{\mathbb{H}}(\cdot)|^{Q-2}\nabla_{\mathbb{H}}(\cdot)\bigr)$ denotes the $Q$-subLaplacian on the Heisenberg group $\mathbb{H}^N$, $Q=2N+2$ is the homogeneous dimension, $\Omega \subset \mathbb{H}^N$ is a smooth bounded domain, $\lambda>0$, $\alpha > \frac{1}{2}$, and $f$ has critical or subcritical exponential growth of order $\exp\bigl(\beta|t|^{2\alpha Q/(Q-1)}\bigr)$. We prove three results: the existence of a nontrivial positive weak solution in the critical case for all large $\lambda$, and the existence of a least-energy nodal solution with exactly two nodal domains, under subcritical and critical exponential growth. A change of variables $u=g(v)$ reduces the problem to a quasilinear problem whose energy functional is of class $C^1$; the exponent $2\alpha Q/(Q-1)$ arises from the growth of $g$. We handled the exponential growth using the sharp Moser-Trudinger inequality of Cohn and Lu. The positive solution is obtained by the mountain pass theorem and the nodal solutions by minimization on a nodal Nehari set.

This paper investigates a class of nonlinear elliptic obstacle problems involving measure data, where the nonlinearity depends on the variable of the possibly unbounded solution itself. We establish pointwise gradient estimates for the solutions in terms of Riesz potentials.

In this paper, we consider Nakao's problem in one space dimension, which couples a semilinear damped wave equation with a semilinear wave equation. For compactly supported energy data satisfying suitable integral sign conditions, we establish sharp lifespan estimates for all powers $p,q>1$. The estimates are classified according to whether the total integral of the initial speed in the wave equation is positive or zero.

We show anchored versions of the Nash inequality for discrete non-local divergence-form operators with degenerate weights. They allow to control the $L^{2}$-norm of a function by Dirichlet forms that are not uniformly elliptic. We then use them to provide on-diagonal heat kernel upper bounds for a class of random conductance models with degenerate jump rates allowing long-range jumps. The results are established on a class of graphs including the integer lattice and possibly correlated supercritical percolation clusters.

This article investigates the existence of closed, star-shaped hypersurfaces for a class of Hessian curvature equations. By combining a priori estimates with the continuity method, we establish the existence and uniqueness of $k$-convex hypersurfaces for both nonhomogeneous and homogeneous Hessian curvature equations, and by establishing a constant rank theorem, we prove that the resulting $k$-convex hypersurfaces are strictly convex.

This is the second part of a two-paper sequence establishing the global existence of $3D$ relativistic Vlasov-Maxwell system (RVM) for arbitrarily large smooth localized initial data with cylindrical symmetry. In Part II, independently of the results in Part I, we establish two sets of pointwise estimates for the localized electromagnetic field. This is achieved by exploiting cylindrical symmetry, conservation laws, smoothing effects, and structural observations of the electromagnetic field, including its null structure, double null structure, and the magnetic field dichotomy. The first set provides a uniform upper bound for the localized electromagnetic field. The second set offers an upper bound that depends on the distance from the z-axis, providing better control when the electromagnetic field is located further away from the $z$-axis. These two sets of estimates are fundamental to the iterative smoothing scheme (ISS), which is originated and inspired by the work of Klainerman-Staffilani(2002). These two sets were treated as black boxes in Part I. A defining feature of these estimates is that they are assessed concerning the magnitude of the oscillation phase over characteristic time, effectively quantifying the smoothing effect. It follows from these estimates that cases without any smoothing effect are error terms. Addressing this constitutes the initial challenge in the bootstrap argument.

We study strongly indefinite nonlinear Dirac functionals in the lowest partial-wave channel with radial coefficients in $\mathbb R^3$. The symmetric model has the exact power nonlinearity $F_0(x,\psi)=b(|x|)|\psi|^p/p$, where $2<p<3$ and the radial profile $b$ is bounded, continuous, and uniformly positive. Compactness in this finite-angular-mode spinor channel and radial approximation-number estimates give quadratic growth of the symmetric minimax levels. For a non-even localized perturbation of order $1<\tau<p/2$, we impose a semiconvexity condition whose negative curvature is strictly smaller than the spectral gap. Together with the negative quadratic part and the convex exact-power core, this makes every negative spectral fiber uniformly strongly concave. Maximizing along that fiber reduces the problem to a $C^1$ path on the positive spectral space. The polynomial Chambers–Ghoussoub–Bolle deformation theorem then yields infinitely many high-energy critical points of the restricted partial-wave functional; under the channel-invariance hypothesis these are weak solutions of the full Dirac equation. Convex perturbing primitives are covered without a smallness restriction on their amplitude. The same reduction applies after an even Hermitian quadratic term is absorbed into a renormalized Dirac operator, provided the renormalized operator has a gap at zero and the remaining perturbation satisfies the corresponding semiconvexity bound.

Stein variational gradient descent (SVGD) transports interacting particles toward a target distribution through deterministic kernelized dynamics. Singular Riesz kernels are attractive because they can provide quantitative population-level convergence, but at the finite-particle level the corresponding Stein energy has infinite self-interaction. We study periodic Riesz SVGD with self-interaction removed and prove a many-particle, long-time sampling theorem. Throughout the range in which the singular Stein energy is locally integrable, under a uniform bound on the initial relative entropy per particle, the time-averaged empirical-measure law converges weakly to the point mass \(\delta_\pi\) at the target as the particle number and any diverging averaging horizon tend to infinity. We also show that the empirical-measure laws induced by invariant particle laws of finite relative entropy converge weakly to \(\delta_\pi\), without a uniform entropy bound. Below the logarithmic singularity threshold, we obtain an explicit algebraic finite-particle error bound. These results extend the joint-entropy approach for smooth-kernel SVGD to singular interactions.

I present an overview of different phases used in the literature to describe the phase of travelling waves in reaction-diffusion equations. The discussion is tailored to understand the effectiveness of these different phase descriptions in a stochastic setting in 1D. The explicit computation of the autonomous isochronal phase for the Nagumo equation is a new result.

We consider hyperbolic orthogonal ring patterns as introduced in arXiv:2409.06573 and focus on their characterization by uniformizing variables at the centers of the rings. Given a smooth solution of the sinh-Gordon equation $\Delta u -\sinh(2u)=0$, we restrict to a compact subset of its domain and discretize it by square grid lattices with edge length $\varepsilon$. Taking the values of $u$ as Dirichlet boundary conditions, we prove that the corresponding uniformizing variables $u^\varepsilon$ of the hyperbolic ring patterns converge to $u$ in $C^\infty$ with error of order $\varepsilon^2$, given that the pairs of rings suitably converge to circles. As a consequence we deduce that the hyperbolic orthogonal ring patterns converge to a harmonic map to the hyperbolic plane.

We prove that $\mu$ is a semiclassical measure associated to a sequence of eigenfunctions of energy $E<0$ of the attractive Coulomb operator if and only if $\mu$ is a probability measure on the energy (hyper)surface $\Sigma_E$ invariant under the regularized Kepler flow due to Moser. The converse was shown in recent work by the author, and the present article proves the other direction (as well as an independent proof of the converse). We prove the main theorem for a general symbol class allowing certain non-decay at infinity, which implies that semiclassical measure mass entirely reflects off of the origin. In the special case of semiclassical measures of sequences of eigenfunctions of the exact Coulomb operator, this article solves an open problem posed by Keraani. The main tools include the celebrated Moser-Fock map along with a technical operator extension lemma in $\Psi_{\hbar}^0(\mathbb{S}^d)$, which utilizes standard eigenfunction concentration bounds.

Wyman and Xi (2023) proved a Kuznetsov-type theorem for the submanifold counting function associated with Laplace–Beltrami eigenvalues. We refine the results of Wyman and Xi, improving their asymptotic: we remove a constant in the codimension-one case, and we prove a new asymptotic in the case that the set of looping times has measure zero.

Solutions of the slightly compressible shallow water equations on a rapidly rotating sphere are shown to be bounded uniformly when ratio of the Froude number to the Rossby number is bounded. Moreover, in the singular limit in which the ratio of those parameters remains fixed while they both tend to zero, solutions with well-prepared initial data tend to corresponding solutions of limit equations. A convergence result is also obtained for the three-scale singular limit in which the Froude number tends to zero faster than the Rossby number.

In this article, we study scattering and inverse scattering problems for 3D nonlinear Schrödinger equations of the form, $$ \mathrm{i}\partial_t u + \Delta u = a(x)|u|^2u.$$ For $a(x)$ in a suitable class, such equations are globally well-posed and admit a large-data scattering theory. In this work, we prove that the large-data scattering behavior uniquely determines the inhomogeneity $a(x)$.

We study the singular limit of a nonlocal conservation law incorporating both non-locality in density and velocity. We derive classical solutions of the underlying Burgers equation as a singular limit. The derivation is valid up until the maximal time before formation of shocks. The main result answers a question raised in [10].

We study the non-equilibrium steady state of a weakly anharmonic chain of $N$ oscillators driven at its boundary by Langevin thermostats at unequal temperatures. Under a perturbative weak-anharmonicity condition, we prove a full-gradient logarithmic Sobolev inequality whose constant is independent of the chain length $N$. For homogeneous pinned chains, an additional quantitative regularity assumption yields a boundary space-time logarithmic Sobolev inequality and relative-entropy decay on the same $O(N^3)$ relaxation time scale as the harmonic chain. The proof extracts a finite-dimensional Gaussian component from the boundary noise and compares conditional terminal-state laws by a change of variables. The estimates are uniform over bounded positive temperatures and require no near-equilibrium assumption on their difference.

We investigate traveling wave fronts for an acid-mediated tumor invasion model with density-dependent degenerate diffusion. The model is a partially diffusive PDE–ODE system of Gatenby–Gawlinski type, in which the tumor diffusion coefficient $D(U)$ is allowed to be a general decreasing function satisfying $D(1)=0$. This degeneracy causes the traveling wave equation for the tumor component to lose uniform ellipticity near the healthy state, and hence standard arguments for nondegenerate reaction diffusion systems are not directly applicable. To overcome this difficulty, we introduce a nonlinear change of variables which removes the degeneracy from the highest-order term of the tumor equation. For each fixed admissible tumor profile, the acid profile is represented by a Green kernel, while the healthy-tissue profile is obtained from an explicit integral formula. The transformed tumor profile is then constructed as the stationary limit of a uniformly parabolic auxiliary problem. By combining comparison principles, local Schauder estimates, carefully chosen super- and sub-solutions, and the Schauder fixed-point theorem, we prove the existence of traveling wave fronts for every wave speed $\theta\ge 2\sqrt{rD(0)}$. The resulting wave connects the tumor-dominant state $(0,1,1)$ at $z=-\infty$ to the healthy state $(1,0,0)$ at $z=+\infty$. We further establish strict pointwise bounds, monotonicity of all wave components, and one-sided exponential asymptotic estimates in both the transformed variable and the original traveling-wave variable.

We show short-time existence and uniqueness up to reparametrization for a large class of higher-order geometric flows of curves and networks obtained as $L^2$-gradient flow of higher-order functionals involving derivatives of the curvature. Additionally, we prove a long-time existence result via energy methods.

We establish all order long-time asymptotic expansions for heat kernels on nilpotent coverings and for prime closed geodesics in fixed central classes of nilpotent quotients of compact hyperbolic surfaces. The exact lattice-side input is the finite-dimensional rational Floquet-Bloch theory of the companion paper: rational Kirillov restrictions give exact finite-dimensional fibers, and a generalized Pytlik functional gives exact Fourier-inversion and normalized-trace identities. At a rational parameter $p/q$ the decomposition is exact, and the fluctuation of the fiber integrand is controlled only by $q$. Hence the large-denominator comparison with the smooth Kirillov or Schrödinger normal form is uniform on the rational support of the Pytlik functional; irrational parameters do not enter the rigorous trace argument. For general nilpotent models, coefficient-weighted spectral sums are justified to every fixed order by positive Rockland estimates, the Plancherel-Mellin formula, and a trace-level order-balance argument. In contrast with approaches which usually give leading terms or integrated Edgeworth-type asymptotics, the method gives genuinely local, pointwise higher-order heat-kernel expansions. The same representation-theoretic quantity governs the leading term in the closed-geodesic asymptotics, producing a nilpotent Chebotarev-type phenomenon. The Heisenberg model is computed to the first correction term, and the Engel model is represented through the resolvent and heat-kernel calculus of the quartic oscillato

We study the Kac particle system associated with the spatially homogeneous Boltzmann equation with non-cutoff collision kernels in the moderately soft potential regime $-1<\gamma<0$. We prove uniqueness in law for the particle system and, in particular, establish an explicit convergence rate from the empirical measure of the Kac particle system to the weak solution of the Boltzmann equation in Wasserstein-2 distance, assuming that the initial datum $f_0$ has finite Fisher information and polynomial moments. To the best of our knowledge, this provides the first quantitative convergence rate for the Kac particle system in the soft potential setting.

We construct a short-time parametrix for the Fokker–Planck semigroup in Euclidean space. Among possible applications, we obtain smoothing and localization properties of the semigroup, the derivation of an approximate short-time polar decomposition for the semigroup, the construction of a parametrix of the resolvent, pseudospectral estimates, and estimates of the asymptotics of the number of eigenvalues of the Fokker–Planck operator. As a step in the proofs, we introduce a class of operators, which we call subsectorial operators, to which the Fokker–Planck operator belongs, and describe some of their functional analytic and spectral properties.

The article focuses on the inviscid and non-resistive limit of the incompressible magnetohydrodynamics (MHD) equations in a bounded domain. We derive sufficient conditions for the limit of Leray-Hopf solutions of the MHD system, which converges to the weak solutions of the ideal MHD system when $\mu,\nu\rightarrow0$, under the assumptions of uniform interior regularity and uniform equi-continuity at the boundary. Therefore, we establish an analogue of the Kato-type criterion to show that the energy dissipation rate is independent of the coefficients of viscosity and resistivity. Moreover, to analyze the different decay rates for the viscosity and resistivity, we obtain the limits of infinite and vanishing magnetic Prandtl number ($Pm=\frac{\mu}{\nu}$) for the incompressible MHD system. To the best of the authors' knowledge, this is the first rigorous result from a mathematical viewpoint on the limit of the magnetic Prandtl number.