Papers

We are going to prove that energy minimizing harmonic maps from a domain $\Omega$ inside an ${\rm RCD}(K,N)$ whose image lies in a small ball inside a ${\rm CAT}(\kappa)$ space are locally Lipschitz continuous. This completes the picture concerning regularity of harmonic maps between singular spaces and justifies a variant of the Bochner-Eells-Sampson inequality between singular spaces.

We study the interior regularity of generalized solutions to the Monge–Ampère type equation governing optimal transportation for the relativistic cost $c(x,y)=\sqrt{a^2-|x-y|^2}$ on $\mathbb{R}^n$. We construct an explicit one-parameter family of radially structured generalized solutions on a ball and exhibit, among them, a solution that is of class $C^{1,\frac{1}{2n-1}}$ but of no better Hölder class: it fails to belong to $C^{1,\beta}$ for every $\beta>\frac{1}{2n-1}$. The construction reduces the equation to a planar autonomous system whose phase variable $s=\dot r$ vanishes to order $(2n-1)$ in the base variable. As an application, in dimension two we transfer the construction to the special Lagrangian curvature equation: for every phase $\Theta\in(0,\pi/2)$ we produce a sequence of smooth graphical solutions converging uniformly to a limit of class exactly $C^{1,1/3}$. Consequently the two-dimensional special Lagrangian curvature equation admits no pure interior $C^{1,\beta}$ estimate for any $\beta>\frac{1}{3}$.

This paper is concerned with Remez-type inequalities and their applications in observability inequality. Our aim is twofold. First, we establish the following fractal Remez's inequality on the unit sphere $\mathbb{S}^{n-1}$ \begin{align*} \sup_{\mathbb{S}^{n-1}} |p|\le C(M,N,n,\delta)\sup_{M} |p|, \end{align*} where $M \subset \mathbb{S}^{n-1}$ ($n \ge 2$) is a fractal set of positive $(n-2+\delta)$-Hausdorff content for arbitrary $\delta \in (0,1)$, and $p$ is a spherical polynomial of degree at most $N\in \mathbb{Z}^+$. Second, building upon this fractal framework, we establish sharp observability inequalities for the heat equation on the sphere, again valid for all $\delta\in (0, 1)$, which improve the result of Burq and Moyano [J. Eur. Math. Soc. (JEMS), 25 (4) (2023)] in the spherical setting. Furthermore, as an additional application, we prove a lower-dimensional observability inequality for the heat equation with super-quadratic potentials $V(x) = |x|^{2m}$ ($m \in \mathbb{Z}^+, m\ge 2$) on the whole space $\mathbb{R}^n$.

We prove that the full group \(\SL(3,\R)\) occurs as the set of one-particle deformation gradients of periodic, unforced, single-shell solutions of both the three-dimensional Euler and Navier–Stokes equations. More precisely, given a particle label \(p\in\T^3\), a time \(T>0\), and \(F_*\in\SL(3,\R)\), every sufficiently large odd integer \(N\) admits a real-analytic curl eigenfield \(W_N\) with \(\operatorname{curl}W_N=NW_N\). With an explicit scalar amplitude, \(W_N\) yields a steady Euler solution; with an explicit exponentially decaying amplitude, it yields a Navier–Stokes solution for any positive viscosity, and in both cases \(\nabla_aX(p,T)=F_*\). The lifted particle trajectory is an embedded analytic arc with nowhere-vanishing velocity. The construction combines global trace-free symmetric-matrix control on \(\SL(3,\R)\), a Beltrami Cauchy problem along the controlled arc, Runge approximation, inverse localization on the torus, and a finite-dimensional endpoint correction. We also classify finite material-direction systems that determine every trace-free strain after arbitrary volume-preserving deformation. In dimension \(n\), this congruence-robust property holds exactly when the associated rank-one projectors span \(\Sym(n)\); hence the sharp number of scalar channels is \(n(n+1)/2\). In dimension three, among minimal systems the undeformed outer-product lower bound is maximized exactly by the six axes of a regular icosahedron. The realization theorem shows that the full deformation-group quantifier in this sensing result is dynamically attained within the rigid class above.

An important problem in neuro-engineering is the stabilization of neural fields. In applications, it is often assumed that the actuator placement can be chosen arbitrarily. In this work, we investigate the stabilizability of controlled Amari-type neural fields where the control input is prescribed to act only on a fixed subset of the neural field. We show that the linearized neural field is open-loop stabilizable under suitable assumptions on the interaction strength and a mild relative density assumption on the control set. Our geometric assumption requires that the volume of each cube intersected with the control set must be bounded below. As a consequence, we derive closed-loop stabilizability of the neural fields, under sensor/actuator placement constraints. Numerical simulations are used to illustrate the results and obtain empirical estimates on the control cost.

We study the qualitative behavior of solutions of Grad-Shafranov type equations arising in plasma physics with general differential operators and general nonlinearities. In particular, we extend recent estimates about threshold values for uniqueness, monotonicity and non-existence of the free boundary. The argument is based on a refined spectral analysis for weighted non-local problems together with comparison techniques and level set analysis.

This paper introduces a Coulomb-corrected labeled energy for smooth compactly supported solutions of the three-dimensional repulsive Vlasov–Poisson system. The resulting identity replaces the direct electric-field term along a characteristic by a weighted singular integral, which can be estimated more effectively. Thus this structure yields improved bounds on the velocity support and the decay of the electric field.

This paper establishes a critical space–time density estimate for the three-dimensional Vlasov–Poisson system in the whole space and on the torus. This estimate has two consequences. First, every nonnegative weak solution with finite initial kinetic energy in the bounded finite-energy weak-solution class conserves the total energy. This gives a positive answer, within this class, to the energy-conservation question raised by Lions and Perthame. Second, in the periodic repulsive case, it yields propagation of every velocity moment of order \(k>2\), thereby closing the previously unresolved interval \(2<k\leq3\).

We establish sharp estimates for the discrete optimal constant of the fractional Hardy Inequality in dimension $N\geq 1$, with fractional exponent $s\in \left(0,\min\left\{1,\frac{N}{2}\right\}\right)$. The convergence rates that we establish take place for the Galerkin approximation with piecewise linear elements, when the computations are carried out in a bounded, convex and smooth domain containing the origin, for which we employ a quasi-uniform and regular mesh.

The compressible Euler-Riesz equations arise in the modeling of a wide range of physical phenomena, including stellar dynamics, plasma physics, and mathematical biology. In this paper, we investigate the nonlinear stability and instability of steady states for the multidimensional compressible Euler-Riesz equations under general pressure laws. In the polytropic case, we establish the nonlinear instability of steady states in the mass-supercritical regime for attractive potentials; this is achieved by analyzing the concavity of the free energy along mass-preserving dilations. At the mass-critical exponent, we show that, for any steady state, there exist solutions that start arbitrarily close to it, but develop growing support. For general pressure laws, we employ a concentration-compactness approach to prove the existence of energy minimizers and establish the nonlinear stability of steady states. Moreover, we quantify the finite-time stability by deriving a relative entropy bound for finite-energy solutions, without requiring uniform pointwise upper and lower bounds on the density. We further exploit the convexity of the second moment to obtain quantitative growth estimates for solutions with positive energy, thereby proving the local nature of the stability result. Finally, we prove the global existence of finite-energy weak solutions to the compressible Euler-Riesz equations with spherical symmetry for general pressure laws via the compensated compactness method, thereby yielding unconditional stability around steady states within the class of weak solutions. The approach developed in this paper should be useful for solving other nonlinear partial differential equations involving similar difficulties.

We consider the dissipative generalized surface quasi-geostrophic equations with a dispersive forcing term and study the behavior of solutions in the limit of vanishing viscosity. Using the approach of relative energy inequality, we prove convergence results in different dispersive regimes that extend the results obtained by Kosloff et al. (Appl. Math. Lett. (2019) 88 243-249) to more singular active scalar equations.

We prove a local weak Harnack inequality for nonnegative weak super-solutions to the anisotropic Trudinger equation. As an application, we show a proof of the Harnack inequality that bypasses any sort of Krylov-Safonov covering argument. We further develop an analysis of global sub-potential lower bounds which ultimately leads to the identification of a Barenblatt-type profile.

We consider the multi-dimensional Hamer model for radiating gases in its coupled hyperbolic–elliptic formulation. By means of energy estimates, we establish the global well-posedness for small initial data in hybrid Besov spaces with distinct regularity exponents at low and high frequencies. This framework enables us to relax the regularity assumptions required in \cite{Duan_Klem_Zhu_2010,Duan_Ruan_Zhu_2012}. In addition, we establish optimal time-decay estimates for solutions with initial data in the critical Besov space $\dot{B}_{2,\infty}^{-d/2}(\mathbb{R}^d)$, thus extending previous results obtained under the stronger assumption $L^1(\mathbb{R}^d)$. We discuss the optimality of these decay rates and derive improved decay rates under a zero-mass cancellation condition, corresponding to initial data in the larger negative Besov space $\dot B^{-d/2-1}_{2,\infty}(\mathbb R^d)$.

Consider a nonnegative harmonic function $u$ on a domain $D\subset\mathbb{R}^{d-1}$. We establish boundedness of Jones' variational integral on an ultradense, $d-1$ dimensional subset of the boundary of $C^{1,\mathrm{Dini}}$ domains. This result was previously only known for the unit ball.

We study the dynamics of periodic waves in a damped-driven nonlinear Schrödinger (NLS) equation perturbed by a small spatially dependent transport term. This equation arises in nonlinear optics as a model for light propagation in a passive optical cavity driven by a bichromatic laser source. Our main result shows that, for initial data close to a stable stationary wave of the translation-invariant unperturbed problem, the solution of the perturbed problem remains close to translated copies of this wave, with the translation parameter evolving according to an effective ordinary differential equation. If this effective equation admits a stable equilibrium, we prove that the solution of the perturbed NLS converges to a stationary state associated with that equilibrium. In particular, the asymptotically selected state attracts solutions with initial data that are not necessarily close to it, showing that its basin of attraction extends far beyond a small neighborhood of the state. The analysis is complicated by the loss of derivatives caused by the heterogeneous transport and combines the renormalization group method with a refined bootstrap argument that exploits parity-induced cancellations of quadratic nonlinear terms. Numerical simulations illustrate the analytical results.

We investigate the long-time behaviour of nonlinear kinetic Fokker–Planck equations with porous medium diffusion in a non-perturbative setting. Under a priori conditional bounds on macroscopic quantities, we establish exponential convergence to equilibrium in $L^1$. These bounds are automatically satisfied if the initial data is trapped between two global equilibrium profiles. Our approach combines the entropy-entropy dissipation structure and some techniques from $L^2$-hypocoercivity.

We construct segregated bubbling solutions for a two-component critical Schrödinger system of Brezis–Nirenberg type in a smooth bounded domain $\Omega\subset\mathbb R^N$, $N\geq5$, with any fixed competitive coupling $\beta<0$. Suppose that the Robin function has two distinct prescribed critical points, each satisfying a local degree condition. For every sufficiently small $\epsilon>0$, the system admits a nonnegative weak solution with both components nontrivial. Each component has a single-bubble profile and concentrates at one of the prescribed points. Each component also vanishes identically in a ball centered at the other concentration point; after rescaling by the natural bubble length, the radius of this ball tends to infinity. The main obstruction is that $p=N/(N-2)\in(1,2)$, so the gradient of the interaction potential $(s,t)\mapsto |s|^p|t|^p$ is not differentiable when one component vanishes and the other is nonzero. Hence the usual global Lyapunov–Schmidt reduction cannot be applied directly. We first solve a nonlinear exterior problem variationally. The resulting dead cores remove the cross-component coupling from the inner localization regions, where a projected critical reduction can then be carried out. The remaining scale and center equations are solved by Brouwer degree theory.

This paper is concerned with the spatial propagation dynamics of time-delayed nonlocal diffusion equations in shifting habitats. We employ the theory of viscosity solutions for Hamilton-Jacobi equations to provide a complete classification of the spreading speeds. In particular, we derive variational formulas that explicitly characterize how these speeds depend on the decay rate of the initial data, the habitat shifting speed, and the maturation delay across three regimes: locally determined, nonlocally selected, and locked. We reveal a distinct directional asymmetry in the nonlocal selection mechanism and a novel delay-insensitivity phenomenon. We also establish the threshold conditions under which the habitat locking effect eliminates the classical decelerating role of time delay.

We investigate a nonlinear quantum Fokker–Planck equation with self-consistent collision frequency, bulk velocity, and temperature. In contrast to quantum Fokker–Planck equations with prescribed diffusion and friction coefficients, the macroscopic quantities are nonlinear functionals of the distribution function. The equation preserves mass, momentum, and kinetic energy, admits a quantum entropy dissipation structure, and propagates the Pauli admissible range in the fermionic case. Its collision operator is also formally connected to the quantum Landau equation. For the Cauchy problem in the three-dimensional whole space, we prove the global-in-time existence and uniqueness of strong solutions near a global quantum equilibrium. The proof is based on a perturbative macro–micro energy method that combines microscopic coercivity, estimates for nonlinear velocity moments, and a macroscopic dissipation argument. We further establish the propagation of nonnegativity and the fermionic Pauli upper bound. Under an additional negative Sobolev assumption on the initial perturbation, we obtain algebraic decay rates toward equilibrium.

This paper investigates the convexity structure and contact set of the $Q$-tensor flow with anisotropic Landau–de Gennes elastic energy and a singular entropy potential $f(Q)$. The flow naturally exhibits a coercivity property, yielding the $H^1$ regularity of $f(Q)$ for almost every time. Combining this $H^1$ regularity with spherical averaging and capacity theory, the contact set $$\mathcal{C}(Q)\triangleq \{x\mid f(Q(x))=\infty\}$$ is characterized. In particular, it is shown to be empty in two spatial dimensions and to have Hausdorff dimension at most one in three spatial dimensions.

We study the Fisher–KPP equation on the half-line under Dirichlet,Neumann, and Robin boundary conditions. For the autonomous logistic equation, we identify bounded stationary profiles converging to $1$ and obtain exponential far-field comparison estimates. We prove local uniform convergence of nontrivial Neumann solutions to $1$. Assuming local uniform convergence of the Robin solution to its stationary profile, we derive asymptotic Neumann–Robin comparison estimates. We then consider small time-periodic Neumann and Robin boundary forcing. Under exponential stability of the homogeneous linearized semigroup, we construct a locally unique small periodic lifted mild solution and obtain a first-order expansion with a uniform $O(\varepsilon^2)$ remainder in $C_0([0,\infty))$.

Chemotactic motion is described by run-and-tumble kinetic models at microscopic scales and by Keller–Segel equations at macroscopic scales. We develop variational loss functionals for estimating the two components $T_0(x)$ and $T_1(x)$ of a turning kernel $T_\epsilon=T_0+\epsilon T_1$, where $T_0$ determines the leading-order turning rate and diffusion, while $T_1$ governs the macroscopic chemotactic drift. Under suitable regularity and data-informativeness assumptions, we establish conditional scale-uniform forward-stability estimates showing that a small loss leads to a small discrepancy between the forward solutions generated by the true and estimated kernels across the kinetic and diffusive regimes. Combined with sparse inversion, the method accurately recovers smooth, nonsmooth, and strongly heterogeneous kernels and remains robust under measurement noise.

We construct quantum Turing patterns in Lindblad lattice dynamics and establish a rigorous theory of their nonlinear order and quantum fluctuations. For an explicit family of completely positive lattice generators with finite-range couplings, the first-moment equations undergo a supercritical instability at a nonzero wave number and admit analytic site- and bond-centered commensurate stripe branches. These branches are locally asymptotically stable in their reflection-fixed period-cell spaces, and projected coherent states exhibit extensive Bragg order on every bounded time interval in the semiclassical limit. We prove $O(N^{-1/2})$ convergence of microscopic covariances to a nonautonomous Gaussian Lyapunov flow, transferring strict partial-transpose uncertainty violations to sufficiently large $N$. In the homogeneous Gaussian sector, a single dimensionless ratio controls both the Turing stability determinant and the logarithmic negativity of opposite momenta, relating wavelength selection directly to quantum entanglement. Differential transport shifts the strongest opposite-momentum correlations from the infrared to the selected Turing scale. Numerical continuation and two-dimensional simulations display stripe, spot, and labyrinth morphologies whose Fourier modes and fluctuation spectra concentrate at the same selected wave numbers.

We extend the Caffarelli-Hardt-Simon perturbation argument for truncated regular minimal cones to the special Lagrangian setting and prove a bridge principle for regular special Lagrangian cones in the spirit of Nathan Smale. Our bridge principle yields a general existence theorem for conically singular special Lagrangian submanifolds with prescribed regular tangent cones: for any finite list of such cones in $\mathbb C^m$ having the same Lagrangian angle and suitably arranged, there exists a connected special Lagrangian submanifold with boundary and isolated conical singularities whose tangent cones at its singularities are precisely the prescribed cones. When applied to the graphical special Lagrangian cone in $\mathbb{C}^5$ recently discovered by Bhattacharya, Orriols, and Skorobogatova, we obtain infinite families of $C^{1,1}$ viscosity solutions to the special Lagrangian equation having any finite number of isolated singularities.

Can a bounded planar domain have a simple Dirichlet spectrum with uniformly separated consecutive eigenvalues? Dimension two is critical: Weyl's law permits both uniform separation and arbitrarily small gaps. We prove that uniform separation is nevertheless exceptional in a natural rough-domain setting, even after multiplicities are removed. Let $D\subset\mathbb{R}^2$ be a bounded domain and, for $\ell\geq 1$, let $\mathcal{C}_\ell(D)$ be the space of nonempty connected open sets $\Omega\subset D$ such that $\overline{D}\setminus\Omega$ has at most $\ell$ connected components, endowed with the complementary-Hausdorff topology. We prove that $\mathcal{C}_\ell(D)$ is completely metrizable and Baire, and that smooth domains are dense in it. If $0<\nu_1(\Omega)<\nu_2(\Omega)<\cdots$ are the distinct Dirichlet eigenvalues, our main result states that \[ \left\{\Omega\in\mathcal{C}_\ell(D):\inf_{m\geq 1}\bigl(\nu_{m+1}(\Omega)-\nu_m(\Omega)\bigr)=0\right\} \] is residual. This statement requires no simplicity assumption. Combining it with our transfer of Micheletti's classical generic-simplicity theorem to $\mathcal{C}_\ell(D)$ shows that a generic domain has simple spectrum and consecutive gaps with zero lower limit. The proof uses Šverák's planar spectral continuity theorem and a local surgery that implants an arbitrarily high pair of close consecutive distinct eigenvalues.

We consider the two-species particle system introduced by Stevens (2000) related to the doubly parabolic Keller-Segel equation. It consists of $N$ cells and of a varying number of chemoattractant particles. Cells diffuse in the plane and follow the (mollified) empirical gradient of concentration of chemoattractant. Chemoattractant particles are produced by cells at some constant rate, diffuse and disappear at some constant rate. We show that when the sensitivity of cells to the chemoattractant is small enough, under some rather weak condition on the family of mollifiers, this system approximates the parabolic-parabolic Keller-Segel equation as $N\to \infty$. We also prove that when $N$ is fixed and when the production rate of chemoattractant particles tends to infinity, this system approximates the (non-Markovian) one-species system introduced Talay-Tomašević (2020) and further studied by the authors (2023).

We study the quasilinear Liouville equation \[ -\Delta_n u=e^u \] on complete noncompact Riemannian manifolds with nonnegative Ricci curvature. Our first result shows that, if a solution $u$ satisfies the optimal logarithmic lower bound \[ u(x)\ge -\frac{n^2}{n-1}\log r(x)+o(\log r(x)) \quad \text{as }r(x)\to+\infty, \] then the underlying manifold is isometric to the Euclidean space and $u$ is a standard bubble solution. Both the leading coefficient and the remainder term in the assumption are sharp. The key ingredient in the proof is the connection between the logarithmic lower bound and a sharp upper bound on the total volume of the solution. We also formulate a conjecture concerning the interaction between the sub-logarithmic decay of solutions and the underlying geometry, and prove it for $n=2$, as well as for $n\ge 3$ under a strengthened assumption.

We prove Pólya's conjectured lower bound for the Neumann eigenvalue counting function of Euclidean balls. If $B_R^d\subset\mathbb R^d$ is the ball of radius $R$, then, for every $d\ge2$, $R>0$, and $E\ge0$, $$ N_{B_R^d}^{<}(E) \ge \frac{\omega_d}{(2\pi)^d}|B_R^d|E^{d/2} = \frac{(R\sqrt E)^d}{2^d\Gamma(\frac d2+1)^2}, $$ where $\omega_d$ is the volume of the unit $d$-ball and $N_{B_R^d}^{<}(E)$ counts Neumann eigenvalues strictly below $E$. Combined with the Dirichlet theorem for balls, this settles both Pólya inequalities for Euclidean balls in every dimension $d\ge2$. In the disk case, the proof replaces a computer-assisted finite-frequency step by explicit Rayleigh–Ritz estimates. In dimensions $d\ge3$, the radial Neumann condition is a Dini condition rather than a derivative-zero Bessel condition. A strict comparison with an auxiliary Robin problem transfers a derivative-zero Bessel phase estimate to the physical Neumann spectrum. The problem then becomes a comparison between a multiplicity-weighted phase staircase and an integral equal to the Weyl term. Variational trial spaces control low frequencies; finitely many radial levels and beta-integral estimates cover the intermediate range; and a uniform phase estimate treats high frequencies. All finite computations for $2\le d\le6$ are printed in the paper. For $d\ge7$, one compact two-parameter estimate is verified in exact rational arithmetic by the ancillary program.

In this paper, we study semilinear elliptic systems with sublinear coupling terms, showing that, under competition-type interactions, solutions typically have dead cores, i.e., they vanish on certain open subsets of the domain. We apply our results to a large class of solutions treated in the literature, for instance to ground states and least energy sign-changing solutions, under Dirichlet boundary conditions or in the whole space. The proofs are based on a general result stating that subsolutions to a certain sublinear equation have dead cores, and on uniform Hölder bounds.

We investigate large solutions of linear and semi-linear equations involving the fractional Laplacian on domains that are becoming unbounded in some, but not all, directions. Solutions that blow up on the boundary of a domain are commonly called large solutions. For local operators, such behaviour arises only in the presence of lower-order nonlinear terms. However, for nonlocal operators, the existence of such solutions is more subtle. In contrast to the local case, the nonlocal nature of the operator gives rise to different boundary blow-up phenomena even in the case of linear equations. In this article, we study the existence and qualitative behaviour of boundary blow-up solutions to fractional linear and semi-linear equations on finite cylindrical domains and employ these properties to construct large solutions on infinite cylinders.