Papers

In convex bounded domains in R^n with n >= 3, we establish interior pointwise upper bounds for the Dirichlet Green's function of elliptic operators in the unit ball B(0,1) in R^n, n >= 3, whose principal part is the Laplacian and which include a drift term that diverges near the boundary like a negative power of the distance with exponent strictly less than 1. This work extends an earlier result for operators with such drifts in the unit ball, and streamlines the proof in particular to adopt it to the question in convex domains.

Motivated by thrombus modeling, we study a modified Navier-Stokes-Cahn-Hilliard system and consider PINN-based numerical illustrations for the modified system. To enable the analysis, we introduce a diffusion-enhanced system for the deformation variable while preserving the associated dissipative energy structure. We prove local well-posedness for this new system. We also present PINN-based numerical illustrations for representative thrombus cases and report residual losses and benchmark errors obtained with Metropolis-Hastings sampling based on the energy decay.

We introduce new kinetic equations to describe epidemics' spread while accounting for individuals' opinions on protective behaviors. Opinion exchanges occur on a social network represented by a graphon, whose choice strongly influences the dynamics and leads to the emergence of complex nonlinear phenomena, like the creation of opinion leaders or the spontaneous formation of epidemic waves. Starting from individual-based interactions, we derive a nonlinear nonlocal Fokker-Planck model involving reaction terms and degenerate drift-diffusion operators, which depend on the underlying graphon. We establish rigorous results of convergence to equilibrium in $L^1$ space, via relative entropy estimates, and in homogeneous Sobolev spaces $\dot{H}^{-s}$, $s \in \big(\frac{1}{2}, 1\big)$, using Fourier-based techniques. We then design a structure-preserving scheme for the coupled opinion-epidemiological system, highlighting graphon effects: opinion leaders supporting protective behaviors limit disease spread, whereas influenceable individuals may shift toward opposing views, worsening epidemics. At last, we introduce a time-dependent quantity analogous to the effective reproduction number, whose oscillations are linked with the formation of epidemic waves. Notably, these waves are not induced by an explicit external forcing but they naturally emerge from the interactions between agents, depending on the connectivity level prescribed by the graphon.

This paper studies plane domains satisfying a quadrature identity with respect to the singular weight $\rho_0(w)=|w|^{-2}$. These are referred to as log-weighted quadrature domains (LQDs). The logarithmic singularity at $w=0$ leads to phenomena not present in the classical theory: in particular, when the domain contains the origin, the associated quadrature data are no longer unique, but are determined only up to a point charge at $0$. A generalized Schwarz function characterization of LQDs is established together with a natural formulation of the inverse problem in the singular setting. In the simply connected case, it is shown that a domain is an LQD if and only if the outer factor of its Riemann map extends to the exponential of a rational function. This characterization yields explicit formulae relating the quadrature function and the Riemann map via the Faber transform, thereby extending earlier formulae from the non-singular theory. Several basic classes of LQDs are also covered, and explicit examples are computed.

We study the GENERIC (General Equation for Non-Equilibrium Reversible Irreversible Coupling) formulation of the nonlinear Vlasov-Fokker-Planck equation from the perspective of gradient flows along trajectories. After pulling back the reversible component, the evolution can be recast as a generalized Wasserstein gradient flow. The associated free energy functional consists of an entropy term and a conservative energy term, while the metric is induced by the Onsager operator. This trajectory based viewpoint shows that the trajectorial rate of free energy dissipation captures the influence of the nonlinear term, an effect that is not directly apparent at the macroscopic level. Finally, partial degeneracy yields a decomposition of the underlying metric space, which in turn enables the derivation of a partial HWI inequality.

We prove global well-posedness and scattering for the 3D Klein-Gordon-Schrödinger system for small radial data in the best known global well-posedness range $(u_0, n_0, n_1)\in L^2\times H^{ -\frac{1}{2} + \epsilon } \times H^{-\frac{3}{2} +\epsilon }$ for any $ \epsilon > 0 $. The proof uses a global-in-time iteration scheme in the adapted function spaces $U^2_\Phi$, radial Strichartz estimates, and bilinear restriction estimates.

We prove mixed-norm estimates for circular averages with respect to $\alpha$-dimensional fractal measures on $\mathbb{R}^2$, using circle tangency bounds when $\alpha \in (0,1]$ and a $\delta$-discretized slicing lemma for fractals when $\alpha \in (1,2]$. The former estimate is sharp, while the latter improves previous results for $\alpha \in (\frac{3}{2},2]$. These estimates can be viewed as X-ray-type extensions of Wolff's and Bourgain's circular maximal functions. As applications, we obtain new exceptional set estimates for the radial integrability of functions in Lebesgue spaces, as well as for the Hölder regularity in time of solutions to the linear wave equation on $\mathbb{R}^2$. The latter results are the first of their kind.

We investigate a continuum Lagrangian $p$-alignment system given by a nonlocal mean-field system of ordinary differential equations for interacting agents with weak initial data. We first establish global well-posedness of the Lagrangian dynamics and derive quantitative flocking estimates. We next construct Eulerian variables from the possibly non-injective Lagrangian flow via pushforward and disintegration, which leads to an Euler–Reynolds–alignment system featuring a nonnegative Reynolds stress and, for $p>2$, a nonlinear defect force induced by microscopic velocity fluctuations. Assuming only heavy-tailed interaction, we then show that these defect terms vanish asymptotically, leading to asymptotic mono-kinetic closure in the long-time limit. In the linear case $p=2$, we further obtain global weak solutions to the Euler–alignment system, including a sharp one-dimensional critical-threshold characterization and a global result in higher dimensions under a large-coupling condition. Finally, we establish a uniform-in-time mean-field stability estimate for the particle Cucker–Smale system in the linear regime and deduce uniform-in-time convergence toward the mono-kinetic Eulerian limit; for general $p\ge2$, we also obtain a finite-time mean-field convergence result toward the associated kinetic/Lagrangian alignment dynamics.

In a previous paper \cite{Itskov-MoSM} we presented a hyperelastic isotropic material model whose stress-strain response is nonlinear even at infinitesimal deformations and cannot thus be linearized. As a result values of Poisson's ratio greater than one half were obtained. In this contribution, we further propose an isotropic strain energy function which is always positive-definite and depending on material constants delivers arbitrary values of Poisson's ratio (except of $-1$) in agreement with the laws of thermodynamics. The model response appears stable and plausible in various deformation states.

We survey analytic and probabilistic approaches to the Harnack inequality for $p$-harmonic functions, $p>1$, with particular attention to the dependence of the constants on $p$ and the dimension. We explain the ideas of Moser iteration, the tug-of-war approach of Luiro, Parviainen and Saksman, and the log-gradient estimate of Kotschwar and Ni. Along the way, we give a quantitative refinement of the probabilistic argument and a direct planar proof.

In this paper, we investigate the asymptotic behavior, as $\beta \to 0$, of positive solutions to the semilinear elliptic Robin problem \begin{equation*} \begin{cases} -\Delta u = u^p, & \text{in } \Omega,\\ u > 0, & \text{in } \Omega,\\ \frac{\partial u}{\partial \nu} + \beta u = 0, & \text{on } \partial \Omega, \end{cases} \end{equation*} where $p \ge 0$, $\beta > 0$, and $\Omega$ is a bounded smooth domain. We will prove that, for all $p\ge0$, the solution $u_\beta$ behaves like a constant as $\beta\to0$. However, the value of this constant is strongly influenced by the value of $p$. Indeed, \begin{itemize} \item if $0 \le p < 1$, $u_\beta$ blows up uniformly in $\Omega$ as $\beta \to 0$. \item if $p=1$ (eigenvalue problem), $u_\beta$ converge to a constant. \item if $p>1$ $u_\beta$ converge uniformly to zero. \end{itemize} In the critical and supercritical regime $p \ge \frac{N+2}{N-2}$, the existence of solutions is no longer guaranteed a priori. In this case, when $\Omega$ is a ball and $0<\beta<\frac{2}{p-1}$ we prove the existence of a radial positive solution.

We investigate the emergence of finite-amplitude non-zonal flows on the sphere $\mathbb{S}^2$ arising from stationary solutions to the 2D Euler equations. By restricting the Laplace-Beltrami eigenspace to the invariant subspace of the tetrahedral symmetry group $\mathbf{T}$, we bypass the $(2l+1)$-dimensional kernel degeneracy, obtaining a scalar Liapunov-Schmidt reduction. We analyze four distinct physical non-linearities: a polynomial model, the sine-Gordon and sinh-Gordon models, and the exponential (Liouville) model. We explicitly derive the bifurcation parameter via spectral projections, proving that the bifurcation topology (subcritical or supercritical) is not a geometric invariant, but is governed by the parity of the nonlinearity and the mass conservation.

In this note, we consider the underdamped Langevin dynamics with invariant measure $\mu(\mathrm{d}x\,\mathrm{d}v) \propto e^{-U(x)-|v|^2/2}\,\mathrm{d}x\,\mathrm{d}v$. Assume that the position marginal $\mu_x(\mathrm{d}x)\propto e^{-U(x)}\,\mathrm{d}x$ satisfies a Poincaré inequality with constant $m>0$, and that $\nabla^2 U\ge -K\,\mathrm{Id}$ for some $K\ge 0$. We revisit the modified $L^2$ method of Dolbeault–Mouhot–Schmeiser, employing a shifted corrector \begin{equation*} A_\alpha=(\alpha- L_{\mathrm o})^{-1}(L_a\Pi_v)^*, \qquad \alpha\ge0, \end{equation*} where ${L}_{\mathrm{o}}=\Delta_x-\nabla U\cdot\nabla_x$ is the overdamped generator, ${L}_a$ is the generator of the Hamiltonian flow, and $\Pi_v$ denotes averaging over the velocity variable. We establish an explicit hypocoercive $L^2$-convergence rate $\Lambda_{\alpha,\gamma}$ for every shift $\alpha\ge0$ and friction coefficient $\gamma>0$, and show that, for each fixed $\gamma$, the rate is maximized at $\alpha=0$. Optimizing further over $\gamma$ gives \begin{equation*} \Lambda_{0,\gamma_*} = \frac{\sqrt m} {2\left(2+\sqrt{2+\frac{2K}{m}} +\sqrt{6+\frac{2K}{m}}\right)}\,, \qquad \text{at} \quad \gamma_*=\sqrt{6m+2K}. \end{equation*} For convex $U$, this recovers the optimal $O(\sqrt m)$ rate.

Given $N\geq 2$ and $\alpha>-1$, we consider the following weighted Liouville-type equation involving the $N$-Laplacian: \begin{equation*} \left\{ \begin{aligned} -& \Delta_N u = |x|^{N\alpha} e^u \quad \text{ in } \mathbb{R}^N && , \\ & \int_{\mathbb{R}^N} |x|^{N\alpha} e^u \, dx < + \infty\,. &&\end{aligned} \right. \end{equation*} Solutions have been completely classified when $N=2$ via complex analysis, and when $\alpha=0$ using Pohozaev identities and an isoperimetric argument. In this paper, we first devise a $P$-function approach to the classification result for all $\alpha>-1$ when $N=2$. Since it is not based on complex analysis, this alternative and more PDE-oriented approach naturally extends to $N\geq 3$ by providing the classification for any $-1<\alpha\leq 0$. In particular, the explicit radial solutions are the unique ones for $-1<\alpha\leq0$ but become degenerate for special values $\alpha_k>0$, a hint that non-radial solutions might arise for $\alpha>0$ as it happens when $N=2$.

We consider the stochastic thin-film equation with linear deterministic and stochastic Itô perturbations. The existence of nonnegative weak martingale solutions on the semi-axis is established, and their asymptotic behavior as $t \to \infty$ is investigated. It is shown that in square mean the $L^\infty$ norm of the solution converges to the spatial mean value of the initial condition, multiplied by a random factor similar to a geometric Wiener process.

We present two characterizations of smooth compact Ricci flow solutions solely in terms of metrics and measures (one of them only works under positive scalar curvature along the flow); thus, provide weak formulations that are generalized to the singular setting in a straightforward manner. These formulations are achieved by weakly formulating super Ricci flows and imposing a saturation condition (solely in terms of metric and measure) to ensure the super Ricci flow inequality is an equality.

We present a generalization of the inverse mapping theorem, where variations of a weaker non-expansiveness property (referred to as property ${\sf A}$) replace the key $\mathsf{C}^1$ condition. We also obtain inverse mapping theorems that can be applied to non-smooth maps. Also as a by-product of the generalized inverse mapping theorem, we prove generalizations of the implicit function theorem and existence and uniqueness theorem of abstract PDE systems as well.

We study how the orientation and elliptic depth of swirl-generated vorticity response affect singularity formation in axisymmetric model equations. For a sign-reversed Hou–Li model on the whole line, the dynamics factorize into two real Riccati–diffusion channels. For every viscosity $\nu>0$, every nontrivial smooth datum in an infinite-dimensional cone of even, nonnegative, monotone channel data develops finite-time amplitude blowup; in the inviscid case, characteristics give the exact blowup time. For the same smooth swirling inviscid datum, the standard response sign is global whereas the reversed sign blows up. We then introduce a three-parameter 5D response family separating transport geometry, response orientation, and elliptic depth. Exact divergence, energy, and scaling identities show that physical three-dimensional incompressibility selects one geometry, weighted 5D incompressibility selects another, and Navier–Stokes scaling makes two elliptic inversions critical. The frozen physical response is critical and dissipative, while a natural one-inversion 5D-solenoidal model is supercritical and energy-pumping. These results identify response orientation and causal depth as load-bearing structure in axisymmetric model dynamics.

We establish resolvent estimates that extend earlier results to a larger class of electric potentials $V\in L^\infty(\mathbb{R}^d;\mathbb{R})$, $d\ge 3$, and magnetic potentials $b\in L^\infty(\mathbb{R}^d;\mathbb{R}^d)$ such that $V(x), b(x)=O_k\left(|x|^{-k}\right)$, $|x|\gg 1$, for every integer $k$. More precisely, we prove estimates for the derivatives of the weighted resolvent of the corresponding magnetic Schrödinger operator, which are uniform with respect to both the spectral parameter and the order of derivation. We also show that these resolvent estimates still hold for the Dirichlet self-adjoint realization of the Schrödinger operator in the exterior of a non-trapping obstacle in $\mathbb{R}^d$, $d\ge 2$, provided the magnetic potential is supposed identically zero. As an application of these resolvent estimates, we obtain the rate of decay of the local energy of solutions to the corresponding wave equation. In particular, we show that for potentials satisfying $|V(x)|+|b(x)|\le Ce^{-c|x|^s}$, $c,C>0$, $0<s<1$, the rate of decay of the local energy is $e^{-c_0t^s}$ with some constant $c_0>0$, where $t\gg 1$ is the time variable.

We solve an inverse initial data problem for the incompressible Navier-Stokes system. The objective is to recover the initial velocity and pressure from lateral boundary observations, without assuming that the time-independent body force is known. To eliminate this unknown force, we differentiate the momentum equation with respect to time and then apply a Legendre polynomial-exponential time-dimensional reduction. This procedure yields a coupled system of elliptic equations for the expansion coefficients. We then construct a contractive map for this reduced system on a suitable admissible set equipped with a Carleman-weighted norm. Its fixed point yields an approximate solution of the time-dimensional reduction model, and the contraction property gives rise to a globally convergent Picard iteration. Finally, we present a numerical algorithm based on this framework and numerical experiments showing accurate reconstructions of the initial velocity and pressure from synthetic boundary data.

We establish the existence of a positive bounded weak solution for a class of Kirchhoff-type $p(\cdot)$-Laplacian problems involving an arbitrary growth and a sandwich-type growth $s(\cdot)\in (\inf p,\sup p)$. This setting leads to substantial analytical difficulties in the variational analysis of the associated energy functional. By combining truncation arguments with a priori estimates, we prove the existence result under suitable assumptions on the data.

Our goal in this paper is twofold. First, we characterize the class of pairwise interactions for which the Seidl conjecture on the structure of optimal plans for the symmetric multimarginal optimal transport problem with one-dimensional marginal holds. This extends previous results by Colombo, De Pascale, and Di Marino [Can. Jou. Math., 67 (2015), https://doi.org/10.4153/CJM-2014-011-x], which treated the case of translation-invariant, convex and decreasing interactions. In particular, our results apply to physically relevant interactions for electrons living on a quantum ring. The second main goal of the paper is to rigorously derive the leading order asymptotics of the adiabatic connection potential for strongly interacting systems. More precisely, we show that for electrons in a quantum ring (or one-dimensional interval), not only the Lieb density functional converges to the optimal transport (or strictly correlated) functional in the semiclassical limit, but also the representing potential converges to a regular Kantorovich potential. As an intermediate step, we also extend previous results on the strongly interacting limit of the Lieb functional to periodic systems in arbitrary dimensions.

We prove a stochastic homogenization result for a class of nonlinear and nonlocal variational problems in domains with many small randomly distributed (bilateral) obstacles. Our model case is a Dirichlet problem for the fractional $p$-Laplacian, $p>1$, where a pinning condition $u=0$ is imposed on the solution in a random collection of small balls whose centers and radii are generated by a stationary marked point process. Such a general obstacle distribution allows for clustering effects to appear with positive probability. Under suitable moment conditions on the obstacle radii, we identify a critical scaling regime in which the fractional $p$-capacity density of the obstacles is asymptotically additive almost surely. In turn, this key property allows us to derive an effective homogenized problem which is formally analogous to the one obtained in the periodic setting or under the assumption of well-separation for the obstacles. The analysis also extends to the case of randomly shaped obstacles and to a broad class of nonlocal interaction kernels. At the methodological level, the paper develops a streamlined proof strategy with several new ingredients, among them the use of Palm measures.

In this paper, we study two local–nonlocal settings for parabolic–elliptic evolution systems. In our problems we have a disjoint partition of the spacial domain $\Omega$ as $\Omega=A\cup B$ and we first consider a local parabolic equation posed in $A$ with a nonlocal elliptic balance equation acting in the complementary subdomain $B$. Next, we reverse the roles and take a local elliptic equation posed in $A$ coupled with a nonlocal parabolic equation acting in $B$. In both models, the interaction between the two regions is driven by a nonlocal transmission term given by a kernel that transfers mass across the interface, giving rise to a mixed local–nonlocal, elliptic–parabolic dynamics. We consider Neumann boundary conditions for both problems. To being our analysis we first establish the existence and uniqueness of solutions using a fixed point argument. Then, we provide a detailed analysis of their qualitative behavior. In particular, we show that the coupling structure induces a natural energy functional whose gradient flow governs the evolution, despite the elliptic–parabolic nature of the system. As it is expected in Neumann settings, we prove that the total mass in the whole domain $\Omega$ is preserved in time. We also analyze the long-time behaviour and obtain decay estimates for the parabolic component, which in turn drive the convergence of the elliptic part to a constant solution. Finally, we prove that the parabolic–elliptic problem under consideration is the limit of a purely parabolic problem when a parameter that controls the speed of the dynamic at which one component evolves goes to zero.

In this paper we introduce a new concept of atoms on discrete sets to develop an advanced method to find a particular solution for higher-order non-homogeneous Cauchy-Euler equations. The proposed method provides also an approximate solution by using approximate roots for the characteristic polynomial of the Cauchy-Euler equation.

We consider the three-dimensional incompressible Euler equations for helical flows without swirl. By adapting gluing techniques, we construct the first smooth multi-vortex solution in the whole space $\mathbb{R}^3$ exhibiting a cluster of collapsing helical filaments, with the associated cross-sectional vorticity remaining compactly supported in $\mathbb{R}^2$ for all times. Our result generalises previous collapsing configurations in $\mathbb{R}^3$ with rapidly decaying vorticity cores, and extends related variational solutions obtained in infinite cylindrical domains.

We rigorously prove the bifurcation of slow-moving pattern interfaces with general direction in a two-dimensional Swift-Hohenberg-type model close to a Turing instability for a large class of nonlinearities. These interfaces describe the invasion of stripe and hexagonal patterns into the spatially homogeneous state and model a possible mechanism for pattern formation, as observed in a wide range of real-world applications. For this, we develop a rigorous framework to establish the existence of such solutions using spatial dynamics and non-standard centre manifold theory. Our approach exploits geometric and algebraic structures generic to $\mathrm{O}(2)$-symmetric pattern-forming systems near a Turing instability, and addresses fundamental technical challenges due to a non-uniform spectral gap around the imaginary axis, quadratic resonances induced by the hexagonal structure, and the high-dimensional phase space of the reduced equations.

We consider a nonlocal curve evolution belonging to a hierarchy of models for the dynamics of an inextensible elastic filament in a 3D Stokes fluid. This model captures the principal part of a full free boundary problem for an elastic filament in Stokes flow. The fluid effects on the filament evolution are encoded in a pseudodifferential force-to-velocity operator which may be regarded as an interpolation between resistive force theory at low wavenumbers and a Stokes boundary value problem at high wavenumbers. Here the curve is considered to be the centerline of a 3D filament with constant cross sectional radius $\epsilon>0$. We show global well-posedness for the curve evolution in the natural energy space. This loosely suggests that the full evolution may be globally well-posed if the large-scale geometry is controlled. Furthermore, we prove convergence to resistive force theory dynamics as $\epsilon\to 0$, which illustrates how resistive force theory emerges from more detailed models.

Recent well-posedness results have identified the Hardy space $L^2_+$ as the natural phase space for continuum Calogero-Moser models, both focusing and defocusing, on the line and on the torus. In this paper, we introduce a symplectic form on this phase space and so are able to realize these models as Hamiltonian systems. Moreover, we demonstrate that previously identified conserved quantities are mutually commuting, reinforcing the notion that these models are completely integrable. We further illustrate the utility of these structures by using them to give a new proof of global well-posedness in the critical space $L^2_+$, under the necessary mass restriction in the focusing case. Our work also brings to light several unforeseen connections: (i) the threshold for well-posedness coincides with that for the nondegeneracy of the symplectic form; (ii) this threshold is connected through Carleman's inequality to the isoperimetric problem in the plane; (iii) the transition from the line to the torus gives rise to a modified dynamical equation.

The propagation of heat and thermal signals in the form of travelling waves is investigated for a nonlinear Maxwell-Cattaneo-Vernotte equation. The exact wave solutions are derived by expressing the thermal conductivity and the relaxation time as polynomial functions of the temperature. This approach enables the identification of suitable degrees of nonlinearity that give rise to soliton solutions. Finally, exact solutions are shown through plots for the values of the selected parameters.