Papers

In this paper, we study the large time behavior of solutions to the linearized Hartree equation near a stable equilibrium which is homogeneous in space and investigate the quantum Landau damping. This allows us to get time decay of the electric field, uniformly in the Planck constant, as well as the uniform-in-time convergence of the quantum density towards the classical density as the Planck constant converges to 0.

We study the stochastic differential equation $$d X_t=b(t,X_t)d t+\sqrt{2}d W_t$$ on $\mathbb R^d$, where $b$ is a time-dependent, divergence-free distributional drift of critical Hölder–Besov regularity $-1$, strengthened by an iterated-logarithmic correction. For every initial probability law, we construct a weak solution by smooth approximation and realize the singular drift as an additive functional. The main analytic ingredient is the Schauder estimate with a logarithmic smallness factor. Combined with uniform logarithmic Krylov estimates and a stochastic substitution formula for distributional test functions, this estimate allows us to apply a Zvonkin transformation and prove uniqueness in law among weak solutions satisfying the corresponding Krylov bounds. For solutions starting from deterministic points, we further show that their time-marginal distributions admit densities satisfying two-sided Aronson-type Gaussian estimates.

We study the stochastic Córdoba–Córdoba–Fontelos equation driven by multiplicative Stratonovich noise. The noise amplitude is allowed to be a pseudo-differential operator whose leading part is nearly skew-adjoint. This class contains classical transport noise and also permits genuinely nonlocal perturbations. We first develop a local-in-time theory for maximal classical solutions in Sobolev spaces, proving existence, uniqueness, and a blow-up criterion. We then consider the special case of Stratonovich transport. For sufficiently large initial nonlocal steepness at a global maximum, we prove finite-time blow-up with arbitrarily high prescribed probability and obtain an explicit upper bound on the lifespan. Finally, on the event that the nonlocal steepness at the transported maximum diverges, we establish a conditional Type-I upper bound. When the terminal Cesàro average of the normalised nonlocal energy converges, we further identify the exact leading-order blow-up rate in terms of its limiting value.

We study instantaneous shrinking of supports for nonnegative solutions of the stochastic partial differential equation \[ \partial_t u=a(t,x)\,\partial_x^2 u + b(t,x)\,\partial_x u + c(t,x)\,u +\sigma(u)\,\xi(t,x), \qquad (t,x)\in(0,\infty)\times\mathbb R, \] where $\xi$ is space-time white noise, the coefficients $a$, $b$, $c$ may be random, and the noise coefficient $\sigma$ vanishes at the origin and is sublinear there. The model case is $\sigma(u)=u^\gamma$ with $\gamma\in(0,1)$. We show that, under a uniqueness-in-law assumption, if the initial datum has a sufficiently light spatial tail, then every nonnegative solution has compact support at every positive time, even though the initial support is not compact. The initial datum may also be a measure, such as a Dirac mass. When $\gamma\in(0,1/2]$, finite initial mass suffices; this covers the super-Brownian case $\gamma=1/2$. When $\gamma\in(1/2,1)$, we identify a polynomial moment condition on the initial state whose order diverges as $\gamma\uparrow1$, quantifying the trade-off between the strength of the noise near zero and the decay of the initial data required for instantaneous shrinking. As a step of independent interest, we establish weak existence of solutions started from measure-valued initial data for non-Lipschitz $\sigma$ and random operators. Our results provide a stochastic counterpart of the instantaneous shrinking phenomenon of Evans and Knerr for deterministic parabolic equations with strong absorption, in which the role of the absorption term is played entirely by the noise.

In this manuscript, we determine the critical exponent for the two-dimensional semilinear wave equation with strong damping. A recent result in D'Abbicco (arXiv, 2026) gives global in-time small data solutions for $p>\frac{10}{3}$, whereas we in the paper prove finite-time blow-up for $2<p\leqslant\frac{10}{3}$ under an averaged sign condition on the initial velocity. Together with the previously known blow-up result for $1<p\leqslant3$, the threshold for the power nonlinearity $|u|^p$ is \begin{align*} p=p_{\mathrm{crit}}=\frac{10}{3}. \end{align*} We prove positivity of the full two-dimensional velocity fundamental solution and construct a positive averaged kernel, which allows us to establish a nonlinear lower-bound iteration in a parabolic region.

In this paper we study the Kirchhoff-Pohozaev equation introduced in \cite{P2} and its constants of motion (see \cite{BoitiManfrin2025}, \cite{BoitiManfrin2026}) on $n$ dimensional tori. We show that these Hamiltonians are all in involution and we prove that they are generated by an infinite list of constants of motion which are all defined and in involution on a fixed phase space. Then we study the Kirchhoff-Pohozaev equation restricted to a finite Fourier support. In dimension $n=1$ we show that such finite dimensional reduction is always completely integrable and provide an analytic Brikhoff normal form in a neighborhood of the origin. We also give sufficient conditions for integrability for $n>1$. We finally show that the formal Birkhoff Normal form of the Kirchhoff-Pohozaev equation is integrable for $n=1$.

In this paper, we prove that for every integer $k\geq2$ and every $c>1$ sufficiently close to $1$, there is $\kappa>0$ such that every sufficiently large subset of $\{1,\ldots,N\}$ of density at least $(\log\log N)^{-\kappa}$ contains \[ x,\quad x+\lfloor n^c\rfloor,\quad x+\lfloor n^c\rfloor^2, \quad\ldots,\quad x+\lfloor n^c\rfloor^k. \] We also prove pointwise almost-everywhere convergence of the associated multiple ergodic averages.

We develop a Caffarelli-Silvestre extension theory for the fully fractional Schrödinger operator $\mathcal{L}^s=(\partial_t-i\Delta_x)^s$, $0<s<1$, nonlocal in space and time, whose Cauchy problem prescribes a past history rather than data at a single time. Every extension theory so far rests on positivity or sectoriality of the generator; here the semigroup is unitary and the symbol changes sign across the characteristic paraboloid, so neither is at hand. We construct the extension nonetheless. Its Poisson kernel, computed explicitly, is oscillatory rather than positive; its Dirichlet-to-Neumann map is $\mathcal{L}^s$; its normalising constant is the Caffarelli-Silvestre constant times the phase $e^{i\frac{\pi s}{2}}$; and the theory undergoes a transition at $s=\frac12$. We then identify the intrinsic Hilbert space of histories, on which the Poisson lifting is an isometry. This closes a circle. A lifted history is precisely an initial datum for the singular Schrödinger equation with nonlinear Neumann interaction of our companion paper, whose well-posedness theory therefore transfers to $\mathcal{L}^su=\mu|u|^{p-1}u$ with prescribed past. With our earlier work on the Bessel operator on a half-line, the three papers form a single programme, of which the present one is the closing step.

We study the hypoellipticity of operators on a product type Lorentzian manifold where the time variable is periodic. In particular, we prove that hypoellipticity holds for such time-periodic equations, with a stronger estimate when the mass parameter or time period lies outside a set of arbitrarily small measure. We consider both compact and noncompact spatial manifolds and provide explicit examples involving the wave operator. The proof relies on a Fourier series decomposition and asymptotics for eigenvalue counting functions.

In 1966, McKean asked whether the entropy production of the Boltzmann equation must be monotone decreasing in time. We show that this is not the case even in the space-homogeneous setting for the Boltzmann collision operator with a constant angular cross section and the kinetic parameter $\gamma \in [0,1]$. This recovers the classical case of hard spheres and the simplest case of Maxwell molecules. Our examples are radially symmetric mixtures of Maxwellians.

We consider the linear cocycles generated by the linearized vorticity equation and by passive scalar advection diffusion, both driven by the two dimensional stochastic Navier-Stokes flow on the torus with non degenerate additive forcing, and we prove uniqueness of the stationary measures for the projective process associated to such linear dynamics. The proof relies on a localized asymptotic strong coupling construction, and on the non-degeneracy of the Malliavin matrix of the projective process. Establishing this non-degeneracy for passive scalar advection poses additional challenges, and requires a proof (based on Cameron-Martin analyticity arguments) that generically the passive scalar is nowhere one-dimensional.

We study the warm-ion Euler–Poisson system with Maxwell–Boltzmann electrons on the cylinder $\R\times\T$ in the quasineutral regime. Taking a smooth expanding planar simple wave of the effective Euler system as the reference state, we construct an even Debye expansion through arbitrary finite order $M$ with a residual of $O(\eps^{2M+2})$. We rigorously establish nonlinear stability on every fixed interval $[t_0,T]$ ($t_0\ge0$), achieving a lifespan and energy constants strictly independent of the Debye length $0<\eps\le\eps_0$. To overcome the singular scaling of the electric field, we develop a novel compensated energy topology that couples the warm-ion symmetrizer to the time-differentiated nonlinear Poisson constraint. By integrating the top-order electric work directly into the time derivative of a weighted energy functional, this mechanism controls the potential in $H^s$ and its gradient in $\eps H^s$, completely eliminating the $\eps^{-1}$ loss typically encountered in the momentum equation. This framework successfully governs distinct neutral end states, captures genuinely two-dimensional rotational perturbations, and yields an arbitrary-order quasineutral asymptotic expansion for prepared data, providing a critical analytical foundation for the geometric theory of multidimensional quasineutral rarefactions.

In this note, we prove the existence of smooth complete admissible hypersurfaces in hyperbolic space with constant higher order mean curvature and a prescribed asymptotic boundary at infinity. The result holds for a large part of indices in the subcritical range where the concavity inequality loses its effectiveness. We supply with new arguments to reduce the proof to a semi-convex situation in which case the concavity inequality can play a role.

Let $H\in C^\infty(\mathbb R^n\times\mathbb T^n)$ be a periodic Tonelli Hamiltonian with critical value $c$. For each $k\in\mathbb N$, let $u_k$ be the normalized minimizer of the variational functional introduced by Evans[7], \[ I_k[w]=\int_{\mathbb T^n} e^{kH(Dw,x)}\,dx, \qquad \int_{\mathbb T^n}w\,dx=0. \] If $u_\infty$ is a uniform limit of a subsequence of $\{u_k\}$ and the Mather quotient $({A}_M,\delta_M)$ satisfies $H^1( A_M,\delta_M)=0$, then $u_\infty$ is a critical subsolution that is strict outside ${{A}}$ and \[ {A} = \{x\in\mathbb T^n\,:\,Du_\infty(x)\ \text{exists and }H(Du_\infty(x),x)=c\}=\{x\in\mathbb T^n\,:\,u_\infty(x)=u_{-}(x)\}, \] where ${A}$ is the projected Aubry set and $u_{-}$ is the backward weak KAM solution associated with $u_\infty$. In particular, by the theorem of Fathi–Figalli–Rifford[10], this conclusion holds for all smooth Tonelli Hamiltonians on $\mathbb T^n$ when $n\leq3$. This characterization also suggests a natural numerical localization principle for approximating the entire Aubry set through near-contact sets between $u_k$ and its large-time backward Lax–Oleinik evolution.

For every $n\ge3$, we show, by a family of explicit entire real-analytic solutions, that, for every subcritical phase and every $0<\alpha\le1$, oscillation-based interior $C^\alpha$ estimates fail for the special Lagrangian equation in $\mathbb{R}^n$. In fact, this family has no common interior modulus of continuity.

We consider the three-dimensional full compressible magnetohydrodynamic(MHD) system on the periodic torus $\mathbb T^3$ in the regime where the only dissipative mechanism acting on the system is the viscosity of the fluid: the magnetic field is non-resistive and the flow is non-heat-conducting. We prove that this system admits a unique global smooth solution, together with explicit algebraic decay rates, provided that the perturbation $(\mathbf u_0,\,P_0-\bar P,\,\mathbf H_0-\mathbf n)$ of the equilibrium state $(\mathbf 0,\bar P,\mathbf n)$ is sufficiently small in a high-order Sobolev space and the background magnetic field $\mathbf n\in\mathbb R^3$ satisfies a Diophantine condition. No smallness whatsoever is imposed on the initial density: it is only required to be bounded away from vacuum and from infinity, and may exhibit arbitrarily large variations. The proof uncovers a hidden dissipation mechanism. Although neither the density, nor the pressure, nor the magnetic field is endowed with any diffusion or damping of its own, the coupling of these quantities with the velocity through the background field $\mathbf n$, combined with a Poincaré-type inequality of Diophantine origin, generates effective dissipation for both the pressure and the magnetic field perturbations. The large variations of the density are handled by a two-tier energy argument, in which weighted time-decay estimates for the intermediate-order energy compensate exactly for the linear-in-time growth of the highest-order norm of the density.

The classical Heisenberg Pauli Weyl (HPW) inequality exhibits a strong rigidity phenomenon on Riemannian manifolds i.e. on Cartan Hadamard manifolds and those with non-negative Ricci curvature, the validity of the Euclidean HPW inequality or the existence of extremizers strictly forces the manifold to be isometric to Euclidean space, $\mathbb{R}^n$. This geometric discrepancy motivates the study of curvature dependent corrections and their associated stability properties. In this article, we investigate the geometric stability of the HPW inequality. Specifically, given a sequence of pointed Riemannian manifolds and appropriately normalized functions with a vanishing HPW deficit, we address whether the sequence converges to the corresponding model space in the pointed Gromov Hausdorff topology. We prove that for pinched Cartan Hadamard manifolds with sectional curvature bounded above by $c < 0$, the manifolds converge to the model hyperbolic space $\mathbb{H}^n_c$. In the non-negative Ricci curvature setting, we establish convergence to $\mathbb{R}^n$, up to a metric rescaling factor governed by the sequence's moment term. Consequently, we deduce that the known HPW inequality formulated via the asymptotic volume ratio (AVR) is suboptimal for non negatively Ricci curved manifolds not isometric to Euclidean space. To resolve this, we introduce a curvature corrected HPW inequality for this setting, analogous to the Cartan Hadamard case. Finally, we establish quantitative rigidity estimates in both curvature regimes, demonstrating that the HPW deficit when evaluated at Gaussian profiles which explicitly controls an appropriately defined distance to the respective model space.

We prove that semi-strong convergence to a Bose–Einstein equilibrium below the critical temperature implies finite-time Bose–Einstein condensation (BEC) for conservative isotropic measure solutions of the spatially homogeneous quantum Boltzmann equation with low-momentum scattering degeneracy exponent $0\leq\eta<1$. For every admissible initial measure with $\overline T/\overline T_c<1$, there exists a conservative solution that has a positive zero-energy atom after a finite time without any assumption of initial concentration near zero energy. We also show that it persists and converges to the equilibrium condensate mass, while the full solution converges strongly. The proof introduces a method to obtain condensation from relaxation estimate to Bose–Einstein equilibrium. The relaxation estimate first gives a fixed amount of mass below a small energy level. We then use collision estimates at smaller and smaller energy scales and add their effects in a weighted sum functional with a fixed upper bound. If no atom forms at zero energy, enough mass remains in the positive-energy shells to force this sum to exceed its upper bound in finite time, which leads to condensation in finite time.

In this paper, we study the fundamental solution of the mixed local-nonlocal operator $ -\Delta+(-\Delta)^s,$ $ 0<s<1. $ We derive precise asymptotic expansions of the fundamental solution and its gradient, up to the first nontrivial correction term, both near the origin and at infinity. The expansions describe the transition between the local and nonlocal diffusion scales and include the critical regimes in which logarithmic terms occur. As applications of these asymptotics, we establish a distributional Bôcher-type theorem for nonnegative supersolutions with an isolated singularity. We further obtain quantitative positive and antisymmetric maximum principles on punctured balls.

A diblock copolymer is a soft-matter composed of two chemically distinct block of repeating monomers covalently bonded together at an end-to-end junction to form a single polymer chain. In this paper, we establish the existence of infinitely many smooth periodic unbounded domain patterns of Delaunay-type in $\mathbb{R}^3$ that optimize the energy distribution in diblock copolymer melts. We emphasize that pattern domains at the equilibrium correspond to stationary sets of the screened Ohta–Kawasaki free energy functional \begin{align*} \mathcal{P}_\gamma(\Omega) := |\partial\Omega| + \gamma \int_{\Omega}\int_{\Omega} G_{\kappa}(|x-y|) \,\mathrm{d}x\mathrm{d}y, \end{align*} where $\gamma>0$, $\kappa>0$ and $G_\kappa(r)=\frac{1}{r} e^{-\kappa r}$ is the repulisive Yukawa potential. Equivalently, these equilibria satisfy the corresponding Euler–Lagrange equation \begin{align*} \mathcal{H}_\Omega (x):= H_{\partial\Omega}(x) + \gamma \int_{\Omega} G_{\kappa}(|x-y|) \mathrm{d}y = Const \quad \text{on } \partial\Omega, \end{align*} where $H_{\partial\Omega}$ denotes the mean curvature of the surface $\partial\Omega$. By analyzing the linearization of $\Omega \mapsto \mathcal{H}_\Omega$ around flat cylinders and applying the Crandall–Rabinowitz bifurcation theorem, for any $\kappa > 0$ and sufficiently small $\gamma > 0$, we prove the existence of non-trivial, $2\pi$-periodic Delaunay-type equilibrium cylinder interfaces with shapes close to a Delaunay unduloid surface of constant mean curvature.

We establish codimension-independent mass bounds for geometrically confined area-minimizing rectifiable currents. In Euclidean space, we combine the confined-volume doubling theorem of Colding–Minicozzi with a current-theoretic squashing argument. This gives an affirmative answer to Lin's interior mass-bound problem for every algebraic projection multiplicity $Q$: the interior mass is bounded by $C(n)Q$, without an a priori mass bound at a larger scale. This result yields a degree-one Bernstein-type rigidity result under sublinear confinement. For the hyperbolic application, we make the curvature modification of the fixed-scale argument needed in a thin tubular neighborhood of a totally geodesic copy of $\mathbb{H}^n$. Combining this auxiliary estimate with a localized squashing estimate removes the doubly exponential local mass-growth condition from the boundary regularity results in [13].

We prove global well-posedness for the two-dimensional infinite-depth gravity water wave system coupled to a pair of point vortices of opposite strengths. The vortex dipole is initially placed deep below the free surface and oriented so that its leading motion is away from the interface. We show that the vortices retreat almost linearly, and that the velocity induced on the free boundary is therefore integrable in time. Combining this decay with Wu's cubic formulation, together with a transition-of-derivatives and localization argument, we obtain global bounds for small localized perturbations of the flat surface.

We study the singular limit of a family of linear degenerate interface transmission problems arising from a regularization procedure in the newly proposed Two-Parameter Diffuse Domain Method (DDM2p). For $\alpha>0$, the regularized problem admits a strictly convex variational formulation on $H^{1}(\Omega)$. In the limit $\alpha\to0$, the problem degenerates to a weakly coupled interface system with a nonstandard energy structure. To characterize the limit, we introduce a closed Hilbert subspace $\mathcal{H}\subset H^{1}(\Omega)$, defined through an auxiliary Helmholtz problem on an annular subdomain $\Omega_2\subset \Omega$, and identify the limiting energy functional $\mathcal{E}_{0}$ on $\mathcal{H}$. We prove that the regularized energies $\mathcal{E}_{\alpha}$ $\Gamma$-converge to $\mathcal{E}_{0}$ in the strong $L^{2}(\Omega)$ topology, using the standard framework. Consequently, minimizers of $\mathcal{E}_{\alpha}$ converge to the unique minimizer of $\mathcal{E}_{0}$, which is shown to be equivalent to the solution of the limiting interface problem. We further prove strong convergence $u_{\alpha}\to u_{0}$ in $H^{1}(\Omega)$ and establish an $O(\alpha)$ convergence rate. Numerical experiments in one spatial dimension confirm the predicted first-order convergence rate and suggest that this rate is sharp.

Isogeometric Analysis (IgA) uses the same spline functions to represent the computational domain and to approximate the solution. This allows exact geometry descriptions, but the resulting mass and stiffness matrices are expensive to assemble and to store, especially in three dimensions. We present a projection-based low-rank approach for assembling the mass and stiffness tensors of orientation-preserving tensor-product B-spline geometries. For the mass tensor, we exploit the polynomial structure of the determinant of the Jacobian of the geometry map and represent it in reduced spline product spaces by univariate coefficient transfer operators; with exact quadrature and without truncation, the resulting low-rank tensor is an exact reformulation of the standard Galerkin mass tensor. For the stiffness tensor, we split the rational weight function into a polynomial numerator, again represented in reduced spline product spaces, and the reciprocal determinant, which is in general not a spline function and is therefore approximated by an $L^2$-projection onto a tensor-product spline space. Both constructions are carried out entirely in the tensor-train (TT) format, with the projection system solved by the alternating minimal energy (AMEn) method, so that full high-order coefficient tensors are never formed and the multidimensional integrals reduce to univariate integrals and contracted products. The method is implemented in MATLAB using GeoPDEs and the TT-Toolbox. Numerical experiments show that it is competitive with full assembly and with the interpolation-based low-rank method, and that it applies in two situations in which interpolation is problematic: a singular interpolation system and nearly singular geometries. The construction is restricted to orientation-preserving tensor-product B-spline geometries and does not cover NURBS.

We prove an obstruction to quasi-invariance of Gaussian fields under divergence free transport flows on Riemannian manifolds. For the centered Gaussian field with covariance $(1-\Delta_g)^{-\alpha}$, $\alpha>\frac{d}{2}$, quasi-invariance under the transport flow is equivalent to invariance. This happens precisely when the flow acts by isometries, or equivalently when the underlying vector field is Killing. The proof leverages the Feldman–Hájek theorem and utilizes localized pseudo-differential computations. On $\mathbb R^d$ this leaves only rigid motions, while on flat tori this leaves only translations.

In this paper, we study the asymptotic Plateau problem in hyperbolic space for hypersurfaces of constant $H_k$-curvature. We prove the existence of a smooth complete $k$-convex hypersurface in $\mathbb{H}^{n+1}$ satisfying \[ H_k(\kappa)=\sigma, \qquad \sigma\in(0,1), \] with prescribed asymptotic boundary at infinity. In particular, our result extends the range of the constant $\sigma$ in the existence theorem of Guan and Spruck [J. Eur. Math. Soc. 12 (2010), no. 3, 797–817] for $H_{k}$ curvature to the full interval $(0,1)$.

Characterizations of pointwise Gaussian upper bounds on graphs with possibly unbounded geometry in terms of localized functional inequalities contain errors depending on the vertex degree. We introduce a new space-time averaged form of the heat semigroup in terms of time-averaged $\ell^{q/(q-1)}-\ell^q$-estimates and obtain Gaussian upper bounds from large-scale Faber-Krahn inequalities which avoid such errors. A main analytic ingredient is an integrated version of Davies' method which yields off-diagonal estimates for this averaged quantity, and which tends to pointwise Gaussian bounds as time tends to infinity. Conversely, on large scales volume doubling and Gaussian bounds on this averaged heat semigroup norms imply relative Faber-Krahn inequalities for subsets of prescribed relative measure. Our proof is based on a lower bound for Dirichlet eigenvalues in terms of these averages. The Faber-Krahn dimension is scale-dependent, but converges to the doubling dimension for increasing radii. This gives a characterization of asymptotic Gaussian heat kernel behavior in terms of Faber-Krahn inequalities.

We establish local existence and uniqueness of solutions for the ideal incompressible magnetohydrodynamics system posed on $[0,T]\times\mathbb{R}^n$, $n\ge2$, with a nonzero constant initial magnetic field $\mathbf{B}_0$ and arbitrary divergence-free velocity data $v_0\in H^s$, in the range $(n+1)/2<s\le n/2+1$. The proof uses a Lagrangian wave–Hodge reformulation and exploits an Alfvén null–structure hidden in the pressure forcing. In particular, the constructed Eulerian solutions are induced by a bi-Lipschitz measure-preserving flow map. Zhang first identified this null-structure in \cite{Zhang2024}; the present work provides a self-contained bridge from that Lagrangian theory to the Eulerian Cauchy problem.

In this article, we introduce a new distance function between hypersurfaces with free boundary. We show that our new quantity, which we call twisted Fermi distance, is monotone under mean curvature flow with free boundary. This overcomes the stumbling block that monotonicity of the usual distance function can fail for non-convex domains, and has several applications. Most importantly, we generalize the avoidance principle for free boundary Brakke flows, recently established by the first author for convex domains, to arbitrary domains. Using this, we then show that all results from our recent joint work, including the mean-convex neighborhood theorem and the uniqueness theorem for free boundary flows through cylindrical singularities, can be generalized to arbitrary domains without any convexity assumptions as well.

We consider the Vlasov-Maxwell system with a repulsive point charge. Under a spherical symmetry condition, this reduces to the relativistic Vlasov-Poisson system with a point charge. For this system, the linear characteristics solve the equations of motion for the relativistic Kepler problem. We define action-angle variables for the linearized equation, and then study the nonlinear system in these coordinates. Within this framework, we prove global existence of classical solutions for small initial data and establish their modified scattering behavior. The proof makes use of the ideas developed in the non-relativistic case.