Papers

In this paper, we study the following linearly coupled Kirchhoff-Choquard system in $\mathbb{R}^3$: \begin{align*}\left\{\begin{array}{l} &-\left(a_1+b_1\int_{\mathbb{R}^3}|\nabla u|^2\,dx\right)\Delta u+V_1(x)u=\mu(I_{\alpha}*|u|^p)|u|^{p-2}u+\lambda v,\quad x \in \mathbb{R}^3,\cr &-\left(a_2+b_2\int_{\mathbb{R}^3}|\nabla v|^2\,dx\right)\Delta v+V_2(x)v=\nu(I_{\alpha}*|v|^q)|v|^{q-2}v+\lambda u,\quad x \in \mathbb{R}^3,\cr &u, v \in H^1(\mathbb{R}^3), \end{array}\right. \end{align*} where $a_1, a_2, b_1, b_2, \lambda, \mu,$ and $\nu$ are positive constants. When the potentials are constant functions, the author previously proved the existence of positive ground state solutions in the following cases: the noncritical case $\frac{3+\alpha}{3}<p\le q<3+\alpha$, the upper half critical case $\frac{3+\alpha}{3}<p<q=3+\alpha$, and the lower half critical case $\frac{3+\alpha}{3}=p<q<3+\alpha$, by using the Nehari-Pohozaev manifold method (NoDEA Nonlinear Differential Equations Appl.33(2026)). In the present paper, we extend these results to the case of nonconstant potentials. Under suitable assumptions on $V_1(x)$, $V_2(x)$, and $\lambda$, we prove the existence of nontrivial ground state solutions. In the noncritical and upper half critical cases, the main tools are Jeanjean's monotonicity trick and a global compactness lemma. For these cases, we establish a refined version of the splitting lemma by relaxing the standard growth assumption at the origin from $o(|t|)$ to the optimal $o(|t|^{\alpha/3})$, thereby significantly broadening the applicable class of nonlinearities. In contrast, for the lower half critical case $p=\frac{3+\alpha}{3}$, the splitting lemma is no longer valid. To overcome this essential difficulty, we obtain a ground state solution directly as a minimizer on the Nehari-Pohozaev manifold by imposing a slightly stronger condition on the potentials.

Local existence and uniqueness of smooth solutions are established for the three-dimensional compressible primitive equations with constant viscosity and a physical vacuum free boundary. We employ a Nash-Moser approach to address two distinct derivative losses, one arising from the free boundary geometry in the transformed viscosity and the other from the diagnostic reconstruction of the vertical velocity. We handle the first through Alinhac's good velocity unknown and the second through a modified height. Combining these two corrections with weighted energy estimates adapted to the vacuum degeneracy yields the tame estimates required for the iteration.

We investigate the existence of a one-bubble solution with maximum points escaping to infinity for the energy-critical heat equation in dimension five. The key point of the construction is a refined inner linear theory as in \cite[Section 8]{Wei-Zhang-Zhou2022LLG}.

In this work, we study the connection between the bidomain model of cardiac electrophysiology and the Kirchhoff Network model (KNM), that was recently introduced by Jäger and Tveito. In the KNM, each cell of the cardiac tissue is represented as a node in a discrete network and the dynamics of the electric potentials in the heart are described by a system of ordinary differential equations on the nodes. We first study the well-posedness of the KNM and then prove that, when the nodes are situated on the vertices of a regular lattice and the mesh size $\varepsilon$ tends to zero, the solutions of the KNM system at cell-size $\varepsilon$ converge to the solution of the bidomain model, along appropriate subsequences. Regarding the convergence, we adopt the framework presented by Pennacchio, Savaré, and Colli Franzone in "Multiscale modeling for the electrical activity of the heart", 2005, which is based on (a) the time-discretization of the evolution equations of the bidomain and $\varepsilon$-KNM systems via a semi-implicit Euler scheme, (b) the identification of the time-discrete solutions as extremal points of a Minimizing Movement scheme, and (c) the $\Gamma$-convergence of the Minimizing Movements functionals.

We investigate the optimal endpoint $L^2$ restriction estimates of Laplace eigenfunctions on submanifolds of codimension 2 in a smooth closed Riemannian manifold $M$. For every fixed smooth codimension-two submanifold $\Sigma$, we prove a little-o improvement $o(\lambda^{1/2}\sqrt{\log\lambda})$ on the classical estimate $O(\lambda^{1/2}\sqrt{\log\lambda})$ of Burq–Gérard–Tzvetkov and Hu. Our proof uses the Bargmann transform and Tataru's phase-space representation to reduce the problem to Stein–Street's estimate for singular Radon transforms. The three-dimensional case can be handled directly by Ricci–Stein's estimate. Moreover, we construct explicit examples to show that the little-o improvement is optimal in general. These are closely related to earlier counterexamples for endpoint Strichartz estimates by Montgomery–Smith and Carbery–Hofmann. In particular, we establish the log-free estimate $O(\lambda^{1/2})$ when $(M,g)$ and $\Sigma$ are real analytic, whereas this estimate fails in general in the smooth setting.

We study the existence and vanishing viscosity limit of steady incompressible Navier-Stokes flows in a horizontally periodic strip with no slip boundary conditions and midline symmetric forcing. Taking a Poiseuille flow as the leading order Euler profile, we construct higher order symmetric approximate solutions and establish the existence of a symmetric solution to the associated nonlinear error equation. Under suitable compatibility conditions on the forcing, this construction yields a family of midline symmetric steady Navier-Stokes solutions that converges to the prescribed Poiseuille flow at the optimal rate.

We prove a Pogorelov interior estimate for strictly plurisubharmonic solutions of the complex Monge-Ampère equation $\det(u_{i\bar{j}})=1$ with homogeneous Dirichlet data under the condition that, for some constant $\kappa\geq 1$, $(\kappa u_{i\bar{j}}-u_{is}u^{s\bar{t}}u_{\overline{tj}})$ is non-negative definite. For $\kappa=1$, this gives the estimate for real convex solutions. The estimate depends only on the dimension, $\kappa$, and the $C^0$ and $C^1$ norms of the solution. As an application, every smooth entire convex solution of $\det(u_{i\bar{j}})=1$ on $\mathbb{C}^n$ is a real quadratic polynomial. The same conclusion holds if real convexity is assumed only outside a compact subset of $\mathbb{C}^n$.

We investigate the large-time behavior of small perturbations of stable constant steady states for an attraction-repulsion chemotaxis system in $n$-dimensional Euclidean space. We consider integrable perturbations in one space dimension and, in higher dimensions, perturbations belonging to Lebesgue spaces with suitable exponents. We first establish decay estimates throughout the full admissible range of Lebesgue exponents and show that the nonlinear perturbation is asymptotically approximated by the corresponding linearized evolution. We then identify the leading term of the linearized evolution as an effective heat flow whose diffusion coefficient is explicitly determined by the parameters of the system. Consequently, when the integrability exponent is strictly smaller than the space dimension, the leading asymptotic profile is governed by the heat equation. At the critical endpoint, the difference between the nonlinear perturbation and the effective heat flow vanishes under the natural parabolic scaling. In the one-dimensional integrable case, this yields a Gaussian profile determined by the total mass of the initial perturbation.

We prove a form of the fundamental theorem of vector calculus for Hölder continuous vector fields with no assumption on the derivatives that exploits Schauder spaces with negative exponents. Then by developing an idea of von Wahl in the context of Lebesgue spaces, we prove a form of the Helmholtz Weyl decomposition Theorem for spaces of Hölder continuous functions, with explicit formulas for the projection operators.

In this paper, we study the sharp Sobolev-type inequality associated with the Born–Infeld energy on the lattice graph \(\mathbb Z^N\), \(N\geq3\): \[ \frac12\sum_{x\in\mathbb Z^N}\sum_{y\sim x} \left(1-\sqrt{1-|\nabla_{xy}u|^2}\right) \geq C_{N,\alpha} \left(\sum_{x\in\mathbb Z^N}|u(x)|^\alpha\right)^{\frac{N}{N+\alpha}}, \] for \(u\in D^{1,2}(\mathbb Z^N)\cap\ell^\alpha(\mathbb Z^N) \) satisfying \(|\nabla_{xy}u|\leq1\) for every \(x\sim y\), where \(C_{N,\alpha}\) denotes the optimal constant. We determine the exact positivity threshold and prove that \(C_{N,\alpha}>0\) if and only if \(\alpha\geq2^*:=2N/(N-2)\). At the Sobolev critical exponent \(\alpha=2^*\), we identify the optimal constant as \[ C_{N,2^*}=\frac12\mathcal S_2, \] where \(\mathcal S_2\) is the optimal discrete Sobolev constant, and show that it is not attained. In the supercritical regime, there exists \(\varepsilon_0=\varepsilon_0(N)>0\) such that \(C_{N,\alpha}\) is attained for every \(2^*<\alpha<2^*+\varepsilon_0\), with a nonnegative Schwarz symmetric extremal function. The main compactness difficulties stem from the lack of a suitable scaling on \(\mathbb Z^N\) and from the nonhomogeneity of the Born–Infeld energy. We overcome them by combining discrete Schwarz rearrangement with a \(Q_1\) discrete-to-continuum comparison and by establishing a strict separation as the \(\ell^\alpha\)-norm tends to infinity.

Focusing on the pullback asymptotic behavior exhibited by the system, this paper studies the three-dimensional (3D) non-autonomous Boussinesq equations with fractional Laplacian. For $ \alpha > \frac{5}{4} $ and $ \beta > \frac{5}{8} $, by adopting the $ \ell $-trajectory method introduced in \cite{MP2002}, we demonstrate the existence of both a finite-dimensional pullback attractor and a pullback exponential attractor for the process $ \{L(t, \tau)\}_{t \geq \tau} $ that corresponds to \cref{eq0101} in the phase space $ X_{\ell} $. Furthermore, we establish the existence of a pullback exponential attractor for the process $ \{U(t, \tau)\}_{t \geq \tau} $ within the original phase space $ H_\sigma^\alpha(\Omega) \times H^{\beta}(\Omega) $.

We establish explicit sufficient conditions for multiplicity of positive solutions of the Hénon equation in the unit ball and for uniqueness in one dimension. In the Sobolev-subcritical range, a spherical harmonic function argument shows that the radial solution has Morse index greater than one and therefore differs from a ground state. In even dimensions, a degree-four perturbation in a symmetry and monotonicity cone yields a sufficient condition for nonradiality in the supercritical range. The one-dimensional uniqueness criterion follows from a comparison of weighted Dirichlet eigenvalues and a concavity estimate.

We study an optimal control problem for a two-equations Keller-Segel system on a bounded domain with Neumann boundary conditions, where the chemotactic sensitivity is modulated by a control $f(x,t)$ acting on the taxis flux $\nabla\!\cdot(f\,u\,\nabla v)$. We prove the well-posedness of the state equation under low regularity assumptions. The analysis is based on the study of an $\varepsilon$-regularized problem, the application of Schauder's fixed point theorem, $\varepsilon$-independent, positivity and a priori estimates, the passage to the limit as $\varepsilon \to 0$, and a uniqueness argument. For a cost functional combining trajectory tracking, taxis penalization, control regularization, and terminal objectives, we establish the existence of optimal controls and derive the first-order optimality conditions through a Lagrange multiplier framework, leading to an adjoint system and a variational inequality for the optimal control. A numerical scheme based on spline collocation in space and Runge-Kutta time-stepping is set and implemented. Numerical experiments are performed to simulate the free and controlled dynamics in conservative and non-conservative cases.

We provide a unified description of a stable variational family of traveling vortex dipoles for the two-dimensional incompressible Euler equations, connecting the classical Lamb dipole to asymptotically radial dipoles in the large-impulse regime. We show that this family undergoes a sharp transition from the Lamb dipole, where the mass constraint is inactive, to non-explicit traveling waves for which the mass constraint becomes active, and determine the critical value exactly. We further prove that the variational solutions are unique up to translation both near the Lamb dipole and in the large-impulse regime. As a consequence, the corresponding individual traveling waves are orbitally stable. Our results provide a unified picture of these two distinct asymptotic regimes within a single variational family of traveling vortex dipoles.

This paper presents a study of dispersion for the two-dimensional Friedlander model with Neumann and Robin boundary conditions, thereby extending previous works treating only Dirichlet boundary conditions. By considering an Airy-type special function, we obtain a parametrix construction similar to the Dirichlet case but with additional terms requiring careful control. By studying the dispersion of this parametrix, we obtain the same dispersive estimates as in previous results for the Dirichlet case, and therefore the same Strichartz estimates.

The core-diagonal closure of Bothe and Druet identifies multicomponent Darken diffusion as a thermodynamically consistent Maxwell-Stefan model in which the constitutive data are species tracer diffusivities. Building on the Stieltjes-coordinate and De Giorgi regularity framework introduced in our recent preprint for additive Maxwell-Stefan frictions, we prove global maximal-$L^p$ strong well-posedness, without smallness assumptions, for the affine Darken class with composition-dependent scalar coefficients and uniformly positive tracer diffusivities. We show that this is exactly the maximal Darken class preserving the fixed Stieltjes eigenframe underlying the global argument. For general smooth, uniformly positive Darken tracer diffusivities we also derive the direct constrained flux form, exact mixing-entropy symmetrization, Fisher dissipation, composition-uniform normal ellipticity, and local strong well-posedness. Solutions in the affine class remain uniformly separated from the simplex boundary, become classical for positive times, and converge exponentially to equilibrium. Constant multiplicative frictions, including repeated factors, are covered by grouping species with equal factors.

We consider interaction functionals of the form \begin{equation*} \nu\mapsto {\mathscr E}(\nu)=\int\_{{\mathbb R}^N}\int\_{{\mathbb R}^N} W(x-y)\intd\nu(y)\intd\nu(x)+\int\_{{\mathbb R}^N} V(x)\intd\nu(x)\text{,} \end{equation*} involving an anisotropic kernel $W$ and a general confinement potential $V$. Under standard assumptions on $W$ and its Fourier transform, we show that the minimizer of ${\mathscr E}$ has $L^2$ density. Our argument relies on a careful analysis of nonlocal variational inequalities, which may be of independent interest.

The kinetic Motsch-Tadmor (in short, KMT) model is a kinetic flocking model with a normalized communication weight. In this paper, we study the emergent dynamics of the phase-spatially extended KMT model. We first establish a global well-posedness theory in the fully noncompact spatial-velocity setting. To this end, we introduce a direct Lagrangian formulation in which the unbounded part of the initial velocity distribution is separated from an interaction-generated bounded remainder. This decomposition allows us to construct global Lagrangian weak solutions without truncating the velocity distribution and to propagate finite phase-space moments. We then investigate the long-time collective behavior of the resulting solutions. When the initial velocity support is compact while the spatial support is allowed to be noncompact, a time-varying effective-region argument yields a uniform contraction mechanism for the normalized interaction and leads to exponential weak support flocking. When both the spatial and velocity supports are noncompact, support-level flocking is in general impossible. Nevertheless, the same effective-region mechanism, combined with the Lagrangian decomposition and a bootstrap argument for the interaction-generated remainder, yields exponential weak moment flocking. In particular, pairwise spatial moments remain uniformly controlled while velocity fluctuations converge exponentially to zero. These results provide a unified framework for the well-posedness and flocking dynamics of the KMT model beyond the compact-support regime.

We study wetting on a doubly periodic rough surface in three dimensions. Although the liquid-vapor interface meets the solid at the local Young's angle, microscale roughness can cause the macroscopic apparent angle to differ substantially from this value. We formulate the directional contact angle hysteresis (CAH) interval in terms of apparent angles associated with pinned microscopic configurations. To approximate its receding and advancing endpoints, we evolve capillary mean curvature flow (CMCF) toward extremal stationary states. Computing these states is difficult because the pinning that produces hysteresis is generated at the scale of the roughness, whereas the apparent angle is only meaningful at the macroscopic scale, so a single uniform grid must resolve both. We therefore introduce a two-scale alternating (TSA) method based on a Schwarz decomposition: a Merriman-Bence-Osher (MBO) diffusion-generated scheme resolves the contact-line near region, while a linearized minimal-surface problem updates the far region. For an idealized reference iteration, we prove decay of an approximate interfacial energy. Numerical experiments on a representative doubly periodic surface show a strongly anisotropic CAH interval whose width varies by more than a factor of three with contact-line orientation and changes sharply near the diagonal directions. Because a stationary droplet must meet the solid at an apparent angle inside this interval, the computed anisotropy constrains which macroscopic wetted regions the surface can support; it is consistent with a square-like stationary droplet whose sides align with the diagonal directions.

We study an infinite-horizon stochastic control problem for the optimal exploitation of an exhaustible resource with unknown total reserves. Information is generated both endogenously through continued extraction without depletion and exogenously through an external information flow. This interaction makes the natural problem non-Markovian and time-inconsistent. We show that it nevertheless admits an equivalent time-consistent formulation with the same optimal controls. The associated value function is characterized as the unique viscosity solution of a stochastic Hamilton–Jacobi equation with random coefficients. We prove comparison on the infinite horizon through a Snell-envelope-based strict-contact argument and establish uniqueness by an independent Brownian regularization and a BSDE correction, avoiding piecewise Markovian approximations. Finally, we identify the deterministic benchmark and show that, under persistent reserve uncertainty, the rescaled stochastic value function converges to a pathwise deterministic control problem, with the long-run optimal extraction rate determined by the asymptotic hazard rate of the limiting reserve distribution.

We introduce several notions of resonance for systems exhibiting weak or partial dissipation: $\dot u=Au+f(t)$, with $A$ the generator of a strongly stable semigroup $S(t)$ on a Hilbert space $H$. We refer to resonance as the phenomenon where a time-periodic $f$ may yield an unbounded $u$, classically occurring when $A$ has imaginary eigenvalues. In infinite dimensions, however, unbounded growth may depend on topological choices, and several “resonances" may occur. Classically, existence of $T$-periodic $u$ under $T$-periodic $f$ is equivalent to a range condition: $\mathcal{R}(I-S(T))=H$. Consequently, its failure permits unbounded solution growth and we elucidate that connection through properties of $S(t)$. We describe a hierarchy of range failures, and provide an associated resonance taxonomy. The growth rate of ~$||(A-\lambda I)^{-1}||_H$ on ~$i\R$ dictates a regularity gap between $u$ and $f$, and, for rapid growth in $\lambda$, that gap can be infinite. In this case, we demonstrate implications for periodic solvability, and a mechanism for constructing smooth resonant forces. The theory developed here is motivated by (and demonstrated for) a hyperbolic-parabolic system, where the latter component provides the only system dissipation. Such dynamics are a simplification of fluid-structure phenomena, which demonstrate strong stability but do not provide unconditional periodic well-posedness. Though point-spectral resonance is ruled out, we note that $\mathcal R(I-S(T))\neq H$. Our main result here shows that, for a carefully constructed geometry, a classical heat-wave system possesses periods $T$ which yield infinite derivative loss and, via supporting results, yield resonance with smooth forcing.

The finite-time blow-up construction for the forced Navier-Stokes equations released by OpenAI in Sep. 2026 (OpenAI 2026) has as its object an axisymmetric swirl core in anisotropic similarity variables. Its geometry is that of an exact steady swirl between rotating porous cylinders found by Gumerov and Duraiswami in 1998 (GD1998). We find that the GD1998 boundary value problem does not recast into the similarity variables and that its generalization is a two-dimensional profile problem between porous walls held at fixed similarity radii, second order in the radial variable X and first order in the time-like axial variable $\eta$. We solve it by tensor Chebyshev collocation with a pinned pressure gauge, complex-step Newton and arclength continuation, verify it three ways, and sweep the radial Reynolds number, the wall swirl and a symmetry-breaking datum. Below inflow 9 the solution is one smooth branch; above it the symmetric branch folds at a critical swirl $F^*=0.41V_0^2$. The Dirichlet axis problem has no resolution-stable solution; the core must be a Cauchy problem from the axis. Its moment identities cannot be met by a core symmetric about the dividing plane: the core is an axial through-flow, as its authors chose it; a non-symmetric core meets them to 0.2%. The cone condition is Rayleigh's criterion with axial shear and requires radii of order $10^{20}$. Dynamic rescaling shows the profile to be an attractor at weak inflow and toward the fold, a saddle-node; at moderate inflow and strong swirl the spectrum does not converge in the axial resolution and is left open. A real fluid cavitates (water) or shocks (air) first; the GD1998 chamber is proposed as the experiment. Nothing found suggests the mechanism is reachable in a flow one computes or builds; the forced theorem, and this study, leave the unforced equations of practice as they were. Code, tests and log accompany the paper.

We prove that every smooth properly embedded free-boundary minimal annulus in the unit ball is congruent to the critical catenoid. We also classify positive solutions of $(\Delta_{\mathbb{S}^2}+2)u=0$ on smooth spherical annuli with $u=0$ and $|\nabla u|=1$ on the boundary; their one-homogeneous extensions are the axially symmetric Alt-Caffarelli cones. Lastly, we also treat the case of free-boundary minimal annuli in spherical caps.

We study the singular limit $p \downarrow 1$ for positive radial entire solutions $u_p$ of $$ -\Delta_p u_p=F_p\left(|x|, u_p,\left|\nabla u_p\right|\right) u_p^{-\beta_p} \quad \text { in } \mathbb{R}^n, $$ where $n \geq 2,1<p<\min \{2, \sqrt{n}\}$, and $0 \leq \beta_p \leq p-1$. Assuming that $$ F_p \rightarrow F_1 \quad \text { locally uniformly }, \quad \frac{\beta_p}{p-1} \rightarrow \mathbf{c} \in[0,1] \quad \text { as } p \downarrow 1, $$ we prove that, after passing to a subsequence, $u_{p_j} \rightarrow u$ in $C_{\text {loc }}\left(\mathbb{R}^n\right)$ and $\nabla u_{p_j} \stackrel{*}{\rightharpoonup} \nabla u$ in $L_{\text {loc }}^{\infty}\left(\mathbb{R}^n ; \mathbb{R}^n\right)$, where $u \in W^{1, \infty}\left(\mathbb{R}^n\right)$ is radial, and solves $$ -\Delta_1 u=\mathtt{K}(|x|) \quad \text { in } \mathbb{R}^n . $$ On $\left\{x \in \mathbb{R}^n \mid \mathtt{G}(|x|)<1\right\}$, we prove that $\mathtt{K}(|x|)=w^{-\mathbf{c}} F_1(|x|, u(x), 0)$ and that $\nabla u=0$ a.e., where $u_{p_i}^{p_i-1} \rightarrow w$. On the other hand, we show that if $\rho \mapsto F_1(r, t, \rho)$ is affine for every $r$ and $t$, then $\mathtt{K}(|x|)=$ $w^{-\mathbf{c}} F_1(|x|, u(x),|\nabla u(x)|)$ a.e. in $\mathbb{R}^n$. For $n=1$ and $1<p<2$, we study $$ \frac{\mathrm{d}}{\mathrm{~d} x}\left(\left|u_p^{\prime}\right|^{p-2} u_p^{\prime}\right)=\mathtt{F}_p\left(|x|, u_p,\left|u_p^{\prime}\right|\right) u_p^{-\beta_p} \quad \text { in } \mathbb{R} $$ and obtain analogous results as $p \downarrow 1$.

We extend DiBenedetto's intrinsic parabolic Harnack estimates to a class of integro-differential equations with measurable kernels modeled on the fractional $p$-Laplacian. More remarkably, we establish intrinsic, representation type nonlocal estimates and reveal that nonnegative local solutions abide by a strong maximum principle even for $p>2$, a property unique to nonlocal problems. Analogous results hold true for nonlocal equations of porous medium type.

We derive an explicit description of the asymptotic singular behavior of 2D Laplace and Stokes solutions as the distance between a Lipschitz sharp tip and a smooth boundary vanishes. While classical lubrication theory for smoother boundaries relies on the anisotropy of the gap to perform dimensional reduction, the isotropic scaling imposed by a corner singularity requires a different approach. We introduce a stability-based method that allows us to construct the leading-order asymptotics of these solutions in the gap. As an application, we determine the asymptotic behavior of the Stokes resistance matrix and the pressure field.

We study multidimensional compressible fluid-particle systems at critical regularity, in which a carrier fluid and a particle phase with Fokker-Planck diffusion are coupled through a drag force. We prove the existence and uniqueness of strong solutions for the Cauchy problems of the Navier-Stokes-Vlasov-Fokker-Planck and Euler-Vlasov-Fokker-Planck systems near equilibrium in their respective critical Besov spaces. Moreover, we establish regularity estimates for the Navier-Stokes-Vlasov-Fokker-Planck system uniform with respect to the common viscosity parameter $\mu=\lambda=\varepsilon$ and justify the global-in-time vanishing-viscosity limit with the convergence rate $\mathcal O(\varepsilon)$. Finally, under an additional lower-order Besov assumption on the initial data, we obtain optimal time-decay estimates for both systems and derive enhanced decay rates for the relative velocity and the microscopic part of the distribution function.

Let $\Th(\alpha,L)=\sum_{v\in L}e^{-\pi\alpha|v|^2}$ for $\alpha>0$ and $E(L,s)=\sum_{v\in L\setminus\{0\}}|v|^{-2s}$ for $s>3/2$ be the theta and Epstein zeta functions associated to the lattice $L$, respectively. We are particularly interested in physically relevant dimension three. Fix the covolume of the lattice $L$ to $1$. Up to an orthogonal transformation, we prove that \begin{equation}\nonumber \argmin_{|L|=1}\Th(\alpha,L)= \begin{cases} \boldsymbol{\mathrm{FCC}}\;\;\mathrm{lattice},\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;&\text{if}\;\; \alpha>1, \boldsymbol{\mathrm{BCC}}\;\;\mathrm{lattice}, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;&\text{if}\;\; \alpha<1, \boldsymbol{\mathrm{FCC}}\;\;\mathrm{or}\;\; \boldsymbol{\mathrm{BCC}}\;\;\mathrm{lattice},\;\;&\text{if}\;\;\alpha=1, \end{cases} \end{equation} and \begin{equation}\nonumber \argmin_{|L|=1}E(L,s)= \boldsymbol{\mathrm{FCC}}\;\;\mathrm{lattice},\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\text{if}\;\; s>3/2. \end{equation} Therefore, we prove the Sarnak–Strömbergsson conjecture \cite[Inequalities (43)–(44), Section 5]{SS}.

We construct singular weak solutions to scalar, linear, uniformly parabolic equations with complex coefficients in one space dimension. Our examples show that the space-time integrability provided by the energy estimates is sharp: for every $p>6$, there exists such an equation admitting an energy solution that fails to be $p$-integrable on a compact subset of the space-time domain. In particular, the local boundedness of weak solutions from the De Giorgi–Nash–Moser theory fails for complex coefficients already in one space dimension.

We prove a spectral gap inequality for Gaussian measures modified by singular attractive pair interactions. The spectral gap constant is explicit and uniform in the number of particles and the regularization parameter. As an application, we use a corresponding spectral gap inequality for a cutoff interaction weight to obtain two-sided bounds on the transition density of the attractive logarithmic gas.