Papers

We study overdetermined problems for normalized sum Hessian equations and their quotients in the space forms. The equations and boundary conditions imply admissibility and ellipticity. For quotient equations, we prove that the domain is a geodesic ball if it is strictly star-shaped, or if the Dirichlet value satisfies an upper bound. We also prove the corresponding result for sum Hessian equations without star-shapedness. The proofs use two auxiliary functions and a Rellich–Pohožaev identity.

For a prescribed classical profile and a fixed positive regularization parameter, we construct an Itô diffusion whose interior generator reproduces the regularized $p(x)$-Laplacian when applied to that profile. We determine the exact tangent spectra of the unregularized and regularized fluxes, characterize their behavior at zero gradient, and establish quantitative ellipticity bounds on bounded gradient ranges. The full divergence expansion identifies a covariance equal to twice the gradient derivative of the regularized flux, evaluated at the profile gradient, and a logarithmic drift generated by spatial variation of the variable exponent $p(x)$. Under explicit classical regularity assumptions, we establish boundedness and local Lipschitz continuity of the coefficients and prove strong existence and pathwise uniqueness up to domain exit. We identify the interior generator of the boundary-stopped process and verify the profile-reproduction identity. Further consequences include initial-point stability for a fixed global extension, localized martingale identities, and Dynkin's formula at bounded stopping times. An explicit nonconstant exponent example illustrates the construction at a critical point, and the coefficient selection is shown to be canonical under the flux-based prescriptions rather than determined uniquely by profile reproduction alone.

We study spectral zeta densities associated with complex powers of principally scalar Laplace-type operators on pseudo-Riemannian scattering spaces of mixed signature and arbitrary dimension. The construction extends, beyond the even-dimensional Lorentzian case, the local analysis of Feynman complex powers developed by Dang and Wrochna. Namely, for $\varepsilon > 0$, we consider complex powers $(P\pm i\varepsilon)^{-\alpha}$ and prove that the diagonal trace densities admit meromorphic continuation. We relate their poles to local geometric invariants, in particular to the pseudo-Riemannian scalar curvature. Our proof combines uniform Feynman resolvent estimates with a direct analysis of the Hadamard parametrix in pseudo-Riemannian signature. Beyond the Lorentzian case, the use of causal convexity is replaced by a geometric no-return argument.

We give a proof of the $\Gamma$-liminf inequality for the Maz'ya–Shaposhni\-kova limit, as $s\to0^+$, of the fractional Sobolev seminorm, based on spherical rearrangements. The key point is that the rearrangements of a weakly convergent sequence are monotone radial profiles, which converge strongly by Helly's theorem; since the energy does not increase under rearrangement (in the limit), this reduces the liminf inequality to a pointwise convergence statement for radially decreasing functions. As an application, we treat truncated energies, in which the interaction is restricted to a ball of radius $r_s\to\infty$. Assuming $r_s^{-2s}\to\lambda\in[0,1]$, we prove that the scaled energies $\Gamma$-converge, with respect to weak $L^2$-convergence, to $$ (1-\lambda)N\omega_N\|u\|_{L^2}^2, $$ where $\omega_N$ is the measure of the unit ball of $\mathbb R^N$. This identifies the contribution of the long-range interactions and gives a continuous interpolation between the classical Maz'ya–Shaposhnikova limit ($\lambda=0$) and the vanishing-energy regime ($\lambda=1$).

We study the homogenization of an elliptic eigenvalue problem in a periodically perforated domain, where the spectral parameter appears both in the equation, through a bulk density $q$, and in a Steklov condition on the boundaries of the holes, through a surface density $\rho$, and where the resulting bulk–surface spectral measure $q\,dy+\rho\,dS_y$ is indefinite. We show that the asymptotic behavior of the spectrum depends essentially on whether the combined mean $m_{\rm eff}=\int_{Y^*}q\,dy+\int_\Gamma\rho\,dS_y$ is positive, negative, or zero. In all three cases, we prove convergence of eigenvalue clusters with multiplicity and of eigenspaces and, under additional regularity, $O(\varepsilon)$ estimates for the shifted or rescaled eigenvalues, spectral projectors, and aligned eigenspaces.

We investigate singularities of radial solutions of the classical two-phase and one-phase Stefan problem. In a remarkable result, M.A. Herrero and J.J.L. Velazquez show existence of a solution such that the free boundary described by $g(t)$ satisfies in dimension $n\geq 3$ $$g(t)=c_n \sqrt{t} |\log t|^{\frac{-1}{n - 2}} (1 + o(1)), t\to 0,$$ and they conjecture that these asymptotics hold in general. Here we prove the conjecture for all radial solutions, which means complete asymptotic rigidity in that class. In order to obtain these precise asymptotics for the two-phase Stefan problem, for which frequency formulas and monotonicity formulas are unknown, we apply the Potential Reduction to ODE method introduced in arXiv:2608.18913 for the Neumann Bernoulli problem. The philosophy of that method is that the asymptotic shape of the free surface is governed by a precise ordinary differential equation. The fact that we can apply it in this paper to an equation of different type with quasilinear principal part, which is in our regime not the perturbation of a linear operator, confirms the versatility of the method.

This work studies the asymptotic behavior of a class of viscous Hamilton–Jacobi equations with oscillatory periodic potentials, already studied by other authors in the PDE framework, and described in terms of an effective Hamiltonian. A novel variational approach is introduced, based on the classical stochastic control representation involving a Brownian motion, which allows the problem to be reformulated in terms of functionals defined on probability spaces. The core result is the identification of a homogenized Lagrangian obtained via a $\Gamma$-convergence procedure adapted to random processes, where filtrations and Brownian motions are treated as variables. The effective Hamiltonian is then recovered as the Legendre transform of this homogenized Lagrangian. The analysis provides both lower and upper bounds by means of blow-up arguments and the construction of stochastic correctors, yielding a complete characterization of the homogenized limit. This approach allows for a stability result with respect to the addition of some localized defects embedded in the periodic background.

For $d\ge3$, let $\lambda_d$ denote the sharp constant in the nearest-neighbor Hardy inequality on $\mathbb Z^d$ with Euclidean inverse-square weight for functions vanishing at the origin. Recent work established that the continuous Hardy coefficient \[ C_d=\frac{(d-2)^2}{4} \] is sharp in dimensions three and four, while it is not sharp for $d\ge9$. We prove $\lambda_d=C_d$ for the remaining cases $5\le d\le8$. The proof uses reciprocal edge fields and pointwise estimates for the associated vertex weights. The exterior estimates follow from polynomial inequalities, and the weights near the origin are treated by explicit estimates and local corrections.

We study three affine variational quantities built on the affine $L^p$ energy of Lutwak, Yang and Zhang: the affine Poincaré–Sobolev constants $\Lambda^{\mathcal{A}}_{p,q}(\Omega)$ of a bounded open set, the associated affine torsional rigidity, and the affine $p$-capacity of a compact set. We show that ellipsoids are the only extremal sets in their corresponding isoperimetric inequalities, and that the affine capacity of a compact set of positive volume is attained by a potential solving the affine $p$-Laplace equation. We also prove an affine Kohler-Jobin inequality: among sets with prescribed affine torsional rigidity, only ellipsoids minimize $\Lambda^{\mathcal{A}}_{p,q}$. Following the approach of Haddad, Jiménez and Montenegro to affine Sobolev inequalities, our methods rest on the $L_p$ Busemann–Petty centroid inequality. Proving a stability version of it, we obtain quantitative forms of the isoperimetric inequalities for $\Lambda^{\mathcal{A}}_{p,q}$, the affine torsional rigidity and the affine $p$-capacity, together with a stability result for the affine Kohler-Jobin inequality on convex sets.

We construct Anderson-type operators in the full subcritical regime via a SPDE argument. Our work encompasses perturbations by a Gaussian noise of the fractional Laplacian on the torus or on the full space, and of the harmonic oscillator on the full space. In every case, we are able to derive an optimal estimate on the difference between the solution and the deterministic heat flow, which allows us to get a Weyl law for the fractional case on the torus and for the harmonic oscillator case, and so in the full subcritical regime. For the non-translation-invariant operator, the main novelty is that we develop a general argument in order to prove how to renormalise the operator with some constants. The SPDE argument uses the flow approach introduced by Duch. To construct the operators, we use the Hille-Yosida theorem for the fractional Laplacian on the torus and the harmonic oscillator, and the Klein-Landau theorem for the fractional Laplacian in the full space.

We study a log-Hardy-Littlewood-Sobolev (log-HLS) inequality under constraints and a naturally induced Wasserstein flow. The constrained log-HLS inequality is dual, in a variational sense, to the constrained Trudinger-Moser-Onofri-Aubin inequality. The corresponding gradient-flow formulation leads to a parabolic-elliptic Patlak-Keller-Segel (PKS) system whose global dynamics exhibit a dichotomy between global existence and finite-time blow-up depending on the initial mass. Remarkably, the constraint raises the critical mass threshold of the flow by a factor of two compared with the unconstrained setting. This provides a distinct mechanism for suppressing blow-up from those arising in related formulations of the PKS-system, where blow-up is mitigated through additional effects such as a forcing term, coupling with a fluid equation, or the ambient flow generated by harmonic perturbations of the Newtonian potential.

We determine the sharp uniform constant in the quartic stability estimate for the logarithmic Sobolev inequality on finite cycles. We equip $C_{n}=\mathbb{Z}/n\mathbb{Z}$ with the uniform probability measure, and we let $\mathcal{E}_{n}$ be the Dirichlet form of the simple random walk on $C_{n}$, normalized so that its spectral gap is one. Frank and Ivanisvili recently proved the sharp logarithmic Sobolev inequality $2\mathcal{E}_{n}(u)\geq \mathrm{Ent}(u^{2})$ for all $n\geq 4$. We prove that $2\mathcal{E}_{n}(u)-\mathrm{Ent}(u^{2})\geq \frac{1}{3}\Vert u-1\Vert _{2}^{4}$ for all $n\geq 4$ and all nonnegative $u$ with $\Vert u\Vert _{2}=1$. This is the sharp form, uniformly in $n$, of the quartic stability estimate of Xie and Zhang, who obtained the constant $\frac{1}{12}$. The same estimate holds on the circle and for the word-length energy on $C_{n}$. Among these models, the four-cycle is the only extremal one: on $C_{4}$ the constant $\frac{1}{3}$ is sharp but not attained, while on $C_{n}$ with $n\geq 5$ and on the circle the sharp constant is at least $\frac{1}{3}+c$, where $c>0$ does not depend on $n$. We also give explicit upper bounds for these sharp constants. Our second main result is a stability estimate for the cubic Sobolev inequality of Frank and Ivanisvili, which is the main step in their proof. It gives the quantitative version of this inequality expected by Frank and Ivanisvili, with the optimal powers of $\Vert u-1\Vert _{2}$: four on $C_{n}$ with $n\geq 5$ and on the circle, and six on $C_{4}$. Two key new ingredients of the proofs are a bound for the first Fourier coefficient of $u$ in terms of its mean, which holds because $u$ is nonnegative, and a sharp stability estimate for Gross's two-point logarithmic Sobolev inequality, with sharp constants in both the fourth and the sixth powers of the distance.

Let $\mu$ be a doubling Radon measure in $\mathbb R^d$ with support $X$. We prove a local and quantitative form of the Bennewitz–Lewis theorem in this setting: if a Radon measure $\nu$ satisfies the Bennewitz–Lewis condition on a bounded relatively open set $V\subset X$ with a small enough constant, then $\nu|_V\ll\mu|_V$ and its density satisfies a reverse Hölder inequality on every compact subset $K\subset V$. Along the way we develop the dyadic toolkit these arguments run on — triple cubes defined through neighbours, the basis $\mathcal{D}\cup3\mathcal{D}$, and a Whitney decomposition — and we collect the Muckenhoupt-type classes of weights over that basis. The same machinery yields a local Gehring lemma for enlarged balls: both theorems are deduced from a single bootstrapping lemma, stated once for $p=1$ and $p>1$.

We study the two-dimensional periodic resistive compressible magnetohydrodynamic equations with constant shear viscosity and magnetic diffusivity, bulk viscosity $\rho^\beta$, and pressure $\rho^\gamma$. For every $\beta>1$ and $\gamma>1$, we construct global finite-energy renormalized weak solutions from strictly positive bounded $W^{1,q}$ densities, $q>2$, and arbitrary $H^1$ velocity and magnetic data. The construction combines approximation-independent finite density moments, generalized effective-flux identities, and a joint logarithmic renormalization of the density and its entropy defect. An augmented relative energy containing the $\beta$-power density potential yields weak–strong uniqueness against positive $H^2$ strong solutions for the same full pressure range. We also establish global strong existence for a large-background family when $\gamma>\max\{5-2\beta,\beta+1\}$. Writing $m=\int\rho_0$, the density perturbation is bounded in the scaled norm $m^{(\gamma-3)/2}\|\rho_0-m\|_{H^2}$, while the fixed $H^2$ velocity and magnetic norms need not be small. For sufficiently large $m$, frequency-dependent acoustic dissipation, staged vorticity weights, and a separate initial-layer flux estimate close a simultaneous bootstrap. The admissible pressure exponents have infimum $7/3$, with the endpoint excluded. For $\beta>2$ the relevant damping rate is the minimum of the viscous and pressure-relaxation rates; keeping both rates extends the argument without an upper restriction on $\beta$. The strong theorem has no upper restriction on $\gamma$ under this scaled density hypothesis; for fixed additive density perturbations it applies in the stated lower range up to $\gamma=3$.

We consider the initial value problem of the semilinear heat equation in the Fujita supercritical range for which global-in-time solutions may exist. The main result shows the existence of a separating norm $\|\cdot\|_X$ in the sense that, for nonnegative initial data $u_0\in L^\infty(\mathbf{R}^n)$, the smallness of $\|u_0\|_X$ implies global-in-time existence, whereas the largeness of $\|u_0\|_X$ implies finite-time blow-up. The separating norm is unique up to equivalence on nonnegative data.

We investigate the global existence and energy decay properties of a weakly coupled evolution system consisting of two wave equations in a bounded domain. The system exhibits a unique structural asymmetry: a nonlinear logarithmic damping term acts exclusively as a friction-like feedback mechanism on the first wave equation, while the second equation remains completely undamped and relies solely on linear cross-coupling to transfer and dissipate its energy. We establish two primary objectives. First, by utilizing the Faedo-Galerkin approximation scheme in conjunction with the structural monotonicity properties of the non-Lipschitz logarithmic nonlinearity, we prove a rigorous global existence result for strong solutions. Second, we provide a rigorous mathematical proof demonstrating the stability and asymptotic energy decay of the solutions under suitable constraints on the cross-coupling parameter. These results provide new insights into the stabilization mechanisms of coupled hyperbolic systems under degenerate nonlinear damping.

We construct new classical global solutions of the one-phase Bernoulli free boundary problem in $\mathbb{R}^3$ whose positive phases are periodic perturbations of the product of the Hauswirth–Hélein–Pacard double hairpin domain with a line. In contrast with previous constructions of undulating cylinders, the free boundary of the base solution is not compact modulo translations. More generally, we show that the same scheme yields periodic solutions in $\mathbb{R}^{n+1}$, $n \geq 3$, bifurcating from product solutions $U(x,t)=u(x)$, when the base $u:\mathbb{R}^n\to [0,\infty)$ has nontrivial finite Morse index, is nondegenerate within a symmetry class, and satisfies certain natural geometric conditions at infinity. In dimensions $n\in\{3,4\}$, where the stable Bernstein theorem is available, the recent classification of finite Morse index solutions by Fernández-Real, Florit-Simon and Serra shows that every such solution satisfies these geometric conditions. Thus, the existence of undulating solutions in $\mathbb{R}^{4}$ and $\mathbb{R}^{5}$ bifurcating from their respective products reduces to a nondegeneracy property of the base solution. In order to confirm that the perturbed solutions are positive, we prove that finite Morse index solutions with noncompact free boundary satisfy a strong distance nondegeneracy estimate.

We prove that no single rank ordered adversary strategy is globally optimal for the prediction with expert advice problem with six or more experts, in both the geometric stopping and finite time horizon settings. The proof is based on establishing a leading order correction when one expert moves far ahead of the others. This allows us to connect optimal strategies between $n$ and $j<n$ experts and utilize recent results on the exact optimality set for the five expert problem.

In this paper, we study the existence of ground states for two-component attractive Bose–Einstein condensates with displaced harmonic trapping potentials \[ V_1(x)=|x-x_1|^2 \ \text{and}\ V_2(x)=|x-x_2|^2, \] where $x_1\neq x_2\in\mathbb R^2$. For intraspecies interactions $a_1,a_2\in (0,a^*)$, we focus on the critical interspecies coupling \[ \beta=\beta^*:=a^*+\sqrt{(a^*-a_1)(a^*-a_2)} \] and prove that there exists a separation threshold $d_c\in(0,\infty)$ such that the corresponding constrained minimization problem admits no minimizer when $0<|x_1-x_2|<d_c$, whereas it admits a minimizer when $|x_1-x_2|\geq d_c$. The threshold is given by $d_c=\Lambda_*^{-1/4}$, where $\Lambda_*$ is characterized by an auxiliary variational problem and attained. Our results extend the previous work of Guo et al. [J. Funct. Anal. 276 (2019)] and complete the existence classification of ground states for two-component attractive Bose–Einstein condensates with harmonic trapping potentials.

We prove a geometric hierarchy of Fourier layer energies for the weakly nonlinear cubic Schrodinger equation on $\mathbb T_\pi\times\mathbb T_1$, retaining its nontrivial resonant dynamics. For initial Fourier support in a rectangle, form nested rectangles with half-widths $(2j+1)$ times the initial ones; layers are the differences between consecutive rectangles. On $0\le t\le T\delta^{-1}$, the energy in layer $j$, measured by its squared $H^1$ norm, is at most $C_N\delta^{2j}$ for $1\le j\le N$; the energy outside the $N$th rectangle is at most $C_N\delta^{2N+2}$. These bounds hold at every fixed depth $N$ under a uniform $H^s$ bound, $s>2$, with constants depending on $N$. The Sobolev bound holds for every fixed $T$ in the defocusing case. We also obtain layer estimates for support bounded in one coordinate and for geometrically decaying initial Fourier tails. The proof uses a filtered Poincare-Dulac normal form: input layers select which nonresonant interactions are removed, while the same layer weights control the interactions left in the equation. Finite coordinate changes and stability estimates transfer this hierarchy to the exact solution.

We prove the existence of infinite time bubble tower solutions for energy critical $k$-equivariant Schrödinger maps $\mathbb{R}^2\to \mathbb{S}^2$ with $k\geq 3$. More precisely, for any integers $k\geq 3$ and $N\geq 1$, we construct a $k$-equivariant solution that is global in one time direction and decomposes asymptotically into a superposition of $N$ bubbles with scales $\lambda_j(t)\sim|t|^{-\beta_j}$, where $\beta_j=\frac{1}{2}\left(\left(\frac{k}{k-2}\right)^{N-j}-1\right)$. Our approach is a backward construction combined with modulation analysis and high order energy estimates. We use Darboux coordinates adapted to the multi bubble profile to express the remainder in a complex scalar form. A key ingredient is the construction of a correction that cancels the leading effect of the nonlinear defect.

The ideal magnetohydrodynamic (MHD) system that governs a conducting plasma—the Maxwell equations together with the conservation laws of mass, momentum and energy—is the same local balance structure that describes hot accretion flows around black holes. We introduce a uniform four-dimensional spacetime dilation \[ \Phi_L:\;(t',x',y',z')\mapsto (t,x,y,z)=(Lt',Lx',Ly',Lz'),\qquad L>1, \] and show that it is a smooth, orientation-preserving $C^\infty$ diffeomorphism of flat spacetime. Pulling the MHD conservation laws back through $\Phi_L$, we prove that the closed ideal-MHD system is form-invariant (covariant), with the current density rescaling as $\bm j=\bm j'/L$, the magnetic diffusivity as $\eta'=\eta/L$, and external sources rescaled by $L$. The transformation realises three physically distinct effects on the perturbed flow: a linearization to the tangent-space (linearised MHD) system, a movie slow-motion dilation of dissipative and dynamical timescales, and an electron-microscope magnification of small-scale structure. Taking the formal limit $L\to\infty$ localises the dynamics on a neighbourhood of a horizon: resistivity vanishes, the asymptotic time derivative freezes, and all transient multipolar structure of the infalling plasma redshifts away. The remaining stationary, axisymmetric, asymptotically flat electrovacuum is uniquely the Kerr–Newman geometry, so that the exterior of the compact object is characterised only by its mass $M$, angular momentum $J$ and charge $Q$. The result provides a local, conservation-law-based perspective on the black hole no-hair theorem that is complementary to the classical Israel–Carter– Robinson theorems and to contemporary gravitational-wave and Event Horizon Telescope tests.

In this paper, we study radial normalized solutions to nonlinear Schrödinger equations in the unit ball \[ -\Delta u+\lambda u=|u|^{p-1}u, \qquad \int_{B_1}|u|^2\,dx=\rho, \] where $\lambda$ is unknown and $p\ge p_c:=\frac{N+2}{N-2}$. Soave \cite{Soave} proved that for every $1<p<p_c$ every positive mass is attained by infinitely many radial solutions and explicitly raised the question of whether every mass is attained in the Sobolev critical and supercritical regimes. We answer this question negatively in the radial class: for every $p\ge p_c$, the set of masses attained by nontrivial radial solutions is an interval with finite positive supremum. In particular, at $p=p_c$ the set of attainable masses is $(0,\rho^*_{\rm rad})$ or $(0,\rho^*_{\rm rad}]$, with $0<\rho^*_{\rm rad}<\infty$, and the same finite threshold alternative holds for every $p>p_c$. Thus, in the radial class, the subcritical all masses phenomenon fails throughout the full Sobolev critical and supercritical range.

In this paper, we study multidimensional perturbations of radially symmetric rarefactive flows for the compressible Euler equations. No assumptions on the smallness of the background are imposed. We derive lower bounds on both the classical lifespan and stability timespan of the perturbed solution. These lower bounds are demonstrated to tend to infinity as the size of the perturbation approaches zero. Furthermore, owing to the rarefactive nature of the background, the resulting estimate is shown to be substantially longer than the corresponding estimate for a general classical background with bounded first derivatives.

We study the stochastic heat equation on a bounded interval with zero Dirichlet boundary conditions, driven by symmetric $\alpha$-stable space–time Lévy white noise, where $1<\alpha<2$. Both the drift and noise coefficients are continuous and have at most linear growth. For bounded deterministic initial data and every $r>1/2$, we construct a weak solution with càdlàg paths in the negative Sobolev space $\mathbb{H}_{-r}$. We also construct a predictable mild solution on the same stochastic basis and with the same noise. At each fixed deterministic time, this mild solution is a density of the distribution-valued weak solution.

We study the initial–boundary value problem for the one-dimensional full compressible Navier–Stokes system whose viscosity and heat conductivity are given by $\mu(\theta)=\mu\theta^\alpha$ and $\kappa(\theta)=\kappa\theta^\beta$, respectively. The initial density is nonnegative and is allowed to vanish. For $\alpha>2$ and $\beta>2\alpha+4$, we prove the existence of a global strong solution for arbitrarily large compatible $H^2$ data, provided that the prescribed boundary temperature is sufficiently high relative to their size. No velocity or temperature perturbation is required to shrink as the boundary temperature tends to infinity. A shifted initial-layer weight and an exact symmetrization of the two material-derivative equations separate the fast thermal mode from the momentum mode. The resulting estimates are uniform with respect to the positive lower bound used in the approximation and therefore survive the limit to genuine vacuum. The proof further combines a relative-entropy identity centered at the conserved mean density, the physical effective viscous flux, a Zlotnik-type estimate along particle trajectories, and finite-time transport estimates for the density derivatives. The density remains uniformly bounded and the temperature stays uniformly comparable with the prescribed boundary temperature. After the initial layer we establish vacuum-compatible exponential relaxation: the density converges to its conserved mean in every finite $L^p$, the velocity and temperature perturbation decay in $H^1$, and the weighted material-derivative energy decays at rate $c\bar\theta^{1-\alpha}$.

We investigate the inhomogeneous incompressible Navier-Stokes system in the 2D global domain with zero initial total momentum, allowing vacuum and densities with finite moments but not necessarily compact support. Exploiting the cancellation structure inherent in the 2D equations along with spatial moment bounds on the density, we derive that the inertial force belongs to the Hardy space and establish weighted Stokes estimates through endpoint Hardy-Littlewood-Sobolev inequalities. These estimates identify the far-field velocity and control the remainder, leading to uniform exponential decay of the velocity throughout global domains. Combined with higher order regularity bounds, we establish the global well-posedness of strong solutions for the density-dependent viscosity case, and prove the uniform regularity of flow maps for measurable density patches in the constant viscosity model.

We prove strong ill-posedness for the two-dimensional incompressible Euler equation in the endpoint velocity space $W^{3,1}_\sigma(\mathbb R^2)$. Given any compactly supported smooth divergence-free background and any $\delta>0$, we construct a single compactly supported divergence-free datum within $\delta$ of the background in $W^{3,1}$. The perturbation can be localized inside any prescribed nonempty open neighborhood of the background support while remaining disjoint from that support. The associated unique global finite-energy solution $u$ satisfies ${\rm curl} u\in C([0,\infty);W^{2,1}(\mathbb R^2))$, but \[ \operatorname*{ess\,sup}_{0<t<T}\|u(t)\|_{W^{3,1}}=\infty \qquad\text{for every }T>0. \] The proof combines localized hyperbolic insertion, the Bonami–Poornima nonmultiplier mechanism, and many-packet dilution. Moment cancellation and positive material separation control the nonlocal Biot–Savart interactions, allowing a nested construction inside one fixed compact region. Without the compact-support requirement on the limiting datum, a complementary far-field argument yields a dense $G_\delta$ set of data whose solutions leave $W^{3,1}$ at every positive rational time and on a residual set of times, with unbounded essential supremum on every nonempty open time interval.

Let $(M^n,g)$ be a compact Riemannian manifold of dimension $n\ge 2$, possibly with nonempty convex $C^2$ boundary. We study the first nonzero eigenvalue $\lambda_1(M)$ of the Laplace–Beltrami operator, subject to the Neumann boundary condition when $\partial M\neq\emptyset$. Given a reference scalar curvature $R_E>0$, set $K_E:=R_E/(n(n-1))$, and define the traceless Ricci tensor $\mathring{\Ric}:=\Ric-\frac{R}{n}g$ and the Einstein deviation \[ \Theta:=|\mathring{\Ric}|+\frac{|R-R_E|}{n},\qquad \kappa_{\mathrm{Ein}}(p):=\left(\frac{1}{\vol(M)}\int_M\Theta^p\,dV\right)^{1/p}, \quad p>n/2. \] We say that $M$ is $\varepsilon$-nearly Einstein if $\kappa_{\mathrm{Ein}}(p)<\varepsilon$. The main result states that for every $\alpha\in(0,1)$ there exists $\varepsilon_0=\varepsilon_0(n,p,\alpha,D,R_E)>0$ such that if $\diam(M)\le D$ and $\kappa_{\mathrm{Ein}}(p)<\varepsilon_0$, then \[ \lambda_1(M)\ge \alpha\,\lambda_1(n,K_E,D), \] where $\lambda_1(n,K,D)$ is the first nonzero eigenvalue of the Bakry–Qian one-dimensional model problem with curvature parameter $K$, with the convention $\lambda_1(n,K,D)=nK$ when $K>0$ and $D\ge\pi/\sqrt{K}$. The proof combines the Bochner formula for the decomposition $\Ric=(n-1)K_Eg+\theta g+\mathring{\Ric}$ with an auxiliary function $J$ satisfying \[ \Delta J-\tau\frac{|\nabla J|^2}{J}-2J\Theta=-\sigma J \] and Neumann boundary conditions, together with Moser iteration and a maximum principle. As $\kappa_{\mathrm{Ein}}(p)\to0$, the estimate recovers the classical Bakry–Qian theorem.

We establish global well-posedness of classical solutions to the two-dimensional primitive equations with fractional horizontal dissipation for arbitrarily large initial data in the full subcritical range $1<\alpha\leq2$. Together with the known ill-posedness results for $0\leq\alpha<1$, this establishes the sharp dissipation threshold for large-data global well-posedness in the corresponding solution framework. The key ingredient is a hydrostatic energy estimate obtained by splitting the nonlinear energy into symmetric and antisymmetric parts and exploiting the commutator structure of the latter. Using anisotropy and incompressibility, we bound the nonlinear energy by the $L^\infty$ norm of the hydrostatic vorticity times a quadratic velocity norm with only one-half additional horizontal derivative. The vorticity maximum principle then yields enhanced velocity bounds, which close the vorticity estimates and verify the continuation criterion throughout the subcritical regime.