Papers

We establish sharp mixed spectral Barron regularity for eigenfunctions of molecular Coulomb Hamiltonians. The mixed norm is a Fourier $L^1$ norm with one isotropic weight and coordinate-product weights, and therefore detects regularity invisible to the isotropic Barron scale. For a nonempty set $I$ of electron indices on which the wave function is antisymmetric, we derive an explicit admissible region for the isotropic order $s$ and the coordinate orders $\alpha,\beta$. This region is optimal as a uniform statement over the class of clamped-nuclei Coulomb Hamiltonians. For fixed-spin components with two occupied spin blocks, it reduces to $s+\alpha+\beta<1$; in the fully spin-polarized class it reduces to $s+\alpha<1$. In particular, if $\mathcal I_\sigma$ denotes the family of occupied same-spin blocks determined by $\sigma$, then every fixed-spin spatial component $\psi_\sigma$ satisfies, for every $0\leq\alpha<1$, \[ \left(\sum_{I\in\mathcal I_\sigma}\prod_{i\in I}\langle\xi_i\rangle^\alpha\right)\widehat{\psi_\sigma}\in L^1(\mathbb{R}^{3N}). \] For a fully spin-polarized state, $\mathcal I_\sigma=\{\{1,\ldots,N\}\}$.

We prove uniform-in-$N$ partition-function estimates at all positive temperatures for the centered modulated energy of logarithmic, Riesz, and Bessel–Riesz kernels on $\mathbb R^d$ in the locally square-integrable range $0\le s<d/2$. They yield finite-particle exponential integrability, quadratic behavior at small parameter, explicit kernel-dependent growth at large parameter, and uniform entropy control of the associated Gibbs measures. As applications, we obtain uniform Rényi-divergence and relative-entropy bounds for interacting Gibbs equilibria and entropic mean-field convergence near thermal equilibrium; the optimal $N^{-1}$ normalized relative-entropy rate for the three-dimensional Coulomb flow; and an $N^{-1/2}$ Gaussian approximation for fixed-time finite-dimensional fluctuations. Finally, the modulated energy converges to a random variable in the second Wiener chaos, and for every positive parameter the partition functions converge to its Laplace transform, represented by a Carleman–Fredholm determinant. This formula identifies the sharp small-parameter behavior and, when the reference density is bounded below on a ball, the sharp large-parameter growth.

Let $1 \leq q \le 2$ and denote by $2 \leq q'$ its corresponding conjugate exponent. We prove the continuity of solutions $u \in W^{1,(\frac{n}{n-1},q')}(B^n, \mathbb{R}^m)$ to the critical elliptic system $-\Delta u = \Omega \cdot \nabla u$ in dimension $n \ge 3$, where the potential $\Omega \in L^{(n,q)}(B^n, \mathfrak{so}(m) \otimes \wedge^1)$ is antisymmetric. First, we construct $P \in W^{1,(n,q)}(B^n, \mathrm{SO}(m))$ such that the PDE can be rewritten as $-\operatorname{div}(P^{-1}du) = \ast d\xi \cdot P^{-1}du$, which is nearly a Jacobian structure up to the rotation $P$. Second, we provide a Rivière's $\mathrm{GL}(m)$-Gauge in order to establish a "full" $(A,B)$-conservation law, i.e. $-\operatorname{div}(Adu)=d^\ast B \cdot du$. We show that the assumption on $\Omega \in L^{(n,q)} (B^n, \mathfrak{so}(m) \otimes \wedge^1) $ for $q\leq 2$ is optimal.

We investigate the eigenvalue of a generalized Monge-Ampère equation. We proved that the eigenvalue of the equation on the Lp Minkowksi addition of two convex bodies satisfies the Lp Brunn-Minkowski inequality.

We study asymptotic properties of critical points of the two-dimensional vectorial Allen-Cahn energy with finitely many non-degenerate wells, subject to a homogeneous Neumann boundary condition. For any sequence with uniformly bounded energy, we prove that the limiting full and potential measures are supported on a closed countably 1-rectifiable set up to the boundary and satisfy the discrepancy relations. The potential measure defines a free-boundary stationary rectifiable varifold. Boundary mass may occur: weights are constant on regular boundary arcs, finite-type junctions obey projected balance, and the sole non-boundary branch meets the boundary orthogonally. This provides a Neumann-boundary extension of Bethuel's planar interior theory. The key idea is to read the boundary geometry from the limiting stress-energy tensor, which identifies the potential-energy measure as the stationary interfacial measure even in the presence of boundary concentration.

We develop a rigorous analytical framework for a class of physiologically structured population models with two internal state variables, nonlocal ecological feedbacks, dynamic resources, inter-zone transfer, and selective harvesting. The full model is a coupled nonlinear PDE–ODE transport–renewal system with endogenous inflow at the recruitment boundary, a setting in which transport, nonlocal dependence, and boundary renewal interact at the same level. For this full nonautonomous multi-zone system, we prove finite-horizon well-posedness in a positive $L^{1}$-based state space, including global existence on arbitrary bounded time intervals, uniqueness, nonnegativity, and continuous dependence on initial data, environmental forcing, and harvesting effort. We then isolate an autonomous single-zone reduction at extinction and construct a positive compact next-generation operator on the recruit space. In a further nonlinear stationary reduction, we prove that supercriticality of the basic reproduction number $\mathcal R_{0}>1$ yields existence of a nontrivial stationary state under a parametrized compact-operator hypothesis encoding density-dependent renewal feedback. Finally, for a finite-horizon harvest objective over a compact Lipschitz-regular admissible class, we establish existence of an optimal control. The results separate what can be proved for the full climate-explicit system from what can be justified only after autonomous reduction, thereby clarifying the mathematical scope of threshold and control theory for structured populations.

We establish Aleksandrov–Bakelman–Pucci-type estimates for viscosity subsolutions of Poisson equations associated with the weighted $1$-Laplacian and, more generally, with fully nonlinear operators acting on the Hessian in directions orthogonal to the gradient. A key feature of the proof is that the quasiconcave envelope plays the role of the concave envelope in the classical ABP estimate.

Let \(n\ge1\), \(0<s<1\), \(c>0\), and let \(A\) be a positive definite symmetric matrix. We prove that the unique decaying viscosity solution of \[ \min\{u,\,(-\Delta)^s u-(c-\langle Ax,x\rangle)\}=0 \qquad\text{in }\mathbb R^n \] has the form \[ u(x)=K\max\{1-\langle Bx,x\rangle,0\}^{1+s} \] for some \(K>0\) and some positive definite symmetric matrix \(B\). In particular, its positivity set is an ellipsoid. For \(s=1/2\) and \(n\ge2\), this result, combined with the classification of Fernández-Real and Yu, implies that cubic global thin obstacle solutions with nonempty bounded positivity set have ellipsoidal positivity sets. This proves the conjecture of Fernández-Real and Yu in the nonempty bounded-positivity case.

We study the diffusive limit of the steady neutron transport equation with velocity-dependent in-flow boundary data in polygonal domains. On a bounded convex polygon, we construct an approximation by the interior solution, flat side layers, and wedge layers, and prove an algebraic error estimate in $L^\infty$. We also give a complete formulation and well-posedness theory for the kinetic wedge layer, including its algebraic decay. The proof combines a characteristic stability estimate, a weighted Mellin mapping theorem, a two-depth construction and matching scheme, and a shifted superharmonic barrier for the wedge corrector. We also prove the $L^2$ diffusive limit at rate $O(\varepsilon^{1/2})$ on general bounded simple polygons, including those with reentrant vertices.

In this paper, we study the evolution of nearly radial concentrated vortices for the incompressible Euler equation in a bounded planar domain. We prove that if a single vortex is initially concentrated near a strict local minimum point of the Robin function of the domain and is close, up to translation, to its symmetric decreasing rearrangement, then both its shape and location remain uniformly controlled for all time. We also establish an analogous result for a pair of opposite-sign vortices near a strict local minimum point of the corresponding Kirchhoff–Routh function. No symmetry is imposed on the domain or the initial data, and the initial data need not be close to any steady state. To prove these results, we develop a new Lyapunov mechanism for the evolution of vorticity governed by the Euler equation starting from such initial data, providing quantitative control of both vortex shape and location. Specifically, we combine kinetic-energy conservation and vorticity equimeasurability with several fixed-time estimates to obtain a conditional estimate, and then use a set-valued first-exit argument to propagate it globally in time.

We study the initial boundary value problem of the two-dimensional compressible nematic liquid crystal flow with the shear viscosity $\mu$ being a positive constant and bulk viscosity $\lambda$ being a power function of density with the power exponent $\beta$. Under the condition $\beta>1$, we establish the global existence and large time behavior of the radially symmetric strong solutions to this Vaigant–Kazhikhov type model of simplified compressible Ericksen-Leslie system of Dirichlet boundary conditions for the velocity and Neumann boundary ones for the director with arbitrary large data and vacuum. This work improves the results of Zhong and Zhou (Math. Ann. 390, 2024; J. Math. Pures Appl., 212, 2026) for general 2D domains by removing the geometric angel condition on the director and relaxing the constrain on $\beta$ from $\beta>4/3$ to $\beta>1$. The key ingredient is that the rigidity mechanism arising from the radial symmetry of director prevents concentration phenomena in the transported harmonic heat flow.

The magnetized Vlasov–Poisson equation is a fundamental kinetic model for collisionless plasmas. We establish its quasi-neutral limit in two dimensions in the presence of a spatially inhomogeneous and time-dependent external magnetic field. In the same setting, we also establish the combined mean-field and quasi-neutral limit of the associated repulsive Coulomb particle system, thereby extending the results of \cite{han2021newton}. Both limits lead to the incompressible Euler equations with Lorentz forcing. A key structural feature is the pointwise skew-symmetry of the Lorentz operator, which produces exact cancellations in the modulated-energy estimates and is crucial for controlling the limiting dynamics. We further derive a closed vorticity formulation in which the sum of the fluid vorticity and the magnetic field is transported by the flow, coupled to an evolution equation for the spatial mean velocity. Finally, we establish global well-posedness of the resulting limiting system in the Lipschitz class.

We prove uniqueness of finite-time geometric endpoints for the Möbius-invariant Willmore flow in $\mathbb{S}^3$ under uniform quantitative nonumbilicity. The multiplicity-counting varifolds converge, without reparametrization or Möbius renormalization, to a unique integral two-varifold. An intrinsic transport estimate gives quantitative total-variation convergence of the induced area measures on the fixed domain and bounded-Lipschitz Cauchy control of their pushforwards. Together with Allard compactness and rectifiability, this upgrades subsequential compactness to full-trajectory varifold convergence. The limit has generalized Euclidean mean curvature in $L^2$ with the natural endpoint lower-semicontinuity bound. For finite maximal trajectories with initial energy at most $8\pi$, Jakob's subsequential alternative becomes sequence independent: the limit is zero, or it has unit density and embedded Lipschitz support of genus zero or one. For Hopf-torus trajectories under the same energy bound, nonumbilicity is automatic and the anchored constant-speed profiles converge weakly in $W^{2,2}$ and strongly in $W^{1,2}$ and $C^{1,\alpha}$ for every $\alpha<\frac{1}{2}$. At infinite time, the same method yields a unique limit under an additional finite-dissipation-length condition.

We study operators acting on functions defined on the half-space, with methods inspired by sparse domination. Using the specific link between dyadic grids on the boundary and Whitney regions on the half-space, we obtain a very precise form domination with model operators of Hardy type. We introduce mixed-norm off-diagonal estimates that measures both tangential and transversal decay without requiring pointwise control in the transversal variable, substantially weakening the assumptions used in earlier tent space theories. This allows us to prove optimal weighted tent space extrapolation from local boundedness plus off-diagonal decay. The resulting framework yields new bounds for operators arising from non-autonomous elliptic and parabolic PDEs with non-smooth coefficients.

We develop a boundary blow-up scheme for the three-dimensional nonstationary Navier–Stokes equations in domains whose local boundary graphs belong to $W^{2-1/P_b,P_b}(\mathbb R^2)$ with $P_b>3$. The argument is designed to remove the $P_b>15/4$ restriction in the boundary partial-regularity theorem of Breit. After a rigid rotation to the tangent plane and parabolic rescaling, such a graph becomes small simultaneously in the multiplier classes $\mathcal M^{4/3,3/2}$ and $\mathcal M^{16/15,15/14}$ at the rate $r^{1-3/P_b}$. A two-parameter contradiction argument then sends both the fluid excess and the geometric multiplier norm to zero, producing the standard flat Stokes system in the limit. At the critical pressure pair $(5/3,15/14)$, a localized flat-Stokes decomposition separates a strongly vanishing error pressure from forced and homogeneous pressures. Their decay exponents are respectively $6-15/P_f$ and $12/5-9/Q$, where $P_f>5/2$ and $Q>15/4$. Rough-coefficient errors are absorbed only at the critical multiplier level; higher spatial integrability is invoked only for flat Stokes problems. The resulting Campanato iteration gives boundary Hölder regularity outside a relatively closed set of zero parabolic $5/3$-dimensional Hausdorff measure.

We study a two-parameter first-order hyperbolic relaxation approximation of the incompressible Navier–Stokes equations on the two-dimensional torus, under the derivative-level large-perturbation assumptions of Huang, Rohde and Zhang. Building on the auxiliary-system approach used in earlier pressure-convergence results, we introduce a non-autonomous linear corrector that carries the complete initial discrepancy, the linearized convection and the smooth consistency forcing. This corrector describes the full linear response; it is not asserted to be a purely acoustic component. An exact compensated energy identity, including a time integration by parts in the linearized convection term, yields an integrated gradient estimate for the velocity corrector. The two-dimensional Ladyzhenskaya inequality then bounds its quadratic self-interaction by $O(\delta^2)$ in squared space-time $L^2$ norm. Consequently, the nonlinear remainder has weighted energy $O(\delta^2)$, and its pressure and velocity components satisfy the $L^\infty(0,T;L^2)$ bounds $C_T\delta/\sqrt{\epsilon}$ and $C_T\delta$, respectively. In the sufficient parameter window $\delta^2\ll\epsilon\leq\mu_*\delta$, the pressure after subtraction of the linear corrector converges strongly to the incompressible pressure. A mean-zero Fourier-mode construction, transferred to the nonlinear system by the remainder estimate, shows that the uncorrected pressure need not converge even on time intervals bounded away from zero.

Let $\lambda_1(\Omega)$ and $T(\Omega)$ denote the first Dirichlet eigenvalue and torsional rigidity of a bounded convex domain $\Omega\subset\mathbb{R}^2$, and let $M=\max_\Omega u$, where $u$ is the torsion function. We prove the sharp inequalities $\pi^2/24<\lambda_1(\Omega)T(\Omega)/|\Omega|<\pi^2/12$. This proves the two-dimensional case of Conjecture 4.2 proposed by van den Berg, Buttazzo, and Pratelli. Its planar formulation was later restated as Conjecture 1.1 by Bañuelos and Mariano, who proved it for triangles and rectangles. The lower bound follows by combining Payne's strict estimate for $\lambda_1M$ with the sharp torsion-efficiency inequality $T(\Omega)\geq |\Omega|M/3$. For smooth strictly convex domains, we use an Airy stress potential to construct a convex body whose surface-area measure is the torsional first-variation measure. A sharp one-dimensional inequality for directional maximum profiles then yields a stronger geometric containment. For the upper bound, we factor the Pólya functional through the second torsion moment. A level-set torsion–perimeter inequality gives the factor $5/6$, while a sharp weighted one-dimensional estimate gives the factor $\pi^2/10$. Collapsing triangles and elongating rectangles show that both endpoint constants, as well as the efficiency constant $1/3$, are sharp.

Time-harmonic Maxwell scattering by a perfectly conducting obstacle is governed by the electric field integral equation (EFIE), whose natural energy space is $\boldsymbol H^{-1/2}(\mathrm{div}_\Gamma,\Gamma)$. For countably parametrized surface deformations, both the operator and its energy space depend on the geometry. We establish operator-valued shape holomorphy of the EFIE operator on a fixed reference space and derive dimension-independent sparse approximation rates. The main analytical difficulty is a fractional mapping property absent from existing $L^2$-based operator-valued shape-holomorphy theory for weakly singular kernels: the Maxwell graph norm requires uniform holomorphy of the complex-deformed scalar single layer as an operator from $H^{-1/2}$ to $H^{1/2}$. We prove this one-order smoothing by realizing the Laplace principal part as the trace of a uniformly sectorial complex-coefficient divergence-form problem on a fixed real ambient domain. Combined with a surface contravariant Piola transformation, which identifies the geometry-dependent Maxwell trace spaces and cancels the surface Jacobians in the pulled-back EFIE exactly, this yields $(\mathbf b,p,\varepsilon)$-holomorphy of the EFIE operator family for $\ell^p$-summable shape deformations, $0<p<1$. Under pointwise exclusion of interior electric resonances, parameter-uniform invertibility is derived rather than assumed, and the surface current and far field inherit the same parametric regularity. Their Legendre coefficients are $\ell^p$-summable, yielding dimension-independent best $N$-term approximation rates. The operator-level results also transfer to Galerkin boundary element operators on a single reference mesh, with constants independent of the discretization dimension, and provide reusable surrogates for multiple incident fields and bounded linear observables.

We construct, for every $d\ge 5$, a nonnegative radial potential $V\in L^\infty_{\mathrm{loc}}(\mathbb{R}^d\setminus\{0\})\cap L^1(\mathbb{R}^d)$ for which $V(-\Delta\dotplus V)^{-1}$ is not bounded on $L^2(\mathbb{R}^d)$.

We study quantitative stability for the fractional Yamabe functional introduced by González and Qing. Its minimizers correspond to conformal metrics with constant fractional curvature. We prove that the deficit of the fractional Yamabe functional controls, with a suitable power, the distance to the set of minimizers. Our approach is based on a Lyapunov–Schmidt reduction for the nonlocal functional, together with the extension characterization of the fractional conformal Laplacian, which provides the regularity theory needed for the local analysis. As a consequence, we also obtain an analogous quantitative stability estimate for the associated extension functional.

We investigate the discrete spectrum of finitely many point interactions for Neumann and Robin Laplacians on special unbounded and exterior $C^{1,1}$ domains in dimensions two and three. The operators are realized as self-adjoint extensions through an ordinary boundary triple whose gamma field and Weyl matrix are constructed from the Robin Green kernel. Eigenvalues below the background spectrum are characterized by the Weyl matrix, yielding an exact finite-dimensional counting formula. If the background operator is non-negative, this also gives the number of negative eigenvalues without assuming a finite zero-energy limit of the Weyl matrix. In the one-centre case we identify the critical coupling and prove that the unique eigenvalue branch is real analytic, strictly increasing, and strictly concave. For scalar multicentre couplings $\Theta=\alpha I_N$, sufficiently strong attraction produces exactly $N$ eigenvalues below the background spectrum, and all of them have universal leading asymptotics coinciding with the whole-space laws. If the bottom of the background spectrum is an isolated eigenvalue, the branch exists for every finite coupling and we determine its leading decoupling asymptotics as the coupling tends to $+\infty$. Explicit exterior-sphere and exterior-disk models illustrate the critical couplings and their threshold behaviour.

The initial-to-final-state inverse problem for the Schrödinger equation consists in determining uniquely the Hamiltonian that generates the evolution, assuming the knowledge of the initial-to-final-state map. This functional maps each initial state $f\in L^2(\mathbb{R}^n)$ to the corresponding final state at a fixed time $T$. The problem was formulated by Caro and Ruiz in the case of Hamiltonians arising from time-dependent bounded electric potentials that exhibit super-exponential decay at infinity. Caro, Parissis and the authors of this article established that uniqueness also holds for time-independent bounded potentials with super-linear decay at infinity. In this paper, we quantify the above uniqueness results establishing that the potentials are stable under small changes of the initial-to-final-state maps. In the case of time-dependent potentials we get a stability of logarithmic type. A notable improvement is achieved when the potentials are time-independent, where under this assumption we prove Hölder stability estimates.

We establish higher-order Gaussian upper bounds for the heat semigroups associated with a broad class of sectorial maximally subelliptic quadratic forms on manifolds with boundary. Near non-characteristic boundary points and in the interior, we obtain pointwise bounds for all mixed derivatives of the heat kernel in time and along the Hörmander vector fields, expressed in the associated Carnot–Carathéodory geometry. The results apply to operators of arbitrary even order, systems, nonsymmetric forms, and boundary conditions beyond the Dirichlet case.

In 2023, Wyman and Xi asked “Can you hear your location on a manifold?” We pose a related question, asking if probabilistic data can be used to recover location within a manifold. More precisely, we ask if location can be recovered from the distribution of exit times of Brownian sample paths. We show that exit time moments determine location up to symmetry for a variety of two-dimensional domains. Beginning with (convex) domains whose boundaries are conic curves, we show that in elliptic, parabolic, and hyperbolic domains, the first two exit time moments are sufficient to detect location up to symmetry. We establish similar results in unbounded wedge domains and equilateral triangular domains. We show that the full sequence of exit time moments determines location in rectangular domains. We end with two location detection results on general planar domains. The key tool driving our work is a “Poisson hierarchy,” which establishes a connection between exit time moments of Brownian motion and solutions to a family of PDE problems.

We consider the problem of finding critical domains $\Omega\subset{\mathbb R}^3$ for the energy functional $$ \mathcal E (\Omega) = {\rm Per}\,(\Omega) + \frac 12 \iint_{\Omega\times \Omega } \frac{dx\,dy}{|x-y|} $$ under the volume constraint $|\Omega|=V$. We look for smooth, embedded, compact surfaces $\partial\Omega$ that solve this problem. We construct an axially symmetric, non-minimizing solution that, for a sufficiently small $V>0$, resembles the union of two balls with volume $V/2$ connected by a tiny, approximately catenoidal neck with a width of the order $V^{\frac 43}$.

We establish finite-energy Cauchy stability for vacuum and matter spacetimes with $T^2$ symmetry on $T^3$, under general constitutive equations satisfying hyperbolicity and mild asymptotic conditions at vacuum and large mass-energy density. First, using the area of the symmetry orbits as a time function, we introduce the JKL formulation of Einstein areal flows}, a first-order evolution-constraint system coupling a wave-map structure for the geometry to hyperbolic matter balance laws and weighted transport equations for the twists and the momentum tangent to the symmetry orbits. Second, we also define two classes of hyperbolic Einstein-Navier-Stokes models, constructed from (as we call them) a Navier-Stokes potential and a relaxation rate map. The proposed particle-production model has divergence-free matter stress and non-negative particle-number production, while the proposed dissipated-energy model conserves particle number and dissipates fluid mass-energy, while an auxiliary stress restores the total stress-energy conservation required by the contracted Bianchi identity. For regular finite-energy (Einstein, Euler, Navier-Stokes) flows, we establish maximum and invariant-domain principles, geometric and energy monotonicity formulas, spacelike and timelike energy estimates, and total-variation estimates. Third, together, these estimates yield finite-energy Cauchy stability in the future expanding regime and, in the contracting regime, until the spacelike volume collapses. They are independent of the viscosity (or relaxation) mechanisms and cover vacuum, unbounded mass-energy density, and arbitrary finite rapidity. They also extend to weakly regular Einstein-Euler areal flows satisfying a reference mathematical entropy inequality.

We prove interior Hölder regularity for solutions of the stationary kinetic Fokker-Planck equation in non-divergence form in space dimension one. The estimate is independent of any small-oscillation condition on the diffusion coefficient.

We prove that when the J-equation does not admit a smooth solution, there is always only a finite number of obstructing optimally destabilizing curves and divisors on compact Kähler manifolds of any dimension. For threefolds, this establishes the unconditional finiteness of all optimally destabilizing subvarieties. We moreover characterize the union of these optimal subvarieties, proving that they are contained in the locus where a natural continuity path loses local smooth compactness. Finally, to address subvarieties that are neither curves nor divisors, we prove a rigidity criterion which prevents optimal cycles from moving in compact one-parameter families, with further consequences for their deformation and symmetry properties.

Fushchych and Shtelen defined approximate symmetry by expanding the solution of a perturbed differential equation in the small parameter, replacing the original equation by a triangular system for the expansion coefficients. We extend this construction to arbitrary order. A coefficient-map formulation yields the order-$p$ Fushchych–Shtelen (FS) system and establishes, under explicit hypotheses, a correspondence with the Baikov–Gazizov–Ibragimov (BGI) method. The two frameworks are compared for a perturbed cubic wave equation. Up to equivalences, we classify the nonlinearities admitting an FS dilation through second order. The classification consists of a generic power family and logarithmic branches at exceptional exponents. Among perturbations nontrivial at first order, vertical BGI continuations select a proper subfamily of one FS branch, and a second common branch arises when the perturbation first enters at second order; the two methods are related but not interchangeable. Joint reduction by the Lorentz algebra and the inherited FS dilation integrates all resulting FS normal forms; the explicit solutions of the 1989 Fushchych–Shtelen letter are recovered as the first-order members of this reduction. In the part common to both frameworks, the logarithmic corrections arise from the expansion of a power law with a perturbation-dependent exponent. Finally, periodic travelling waves are used to examine the long-scale validity of the FS expansion. The first correction is obtained by quadrature, the second-order secular structure is isolated, and phase renormalization produces bounded order-consistent approximations. For integer-power members of the generic family, the amplitude dependence of the corrected wavenumber agrees with the scaling weights obtained from the symmetry classification.

We consider the spatially inhomogeneous Landau-Fermi-Dirac equation with Coulomb potential, a quantum modification of the classical Landau equation for fermions. Mathematically, the Pauli exclusion principle manifests as an additional a priori $L^\infty$-bound for solutions. Using this bound, we propagate polynomial decay in velocity, yielding unconditional upper bounds on the local mass and energy densities, thereby ruling out the possibility of implosions in the hydrodynamic quantities. Combining this estimate with a modified mass spreading method that yields desaturation, we deduce the existence of global-in-time classical solutions for rough initial data with polynomial decay in velocity. This result stands in stark contrast to the theory for the classical Landau and Boltzmann equations, for which no comparable nonperturbative global existence result is known despite sustained effort. Our treatment is almost entirely self-contained, using only robust, generic estimates for linear kinetic equations. In particular, our proof of local existence, in contrast to prior works, more closely mirrors the theory for parabolic equations using weak solutions and simpler function spaces. It may provide a concise roadmap to organizing, adapting, and applying the various linear and nonlinear estimates to obtain well-posedness for kinetic equations.