Papers

We establish the strict Dirichlet Pólya inequality on every bounded simply connected planar domain. The proof uses a unitary boundary transport formulation of the Helmholtz equation and relates Dirichlet resonances to unitary eigenphase crossings through the Dirichlet-to-Neumann map. An exact area identity yields the sharp Weyl coefficient, while a Ky Fan estimate leaves a strictly positive remainder in the counting estimate. The general case follows by conformal exhaustion.

We establish a principle that converts qualitative regularity into uniform interior estimates for degenerate equations. For the $p$-Laplace equation, $p>2$, we prove that local $C^{1,\gamma}$ regularity of every solution, without any uniform bound on the corresponding seminorms, implies uniform $C^{1,\beta}$ estimates for every $0<\beta<\gamma$. Thus, the optimal qualitative and quantitative gradient Hölder exponents coincide as suprema in every dimension. In the plane, this yields estimates controlled by the $L^p$ norm of the solution for every exponent below the sharp Iwaniec–Manfredi threshold. Applied to Xu's recent qualitative $C^{1,1/3}$ regularity for planar infinity-harmonic functions, we obtain uniform interior $C^{1,\beta}$ estimates for every $0<\beta<1/3$. For the nonlocal superconductivity equation $|Du|(-\Delta)^s u=0$, with $1/2<s<1$, we prove that pointwise differentiability of all solutions with integrable tails implies uniform interior $C^{1,\alpha}$ estimates for bounded solutions, for some $\alpha>0$. The proofs combine compactness for shifted equations with an improvement-of-flatness iteration at variable scales.

In this paper, we develop an abstract framework for the theory of uniform exponential attractors in non-autonomous dynamical systems. Our construction employs time-dependent absorbing sets while ensuring exponential attraction among the constructed sets. Moreover, we show that the resulting exponential attractors are uniform-that is, they remain stable under perturbations. As an application of this abstract theory, we study the weakly damped hyperbolic equation with sub-quintic nonlinearity. Using the proposed approach, we construct exponential attractors for the process generated by the Shatah-Struwe solution, which satisfies both energy and Strichartz estimates. Furthermore, we establish the Holder continuity of these exponential attractors with respect to the external forcing term.

We prove that, for $2\leq p<n$, the optimal $L^p$-Sobolev constant on every complete $n$-dimensional Euclidean minimal immersion without boundary is the Euclidean Aubin–Talenti constant. The proof combines optimal transport to a suitably chosen target with a two-kernel comparison linking the tangential transport moment to deficits in the normal-fibre and Jacobian estimates.

We study Fourier–Laplace transforms on null phases arising in passive inverse problems. Densities with codimension-one dependence in oblique cylinders are first considered. Integration in the distinguished direction reduces the data to an entire transform on a real-analytic maximally totally real submanifold. Holomorphic uniqueness then gives injectivity. A fixed transverse polytope swept along a polygonal curve, forming a polygonal cylinder, is next considered. The density is assumed to be polynomial in the material transverse variables on each swept piece. A weighted Brion–Barvinok expansion separates the vertex terms and the successive pieces. The same representation gives a finite-dimensional Lipschitz estimate when the polynomial degree is fixed. We apply these transform results to passive wave and heat equations in arbitrary dimension. The initial state is zero outside the observation domain. Equality of the Laplace-domain boundary data determines the quantity of the initial data divided by the speed squared in the acoustic model and the quantity of the initial data divided by the coefficient of thermal conductivity in the parabolic model. The question of what remains true when only weighted generalized boundary data on a finite interval \([0,T]\) are available is then studied. These data determine the harmonic moments of the quotient up to the weighted terminal residual. This identity gives a Hölder estimate in an oblique cylinder and a Lipschitz estimate in a polygonal cylinder. Finally, a continuous generating curve is approximated by a polygonal curve and a continuous material density by piecewise transverse polynomials. The resulting reconstruction estimate separates measurement noise, density approximation error, geometric error, and the condition number of the finite sampling matrix.

Practical ship design requires localized refinement and performance evaluation. This paper presents a reward-directed diffusion framework for localized hull optimization. Parametric masking during dataset generation restricts geometric changes to selected variables, including the bulbous bow and forebody, while preserving full-hull continuity. Surrogate-assisted sequential Monte Carlo sampling targets reduced predicted calm-water resistance, with stability retained through training data and displacement matched during screening. An automated viscous free-surface CFD pipeline, validated against measurements for two benchmark container ships, evaluates a generated variant and GenHull E, a hull refined by a naval architect within the same representation. At matched KRISO Container Ship (KCS) displacement, the generated variant has comparable total resistance at $F_n=0.195$ but 45-50% higher resistance at $F_n=0.250$ and $0.260$, missed by empirical screening. GenHull E achieves 6-8% lower total resistance up to $F_n=0.240$ but higher resistance above $F_n=0.250$ as opposing friction and pressure trends reverse the gains. Aggregating resistance across this crossover can conceal tradeoffs that emerge when friction and pressure components are evaluated separately. The assessment reveals an inability to represent a faired bulbous bow and a reward lacking viscous pressure and stability terms. These findings show how high-fidelity assessment informs improved representations and objectives for physically realizable generative designs.

We prove a sharp Zhong-Yang type lower bound for the first nonzero eigenvalue of the $p$-Laplacian on closed manifolds with small integral Ricci curvature for $p \geq 2$. The proof combines a global elliptic estimate obtained by Sobolev truncation with strict perturbations of one-dimensional comparison profiles.

This paper investigates an abstract viscoelastic wave equation in a separable Hilbert space, in which the damping term depends on a power of the velocity norm. Depending on whether the power exponent is zero or nonzero, this damping corresponds to friction damping or nonlocal damping, respectively; the equation also contains a viscoelastic damping represented by an integral term with a relaxation function. We establish global well-posedness for initial data in \(\mathcal{H}_1\) by combining monotone operator theory with the Faedo–Galerkin method, and prove uniqueness for all exponents greater than \(-1\), with the singular range between \(-1\) and \(0\) handled through the monotonicity rather than the Lipschitz continuity of the nonlocal damping operator. We then study the long-time behavior of the energy via integral inequalities, multiplier techniques, and Nakao's method, and obtain explicit decay estimates showing that the energy decays exponentially in the friction case and polynomially in the nonlocal case. To the best of our knowledge, this is the first work to consider the combined effects of viscoelastic damping and nonlocal damping in such an abstract framework, and it is also the first to analyze nonlocal damping with negative exponents.

In this paper, we determine the propagation rates of the Fisher-KPP equation with free boundaries and fractional diffusion $(-\Delta)^s$ $(0<s<1)$, which was considered in \cite{CDN26} as a model for species spreading, where the density function $u(t,x)$ satisfies the fractional Fisher-KPP equation $u_t+(-\Delta)^su=f(u)$ over a one-dimensional spatial domain $(g(t), h(t))$, representing the expanding population range, with $x=g(t)$ and $x=h(t)$ the free boundaries. It was shown in \cite{CDN26} that the model is well-posed, and its long-time dynamics exhibits a spreading-vanishing dichotomy. We show in this paper that, when spreading occurs, \[ \begin{cases} \lim_{t\to\infty}\frac{h(t)}{t^{1/(2s)}} =\lim_{t\to\infty}\frac{-g(t)}{t^{1/(2s)}} =\left(\frac{2^{1-2s}\mu}{1-2s}\right)^{1/(2s)} & \mbox{ if $s\in(0,1/2)$,}\\[2mm] \lim_{t\to\infty}\frac{h(t)}{t\ln t} =\lim_{t\to\infty}\frac{-g(t)}{t\ln t} =\mu & \mbox{ if $s=1/2$,}\\[2mm] \lim_{t\to\infty}\frac{h(t)}{t} =\lim_{t\to\infty}\frac{-g(t)}{t} =c_0(s) & \mbox{ if $s\in(1/2,1)$}, \end{cases} \] where $c_0(s)\in (0,\infty)$ is uniquely determined by the associated semi-wave problem. The level sets of $u(t,x)$ propagate with the same rates as the free boundaries. The proof uses delicate upper and lower solutions constructed from a smooth profile function in the case $s\in (0, 1/2]$, and from the semi-wave profile when $s\in (1/2, 1)$.

In this paper, we study the transition threshold problem for the 2D Navier-Stokes Boussinesq equations around the Couette flow with a constant background temperature in two domains: $\mathbb{T} \times [-1, 1]$ and $\mathbb{R} \times [-1, 1]$. It was proved that if the sizes of initial perturbations of the vorticity and the temperature in Sobolev spaces are $\nu^{\fr13}$ and $\nu^{\fr56}$, respectively, then this system admits a globally stable solution. Besides, we derive the enhanced dissipation estimate for the vorticity and temperature, as well as the inviscid damping estimate for the velocity. The coupling in the equations poses some difficulties to the analysis. We overcome these challenges and establish the results via the quasi-linear method and energy estimates.

We revisit the construction of compact liquid drop equilibria from Delaunay unduloids in [Arch. Ration. Mech. Anal. 249 (2025), Paper No. 74]. We evaluate exactly the coefficient arising in the final solvability condition of the Lyapunov-Schmidt reduction, identifying it as the product of the axial period and the normalised Delaunay flux. Using this identity and the monotonicity of the period, we establish the conjectured positivity of the coefficient and prove that its product with the period strictly increases with the neck radius. We also derive an exact relation between the volume and the surface area of one period and prove that both strictly decrease with the neck radius. These results extend the construction throughout the embedded unduloid family and yield compact equilibria of every sufficiently large mass, with reference neck radii converging to any prescribed interior value as the mass tends to infinity.

The standard Yee finite-difference time-domain (FDTD) algorithm uses Maxwell's equations to replace first order temporal derivatives with spatial curls. In this work, the relation is extended to arbitrary orders: a Time-Space Theorem establishes the correspondence between higher-order temporal derivatives and successively higher-order spatial curl operators. Based on this relation, an arbitrary-order framework for electromagnetic field approximations is developed. The framework constructs higher temporal orders through higher curl orders and permits different spatial discretizations and curl-estimation techniques, thereby providing a general basis for developing numerical algorithms of arbitrary orders. As one implementation, an arbitrary-order FDTD algorithm is constructed using staggered time and space stepping together with recursive curl estimations. Numerical simulations of an initial-value problem are performed for temporal approximation orders from second to tenth order while the spatial and temporal step sizes are kept fixed. Comparisons with the analytical solution show that the errors decrease rapidly and monotonically with increasing approximation order. These results demonstrate the practical potential of the framework for high-accuracy electromagnetic computation and provide a basis for further development of arbitrary-order numerical methods.

We give a reduction from a smooth-domain theorem excluding interior critical points of first positive Neumann eigenfunctions to a non-strict hot-spots conclusion on arbitrary bounded simply connected planar domains with compact Neumann resolvent and simple first positive eigenvalue. The main analytic step is spectral convergence under exhaustion by conformal images of concentric disks. A reflection extension, conformal invariance of Dirichlet energy, and compactness of the Sobolev embedding give a direct proof. A truncation argument justifies square integrability of the transferred extensions without any boundary regularity. The smooth-domain theorem proved by OpenAI is an external input; the approximation argument is proved here in full. The paper was generated by ChatGPT with prompts from the listed author.

Let $\Theta\subset\mathbb{R}^n$, $n\geq 1$ be compact with upper Minkowski dimension $\beta$, and let $\gamma(x,t,\theta)$ be a family of curves which is bi-Lipschitz in the first variable and Lipschitz in the last two. We prove the maximal estimate $$\left\|\sup_{(\tau,\theta)\in[0,1)\times\Theta}\left|e^{i\tau\Delta}f(\gamma(\,\cdot\,,\tau,\theta))\right|\right\|_{L^2(B^n_1)}\lesssim_s\|f\|_{H^s(\mathbb{R}^n)},$$ for every $s>\tfrac{n(\beta+1)}{2(n+1)}$, in the range $0\le\beta\le n-1$. In the range $n-1<\beta\leq n$ we prove the threshold $s>\min\big\lbrace\frac\beta 2 + \frac 1{2(n+1)},\frac n2\big\rbrace$. The proof relies on a fractal $L^2$ estimate of Du–Zhang. We also use the temporal localization of Lee, the locally constant property, and a counting argument allowed by the Lipschitz properties of the curve.

We study fully nonlinear uniformly elliptic equations that are imposed only when the gradients of solutions lie outside a closed ball and a closed purely one-unrectifiable set that can be unbounded. We prove Harnack inequalities and Hölder regularity estimates for viscosity solutions, with constants independent of the exceptional set. The proofs use the geometry of the contact sets in the cusp and barrier arguments to exclude exceptional gradients. We also provide examples and counterexamples showing the role of pure one-unrectifiability. Materials and proofs are presented in detail to ensure the paper is fully self-contained and accessible to a broad audience.

In this note we prove in a broad analytic set-up a general result characterizing the asymptotic vanishing of the $L^2$-relaxation time of averages in the presence of a large incompressible drift-like term. The broad analytic set-up is analogous to that of Constantin-Kiselev-Ryzhik-Zlatos '08 or of Coti Zelati-Delgadino-Elgindi '20. A crucial role is played by the singular value gap method of Chatterjee '25, further developed by Eberle-Lörler '26.

We prove global existence of finite-energy renormalized weak solutions to the 2D isothermal compressible Navier–Stokes equations in bounded $C^2$ domains with no-slip boundary conditions. The initial data are arbitrary in the natural energy class, allowing vacuum and unbounded density. This resolves the isothermal endpoint $\gamma=1$ of the classical 2D finite-energy renormalized existence theory for power-law pressures. We develop a new endpoint compactness argument based on a truncation-independent integrable bound for centered density oscillations. This argument yields strong $L^1$ compactness of the approximating densities at the natural energy level.

We study a Brownian particle in $\mathbb{R}^d$, $d\geq2$, with drift given by the gradient of a log-correlated Gaussian potential. At weak disorder, we prove convergence to a scaling limit whose law is singular with respect to Brownian motion and whose invariant measure is given by Gaussian multiplicative chaos. The proof uses a renormalization group induction: at each scale, the elliptic operator is well approximated by a Laplacian with effective diffusivity decaying as a power of scale. We compute this exponent exactly in dimension two.

Let $n \geq 3$. We prove that if $j \colon (\Sigma^n,g) \to (\mathbb{R}^N,dx^2)$ is an isometric immersion, then \begin{equation*} \int_\Sigma \left( \lvert \nabla u \rvert^2 + \frac{n(n-2)}{4}\lvert H \rvert^2 u^2 \right) \geq \frac{n(n-2)}{4}\mathrm{Vol}(S^n)^{2/n}\left( \int_\Sigma \lvert u \rvert^{\frac{2n}{n-2}} \right)^{\frac{n-2}{n}} \end{equation*} for all $u \in W^{1,2}(\Sigma)$, and characterize immersions for which equality is realized. Our proof exploits the conformal invariance of the inequality, and hence extends to sharp Michael–Simon inequalities for submanifolds of the sphere, of hyperbolic space, and of other conformally developable manifolds. As an application, we derive Chen–Fenchel–Willmore-type inequalities relating $L^p$-norms of the mean curvature vector to the volume of a compact submanifold of a simply-connected spaceform.

We investigate the best constants associated with several types of Moser–Trudinger inequalities within the framework of the Finsler metric in $\mathbb{R}^n$. By employing a convex-symmetrization-free approach, we first establish both the critical and subcritical anisotropic singular Moser–Trudinger inequalities. We further study anisotropic Moser–Trudinger inequalities in both singular and nonsingular cases under the exact growth condition, highlighting the optimality of the best constant as well as the sharp exponent appearing in the denominator. To further enrich the theory, we introduce quasi-conformal-type transformations adapted to the Finsler metric setting to derive several new variants of such inequalities. We also characterize the attainability and nonattainability of maximizers for a generalized version of anisotropic Moser–Trudinger inequalities with the exact growth and explicitly determine the corresponding supremum values. As an application of these inequalities, we analyze the existence and nonexistence of positive ground state solutions to a class of $n$-Finsler-Laplace equations involving constant potential in $\mathbb{R}^n$, where the nonlinearity exhibits a critical exponential growth at infinity.

We study global regularity and the infinite Prandtl number limit for temperature patches of the two-dimensional nondiffusive Boussinesq system on $\mathbb R^2$, with velocity dissipation $\Lambda^{2\alpha}$, $\frac12<\alpha\leq1$. We establish global well-posedness for $\theta_0\in L^1\cap L^\infty$ and divergence-free $u_0\in W^{1,p}$, $2\leq p<\infty$. Under suitable profile and striated regularity assumptions, we prove global persistence of $C^{1,\gamma}$ and $C^{2,\gamma}$ patch boundaries for every $0<\gamma\leq1$, allowing nonconstant profiles and including the Lipschitz endpoints $W^{2,\infty}$ and $W^{3,\infty}$. The velocity gradient and boundary regularity estimates are uniform in $\mathrm{Pr}\in[1,\infty)$ on each finite time interval. Uniform global $L^r$ velocity bounds hold for suitable finite $r$ when $\alpha<1$, and at $\alpha=1$ for zero-mean initial temperatures with finite first absolute moment. Such bounds fail at $\alpha=1$ for compactly supported nonzero-mean data. For temperature patches, we justify the infinite Prandtl number limit in the original frame when $\alpha<1$ or the initial temperature has zero mean, and in a moving frame otherwise. The limit is the unique normalized global weak solution of the (fractional) Stokes-transport system and, importantly, preserves the patch structure and boundary regularity.

We construct smooth closed embedded minimal surfaces with everywhere negative Gaussian curvature in the unit round four-sphere. More precisely, for every sufficiently large integer $m$, our construction gives a minimal surface $\Sigma_m$ of genus $m^2+1$ such that $\operatorname{Area}(\Sigma_m)\to 4\pi^2$ as $m\to+\infty$. This result yields a complete solution to Problem 101 in Yau's 1982 list of open problems.

The paper considers a system of unidirectional first-order hyperbolic partial differential equations (PDEs) coupled with ordinary differential equations (ODEs), modeling two-phase advective transport processes subject to both boundary and distributed input delays. To address the system defined over two spatial domains, we design a set of three backstepping transformations involving both Volterra- and Fredholm-type integral operators, which introduce eight kernel functions: five associated with the PDE states and three with the ODE states. Since the distributed delayed variable spans the two spatial domains, its transformation must integrate over all state variables, leading to three-dimensional kernel functions with singular initial conditions involving Dirac delta functions that serve as sampling operators to establish mappings between variables of different dimensions. By employing the method of characteristics and successive approximations, we prove the well-posedness of the kernel equations. On this basis, we prove the invertibility of the Fredholm transformation and establish the exponential stability of the closed-loop system in the $L^{\infty}$ norm under the delay-compensated controller. Simulation results for a linear example and a nonlinear screw extrusion model are presented to demonstrate the effectiveness of the proposed method.

We study normalized solutions to the 1-Laplace equation \[ -\mbox{div}\left(\frac{Du}{|Du|}\right)+\lambda \frac{u}{|u|}=f(u)+|u|^{1^*-2}u,\quad\mbox{in}~ \mathbb{R}^N \] subject to the constraint $\int_{\mathbb{R}^N}|u| \, \mbox{d}x=m$, where $N\geq2$, $1^*=\frac{N}{N-1}$ is the critical Sobolev exponent for the embedding from BV into $L^p$, $\lambda, m\in\mathbb{R}^+$ and $f:\mathbb{R}\to\mathbb{R}$ is a continuous nonlinear perturbation. We first establish a Pohozaev-type identity for the equation and use it to study the associated critical energy, and then prove the existence of normalized solutions via nonsmooth critical point theory. A key feature of our approach is the proof of the existence of a Palais-Smale sequence for a Lipschitz continuous functional under a Lipschitz continuous constraint. This is significantly different in methodology from previous problems with $C^2$ constraint.

We prove a single measurement uniqueness result for interior sound-soft obstacles in the planar Schrödinger equation with an unknown surrounding potential. For any two configurations with different obstacles, we construct a boundary datum for which the corresponding Neumann measurements differ. The full Dirichlet-to-Neumann map uniquely determines both the obstacle and the potential. The obstacles may have several connected components, and no convexity or analyticity assumptions are imposed. Neither potential is required to be known near the measurement boundary.

For $s>0$, and $z\in\mathbb H=\{z=x+iy:y>0\}$, define $\theta(s;z)=\sum_{m,n\in\mathbb Z}e^{-\pi s\big(m^2y+\frac{(mx-n)^2}{y}\big)}$ and $ J(z;a,b)=\sum_{m,n\in\mathbb Z}e^{-\pi \big(m^2y+\frac{(mx-n)^2}{y}\big)} \cos\bigl(2\pi(ma+nb)\bigr)$ for the classical and shifted theta functions (Gaussian lattice sums), respectively. We prove that, up to the modular group, there exist three thresholds $0<\alpha_a<\alpha_b<\alpha_c<1$ such that \begin{equation}\aligned\nonumber \Minima_{z\in\mathbb H,\,(a,b)\in\mathbb R^2}\Big(\theta(1;z)+\alpha J(z;a,b)\Big)= \begin{cases} \;\big(e^{i{\pi}/{3}};1/3,1/3\big), &\hbox{if}\;\; \alpha\in[0,\alpha_a),\\ \;\big(e^{i{\pi}/{3}};1/3,1/3\big)\;\hbox{or}\;\big(e^{i{\varphi_{\alpha_a}}};1/2,1/2\big), &\hbox{if}\;\; \alpha=\alpha_a,\\ \;\big(e^{i{\varphi_\alpha}};1/2,1/2\big),\;\varphi_\alpha\in(\varphi_{\alpha_a},\frac{\pi}{2}),\;\; &\hbox{if}\;\; \alpha\in(\alpha_a,\alpha_b),\\ \;\big(i;1/2,1/2\big),\; &\hbox{if}\;\; \alpha\in[\alpha_b,\alpha_c],\\ \;\big(iy_\alpha;1/2,1/2\big),\; y_\alpha\in(1,\sqrt3],\;\;y_1=\sqrt3,\;\;&\hbox{if}\;\; \alpha\in(\alpha_c,1], \end{cases} \endaligned\end{equation} where the complex variable $z$ and vector $(a,b)$, represent the total vortex shapes and relative positions of the two-component Bose gas, respectively. Consequently, this completely proves the conjecture of Mueller and Ho~\cite{Mue2002} (2002) arising in Bose–Einstein condensates.

This article establishes a sharp singular Moser-Trudinger type inequality in a new function space and explores a concentration-compactness principle in the Heisenberg group, inspired by P.\textcolor{blue}{-}L. Lions. We analyze the equivalence of sharp critical and subcritical singular Moser-Trudinger inequalities, focusing on their asymptotic behavior and connections between their suprema. Additionally, we apply the mountain pass theorem to demonstrate the existence of positive ground state solutions for zero mass $(p, Q)$-Laplace subelliptic equations with singular exponential nonlinearity for the following equation: $$ -\Delta_{H,p} u-\Delta_{H,Q} u=\frac{f(\xi,u)}{r(\xi)^\vartheta}\quad\text{in}\quad \mathbb{H}^n, $$ with $1<p<Q$, $\vartheta\in(0,Q)$, $Q=2n+2$, the function $r(\cdot)$ is called the Korányi norm in $\mathbb{H}^n$ and the nonlinearity $f:\mathbb{H}^n\times \mathbb{R}\to \mathbb{R}$ is a Carathéodory function, which behaves like $\exp{(\alpha|s|^{\frac{Q}{Q-1}})}$ as $|s|\to~+\infty$ for some $\alpha>0$.

On any bounded convex domain $\Omega\subset\mathbb{R}^n$ with a $d$-dimensional flat boundary piece $F\subset\partial\Omega$, we consider convex solutions with zero boundary values to the Monge-Ampère equations $$ \det\mathsf{D}^2 u= \mathrm{dist}(\,\cdot\,,\mathrm{Aff}(F))^\alpha,\quad\det\mathsf{D}^2 u=\mathrm{dist}(\,\cdot\,,\partial\Omega)^\alpha,\quad \det\mathsf{D}^2 u=(-u)^\alpha, $$ where $\mathrm{dist}$ stands for Euclidean distance and $\mathrm{Aff}(F)$ is the affine hull of $F$. We show that the threshold of the exponent $\alpha\in\mathbb{R}$ for $u$ to have infinite slope at $F$ is $\alpha\leq 2d-n$. For the first two equations, we also show that no solution exists when $\alpha\leq 2d-2n$. Moreover, when $\Omega$ satisfies the “exterior cone condition with a $d$-dimensional ridge” at $F$, both thresholds are sharp for the first equation, and the infinite-slope threshold is sharp for the last equation.

We study the parameter dependence and transversality properties of trajectories of the controlled ODE $$ U_x = P, \qquad P_x = - f(U) - \beta P + \tilde \alpha(x)U, $$ arising from traveling-wave solutions of a bistable reaction-diffusion equation with control. Optimal profiles minimizing the $L^1$-norm of the control $\tilde \alpha$ are determined by the unstable and stable manifolds $\Gamma_u$, $\Gamma_s$, the critical curve $$ P^*(U) = \sqrt{Uf(U)}, $$ and the associated function $$ F(U) = \frac{dP^*}{dU} + \frac{f(U)}{P^*(U)}. $$ We establish generic transversality properties for these objects with respect to perturbations of the speed $\beta$ and the nonlinear source $f$, together with quantitative estimates on the first- and second-order dependence of the invariant manifolds on these parameters. We also analyze the loss of regularity at the critical speed $\beta^{**}$, where the unstable manifold approaches the strongly stable manifold of the equilibrium $(U^*,0)$. These results provide the ODE ingredients used in the analysis of the structure and regularity of optimal profiles in paper [S. Bianchini and C. Trifone. Traveling profiles and control cost for a PDE describing the evolution of invasive species. 2026. arXiv: 2606.23979 [math.OC]].

We establish a conditional theory of genuinely non-perturbative global dynamics for the cutoff Boltzmann equation with hard potentials on the three-dimensional torus. We prove that any global mild solution satisfying \[ \sup_{t\ge0}\int_{\mathbb R^3}W(v)\sup_{x\in\mathbb T^3}F(t,x,v)\,dv<\infty, \] for a suitable stretched-exponential weight \(W\), enjoys uniform weighted \(L^\infty\) control, is unique in the dissipative mild class, and converges exponentially to the global Maxwellian. In particular, these conclusions hold for arbitrary-amplitude solutions admitting a uniform stretched-exponential velocity upper bound. The key mechanism is a *tail-to-coercivity principle*: uniform velocity-tail control generates positive lower bounds for the local density and nonlinear collision frequency, without imposing macroscopic coercivity a priori. A double-Duhamel positivity argument also generates a Gaussian lower bound at arbitrarily short positive times, allowing initial data with local vacuum. We establish a polynomially weighted counterpart under an additional spatially uniform lower velocity profile, obtaining analogous weighted \(L^\infty\) bounds, uniqueness, and exponential relaxation. Our results provide a low-regularity alternative to conditional theories requiring smoothness and pointwise macroscopic bounds. They reveal how a single velocity-tail condition controls large-data Boltzmann dynamics and suggest a route toward problems with physical boundary conditions.