Papers

This is the second paper in our series of works to develop the min-max theory for free boundary minimal hypersurfaces in Riemannian manifolds with boundary via the Allen-Cahn equation. Building on our previous work for general critical points, we study the boundary behaviour of limit-interface integral varifolds arising from stable (or finite Morse index) solutions to the Allen-Cahn equation with Neumann boundary condition. More precisely, we prove that the limit-interface is a curvature varifold with boundary in the sense of Mantegazza satisfying a generalized stability inequality. Furthermore, we rule out classical tangent cones at the boundary and show that the first variation of the limit-interface varifold is singular with respect to its weight measure. In particular, our result confirms a folklore conjecture that the boundary concentration examples of Malchiodi, Ni and Wei have unbounded Morse index. As will be evident in our forthcoming work, the results obtained in this paper are crucial ingredients towards the optimal boundary regularity in the Allen-Cahn free boundary min-max theory.

This paper develops a one-dimensional model for a mobile viral component propagating through a spatially distributed susceptible population. Random viral motion produces diffusion, whereas a response to the relative gradient of susceptible density generates directed transport. The existence of a positive entire traveling wave is proved when directed transport dominates diffusion, and the limiting case of equal transport and diffusion coefficients is constructed separately. A dimensionless transformation then converts the remaining equation into a logistic equation and supplies closed-form profiles. These formulas identify the unique viral maximum and the far-field behavior. A conservative implicit-explicit finite-difference method provides an independent numerical check. Diffusion is evaluated at the new time level, the gradient-driven flux explicitly at cell interfaces, and susceptible depletion pointwise. Exact traveling-wave values specify the initial data and time-dependent Dirichlet conditions at both endpoints. Thus, the computation tests preservation and translation of the analytical profiles. A calculation with eight hundred spatial subintervals reproduces the localized viral band and the monotone susceptible front. The largest uniform errors are approximately $1.70\cdot10^{-2}$ for viral density and $2.22\cdot10^{-3}$ for susceptible density. The numerical results corroborate the analytical construction and show that the conservative discretization captures the proposed propagation mechanism.

In this paper, we revisit a multi-dimensional nonlocal transport equation of the form \begin{eqnarray*} \partial_t\rho+g\mathcal{R}_a\rho\cdot \nabla\rho= 0,~\rho(\cdot,0)=\rho_{0}, \end{eqnarray*} where the transform $\mathcal{R}_a$ is defined by \begin{eqnarray*} \mathcal{R}_a\rho(x,t)=\frac{\Gamma(\frac{n+1}{2})}{\pi^{\frac{n+1}{2}}}P.V.\int\limits_{\mathbb{R}^n}\Big(\frac{x-y}{|x-y|^{n+1}}-\frac{x-y}{(|x-y|^2+a^2)^{\frac{n+1}{2}}}\Big)\rho(y,t)dy. \end{eqnarray*} This model was first introduced in [W. Zhang, J. Differential Equations 467 (2026) 114275] as a natural generalization of the well-known Kiselev-Sarsam equation proposed in [A. Kiselve and N.A. Sarsam, Nonlinearity 38 (2025) 055018]. By using an improved pointwise inequality for the transform $\mathcal{R}_a$, we obtain a better and more natural version of a nonlinear integral inequality proved in Zhang [31] for the transform $\mathcal{R}_a$. In addition, we also establish a new weighted nonlinear inequality for the transform $\mathcal{R}_a$ implying the finite-time blow-up for a class of smooth radial initial data.

In this paper, we focus on the existence of both global and non-global nonnegative solutions for the abstract semilinear heat equation $u_t - Lu = \sum_{i=1}^k \ell_i(t) f_i(u)$ in $L^p(X, \mu)$. Here $(X,\mu)$ represents a $\sigma$-finite measure space, $L$ is the infinitesimal generator of a sub-Markovian strongly continuous semigroup of bounded linear operators in $L^p(X,\mu)$, and for each $i = 1, \ldots, k$, the functions $\ell_i: [0, \infty) \rightarrow [0, \infty)$ and $f_i: [0, \infty) \rightarrow [0, \infty)$ are continuous. We utilize our findings to apply to the Laplacian operator on sub-Riemannian manifolds and locally finite graphs, obtaining Fujita-type results. Our results extend the recent work of Lenz, Schmidt, and Zimmermann, "Blow-up of nonnegative solutions of an abstract semilinear heat equation" (Calc. Var. Partial Differential Equations, 2023).

The notions of classical two-scale convergence and the periodic unfolding method, which both rely on an additional scale for the representation of periodic micro-oscillations, are in this paper traced back to the wider concept of general two-scale convergence. Moreover, a comparison is made between very weak two-scale convergence and periodic unfolding with respect to sequences bounded in Sobolev spaces, amplified by the scale of the micro-oscillations. It turns out that, for amplified sequences, these approaches are not equivalent in an immediate sense. The paper both serves as a literature overview, with examples and illustrations of the concepts of convergence, and contains new results.

We prove sharp Hardy inequalities on $\mathbb{Q}_p^n$ for the quadratic form of the Vladimirov–Taibleson operator, the $L^q$ norm of its fractional derivative, and a nonlinear difference energy. Under the self-dual Fourier normalization, the sharp quadratic-form constant for $0<\alpha<n$ is $[(1-p^{-(n-\alpha)/2})/(1-p^{-(n+\alpha)/2})]^2$, with a remainder given by the ground state representation. For $1<q<\infty$, $q'=q/(q-1)$, and $0<s<n/q$, a weighted Schur estimate determines the exact $L^q$ norm of the weighted Riesz potential $\|\boldsymbol{x}\|_p^{-s}I_s$, giving the sharp linear Hardy constant $[\Gamma_{p}^{(n)}(n/q'+s)/\Gamma_{p}^{(n)}(n/q')]^q$. The two-point Picone inequality and shell decomposition give the sharp constant $2(1-p^{-n})\sum_{m\ge1}p^{-sqm}(1-p^{-(n/q-s)m})^q$ for the Gagliardo-type energy with kernel $\|\boldsymbol{x}-\boldsymbol{y}\|_p^{-n-sq}$. Annular truncations of the critical powers give explicit minimizing sequences with error estimates. Equality is attained only by the zero function in each stated class. For $q=2$, both $L^q$ inequalities reduce to the quadratic-form inequality. Uniform local constancy makes the difference energy finite for every Bruhat–Schwartz function and every $s>0$. The Hardy inequalities on the full test-function space therefore hold precisely when $sq<n$, without the restriction $s<1$.

Spin-1 Bose-Einstein condensates provide a fundamental platform for exploring quantum magnetism and coherent spin dynamics. Their three Zeeman components are coupled by spin-exchange interactions under conserved particle number and magnetization, giving rise to rich magnetic phases and nonlinear spin structures; a rigorous classification of stationary states therefore clarifies the possible spin configurations and their physical origins. In this paper, as a continuation of our previous work \cite{LLWZ23,LLWZ24,LLWZ26}, we study the classification of solution for the spin-1 Bose-Einstein condensates in $\mathbb{R}$, by applying the non-standard Hirota's bilinear method, the determinant formula and the general left-uniqueness strategy, we present some novel explicit solution and give the complete classification of the solutions. In comparison with the classical Hirota's method, the non-standard Hirota's method needs to introduce auxiliary functions to construct Hirota's bilinear form, which essentially adds additional constraint on the wave functions, so that it possible to deal with more complex equations. As far as we know, this is the first systematic study the complete classification of solutions to the spin-1 Bose-Einstein condensates. Our classification results yields experimentally testable predictions for spin populations, relative phases, and magnetic textures

In this paper, we study the optimal control of a phase field tumor growth model of Cahn–Hilliard type with possibly singular potential in which the often assumed parabolic relaxation of the chemical potential is replaced by a hyperbolic one. We investigate the asymptotic behavior of the control problem as the relaxation parameter $\,\alpha\,$ approaches zero. It is shown that the corresponding state and adjoint state variables of the relaxed problem converge in a suitable sense to their counterparts for the unrelaxed regime where $\alpha=0$. Also, weak limits of optimal controls for the relaxed problem can be shown to be optimal controls for the unrelaxed case. Moreover, every fixed optimal control $\,u^*\,$ for the unrelaxed system turns out to be the limit of a family $\{u^*_\alpha\}_{\alpha>0}$ of minimizers of suitable optimal control problems for the relaxed systems, where the original cost functional is replaced by the so-called “adapted” cost functional. It is also shown that the variational inequalities satisfied by $u^*_\alpha$, which constitute the first-order necessary optimality conditions of the relaxed problems with adapted cost functional, converge to the corresponding variational inequality fulfilled by $u^*$ for the unrelaxed problem with the original (nonadapted) cost functional.

Let $N\in\mathbb N$, $p\in[1,\infty)$, and $Q_0:=(0,1)^N$. In this article, we establish sharp Poincaré inequalities for the nonlocal nonconvex functionals $$ \Phi_{\lambda,p}^\gamma(g;Q_0) :=\lambda^p\iint_{\{(x,y)\in Q_0^2:\,x\ne y,\,|g(x)-g(y)|>\lambda|x-y|^{1+\frac{\gamma}{p}}\}} |x-y|^{\gamma-N}\,dx\,dy, $$ where $g:Q_0\to\mathbb R$ is measurable, $\lambda\in(0,\infty)$, and $\gamma\in\mathbb R$. Specifically, we prove that the estimate $$ \int_{Q_0}\int_{Q_0}|g(x)-g(y)|^p\,dx\,dy \lesssim\lambda^p+\Phi_{\lambda,p}^\gamma(g;Q_0) $$ holds for any such $g$ and $\lambda$ if and only if $\gamma\in(-\infty,-1]$, where the implicit positive constant depends only on $N$, $p$, and $\gamma$. This extends Nguyen's Poincaré inequality for $\gamma=-p$ and provides a useful tool for studying regularity and compactness through these functionals. As applications, we obtain a lower bound for the $\Gamma$-convergence of $\Phi^\gamma_{\lambda,p}$ and related asymptotic characterizations for Sobolev and BV spaces. Moreover, we identify the optimal range of $\gamma$ for which both the Rellich–Kondrachov theorem and a criterion for VMO regularity involving $\Phi^\gamma_{\lambda,p}$ at a fixed threshold hold. These results also answer the question on nonlocal Poincaré inequalities posed by H.-M. Nguyen in [C. R. Math. Acad. Sci. Paris 363 (2025)]. For $\gamma\in(-\infty,-1]$, they further answer questions on Sobolev and BV characterizations posed in the same article and by H. Brezis et al. in [Anal. PDE 17 (2024)].

This paper studies the existence and asymptotic stability of traveling shock waves for the one-dimensional full compressible Navier–Stokes–Poisson equations, which model viscous heat-conducting ions in a self-consistent electrostatic field. We first prove the existence and uniqueness of a smooth shock profile of small amplitude by the center manifold theorem, and then establish its large-time asymptotic stability under suitably small smooth perturbations. The proof is based on an energy estimate for a specially constructed integrated system derived from the full compressible Navier–Stokes–Poisson equations. To carry out this estimate, we introduce suitable anti-derivative variables that incorporate the effects of the self-consistent electrostatic potential and heat conduction, thereby reformulating the original system as a dissipative integrated one. The subsequent energy estimates are completed by carefully handling the induced couplings.

We study the slow motion of phase transition layers for a time-fractional Allen–Cahn equation with a Caputo time derivative of order $\alpha\in(0,1)$. We show that the fractional memory preserves the distinction between exponentially and algebraically slow dynamics associated with non-degenerate and degenerate potentials, respectively. In the non-degenerate case, layered solutions persist on an exponentially long time scale of order $\exp(A/(\alpha\varepsilon))$, $A>0$, while in the degenerate case the corresponding algebraic time scale is modified by the factor $1/\alpha$. Finally, we extend the analysis to nonlinear diffusion operators, considering both the $p$-Laplacian and a Perona–Malik type diffusion, showing that the same fractional slow-motion mechanism persists in these more general settings.

We study the vanishing discount selection problem for critical Hamil\-ton–Jacobi equations posed on ${\mathbb R}^d$ with Hamiltonian $G=H-V$, where $H$ is continuous, periodic in space, convex and superlinear in the momentum, and $V$ is a compactly supported potential of constant sign. For $V\leqslant0$, we prove local uniform convergence of the discounted solutions to a distinguished critical solution, which we characterize in terms of Mather measures and of the solution selected by the unperturbed periodic problem. The proof relies on a decomposition of limiting discounted measures into a retained component and a periodic component accounting for the mass lost at infinity. For $V\geqslant0$, we establish convergence under the additional assumption that the unperturbed selected solution has zero average with respect to every periodic Mather measure. We also give explicit examples in which the discounted solutions converge, but natural formulas for the candidate limit fail to identify the selected solution.

The hydrodynamical Hopf-Rinow problem asks whether a given reconfiguration of an ideal fluid can be attained by an Euler flow. On the flat two-dimensional torus, we give an equivalent criterion for the strong version of this problem (which requires that the fluid motion minimizes the total kinetic energy required) in terms of metric betweenness. Toward this, we prove that sufficiently small normal neighborhoods in the volume-preserving diffeomorphism group of the flat $n$-dimensional torus, $n \geq 2$, equipped with the kinetic energy metric, are geodesically convex; resolving the strong problem locally in the $H^s$ topology. We further establish local bi-Lipschitz equivalence of the intrinsic geodesic distance and the extrinsic $L^2$ distance. Finally, we prove that the squared geodesic distance from the identity admits a $C^{1,1}$ extension from a sufficiently small normal neighborhood to the ambient $L^2$ space.

We prove the Aleksandrov-Bakelman-Pucci maximum principle for $L^n$-viscosity sub/super-solutions of fully nonlinear uniformly elliptic equations in $\Omega\subset\mathbb R^n$ with the drift coefficient $\gamma$ such that $\gamma(\ln(1+\gamma))^{\frac{n-1}{n}}\in L^n(\Omega)$. The approach is based on convex duality and Trudinger-Moser type inequalities.

We prove sharp Hardy-type inequalities in Log-weighted Sobolev spaces. Due to the non-attainability of the sharp constant, we improve these inequalities by adding suitable remainder terms consisting of optimal weights and sharp constants. Consequently, this gives an infinite series improvement in Log-weighted Sobolev spaces. Next, by using these improved Hardy-type inequalities, we also prove an optimal Leray-Trudinger-type improvement. Furthermore, we derive interpolation between Hardy-type inequalities and Leray-Trudinger-type inequalities.

We study quantitative unique continuation for bi-Laplace equations \[ \Delta^2u+W(x)\Delta u+V(x)\cdot\nabla u=0 \] with $C^2$ lower-order coefficients. Using the weighted-frequency function, we establish an $L^\infty$ three-ball inequality whose exponential factor has coefficient dependence $\|W\|_{C^2}^{1/2}+\|V\|_{C^2}^{1/3}$. Under some normalization, we obtain a vanishing-order bound of size $C(\|W\|_{C^2}^{1/2}+\|V\|_{C^2}^{1/3})$.

We improve an earlier characterization of the Sobolev space $\dot W^{1,p}(\mathbb R^d)$, $1<p<\infty$, in terms of the oscillations of a function on balls of varying centers and radii. In contrast to earlier work, finiteness of the supremum is replaced by the formally weaker assumption of finiteness of the liminf, thereby answering a question by Brezis.

This article is devoted to the mathematical analysis of a new Navier–Stokes-$\alpha$ model involving a nonlinear filter equation. The resulting model is governed by a doubly nonlinear parabolic–elliptic coupled system. In our previous work [M.~F.~Cortez and O.~Jarrín, Mathematical study of a new Navier–Stokes-alpha model with nonlinear filter equation – Part I, J. Math. Fluid Mech. 28, no.~12 (2026)], we established the global well-posedness of weak Leray-type solutions and the existence of a global attractor. In the present work, under natural assumptions on $A(\cdot)$, we investigate several additional properties of this model, with particular emphasis on higher-order regularity and its consequences for the long-time dynamics. The regularity analysis constitutes a central and delicate issue, since the nonlinear structure of the elliptic filter and its coupling with the evolution equation preclude a direct application of the standard regularity arguments available for linearly filtered Navier–Stokes-$\alpha$ models. Overcoming this difficulty requires new higher-order estimates for the nonlinear elliptic filter equation, which may also be of independent interest. Among the results obtained, two constitute the main contributions of the article. First, we establish uniform higher-order regularity for the global attractor. This result is then used as a fundamental ingredient in proving an exact determining-modes property for complete trajectories on the attractor: if the projections of two complete trajectories onto a sufficiently large finite-dimensional space of Stokes modes coincide at every time, then the two trajectories coincide identically.

We establish the local well-posedness of strong solutions for a thermodynamically consistent diffuse interface model describing two-phase flows with surfactants, specifically the so-called Model C introduced by Garcke, Lam, and Stinner. Because of the non-standard, mixed-order strongly coupled structure of Model C, the classical Agmon-Douglis-Nirenberg theory and conventional Lopatinskii-Shapiro boundary conditions are not applicable due to divergent scaling behaviors. To overcome this, we develop a tailored resolvent analysis, adapted to the temporal regularity of the source term, in a suitable Hilbert space, which yields an exact cancellation of the highest-order cross-couplings in the estimates. This crucial decoupling enables the rigorous application of the operator-valued Mikhlin multiplier theorem to obtain resolvent estimates on $\mathbb{R}$. Furthermore, we analyze a structural variant, denoted as Model D, which achieves a crucial structural decoupling by transferring the interfacial surfactant energy entirely from the gradient part to the potential part, and we readily establish local well-posedness for strong solutions.

We study global bifurcation for Yamabe-type equations on closed Riemannian manifolds with symmetries. For isometric group actions with interval orbit space and a suitable range of exponents, we prove the existence of pairwise disjoint unbounded continua bifurcating from the positive eigenvalues of the Laplace–Beltrami operator that admit invariant eigenfunctions. All nontrivial solutions in these continua change sign. The results cover critical and some supercritical exponents. On spheres and complex and quaternionic projective spaces, we obtain lower bounds on the number of geometrically distinct continua bifurcating from the same eigenvalue.

In this paper, we study a generalised family of fractional BBM equations on the $d$ dimensional torus $\mathbb{T}^d$ for $d\geq 1$. We establish low regularity local and global well-posedness results on the $H^s(\mathbb{T}^d)$ scale that improve upon the works of Bona and Chen (2003) and Kim and Kwak (2026). We also prove endpoint global well-posedness for the cubic regularised Benjamin-Ono equation in $H^{1/2}(\mathbb{T})$. Furthermore, we obtain a well-posedness result for the generalised fractional BBM in the spaces $L^{k+1}(\mathbb{T}^d)$ where $k$ is order of the polynomial nonlinearity. As an application of this result, we show that for a certain range of values and for a suitable randomisation of the initial data, the model is almost surely globally well-posed in $H^s(\mathbb{T}^d)$ for $s\geq 0$. Lastly, we show that the family exhibits a strong form of ill-posedness in $H^s(\mathbb{T}^d)$ for $s<0$ in the form of an infinite loss of regularity at all initial datum.

We prove that all excited states $-\sin(nx)$, $n\geq2$, of the De Gregorio equation on the torus are nonlinearly unstable in the sense of Hadamard. More precisely, perturbations arbitrarily small in $H^2$ leave a fixed $H^1$ neighborhood of the state. For every $n\geq2$, the linearized operator has a real eigenvalue $\lambda_n\in(0.56,0.6)$. To prove this, we transform the eigenvalue problem into a Fuchsian equation on the complex disk and obtain a matching condition for the eigenvalue. This condition is verified rigorously using interval arithmetic. The nonlinear instability follows from a bootstrap argument in the spirit of Guo, Hallstrom, and Spirn.

Let $u^\varepsilon$ and $u$ be the convex solutions of \[ \det D^2u^\varepsilon=F(x,x/\varepsilon),\qquad \det D^2u=\overline F(x) \] on a bounded convex domain $\Omega$, with the same Dirichlet data. Here, $F$ is uniformly positive and periodic in its second variable, and $\overline F(x)=\int_{\mathbb T^n}F(x,y)\,dy$. For $m=0,1$ and $0<\alpha\leq1$, we prove \[ \|u^\varepsilon-u\|_{L^\infty(\Omega)}\leq C\varepsilon^{m+\alpha},\qquad \forall\,0<\varepsilon\leq1, \] provided that $u\in C^{m+2,\alpha}(\overline\Omega)$ and $F\in C_x^{m,\alpha}(\overline\Omega;C_y^{0,\gamma}(\mathbb T^n))$ for some $0<\gamma<1$. Both exponents are optimal in their respective regularity scales. We also establish the $O(\varepsilon^{\alpha})$ rate for locally weighted periodic Borel measures, with constants independent of the distribution of the microscopic measure.

We study the well-posedness of small BV solutions to the one-dimensional isentropic Euler system on the half-line under inflow and outflow boundary conditions. These boundary conditions are formulated in terms of admissible trace sets determined by Navier–Stokes boundary layers and zero-speed shocks. For both the inflow and outflow problems, we construct small BV solutions taking values in the subsonic region. These solutions are unique and satisfy the quantitative stability estimate \begin{align*} \|U(\cdot, t)-V(\cdot, t)\|_{L^2} \lesssim \sqrt{\|U(\cdot, 0) - V(\cdot, 0)\|_{L^2}} \, , \end{align*} which holds for any such small BV solution $V$ and any $L^\infty$ entropy solution $U$ satisfying the strong trace property and the inflow/outflow boundary conditions. In particular, we emphasize that $U$ may take values outside the subsonic region and need not have bounded variation or be constructed by the front tracking scheme. This establishes the first weak-BV stability and uniqueness theory for inflow and outflow initial-boundary value problems. A key difficulty lies in verifying that the boundary set is compatible with the $a$-contraction framework, and in proving that the front-tracking limit satisfies the prescribed inflow/outflow conditions.

We show that the conformal class of a compact spin manifold with boundary is uniquely determined by its boundary values of harmonic spinors if the metric is locally conformal to a real-analytic metric in some coordinates. More precisely, we consider the boundary conjugation (BC) map for the Dirac operator corresponding to MIT boundary conditions, and prove that if two locally conformally real-analytic (LCRA) metrics have equivalent BC maps over an open subset of the boundary, then the two manifolds are conformally equivalent. The class of LCRA manifolds includes conformally Einstein manifolds, as well as all smooth surfaces. We also show that a real-analytic unitary connection on an LCRA manifold is uniquely determined up to local gauge equivalence by the BC map of its twisted Dirac operator in dimensions greater than 2.

We develop a direct paradifferential approach to nonlinear graph integral operators arising in contour dynamics. The main methodological result is an abstract structure-preserving paralinearization theorem, valid in arbitrary dimension, for a class of homogeneous kernels whose nonlinear dependence on the graph is expressed through normalized finite differences. The theorem covers both critical principal-value kernels and subcritical locally integrable kernels, gives explicit formulas for the principal symbols, and separates the finite-order symbolic contribution generated by the homogeneous singularity from order-zero far-field terms and arbitrarily smoothing remainders. A structural feature of the reduction is the cancellation of the intermediate symbolic order; at the critical endpoint the construction also preserves the exact skew-adjoint structure of the original singular integral. As a main application, we consider the three-dimensional two-phase free-boundary Euler equations with constant interfacial background vorticity. We first derive an autonomous contour-dynamics equation in canonical surface variables and then apply the abstract theorem directly to the singular integral operators generated by the Birkhoff–Rott formulation. This yields the complete paradifferential structure of the system without taking the Dirichlet–Neumann reduction as a starting point. After introducing an Alinhac good unknown and diagonalizing the resulting system, we obtain local well-posedness for small Sobolev perturbations in the stable Kelvin–Rayleigh–Taylor regime, uniformly on compact subsets of the stable parameter set.

We develop a cancellation mechanism for linear second-order kinetic equations in non-divergence form with measurable uniformly elliptic coefficients. We decompose the dynamics into spatial waves and follow two families of velocity paths along which the waves acquire nearly opposite phases. The parabolic Krylov–Safonov theory gives a common lower bound for the two velocity marginals. The corresponding contributions cancel up to a small phase error, producing a contraction \textità la Döblin}. Iterating this contraction yields exponential Fourier decay estimates with enhanced dissipation. We then present two applications. The first contribution is a kinetic Aleksandrov estimate: a maximum principle in which the source is measured in an $L^p$ norm. For rough coefficients $A(t,v)$ independent of position, we obtain the estimate for every $p>2n+1$, where $n$ is the dimension of position and velocity. The Fourier decay also gives spatial smoothness, and parabolic regularity gives Hölder continuity in time and velocity. For autonomous coefficients $a(x,v)$ in dimension one, position serves as time away from zero velocity. A Harnack comparison controls returns to small velocity intervals and yields the estimate for every $p>4$. Both thresholds are optimal among those valid for all ellipticity ratios. The second contribution concerns position on the torus and velocity on the sphere, with measurable uniformly elliptic diffusion coefficients $A(t,v)$ independent of position. We prove enhanced dissipation with the optimal square-root frequency power and Gevrey regularity in position. For autonomous coefficients $A=A(v)$, we also obtain exponential convergence to equilibrium in $L^2$ weighted by the invariant velocity measure, and a quantitative spectral gap. To our knowledge, these are the first results on decay established for laws of kinetic equations with rough coefficients.

We establish comparison Pogorelov estimates for admissible solutions of graphical $\sigma_k$-curvature equations, extending the two-surface estimate of Qiu and Yan for the graphical scalar curvature equation. The comparison graph is assumed only to be $k$-admissible, with a bounded slope and a positive interior gap that vanishes on the boundary. The estimates cover $2\le k<n$ for prescribed data $f(x,u)$, and $n/2\le k<n$ for general data $f(x,u,Du)$. The proof combines the two-surface localization and mixed Gårding comparison of Qiu and Yan with Yan's spectral concavity inequality and an angle-function weight.

Let $M$ be a closed manifold equipped with a collection of smooth real vector fields $X_1,\ldots, X_k$ and a smooth measure $\mu$. Let $P:=\sum_j X_j^\dagger X_j $ be the associated sublaplacian. Under the assumption that each pair of points of $M$ can be connected by a path obtained by joining integral curves of the vector fields, the equation \[ Pu=f \] admits unique zero-average solutions $u$ in the natural energy space, for all zero average data $f\in L^2$. We say that exact regularity holds for the above equation if $u\in H^k(M)$ whenever $f\in H^k(M)$, where $H^k(M)$ is the $L^2$ based Sobolev space of any order $k\in \mathbb{N}$. We investigate how the presence of a characteristic submanifold, namely a submanifold tangent to all vector fields $X_j$, may cause a failure of exact regularity. We prove that this indeed happens for a class of "worm sublaplacians", which are real analogues of Kohn Laplacians on Diederich–Fornaess worm domains, of interest in several complex variables. Our main tool is a scaling lemma inspired by work of Barrett and Christ on the $\bar\partial$-Neumann problem.

In this paper, which originates from Chapter 4 of the author's PhD thesis [1, Chap. 4], we investigate the global well-posedness of a class of partially diffusive hyperbolic systems in critical homogeneous Besov spaces. Our main objective is to extend the analysis developed in our previous work [2] to a functional framework in which the high-frequency com- ponents of the solution are controlled in $L^p$-based spaces with $p > 2$. This broader framework allows us to establish global existence under weaker smallness assumptions on the initial data and provides more precise information on the qualitative behavior of the resulting solutions. In particular, our analysis yields refined regularity. As an application, our global existence theorem applies to the multidimensional compressible Fourier-Navier-Stokes system