Papers

The purpose of the paper is to place Whiting's classical growing mode stability argument, extended to real frequencies by Shlapentokh-Rothman for the scalar wave equation and by Andersson, Ma, Paganini and Whiting in general, in the framework of classical PDE theory. The key steps are: a description of the dual or adjoint modes, a singular phase space pairing argument which is technically executed via the Fourier transform, followed by a standard unique continuation result. One part of our description of the dual modes connects them directly to the standard mode solutions which are smooth over the event horizon. In the zero spin case, there is a geometric interpretation of this: quasinormal modes (for non-zero real frequencies) which are smooth across the future event horizon can be extended as distributional QNM solutions by 0 through the past event horizon. We also describe the behavior of mode solutions at the bifurcate sphere.

We prove mode stability for the Klein-Gordon equation on Kerr-de Sitter black hole spacetimes in the full subextremal range and for the range $m_{\rm KG}^2\in[0,\Lambda\sqrt{2}]$ of squared scalar field masses that includes, in particular, the minimally coupled case $m_{\rm KG}=0$ and the conformally coupled case $m_{\rm KG}^2=\frac{2}{3}\Lambda$ (i.e., the Teukolsky equation for spin $0$). In combination with recent work by Petersen-Vasy, this proves, unconditionally, that sufficiently regular solutions of the scalar wave equation decay exponentially fast to constants, and in fact to zero for nonzero $m_{\rm KG}$.

The huge amount of literature about the divergence equation in bounded Lipschitz domains of $\R^n$ ($n\ge2$) is fairly disconnected and even apparently simple problems remain unsolved. We go several steps further in the knowledge of this equation and of some related inequalities. We prove that among its infinitely many solutions there exists a special one obeying elliptic regularity theory. This solution simplifies the definition of the Bogovskii constant $C_B$ and allows us to prove its attainment in smooth domains. We then obtain a universal lower bound for $C_B$ in any Lipschitz domain as well as a non-minimality criterion. As expected, balls are minimisers as the domain varies, although no symmetrisation technique is used. We also analyse ellipsoids and annuli: for the first we improve the (so far) best asymptotic inequality for thinning domains. Finally, we introduce higher-order Bogovskii constants, which lead to a polyharmonic Stokes problem. Not only regularity theory applies without smoothness of the domain when the source has some vanishing traces, but we also prove that balls are again minimisers among Lipschitz domains with the very same Bogovskii constant. Three main challenging open problems are suggested.

We study steady solutions to the two-dimensional incompressible Euler equations in an infinite channel, whose far-field limits are uniformly non-stagnant shear flows. In the smooth category,for a broad class of prescribed far-field shear profiles, non-shear steady states exist via the construction of two-dimensional solutions of the semilinear elliptic equations of stream function by the min–max method. In the analytic category, we establish a comparison principle for the analytic steady states and we show for a dense family of analytic uniformly non-stagnant shear profiles, every analytic steady state with the prescribed far field must itself be a shear flow. In particular, there are far-field shear profiles which exhibit flexibility in the smooth category but rigidity in the analytic category. Furthermore, the dense rigidity is sharp in the sense that there exists analytic shear profile which admits flexibility in the analytic category.

The motion of a slender vortex in ideal incompressible fluid is a classic problem in hydrodynamics. Formulae for the velocity of a vortex ring go back to work of Kelvin, Helmholtz, Hicks and Dyson in the nineteenth century, while more recently a number of models have been created for simulating the motion of slender tubes of vorticity of general shape. For a vortex tube or filament localised near to a curve C(t), the relevant small parameter to measure slenderness is the tube radius divided by the radius of curvature of C. The present paper revisits this range of classic problems by introducing a coordinate system closely linked to the geometry of vortex surfaces in a slender vortex. The motion of fluid elements in this coordinate system has an action–angle form, and with this the vorticity equation simplifies radically. At the same time, however, most aspects of the shape or evolution of a vortex are thrown into the description of the coordinates, and in particular the corresponding metric and volume form. As the coordinate system is non-orthogonal and time-dependent, tools of differential geometry are most easily used to describe the structure of both vorticity and coordinate system, and a general mathematical framework is set out. This resulting system of equations is taken as far as possible with only the assumption of vortex slenderness in place, but allowing arbitrary motions and distortions of the vortex core and of the curve C(t). The modelling is applied to calculate the motion of a slender vortex ring with arbitrary axial flow, solving directly for the shape of perturbed vorticity surfaces and giving results in agreement with earlier studies. The general framework set up in this paper is suitable for the development of simplified equations for vortex motion and interaction in future studies.

We study the quasilinear Schrödinger equation \begin{align*} -\Delta_{\mathbb{H}} u +V(\xi)u-\Delta_{\mathbb{H}} (\left|u\right|^{2\alpha})\left|u\right|^{2\alpha-2} u= \lambda \left|u\right|^{q-2}u + \left|u\right|^{p-2}u \quad \text{ in } \mathbb{H}^N, \end{align*} where $\Delta_{\mathbb{H}}$ is the Kohn Laplacian on the Heisenberg group $\mathbb{H}^N$, $4\alpha<q<p \leq 2\alpha Q^*$, $\alpha>\frac12$, and $2\alpha Q^{*}$ is the critical exponent, $Q=2N+2$ being the homogeneous dimension and $Q^{*}=\frac{2Q}{Q-2}$. For $p=2\alpha Q^{*}$ and $\lambda>0$, we obtain a nontrivial solution, assuming that the potential is bounded below by a positive constant and is either asymptotically constant from above or invariant under a discrete subgroup of $\mathbb{H}^N$. In the opposite direction, we prove a Pohožaev identity for $\Delta_{\mathbb{H}}$ and combine it with the Nehari identity, obtaining a family of identities from which the quasilinear energy disappears exactly at the exponent $2\alpha Q^{*}$. This yields a nonexistence theorem under a monotonicity condition on the potential with respect to anisotropic dilations and shows that no nontrivial solution exists for $\lambda \leq 0$; the sign of the subcritical perturbation determines solvability. Along the way, we show that every weak solution is bounded and decays exponentially in the Korányi gauge.

We consider the radially symmetric version of the reaction-diffusion equation $u_t-d\Delta u=f(u)$ with a monostable nonlinearity $f$, viewed as a model for the spreading of a species with population range $r<h(t)$ and density $u(t,r)$ ($r=|x|$), where the free boundary $r=h(t)$ is governed by $u(t,h(t))=\delta>0$ and $h'(t)=-d u_r(t,h(t))/\delta$. For the one-dimensional case ($N=1$), Du \cite{DN} proved that when $\delta\in(0,1)$, spreading occurs: $u\to1$ locally uniformly in $\mathbb{R}$, $h(t)\to\infty$, and $\lim_{t\to\infty}[h(t)-c_*t]=\tilde{h}\in\mathbb{R}$ with no logarithmic shift. In the present paper we consider $N\ge2$ and establish a complete trichotomy: spreading for $\delta\in(0,1)$; transition for $\delta=1$, where $u\to1$ uniformly on $[0,h(t)]$ and $h(t)\to h_\infty\in(0,\infty)$; and vanishing for $\delta>1$, where $h(t)\to0$ and $u\to\delta$ uniformly on $[0,h(t)]$. For the spreading regime, by constructing sharp upper and lower solutions, we prove that the solution converges globally to the semi-wave profile and reveal a logarithmic shift of the form $ \lim_{t\to\infty}\big[h(t)-c_*t+c_N(\delta)\log t\big]=\hat{h}\in\mathbb{R}$, with the coefficient $c_N(\delta)>0$ satisfying $ \lim_{\delta\to0}c_N(\delta)=d(N-1)/c_0$, where $d(N-1)/c_0$ is the shift coefficient for the high-dimensional radial pushed-case Cauchy problem. These results reveal the connection to the spreading behavior modeled by the corresponding Cauchy problem.

We establish a new concavity inequality for the elementary symmetric function \(\sigma_k\), which controls the spectral quadratic form arising from the second variation of \(\log\lambda_{\max}\). The proof is based on an easy–hard decomposition of the spectral variables. The easy region is treated using an optimal constrained concavity estimate for \(\sigma_k\), whereas the hard region is analyzed through Gårding-root coordinates and the concavity of the ordered partial sums of the inverse roots. As an application, for \(n/2\leq k<n\), we obtain global curvature estimates for closed star-shaped \(k\)-convex hypersurfaces satisfying \(\sigma_k(\kappa)=f(X,\nu)\) with a general positive right-hand side, together with the corresponding global-to-boundary estimates for Euclidean Hessian equations.

We analyze a thermodynamically consistent diffuse interface model for incompressible two-phase viscous flows with unmatched densities in a smooth bounded domain $\Omega\subset\mathbb{R}^d$ $(d=2,3)$. This model consists of the Navier-Stokes-Voigt equations for the velocity field and a convective Cahn-Hilliard equation with a singular potential for the phase-field variable. The resulting hydrodynamic system is subject to a generalized Navier slip boundary condition for the velocity field and Cahn-Hilliard-type dynamic boundary conditions for the phase-field variable and the chemical potential. With the aid of the Voigt regularization, we establish the existence of global quasi-strong solutions that satisfy an energy equality. This is achieved through a combination of delicate approximation schemes and a semi-Galerkin method.

Let $H_0$ be the standard discrete Laplacian on $\mathbb Z^d$, $d\geq4$, let $R_0(z)=(H_0-z)^{-1}$, and let $p'$ denote the Hölder conjugate of $p$, with $p'=\infty$ when $p=1$. We establish uniform diagonal resolvent estimates from $\ell^p(\mathbb Z^d)$ to $\ell^{p'}(\mathbb Z^d)$ by proving local Fourier-decay bounds for surface measure on the Fermi surfaces of the lattice dispersion relation. At a regular point, the decay is governed by the number of coordinate directions in which the quadratic term vanishes: after a local change of variables, these directions produce cubic one-dimensional phases, while the remaining directions are quadratic. For $d\geq5$, we prove the sharp global range \[ 1\leq p\leq\frac{2(d+2)}{d+5}. \] Away from the threshold energies, this range remains sharp in odd dimensions, whereas for even $d\geq6$ the sharp range improves to \[ 1\leq p\leq\frac{2(2d+5)}{2d+11}. \] In dimension four, we obtain uniform estimates for $1\leq p<4/3$ and an $\ell^{4/3}$-to-$\ell^4$ endpoint estimate with a square-root logarithmic loss. Matching anisotropic Knapp-type constructions give the corresponding necessary conditions. We also derive Birman–Schwinger and Kato smoothing bounds, complex-eigenvalue estimates, and improved thin spectral projection estimates.

We study axisymmetric self-similar solutions to the stationary Navier–Stokes equations in the half-space with the no-slip boundary condition, driven by an axisymmetric (-3)-homogeneous external force. If the tangential curl of the force on the unit sphere is sufficiently small, we prove the existence of a unique small solution; when the force is swirl-free, the solution is automatically swirl-free and unique. For a swirl-free external force $\boldsymbol{F}$, we introduce a scaling parameter $\lambda$ and consider the system with force $\lambda\boldsymbol{F}$; we prove that solutions exist precisely for $\lambda$ in an open interval containing zero. The same approach extends to solid cones with the no-slip boundary condition, where narrower opening angles allow the existence of solutions under larger external forces.

In this paper, we construct the explicit characteristics for the $\lambda$-dissipative solutions ($\lambda\in[0,1]$) to the two-component Hunter–Saxton (2HS) system. Using these characteristics, we provide a comprehensive study of the existence, uniqueness, and asymptotic behavior of these solutions. For the fully dissipative case ($\lambda=1$), uniqueness follows from the absence of outgoing cusps. In the partially dissipative regime ($0<\lambda<1$), outgoing cusps are present. We formulate an exact Eulerian dissipation rule which identifies the energy that has already passed through wave breaking by the intrinsic condition $\rho=0$ and $u_x\geq 2/t$. This rule determines the dissipated part of the energy measure and yields uniqueness. This uniqueness applies to the classical Hunter-Saxton equation and seems to be the first uniqueness result for the general $\lambda$-dissipative solutions. Concerning the large-time dynamics, we show that the density $\rho$ and the singular part of the energy measure decay to zero as $t\to\infty$, indicating that all energy is eventually concentrated in the $u$-component. Moreover, we derive the leading-order asymptotic term, which takes the form of a kink-wave determined by the system's remaining energy. This kink-wave can be explicitly computed from the initial data and the dissipation parameter $\lambda$.

This paper is devoted to studying the strong solutions to the vacuum free boundary problem for the isentropic compressible Navier-Stokes equations with density-dependent viscosities under spherical symmetry, which models the motions of isentropic compressible viscous flows surrounded by vacuum. We construct a class of expanding global solutions when the initial data is a small perturbation of the expanding affine solutions for the adiabatic exponent $\gamma>1$ and the viscosity coefficients proportional to $\rho^{\gamma}$, with $\rho$ being the fluid density. In addition, when the viscosity coefficients tend to zero, the perturbed solutions are proved to converge to the solutions to the compressible Euler equations with an explicit converging rate of viscosity coefficients.

This paper summarises our previous work on a fully $3$D Summation-by-Parts scheme, derived for a class of linear wave equations on hyperboloidal slices on a fixed Minkowski background. The scheme is derived in spherical polar coordinates, and allows having grid points at the origin and on the $z$-axis, despite coordinate singularities, and at infinity, by introducing compactification followed by rescaling, and is proved to be stable. Reducing it to the standard Cauchy problem, or to finite spacelike slices with an outer boundary, will follow a similarly. Second-order accurate finite-difference methods are used to implement this scheme numerically, but higher-order finite-difference or spectral methods could also be used. Kreiss-Oliger dissipation operators are generalized to curvilinear coordinates and are defined everywhere in the domain, including at the boundary points, such that they satisfy the dissipative property in the energy norms. We also propose new norm convergence tests that include all the grid points at all resolutions and produce more accurate results. Promising results are obtained, giving hope for application to fully nonlinear systems, like the Einstein Field Equations, and extracting the resulting gravitational waves free of systematic errors or gauge ambiguities.

We prove Liouville-type theorems for smooth solutions $(u,p)$ of the stationary Navier–Stokes equations in $\mathbb R^3$ satisfying the finite Dirichlet energy condition and the uniform decay condition. Our results give geometric refinements of two recent Osgood-type criteria, namely the relative decay criterion and the weighted integrability criterion formulated in terms of the head pressure $Q=\frac{1}{2}|u|^2+p$. The proof exploits several properties of the head pressure and the scalar triple-product structure of $(u\times\omega)\cdot\nabla Q$. Combined with an Osgood-type representation of the total vorticity energy, this geometric structure yields Liouville-type criteria involving only the tangential interaction of $u$ and $\omega$ over the superlevel sets of $|Q|$. In the relative decay setting, they lead to Osgood-scale smallness conditions that allow subcritical growth beyond the corresponding uniform bounds. In the weighted integrability setting, the previous conditions involving the full velocity and velocity gradient are refined to involve only the tangential components of $u$ and $\omega$ along the level surfaces of $Q$, together with an angular factor measuring their relative orientation.

This paper investigates the global well-posedness of strong solutions to a stress-diffusive Oldroyd-B system in two-dimensional smooth bounded domains in the zero Reynolds number (creeping flow) regime. The stress-diffusion term is kept explicit throughout the paper and is understood as the usual center-of-mass diffusion regularization of the Oldroyd-B constitutive equation. In the corresponding non-diffusive creeping-flow setting, the strongest available result is a Beale-Kato-Majda type breakdown criterion for the three-dimensional Cauchy problem due to Kupferman, Mangoubi and Titi [Commun. Math. Sci. 6 (2008)], and global well-posedness remains open even in two dimensions. We prove the global existence and uniqueness of strong solutions for arbitrarily large H1 initial polymeric stresses satisfying the natural non-negativity condition on the conformation tensor. The result covers the initial-boundary value problem on general smooth bounded domains and shows how the Stokes elliptic structure, the preservation of the non-negativity of the conformation tensor, and the stress diffusion combine to close large-data estimates at the H1 level. We also point out a regularity feature specific to the zero Reynolds number regime: the velocity field gains higher spatial regularity from the elliptic Stokes equation than the polymeric stress tensor.

We study nonlinear Dirac equations on the three-dimensional torus with subcritical Soler-type nonlinearities. The variational problem is strongly indefinite, while the non-coercive Soler potential prevents direct compactness. We recover the Palais–Smale condition by introducing a small coercive perturbation. Refined spectral estimates and a linking argument yield perturbed solutions with uniform energy bounds. Passing to the limit as the perturbation vanishes gives a nontrivial continuously differentiable periodic solution. This existence result holds for every spectral parameter in an explicit interval below the mass.

In this paper, we initiate the study of the Yang-Mills-Higgs-Schrödinger(YMHS) flow, i.e. the Hamiltonian flow of the Yang-Mills-Higgs functional defined on a symplectic fiber bundle. The YMHS flow provides a natural gauge-theoretic extension of the classical Schrödinger flow within the framework of symplectic reduction theory, and generalizes the Chern-Simons-Schrödinger equations to a non-Abelian gauge and non-linear fiber setting. We study its geometric structures and establish the local well-posedness of the corresponding Cauchy problem on compact Riemann surfaces.

We study the Cauchy problem for the defocusing mKdV equation with almost periodic initial data. If the initial data corresponds to a reflectionless Dirac operator which satisfies certain Craig-type conditions on the spectrum, we prove that the Cauchy problem has solution that is almost periodic in spacetime, and that this is the only solution which is locally bounded in a suitable sense. In particular, the result applies to small analytic quasiperiodic initial data with Diophantine frequencies.

We determine the sharp constant in the nearest-neighbor Hardy inequality on $\Z^3$ with the Euclidean inverse-square weight. For every finitely supported function $u:\Z^3\to\C$, we prove \[ \sum_{x\in\Z^3}\sum_{j=1}^3 |u(x+e_j)-u(x)|^2 \geq \frac14\sum_{x\in\Z^3\setminus\{0\}} \frac{|u(x)|^2}{|x|^2}. \] The coefficient $1/4$ is sharp, and equality is not attained by a nonzero finitely supported function. The proof uses an explicit reciprocal edge field and an edgewise completion of squares, together with a concavity argument for the associated vertex weight.

We study travelling shock profiles for dissipative quasilinear hyperbolic balance laws endowed with a convex entropy principle and subject to involutive differential constraints. For systems without constraints, the classical Boillat–Ruggeri result excludes continuous shock profiles whose speed exceeds the largest characteristic velocity at the equilibrium state ahead of the shock. We show that this bound remains unchanged for homogeneous constraints and for nonhomogeneous constraints whose source Jacobian has full row rank at equilibrium; a longitudinal two-species Gaussian ten-moment plasma with Landau collisions and Gauss's law provides a physical example of the latter case. For rank-deficient constraints, travelling-wave compatibility alone leads, after restriction to the joint conservation–constraint manifold, to a quadratic correction determined by a finite-dimensional Lyapunov equation. For $2\times2$ systems with one conservation law, one genuine balance law and one scalar involutive constraint, we prove that a regular noncharacteristic rank-deficient tail cannot sustain a nonzero correction, so that the original Boillat–Ruggeri bound is recovered. The remaining degenerate case is illustrated by an exact propagated two-field model in which the constraint selects the travelling speed and the smooth profile approaches equilibrium algebraically. The same model also admits entropy-admissible Lax composite profiles containing a sub-shock, while a completely smooth connection exists between the same equilibrium states. Numerical entropy evolutions further show that, depending on the smooth compatible initial datum, the first-order model may either remain smooth or dynamically develop a sub-shock.

We study the stabilization of the linearized periodic Whitham–Boussinesq system on the one-dimensional torus. We establish the well-posedness of the conservative and damped dynamics in the natural energy space and describe the spectral structure of the conservative generator, whose frequencies exhibit sublinear growth of order $|k|^{1/2}$ at high frequency. We then prove strong stability of the damped semigroup and obtain a high-frequency resolvent estimate with linear growth in the spectral parameter. Under genuinely localized damping, a family of high-frequency quasimodes provides the matching lower bound and shows that this resolvent growth is optimal. By the Borichev–Tomilov theorem [5], we deduce a $t^{-1}$ decay rate for the semigroup on the domain of the generator, corresponding to a $t^{-2}$ decay rate for the energy. Under the same localization assumption, these polynomial decay rates are optimal.

We present a high-order entropy-stable framework for the compressible Euler and Navier-Stokes equations on moving domains. The space-time mapping describing the domain motion is recast in an arbitrary Lagrangian Eulerian (ALE) formulation, in which the physical inviscid fluxes and the contributions induced by mesh motion are treated in a unified manner. At the continuous level, we prove that the proposed moving-wall boundary conditions are entropy conservative for the Euler equations and entropy stable for the Navier-Stokes equations. The no-slip condition is formulated in terms of the velocity relative to the moving wall, yielding a bounded inviscid contribution to the entropy balance, while the viscous terms contribute only entropy dissipation. Using diagonal norm summation-by-parts (SBP) operators together with appropriate numerical fluxes, these properties are extended to the semi-discrete formulation, resulting in nonlinear stability in the $L^2$ sense. The accuracy, robustness, and scalability of the proposed method are demonstrated in practice through an extensive set of numerical experiments, ranging from canonical two-dimensional verification cases to large-scale turbulent and supersonic simulations involving moving boundaries and fluid-structure interaction. The results confirm the suitability of high-order entropy-stable schemes for complex moving-domain problems across a broad range of flow regimes and multiphysics applications. Because the analysis relies on the SBP property rather than on a particular discretization, the framework naturally extends to a broad class of methods based on diagonal-norm SBP operators, including finite volume, finite element, and flux reconstruction schemes.

We study a nonlocal evolution problem driven by the fractional $1$-Laplacian with a Carathéodory nonlinearity satisfying subcritical growth conditions. We develop a potential well framework to investigate the existence and qualitative behavior of solutions at different initial energy levels. By combining a modified potential well method with Galerkin approximations, we establish the global existence of weak and strong solutions under appropriate conditions on the initial energy and the sign of the associated Nehari functional. To address the singular nature of the fractional $1$-Laplacian, we approximate the problem by a family of fractional $p$-Laplacian equations, derive estimates uniform with respect to $p>1$, and pass to the limit as $p\to 1^{+}$. Furthermore, in the low-dimensional regime $N<2s$, we employ a subdifferential approach to establish the local existence of strong solutions and investigate their qualitative behavior.

The oscillator (or metaplectic) semigroup is classically defined as the two-fold cover of the semigroup of positive complex symplectic matrices. Although geometrically precise, this definition does not provide an intrinsic operator-theoretic characterization of the bounded operators corresponding to positive symplectic matrices. This is in contrast with the real metaplectic group, whose relation with the symplectic group can be expressed directly in terms of the Schrödinger representation of the Heisenberg group through its intertwining property. In the complex setting, such a relation appears in the existing literature mainly in infinitesimal form or on suitable classes of Gaussian functions. In this work we prove that the (complexified) Schrödinger intertwining relation holds for every function in $L^2(\mathbb{R}^d)$. The main point is the converse statement: if an arbitrary complex symplectic matrix $S$ admits a nonzero bounded operator on $L^2(\mathbb{R}^d)$ satisfying this intertwining relation, then $S$ is necessarily positive and the operator coincides, up to a nonzero scalar, with the corresponding element of the oscillator semigroup. Consequently, positive complex symplectic matrices are exactly those admitting a nonzero bounded Schrödinger intertwiner, and the associated intertwining space is one-dimensional. This provides a Schrödinger-representation characterization of the oscillator semigroup, parallel to the classical one for the real metaplectic group.

A celebrated conjecture of P.-L. Lions concerns maximal regularity for viscous Hamilton–Jacobi equations. In this paper, we study the endpoint case. We consider normalized strong solutions of $$-\Delta u+|Du|^\gamma=f$$ in $\mathbb T^d$, where $d\geq 2$, $\gamma>2$, and $f\in L^{q_c}(\mathbb T^d)$ with $q_c=d(\gamma-1)/\gamma$. At this critical exponent, the main difficulty is possible concentration under the critical scaling. Assuming that the source terms form a uniformly equi-integrable subset of $L^{q_c}$, we rule out this concentration and prove maximal $L^{q_c}$ regularity for strong solutions. The proof combines a two-stage blow-up argument with a Liouville rigidity theorem. More precisely, the second blow-up yields a uniform local $L^{\gamma q_c}$-bound for the gradients, while the small drift arising from the first blow-up upgrades weak convergence to strong local compactness, ultimately leading to a contradiction with Liouville rigidity. Finally, we apply the maximal regularity theory for Hamilton–Jacobi equations at the endpoint case to establish the existence of ergodic solutions to defocusing second-order mean-field games systems with critical coupling exponents.

We develop a rigorous mean-field theory for transformer networks that captures two large-scale limits inherent in the architecture: the number of tokens $N\to\infty$ in the input sequence and the number of attention heads $H\to\infty$ in each layer. It further considers an infinite number of layers leading to a time-continuous formulation. The resulting framework couples two interacting mean-field objects: a token distribution $\mu_t\in\mathcal P(\mathbb R^d)$, which evolves through network depth $t\in[0,T]$ according to a McKean–Vlasov transport equation, and the attention-parameter distribution $\rho_s\in\mathcal P(\Theta)$, that evolves through training time $s\ge0$ according to a Wasserstein gradient flow of the empirical risk, with optional entropic or Tikhonov regularization. We establish a comprehensive analytical foundation for the system coupling the transformer and the training dynamics; in particular, we establish global well-posedness of the resulting nonlinear Fokker–Planck system describing the coupled mean-field and training dynamics. Beyond well-posedness, we connect the mean-field formulation to optimization. For shallow, single-layer attention models, we prove exponential convergence to the entropy-regularized global optimum under a log-Sobolev condition. For genuinely deep, compositional transformers, we establish local linear convergence under a Neural Tangent Kernel non-degeneracy condition.

Hessian-free high-resolution (HFHR) dynamics augments underdamped Langevin dynamics (ULD) with reversible position diffusion for sampling problems that arise in machine learning. We establish an explicit quantitative contraction rate for HFHR dynamics under a position Poincaré inequality, weighted Hessian and Laplacian bounds, and a compact Sobolev embedding, where the potential function is not necessarily convex. An adapted time-augmented Poincaré inequality yields an explicit rate that improves upon the contraction rate of the underdamped Langevin dynamics. We also give a weak-solution construction and a self-contained spectral proof of the divergence lemma underlying the argument. For HFHR Monte Carlo (HFHRMC) algorithm, which is based on a discretization scheme of HFHR dynamics, we use a path-space Girsanov argument to obtain a non-asymptotic convergence bound and an explicit iteration complexity in total variation distance. The bounds hold for every $\alpha\geq0$ and $\gamma>0$ and remain regular at the ULD endpoint. Optimizing the iteration complexity bound yields a positive, accuracy-dependent position-diffusion parameter at finite accuracy, while its leading high-accuracy order coincides with that of the optimized ULD endpoint. Our iteration complexity bound improves upon the existing work on HFHR algorithms. Numerical experiments including Bayesian learning problems on real data are provided to illustrate the effect of positive $\alpha$ and its benefit.

It was recently discovered by Boiti–Manfrin that a variant of the Kirchhoff wave equation introduced by Pohozaev admits infinitely many conserved quantities. In this paper, we obtain explicit formulae for such conserved quantities, as well as introduce a generating function for them that is coercive. These tools are then employed to establish a priori bounds, the propagation of equicontinuity, global well-posedness in $\mathcal H^s$ for $s\geq \frac32$, and the existence of global $C_t \mathcal H^1$ solutions.

For $N\ge2$ and $\beta>0$, consider the canonical 2D Coulomb gas ensemble \begin{equation} \mathrm{d}\mathbb{P}_{N,\beta}(Z) =\mathsf{Z}_{N,\beta}^{-1}e^{-\beta|Z|^2/2} \prod_{i<j}|z_i-z_j|^{\beta/N}\mathrm{d} Z \qquad\text{on }\mathbb{C}^N. \end{equation} The factor $N^{-1}$ in the pair exponent places the ensemble at the diffusive, or high-temperature, scale. We prove that, for every fixed inverse temperature $\beta>0$, this Gibbs measure satisfies a logarithmic Sobolev inequality (LSI) on the closed Dirichlet-form domain generated by collision-free compactly supported smooth functions, with a positive constant independent of $N$. The LSI constant may be chosen uniformly when $\beta$ ranges over a compact subset of $(0,\infty)$. A second main result establishes Poincaré and logarithmic Sobolev inequalities for planar Gaussian measures weighted by finite products of positive powers of distances to points, with constants depending only on the total variance of the underlying Gaussian measure and the total exponent, not on the number or locations of the points. To prove the full labeled Poincaré inequality, we decompose variance into permutation-invariant and label-dependent parts, controlled respectively by a weighted $\bar\partial$ estimate and by relative-coordinate conditioning with random transpositions; separately, one-site logarithmic Sobolev inequalities, a conditional entropy inequality, and a repulsive partition estimate give a defective logarithmic Sobolev inequality. Rothaus tightening combines the resulting full labeled Poincaré inequality with the defective logarithmic Sobolev inequality. The two main theorems on their stated closed form domains and all results used in their formal proofs, including those from the literature, have been formalized in Lean 4.