Papers

This paper introduces a multi-class extension of a discrete-velocity kinetic traffic flow model based on a non-local Prigogine-Herman framework. We derive a hyperbolically scaled system of equations from a continuous kinetic formulation describing interactions between different vehicle classes through braking and relaxation terms. The model is then discretized with respect to the velocity variable for an arbitrary number of vehicle classes, and the structural properties of the resulting formulation are analyzed. In particular, we prove hyperbolicity and total linear degeneracy. Due to the non-conservative structure of the model, we employ a path-conservative finite volume scheme for the numerical approximation of the system. Finally, we derive the corresponding diffusively-corrected macroscopic multi-class model, investigate its stability and present numerical simulations on a single-lane road to illustrate the theoretical findings.

We prove that the notion of mass recently introduced by Mazurowski and Yao for continuous metrics coincides with Huisken's isoperimetric mass on smooth Riemannian $3$-manifolds with nonnegative scalar curvature whose metric has sharp $C^0$-asymptotically flat behavior. As a consequence, we deduce a Riemannian Penrose inequality for the mass parameter of asymptotically Schwarzschildian $3$-manifolds.

We investigate ground states of the stationary magnetopolaron in $\mathbb R^3$ with a constant magnetic field. When the strength $|b|$ of the magnetic field is sufficiently small, we prove the uniqueness and nondegeneracy of ground states, up to magnetic translations and phase shifts. Applying the uniqueness result, we further derive rigorously the symmetry and monotonicity of ground states for sufficiently small $|b|>0$. The second-order asymptotic expansion of the ground state energy is also derived as $b\to 0$.

We study wave propagation through a time-modulated thin heterogeneous layer. The layer is assumed to have a thickness of order $\es\ll 1$ and its material properties exhibit rapid oscillations on multiple spatial and temporal scales. We aim to rigorously derive the effective model and the corresponding interface conditions for wave propagation through the limiting interface. To perform the homogenization together with dimension reduction, we generalize the notion of two-scale convergence for thin layer introduced by Neuss-Radu and Jäger (2007) to a multiple space-time scale framework. One of the main analytical difficulties arising in the homogenization analysis is that, in general, a uniform energy estimate cannot be obtained for a wave equation with time-varying coefficients. Therefore, we identify two physically relevant classes of coefficients for which we can derive a uniform energy bound. These include coefficients with traveling wave-type modulations of their properties. Using the energy estimates and the multiscale convergence for thin layer concept, we derive the effective model, which consists of linear wave equations in the bulk domains coupled through a nonstandard jump condition at the interface. This jump condition is governed by a wave-type dynamical equation with effective macroscopic coefficients determined by suitable cell problems of elliptic and hyperbolic type. Finally, we discuss the uniqueness of the effective model.

We study stochastic hypoviscous Navier–Stokes equations on the torus with dissipation $(-\Delta)^\gamma$ for $\gamma \in (\frac{1}{2},1]$ and multiplicative noise. Relying on stochastic maximal regularity results for the linear equation, we establish local well-posedness for this problem in arbitrary dimensions with initial data in a range of scaling-critical Besov spaces. With the aid of $L^q$-energy estimates for the vorticity equation, we also prove global well-posedness of the stochastic hypoviscous Navier–Stokes equation in 2D with linear multiplicative noise.

We prove global existence and modified scattering for solutions to relativistic Vlasov–wave systems without null structure, arising from small distribution functions. When the homogeneity of the force field in the momentum variable is the same as in the Vlasov-Maxwell system, the scalar fields are allowed to be large and our method requires stretched exponential velocity decay. When it matches that of the Einstein-Vlasov system, the scalar fields are required to be small and the distribution function to have slightly super-exponential velocity decay, which in particular includes Maxwellian decay. Moreover, we show that the smallness assumption on the scalar fields cannot in general be removed, even for compactly supported initial data.

We study 2D compressible Euler equations linearized around monotone shear flows $(U(y),0)$ on $\mathbb{T} \times \mathbb{R}$. The shear rate $U'$ is strictly positive, not necessarily close to any constant, and varies sufficiently slowly. For every fixed Mach number $M > 0$, we prove that the density and the irrotational velocity obey algebraic growth bounds, whereas the solenoidal velocity undergoes componentwise inviscid damping. Although a non-uniform shear couples the transverse Fourier frequencies and precludes the full Fourier reduction available for Couette flow, we are still able to recover the Couette rates without loss. The proof hinges on two new ingredients: a time-dependent pseudodifferential energy that restores a coercive structure for the variable-coefficient shear dynamics, and terminal-time-dependent higher- and lower-order weighted energies that capture the long-time effects of shear mixing.

Inverse inequalities are an important tool in the stability and convergence analysis of kernel approximation methods. In a multiscale setting, however, the trial space changes with the kernel scale $\delta$, and inverse estimates for a fixed kernel are not sufficient. The constants must remain controlled as $\delta\to0$. In this paper, we study scale-uniform inverse inequalities for spaces generated by scaled positive definite kernels whose native spaces are Sobolev spaces. We first establish inverse estimates in scale-dependent Sobolev spaces. On bounded domains, this leads to a scale-uniform inverse inequality in standard Sobolev norms with $L^2$ as the weaker norm. The estimate requires only that the separation distance of the centers be bounded above by a fixed multiple of the kernel scale. This allows the kernel scale to decrease more slowly than the separation distance, as is relevant in multiscale refinement. We then establish more general scale-uniform Bernstein inequalities on the whole space. Our result covers a broad range of weaker and stronger Sobolev indices under the same relation between the separation distance and the kernel scale. The proof works directly with the Fourier representation of functions in the kernel space and combines a low-high frequency decomposition with frame estimates for separated exponential polynomials. This avoids the additional scale factors that arise from separate comparisons of scaled and standard Sobolev norms.

We study a free-energy minimization problem for a trapped semiclassical Bose gas with repulsive Coulomb self-interaction. Because the bosonic entropy density has zero recession slope, the natural relaxation is posed over nonnegative finite Radon measures on phase space, and a minimizer may have a singular component. We interpret this component as condensation within the relaxed semiclassical model. We prove existence and uniqueness at every positive temperature, derive the Euler-Lagrange contact condition and an obstacle-type formula for the condensed density, and establish sharp phase boundaries under only the standing continuity assumptions on the trap. At fixed mass there is a unique finite positive critical temperature, with condensation precisely below it. At fixed temperature there is a sharp critical mass, possibly infinite, separating normal and condensed minimizers. For harmonic traps we obtain a sharp confinement-strength dichotomy. We also prove an abstract conditional variational-stability statement for measure-valued curves that conserve mass and satisfy the free-energy inequality.

For connected proper metric spaces $X$, equipped with a locally doubling measure supporting a local $p$-Poincaré inequality, we completely characterize which compact sets $K$ with positive capacity are removable for bounded $p$-harmonic functions, $p>1$. Similar results are proved also for bounded quasiharmonic functions. The characterization is both in geometric and analytic terms. In particular, removability is shown to be equivalent to the validity of a Liouville type theorem in $X\setminus K$. Properties such as local connectedness, sequential annular quasiconvexity, concentration of capacity and $p$-parabolicity are identified as crucial for removability. Along the way, we give a rather elementary proof of the Liouville theorem for quasisuperharmonic functions in $p$-parabolic spaces. Our results apply in particular to manifolds and $\mathbf{R}^n$ equipped with (locally) $p$-admissible weights.

For the surface growth model under molecular beam epitaxy on $\mathbb{R}$, we show that an inverted parabola is a stable steady solution in the sense that any small perturbation of it decays in $L^{\infty}(\mathbb{R})$ at the rate $O(t^{-1/2}\log t)$. The proof is based on the explicit solution operator generated by the linearized perturbation equation. This result partially justifies the merging of two neighboring islands into a bigger one. Moreover, any derivative of its solution is shown to be decaying in $L^{2}(\mathbb{R})$ for sufficiently small initial data.

Since Ilmanen's pioneering work [J. Differential Geom. 38, 417-461, (1993)] it has been a major open problem to understand the sharp-interface limit of systems of coupled Allen-Cahn equations. We prove - for the first time - a global, unconditional convergence result for such a coupled system and show that the limit is a multiphase mean curvature flow in the sense of Brakke.

This paper is concerned with the following semilinear elliptic equation involving the fractional Laplacian: $$(-\Delta)^s u+ g|u|^{p-1}u= \lambda \frac{u}{|x|^{2s}}+f(x),$$ in a bounded domain $\Omega$ of $\mathbb{R}^N\,(N>2s)$, subject to the zero Dirichlet condition in $\mathbb{R}^N\setminus \Omega$, where $0<g\in L_{loc}^1(\Omega)$, $p>1$ and $f\in L^{(p+1)/p}_g(\Omega)$. Under certain integrability condition on $g$, the existence of solution is proven for every $\lambda\in \mathbb{R}$. Moreover, the regularity of solution is also obtained.

Subdiffusion occurs when long trapping or residence times of particles slow down spatial transport and produce a mean squared displacement proportional to $t^\alpha$, with~$0<\alpha<1$. We develop a general renewal–jump framework that combines the internal trapping dynamics with the reinjection of particles with spatial jumps. The internal trapping dynamics enters the macroscopic limit through the small-frequency behavior of a resolvent which limiting solution is not integrable. A growth mass condition determines the parameter~$\alpha$. Then the spatial density converges on the time scale~$\eps^{-2/\alpha}$ to a time-fractional diffusion equation. This criterion provides a common derivation for models with very different internal mechanisms. We first establish the abstract limit by Laplace transform in a general setting. We then show that the classical age-structured renewal model is a direct instance of the framework and apply the same criterion to an internal pathway model governed by a degenerate elliptic operator. Both yield the subdiffusion equation and its correct initial condition. The approach clarifies which microscopic property produces subdiffusion, how the anomalous exponent determines the macroscopic scaling, and why the initial distribution enters the limit through the initial spatial mass under the stated preparation assumptions.

We study the two-dimensional stationary Navier-Stokes equations in an unbounded wedge with small rough perturbations of its angular boundaries. The Jeffery-Hamel flow in the corresponding straight wedge is taken as the effective background flow. Under a suitable small-flux condition, we prove the existence of weak solutions and establish an $H^1$-type energy error estimate of order $O(\epsilon^2)$. For sufficiently small wedge angles, we further derive weighted estimates and improve the squared $L^2$-error to $O(\epsilon^3)$.

Spectral Barron spaces, characterized by an \(L^1\)-based Fourier-Lebesgue norm, have earned significant attention in approximation theory due to their remarkable capacity to represent functions via shallow neural networks with controlled complexity. Meanwhile, recent theoretical advances have firmly established an intrinsic and profound connection between these function spaces and the regularity theory of elliptic partial differential equations. Building upon this foundational interplay, the present work undertakes a systematic and comprehensive investigation into the well-posedness of heat equations formulated within the spectral Barron spaces framework. Specifically, we rigorously establish the core aspects of well-posedness, including the existence, uniqueness, and stability of solutions, under suitable assumptions on the source terms and conductivity coefficients. We also investigate a typical parabolic inverse problem, namely the backward heat equation, for which we derive a logarithmic conditional stability estimate. To the best of our knowledge, this constitutes the first stability estimate for inverse problems within the spectral Barron space setting. Moreover, we extend our analytical results to address the more intricate setting of time-fractional heat equations, which govern anomalous diffusion phenomena and introduce nonlocal temporal memory effects. In this extended context, we provide a characterization of the corresponding heat kernels, deriving decay estimates, regularity properties, thereby enriching the theoretical landscape of evolutionary PDEs within the spectral Barron spaces setting.

We prove local Calderón-Zygmund estimates for distributional solutions to non-uniformly elliptic equations in divergence form modeled on the energy density $a(x)G(|z|)+b(x)H(|z|)$, where $G$ and $H$ are Young functions and $a(\cdot)$, $b(\cdot)$ are nonnegative coefficients, Hölder continuous with exponents $\alpha$ and $\beta$. Either coefficient may vanish, and only their sum is bounded away from zero, so that each phase can dominate or disappear. Under a gap condition that reduces to $q/p\le1+\min\{\alpha,\beta\}/n$ for $G(t)=t^p$ and $H(t)=t^q$, including the borderline case of equality, we show that the energy density of the gradient inherits the Orlicz integrability of the energy density of the datum. The proof combines a fractional differentiability estimate, which gives higher integrability up to the borderline growth, with comparison arguments that freeze the two coefficients in the appropriate order.

We study the Bayesian recovery of the initial state in a semilinear time-fractional subdiffusion equation from noisy random space-time point observations. A rescaled Gaussian prior based on a Whittle–Matérn process is assigned to the unknown initial condition. We prove that the solution has \(\dot H^3\)-regularity when the nonlinearity satisfies a Lipschitz condition in the Sobolev norm. We then establish posterior contraction rates for the prediction error in the \(L^2\)-norm and for the parameter in Sobolev norms. The rates are polynomial in the sample size, with exponent depending on the prior smoothness and the spatial dimension. Moreover, we prove a minimax lower bound by constructing a wavelet-packing set and controlling the Kullback–Leibler divergences.

We study Kirchhoff-Boussinesq-type equations on closed Riemannian manifolds $(M,g)$ of the form \[ \sum_{j=0}^m a_i(-\Delta_g)^m u \pm \text{div}_g(\vert\nabla u\vert_g^{p-2}\nabla u) = f(u),\qquad \text{ on }\ M, \] where $m<\dim M/2$, $(a_0,a_1,\ldots a_m)\in C^\infty_+(M)\times [0,\infty)^{m-1}\times(0,\infty)$, $2<p\leq\frac{2\dim M}{\dim M- 2}$ and $f:\mathbb{R}\rightarrow\mathbb{R}$ is a continuous nonlinearity of superlinear type with subcritical or critical Sobolev growth. We briefly motivate the above equation in the case of $m=2$, as an extension of the stationary Kirchhoff-Boussinesq equation when modeling the dynamics of curved elastic plates. Under some symmetry assumptions, we prove the existence of multiple equivariant solutions when considering critical Sobolev nonlinearities and, for subcritical ones, we also prove the existence of ground-state solutions. As a byproduct, we prove a Gagliardo-Nirenberg interpolation inequality and give several equivalent norms on the higher order Sobolev space $H_g^m(M)$.

We prove interior $C^2$ regularity for convex viscosity solutions of the $3$-Hessian equation $\sigma_3(D^2u)=f(x)$ with $f\in C^{0,1}, \inf f>0$, under a strict $3$-convexity condition on $u$.

We study the compactness of two sets of positive radial solutions to “zero mass” quasi-linear Schrödinger equation \[ -\Delta u-u\Delta(|u|^2)=g(u) \quad\text{in }\mathbb R^N,\quad N\ge5. \] For nonlinearities that are either strictly subcritical or asymptotically critical at infinity, we prove that the set of least energy positive radial solutions is nonempty and compact in the natural space. Moreover, in the strictly subcritical case and under additional assumptions, we show that the set of all finite $L^2$-mass positive radial solutions is compact in \(L^2(\mathbb R^N)\). The proof relies on uniform decay estimates for the corresponding positive radial solutions of the transformed semilinear equation, which yield the required uniform \(L^2\)-tail control.

In this paper we derive the time-dependent correlation functions of the renormalized Hartree NLS equation on the torus $\mathbb T^d$, $d=2,3$, from the corresponding bosonic many-body Gibbs dynamics. In contrast with the 1D problem studied earlier by Fröhlich, Knowles, Schlein and Sohinger, in higher dimensions the scaled quantum particle number is not uniformly bounded and the relevant classical fields require Wick renormalization. Our proof combines convergence of the quantum generators in weighted Hilbert spaces with uniqueness for positive solutions of the limiting Liouville equation that are dominated by the Gibbs measure.

We prove a periodic long-time Boltzmann–Grad limit for hard spheres in every fixed spatial dimension $d\ge4$. Under the assumptions of the main theorem, the rescaled $s$-particle correlations converge in $L^1$ to the corresponding tensor-product Boltzmann profile at rate $\varepsilon^{1/(400d)}$, uniformly for $1\le s\le|\log\varepsilon|$ and $0\le t\le t_{\rm fin}$. In particular, when the activity and weighted solution bounds are fixed, $t_{\rm fin}=O(\log|\log\varepsilon|)$. The componentwise long-bond estimate used in dimensions two and three is insufficient in higher dimension. We replace it by a joint estimate for two connected time sublayers, selecting the first two lower collisions as landing roots. Disjoint landing edges yield a direct tangential frame. When the edges overlap, the newly appearing particle line contains a collision-free chord whose radial relative speed produces a rank-$(d-1)$ positive Schur factor. A positive Jacobi-network argument prevents focusing, while bounded-degree finite-fibre coarea globalizes the estimate for arbitrary nonnegative joint kernels. A first-failure construction repairs periodic double overlaps, and the resulting packet bound closes the exceptional top-layer contribution. A paid epoch restart for sealed complete joint kernels, using the fixed-word multi-landing operator for every fixed integer $k\ge1$, yields the stated time range. Within the regular connected full two-sublayer packet class, every packet with at least $2k-1$ physical lines admits a canonical $k$-birth flag and the operator factor $\varepsilon^{(d-1)(k-1)}\varepsilon_*^{-k(d-1)}$; $k=2$ is the first gain-producing case. A complementary full-chord estimate provides a coarse fallback.

We study graded arrays of highly dispersive subwavelength resonators through their volume-potential matrix reduction. The reduction gives a finite-dimensional nonlinear resonance problem whose diagonal entries encode the local dispersive response and whose off-diagonal entries encode volume-mediated coupling between distinct resonators. We show that monotone material grading turns this matrix into an analytically controllable frequency sorter. When graded detuning dominates coupling in the reduced characteristic matrix, the characteristic vectors localize near individual resonators; under an explicit half-lattice-site localization condition, this localization becomes a monotone frequency-position map. We also prove a constructive inverse-design statement: prescribed monotone target frequencies can be realized by choosing the local dispersive parameters, with robustness under sufficiently small grading errors. Numerical experiments, whose parameters and algorithms are recorded in an appendix, illustrate localized resonances, designed frequency-position maps, disorder stability, spacing dependence, and a resolved physical Lippmann-Schwinger validation beyond the leading logarithmic quadrature model.

In the present work we consider the problem of the homogenization of the Cauchy problem for the equations of the elasticity. We assume the composite material is constructed as a periodic repetition along the chosen axis of the unit cell which models to different materials. The homogenization procedure is based on the operator separation of variables, which gives the new incites for the homogenization of the elasticity equations and provides new way of the homogenization and constructing the homogenized equation.

We consider an aggregation-diffusion energy on the sphere and investigate the stability of its equilibria. The energy consists of a porous-medium type nonlinear entropy $\frac{1}{m-1}\int \rho(x)^m\mathrm{d}S(x)$ with $m>1$, together with an interaction energy modeled by a quadratic interaction potential. The energy generalizes the Onsager free energy with dipolar potential, which models polymer orientation. Our study complements the authors' previous work [Nonlinearity 39 (2026), 055011], where the ground states of the energy functional were investigated. In the current paper we extend the previous results on existence of equilibria for $m>2$ from $d=2$ to arbitrary dimension $d \geq 2$. This allows us to present the bifurcation structure (with respect to the interaction strength) of all equilibria in any dimension $d \geq 2$, for all $m>1$. Furthermore, we provide a complete classification of the stability of all equilibria of the energy, by deriving a criterion for stability that can be checked explicitly. In particular, for $m>2$ we identify a saddle-node bifurcation of strictly supported equilibria, and a subcritical pitchfork bifurcation where the uniform distribution loses stability.

In this paper we prove the following result. Let $\Omega\subset\mathbb R^n, n\geq 3,$ be a bounded, strictly convex, smooth domain and $\varphi: \partial\Omega\rightarrow\mathbb R$ be a smooth function. Then for any $z\in\Omega,$ there exists $c_1=c_1(\Omega, \{z\}, n, \varphi)>0,$ such that if $c\geq c_1$ then the problem: $\sigma_{n-1}(D^2 u)=0$ in $\bar\Omega\setminus\{z\},$ $u|_{\partial\Omega}=\varphi,$ $\lim\limits_{r\rightarrow 0}\sup\limits_{B_r(z)}\frac{u(x)-u(z)}{|x-z|^{(n-2)/(n-1)}}=c,$ admits a smooth solution in $\bar\Omega\setminus\{z\}.$ Moreover, we obtain the optimal a priori estimates for this solution. In particular, we show for any integer $m\geq 0$ there exists $C_m=C_m(m, \Omega, \{z\}, n, \varphi, c)>0$ such that $|D^m u(x)|<C_m|x-z|^{\frac{n-2}{n-1}-m}.$ This work provides the first result demonstrating the existence of smooth solutions to the Dirichlet problem for fully nonlinear elliptic equations with prescribed density in Euclidean space. Previously, only continuous solutions were obtained.

We give a short complex-analytic proof of a square-restricted form of local stable phase retrieval at the Gaussian in one-dimensional Fock space. The main estimate is a coercivity inequality for the map $F\mapsto F^2$: \[ \|F^2-F(0)^2|_{\mathcal{F}^2(\mathbb{C})} \lesssim \inf_{c\in\mathbb{R}}\||F|^2-c\|_{L^2(d \gamma)}. \] The proof uses a weighted derivative norm, two integrations by parts, and a weighted Cauchy inequality. The proof has been completely verified in Lean with the aid of Large Language Models.

We study a thermodynamically consistent non-isothermal phase-field system coupling two order parameters and the inverse temperature through a fully non-diagonal Onsager mobility. The model describes the interaction of mass diffusion, heat conduction, and local phase relaxation while conserving mass and internal energy and producing entropy. Global generalised dissipative weak solutions are constructed using a fully discrete approximation. The discrete scheme conserves mass and internal energy and satisfies a discrete entropy inequality. An availability estimate, together with a dimension-adapted barrier, yields strict positivity at fixed mesh and uniform estimates. Compactness then allow passage to the continuous system. Concentration of the singular part of the internal energy is represented by a non-negative defect measure. The entropy-availability structure yields finite dissipation on the infinite time interval and the existence of stationary $\omega$-limit states. Finally, a relative-entropy argument establishes weak-strong uniqueness whenever the weak and strong solutions remain in a bounded thermodynamic state range.

This paper proves the short-time existence of $G_2$-Laplacian flow for closed, asymptotically conical $G_2$-structures on a complete manifold, shows that this solution is unique among closed solutions, and gives a lower bound for the existence time depending on the initial geometry.