Papers

We construct a branch of travelling periodic `roll wave' solutions to the free-boundary incompressible Navier–Stokes equations on an inclined plane in two dimensions. These solutions bifurcate from a parallel shear flow, under natural assumptions on the related Orr–Sommerfeld equation. Using techniques from analytic global bifurcation theory, we extend the local branch to a global curve of solutions. A key step of the proof is reformulating the problem, including the unknown free boundary, as an elliptic system in the sense of Agmon–Douglis–Nirenberg. Finally, we verify the hypotheses on the Orr–Sommerfeld equation for two regimes: small wavenumber and low Reynolds number.

We investigate a three-dimensional inverse scattering problem for tracking a rigidly moving target from far-field data generated by a single incident field. Extending our recent two-dimensional study, we develop a Bayesian optimization framework for simultaneously tracking the target's location and orientation over successive time steps, with the translational and rotational motions modeled as independent stochastic processes. We derive analytical formulas for the far-field pattern under translations and rotations and use them to design a Bayesian optimization procedure tailored to the tracking problem. We further establish posterior consistency for the underlying probabilistic model. When the target shape is unknown, its shape is identified at the initial time using a fully connected neural network trained on a precomputed dataset. Numerical experiments validate the effectiveness of the proposed framework.

This paper is devoted to the rigidity of weak solutions for anisotropic $N$-Laplacian equations with Neumann or Robin boundary conditions on smooth bounded convex domains of $\mathbb{R}^N$. The anisotropic operator is given by $$a(\xi) = H^{N-1}(\xi)\nabla H(\xi),$$ where $H$ stands for a norm on $\mathbb{R}^N$; this formulation contains the classical $N$-Laplacian as a special case. We establish a key integral inequality involving the anisotropic gradient and the second fundamental form of the domain boundary, which acts as the core technical tool in our proofs. Under natural monotonicity assumptions on the nonlinearity, we prove that all weak solutions to the Neumann boundary problem are constant, without requiring any a priori boundedness assumption on the solution. Furthermore, we extend this rigidity result to Robin boundary value problems by imposing suitable constraints on the boundary nonlinear term. Moreover, our rigidity results remain valid not only on bounded convex domains but also on suitable unbounded domains. By working under substantially weaker assumptions than those previously available, we establish rigidity results that fill the gaps in the existing literature for anisotropic $N$-Laplacian equations with nonlinear boundary conditions and substantially extend the rigidity theory of anisotropic quasilinear elliptic equations at the critical exponent $p=N$.

In the paper we introduce vector-valued dynamic extended graphons in order to obtain the mean-filed limit for certain class of problems with adaptive weights. We impose fairly general assumptions on the matrix of initial connections, which allows us to cover the case of sparse graphs.

We prove Liouville rigidity theorems for translation-invariant real and complex Hessian equations in the viscosity sense, where the PDE is encoded by an admissible set $\mathcal{A}$. The main structural notion is Liouville admissibility, a recursive geometric condition requiring each quotient set to be either boundary compatible or to fall into a terminal class. Our main theorem states that every bounded, globally $C^{0,\alpha}$ entire viscosity solution of \[ \mathrm{Hess}_{\mathbb F}u\in\partial\mathcal{A} \] is constant if and only if $\mathcal{A}$ is Liouville admissible; thus the Liouville-type property is characterized as a geometric property of the admissible set. A central class of examples arises from polarizations of univariate Gårding polynomials satisfying the monotone root sequence condition, producing mixed elementary-symmetric admissible sets and recovering the standard $k$-Hessian equations as monomial cases. The framework also allows anisotropic constructions, including linear pullbacks and intersections of admissible sets.

This paper investigates recovery of an unknown coefficient in a semilinear wave equation defined on a bounded, open, and strictly convex domain in \(\mathbb{R}^{1+n}\) with \(n \ge 2\). We demonstrate that the unknown coefficient \(q\) appearing in the semilinear wave equation \(\square u + q u^m = 0\) with Neumann boundary conditions can be reconstructed with Hölder stability from the linearized Neumann-to-Dirichlet (NtD) map. Our approach combines first-order linearization with the Principle of Inclusion-Exclusion (PIE) identity, and employs geometric optics solutions for wave equations in two distinct regimes: the case \(m=2\) with \(q = q(x)\), and the case \(m \ge 3\) with \(q = q(t,x)\). Furthermore, numerical examples illustrate that the unknown coefficient can also be effectively reconstructed using a neural network-based inversion algorithm within the framework of the least squares method.

On general metric measure spaces, we develop a new well-posedness theory for the signed porous medium equation and its fast diffusion counterpart \[ \partial_t u = \mathcal{L}\left(|u|^{m-1}u\right), \qquad m>0, \] where $\mathcal{L}$ is the associated non-positive self-adjoint operator of a symmetric Dirichlet form. The theory does not rely on a Gelfand triple or compact embeddings; instead, it is built upon the extended Dirichlet space $\mathcal{F}_e$ and auxiliary spaces $V^q:=L^q\cap\mathcal{F}_e$, whose uniform convexity plays a key role in the proof. The proof uses only the existence of the Dirichlet form and its extension; no additional regularity of the form or geometric assumptions on the underlying space are needed. Consequently, the results apply to a wide range of metric measure spaces, including non-smooth fractals.

We prove an Eyring–Kramers law for metastable transitions of the hypoelliptic third-order Langevin diffusion in the low-temperature limit. This diffusion is a three-level Markovian lifting of Langevin dynamics: the Brownian noise acts only on the highest auxiliary variable and reaches the position variable through a third-order H"ormander chain. For a double-well potential with a unique index-one transition saddle, we determine both the Arrhenius exponential scale and the sharp prefactor of the mean transition time. The prefactor is governed by the unique positive unstable rate of the deterministic linearization at the saddle, equivalently the positive root of a cubic polynomial. Our proof combines a weak-capacity framework with a saddle-adapted boundary layer, an explicit Gaussian current calculation, committor localization, and intrawell flatness. Under matched kinetic normalizations, the resulting metastable prefactor is strictly smaller than its underdamped counterpart. A numerical experiment for a one-dimensional double well illustrates the Arrhenius scaling and the predicted prefactor comparison.

For $p\in(1,n]$ and any bounded convex domain $\Omega\subset\mathbb{R}^n$, we prove the sharp inequality \[ \Lambda_p(\Omega):=\frac{W_p(\Omega)}{P(\Omega)V(\Omega)^{p/n}}\geq\omega_n^{-p/n}, \qquad W_p(\Omega)=\int_{\partial\Omega}|x|^p\ dS, \] with equality holding exactly at centered balls. Combining this with the isoperimetric inequality yields the explicit upper bound \[ \sigma_{1,p}(\Omega)\leq \frac{A(n,p)}{r(\Omega^*)^{p-1}}, \] where $\Omega^*$ is a ball having the same perimeter as $\Omega$, and $A(n,p)=1$ for $1<p\leq 2$, $A(n,p)=n^{p/2-1}$ for $2<p\leq n$. When $p=2$, the result recovers the higher-dimensional Weinstock inequality of Bucur et al. [J. Differential Geom. 2021]. We also obtain an explicit upper bound for the first Wentzell eigenvalue of the $p$-Laplacian on convex domains.

We investigate a general class of variational integrals under a structural condition imposed on the double-phase function, recently introduced in~\cite{ADKO2026}. In this setting, the strong Harnack inequality for non-negative local quasi-minimizers is established via an appropriate De Giorgi-type iteration argument. Most notably, the proposed analytical approach in this paper provides a new perspective for deriving Harnack-type inequalities for more general variational functionals under a Muckenhoupt-type structural condition on the double-phase function, without relying on the classical coefficient-freezing strategy based on the Hölder continuity of the modulating coefficient and the balance condition on the growth exponents.

We study the master equation for a second-order mean field game on Carnot tori, which means the generic player can move periodically only along admissible trajectories given by the family of vector fields generating the Carnot group. As examples of sub-Riemannian manifolds, Carnot groups represent a type of non-commutative groups characterized by stratified Lie algebra structures. In order to obtain the well-posedness of the master equation, we analyze the properties of its solution by investigating a degenerate mean field game system for which there exists an equivalent characterization with the master equation. The main part of this paper lies in leveraging the regularity properties of solutions to two classes of linear degenerate parabolic equations and a class of linear degenerate coupled systems to derive the existence of solutions to the master equation. The research in this paper is motivated by \cite{19CDLL,24MMM}.

Below-threshold scattering is studied for the three-dimensional focusing energy-critical Schrödinger equation in the cylindrically symmetric class. For soliton-like compact critical elements, two linked rigidity entrances are isolated: a best fixed-axis window condition for the axial center and a low-frequency tail smallness condition. A shifted localized virial estimate and a finite-mass virial argument after axial zero-momentum normalization exclude these entrances, yielding a conditional cylindrical threshold scattering criterion via compactness reduction.

Higher-order Poisson equations involve an integer power of the Laplacian and a nonzero forcing term, and appear in areas of physics and engineering such as hydrodynamics, structural engineering, and image processing. We introduce a family of mean-value formulas for solutions to higher-order Poisson equations, given in terms of linear combinations of iterated means, with an exact remainder quantified by the oscillation of the forcing term. We also prove a regularity result and a strong converse to the mean-value property, in the sense that merely locally integrable functions satisfying our formulas are regular solutions of the higher-order Poisson equation. In contrast with the homogeneous case, where the mean-value property forces smoothness, the regularity attainable here is dictated by the regularity of the forcing term. Together, our results provide a mean-value characterization of solutions to higher-order Poisson equations.

We prove the existence of positive weak solutions for a quasilinear equation with exponential nonlinearity on arbitrary stratified Lie groups for \emph{horizontal $q$-Laplacians,} also called $q$-sub-Laplacians on these groups. The nonlinearity combines a concave term with an exponential growth that can be subcritical, critical, or supercritical with respect to the Trudinger–Moser inequality for Lorentz spaces on stratified Lie groups. We prove a version of this inequality in this paper. Since the problem is not variational in the natural energy space, classical minimisation or critical-point techniques do not apply directly. Positive solutions to the resulting semilinear equation are then obtained via a carefully designed Galerkin method applied to this setting. Our main theorem recovers previous known results on $\mathbb{R}^n$ in the complete range $q\in (1,\infty),$ and the previous known result on the Heisenberg group $\mathbb{H}^n$ for $q=2n+2,$ and is extended here in the full range $q\in (1,\infty)$. Other particular and fundamental cases, including the Engel group, are discussed.

We study the spatially homogeneous Boltzmann equation with continuous mass exchange on $X=(0,\infty)_m\times\mathbb R^d_v$, with a Grad cut-off hard-potential collision kernel. For bounded continuous symmetric mass-exchange rates, every nonnegative initial datum with finite number, mass, and kinetic energy admits a global nonnegative $L^1$-integral weak solution in $W^{1,\infty}(0,\infty;L^1(X))$ with number and mass conserved, kinetic energy dissipated. If $\int_X (m|v|^2)^{1+\delta} f_0(x)dx<\infty,$ for some $\delta>0$, this higher-energy moment propagates on every finite time interval and kinetic energy is conserved through the constructed solution. Moreover, every energy-dissipating solution propagates any such moment. Under the additional $1+\gamma$ moment assumption, the solution is unique among all energy-dissipating $L^1$-integral weak solutions with the same initial datum. We also establish a local theory for a linearly growing mass-exchange rate. With $H_p(f)=\int_X (1+m+m|v|^2)^pfdx,$ every datum with $H_p(f_0)<\infty$, where $p\ge1+\gamma$ admits a conservative local $H_p$-solution. Moreover, an $H_p$-solution continues across every finite time $T$ for which $H_{1+\gamma}(f)\in L^1(0,T).$ This proof requires no detailed-balance or relative-entropy structures. It is based on a new two-stage bootstrap method. The first stage rules out mass concentration at $m=0$, while the second stage combines this control with collision geometry to establish uniform integrability. These estimates provide the compactness needed for the global solution and for the identification of the nonlinear collision form.

In this paper, we construct counterexamples to the Schauder estimate for kinetic Fokker–Planck equations, disproving \cite[Conjecture~1.3]{HW}. The conjectured estimate would control second velocity derivatives using only velocity Hölder regularity of the coefficients and forcing, with no spatial regularity. We also give a zero-forcing variant, where the same {conditions are imposed on} a bounded zeroth-order coefficient. Finally, in dimensions $d\geq 2$, we give endpoint examples showing that bounded velocity-independent forcing may produce bounded distributional stationary solutions whose second velocity derivatives are not locally bounded.

We give a derivation of the Gibbs measure for the focusing nonlinear Schrödinger equation (NLS) on the circle with rough cut-off. This extends earlier work by Sohinger and the second author, which proved analogous results for smooth cut-offs. Our proof is based on the perturbative expansion developed by Fröhlich, Knowles, Schlein, and Sohinger (2017), and provides an alternative proof of the recent derivation given in Lü, Nam, and Zhu (2026). To prove convergence of the explicit terms, we employ a Wigner measure approach and an inductive argument to overcome the lack of smoothness for the cut-off. In particular, we give a derivation of the Gibbs measure for the focusing quintic NLS with the optimal cut-off from Oh, Sosoe, and Tolomeo (2022).

We investigate a Hele-Shaw type free boundary problem in general spatial dimension. We establish a comparison principle and well-posedness for the problem using a notion of viscosity flows. An important feature of our work is that our hypotheses allow for the free boundary velocity to take negative values.

For a domain $\Omega$ that is a deformation of a unit ball in $\mathbb{R}^n$, we establish the existence of a sequence of local minimizers for the vector Allen-Cahn energy having $n+1$ wells. This sequence converges in the $L^1$ topology to a partition of $\Omega$ whose skeleton is given by a simplex cone that contains an $(n+1)$-junction point. This is accomplished by proving that the partition is an isolated local minimizer of a weighted perimeter problem arising as the associated $\Gamma$-limit of the sequence of Allen-Cahn functionals. The results established in this article generalize those in the author's earlier article with Peter Sternberg (MR5033050), which dealt with the case $n=3$. We also weaken the one crucial assumption from the author's earlier article with Peter Sternberg (MR5033050).

In this paper, we show that the modified mean curvature flow starting from an arbitrary graph in a Fuchsian manifold exists for all time and converges smoothly to an equidistant surface of constant mean curvature as $t\to \infty$. This result generalizes earlier work to the modified mean curvature flow setting and removes the restrictive global gradient bound initially required for the standard mean curvature flow by Huang, Zhou, and the second author \cite{HLZ2020}.

We establish an upper bound on the lifespan of bounded weak solutions to the three-dimensional isentropic compressible Euler equations on $[0,T)\times\mathbb R^3$, under a localized positivity condition on the initial data and the assumption that the essential support of the disturbance propagates into an undisturbed background state no faster than predicted by classical local well-posedness. We further show that any entropy-admissible bounded weak solution with bounded inverse density that persists beyond this upper bound must propagate into the undisturbed region at a strictly super-classical speed and that this accelerated propagation is necessarily accompanied by a jump discontinuity in an associated one-sided $L^\infty$-profile of the solution.

This paper investigates the principal spectral theory of a nonlocal dispersal operator with coupled diffusion and aims to establish a variational characterization of the spectral bound for the case where the system is not strongly coupled. In this setting, a key difficulty arises since the principal eigenfunction may have components that are identically zero, rendering existing generalized eigenvalue methods inapplicable. To overcome this, we reorder the components of the operator using a permutation matrix, thereby decomposing it into suitable suboperators, and characterize the spectral bound of the original operator in terms of the spectral bounds of these suboperators. Building on this principal spectral theory, we provide a variational characterization of the basic reproduction ratio for nonlocal dispersal systems and analyze the dynamical behavior of a class of multi-genotype stem cell regeneration models with epigenetic transitions, both in the presence and absence of gene mutations, without assuming the existence of a principal eigenvalue. Furthermore, we investigate the threshold dynamics when the system is not strongly coupled.

We prove that the centro-affine Poincaré inequality holds with constant $n$ for every $C^{\infty}_+$ unconditional convex body and every smooth function with natural orthogonality conditions.

This article concerns sonic-supersonic jet flows issuing from a two-dimensional straight nozzle described by the steady compressible Euler system under the van der Waals equation of state. The flow state is prescribed at the nozzle exit, while the surrounding medium is assumed to be either a vacuum or a static atmosphere with lower pressure. When the flow reaches the sonic state at the nozzle exit, the governing equations become degenerate hyperbolic and the resulting jet flow problem can be formulated as a degenerate free boundary problem. The analysis is further complicated by singular behavior near the endpoints of the nozzle exit. By employing characteristic decompositions and suitable a priori estimates, we establish the global existence of locally Lipschitz continuous sonic-supersonic jet flows expanding into a vacuum. Moreover, in the presence of a static atmosphere with pressure lower than that at the nozzle exit, we prove the local existence of sonic-supersonic jet flows near the nozzle exit.

We study the Hamiltonian structure of the Flaschka-Newell Painlevé II hierarchy via symmetry reduction of the associated space of meromorphic connections. Building on the realization of this hierarchy as a reduction of the Painlevé IV hierarchy via a $\mathbb{Z}_2$-symmetry, we construct a set of Darboux coordinates adapted to the involution. After a suitable change of trivialization, the symmetry acts diagonally in these coordinates, allowing the fixed-point locus to be explicitly described as a symplectic submanifold. This enables us to derive the reduced Hamiltonians after symmetry, thereby obtaining explicit expressions for the Hamiltonians and the Lax matrices of the Flaschka-Newell Painlevé II hierarchy. This strategy also illustrates how symmetry-adapted canonical Darboux coordinates enable explicit reductions of isomonodromic systems at the level of their underlying symplectic geometry.

We develop the boundary regularity theory for solutions to linear kinetic Fokker-Planck equations with Maxwell boundary conditions. These conditions interpolate between diffuse and specular reflection via an accommodation coefficient $\alpha \in [0,1]$. While existing literature is restricted to the extreme cases $\alpha = 0$ and $\alpha = 1$, we resolve the entire intermediate regime $\alpha \in (0,1)$. Specifically, we show that solutions are Hölder continuous if the coefficients are merely uniformly elliptic. Furthermore, for sufficiently smooth coefficients, we establish boundary regularity of order $\frac{3}{\pi} \arccos(\frac{\alpha}{2}) - 1$ up to the grazing set. This exponent is optimal. Beyond Maxwell conditions, we develop a unified approach that extends to a broad class of reflection boundary conditions, including super-elastic collisions.

We study the Boltzmann equation in the whole space under the radial anharmonic confining potentials $\Phi(x)=|x|^p/p$ for $p>2$ and $\Phi(x)=\langle x\rangle^p/p$ for $1<p<2$. We first classify all positive finite-mass-and-energy entropy-invariant, equivalently zero-entropy-production, solutions. For $p>2$, the nonlinear equilibrium manifold is parametrized by mass, temperature, and the three components of angular momentum; for $1<p<2$, integrability excludes rotating equilibria and only mass and energy remain as equilibrium parameters. We then identify the five-dimensional stationary space of the equation linearized about an arbitrary equilibrium and construct an explicit projection determined by the conserved moments. After normalization, the collision term takes the form $C_Me^{-\widetilde\Phi(x)}\mathsf L$, so its microscopic coercivity degenerates at spatial infinity. A far-field weight-transfer estimate compensates for this degeneracy. After subtracting the stationary projection, the corresponding semigroup solution converges algebraically in exponentially weighted $L^2$ spaces. The rate is governed by the growth exponent $p$ and the gap between the two weights, up to an arbitrarily small loss. For $1<p<2$, the mismatch between the two-dimensional nonlinear equilibrium manifold and the five-dimensional linear stationary space yields a conditional obstruction to nonlinear asymptotic attraction for perturbations carrying nonzero angular momentum.

We study one-dimensional half-harmonic maps from the real line into the circle with prescribed exterior data. We show that, for every positive integer $k$, if two disjoint intervals are sufficiently close, there exist at least $k$ distinct non-constant half-harmonic maps with constant exterior data. More generally, we establish a multiplicity result for boundary data with energy below the critical threshold $2\pi$ by introducing a local relative degree and proving corresponding degree-jump estimates. Working on a single interval and for boundary data arising as traces of finite Blaschke products, we investigate the existence and non-existence of energy minimizers in prescribed degree classes.

In this paper we apply the newly developed theory of extended graphons to a certain class of non-exchangeable, non-conservative problems. We establish the mean-field limit by proving the convergence of the associated generalized weighted empirical measures.

We propose a new Galerkin discretization scheme for wave scattering problems that is based on microlocalised basis functions. We show that the proposed method can be made uniformly accurate for large wavenumbers $k$ with a number of degrees of freedom only scaling as $k^{d-1/2}$, while leading to an essentially sparse linear system. In contrast, finite element methods are known to require a number of degrees of freedom scaling at least as $k^d$ to achieve the same property. A similar method based on a Gabor frame was previously introduced by two of the authors, but it was suffering from severe conditioning issues. In the present work, by replacing the Gabor frame by a Wilson basis, we completely alleviate this problem. We rigorously establish error estimates and condition number bounds for the proposed method, and we provide one-dimensional numerical examples illustrating our theoretical findings.