Papers

We establish the Type I estimate for the scalar curvature of the Kahler-Ricci flow on compact Kahler manifolds, for arbitrary smooth initial metrics, whenever the flow develops finite time singularities. This estimate is a building block for the analytic minimal model program with Ricci flow.

For immersions of a closed $2h$-dimensional manifold $\Sigma$ into $\mathbb R^m$, we consider curvature energies that are invariant under ambient isometries and dilations, with the principal term $\int_{\Sigma} \big|\nabla^{(h-1)} \vec{\mathrm{I\!I}}\big|_g^2\,d\text{vol}_g$ and lower-order submanifold invariants. Following the four-dimensional approach developed in joint work with Bernard, Martino, and Rivière, we prove that every weak critical immersion in the natural Sobolev class $W^{h+1,2}$, whose induced metric and its inverse have $L^\infty$ coefficients, is real-analytic in harmonic coordinates. The proof combines geometric conservation laws, structural identities, and elliptic estimates with critical Sobolev coefficients to obtain Morrey decay and bootstrap to full regularity.

We consider the two-dimensional generalized surface quasi-geostrophic equations in $\mathbb{R}^2$, given by \[\partial_t \theta+u\cdot \nabla \theta=0,\quad u=-\nabla^{\perp}\Lambda^{\beta-2}\theta,\,\beta\in [1,2).\] When $\beta=1$, the equation defines the SQG equation and for $\beta>1$, it defines a family of more singular active scalar equations. We prove that if the interval of existence of the smooth solution to the generalized SQG equations for some $\beta_0\in[1,2)$ is $[0,T]$, then with the same initial data, the interval of existence of the generalized SQG equations for $\beta$ close to $\beta_0$ also contains $[0,T]$. To prove these results, we develop estimates for a conservation law with flux modified around the generalized SQG equations. Furthermore, we also prove some new commutator estimates with bounds uniformly bounded as $\beta\to 1$ that may be of independent interest.

We study the stability for a system of coupled second order evolution equations with indirect general memory-damping without the equal-wave-speeds-type hypothesis in a Hilbert space, where the damping only appears just in one equation, the memory kernel can not necessarily be nonnegative and nonincreasing, and the “wave speeds" implied by the system can be different. Taking advantage of new processing ideas and with the help of the properties of the Generalized Positive Definite Kernel, we overcome difficulties caused by the different “wave speeds", the lack of the decreasing and nonnegative property for the memory kernel and the system has only one equation with damping, and obtain an optimal polynomial stability result for the energy, which covers the previous related polynomial stability results for second order coupled equations (abstract or concrete) in the literature. Moreover, applications of the abstract result are given.

We prove the soliton resolution conjecture for the Calogero–Moser derivative nonlinear Schrödinger equation in the scaling-critical space $L^2_+(\mathbb R)$. For a natural class of initial data including a broad spectral Dini—moment class, global flow–Lax admissible solutions decompose into finitely many explicit modulated solitons plus a dispersive radiation, while finite-time blow-up solutions resolve into quantized zero-carrier $R$-bubbles and a strongly convergent endpoint remainder. This work extends soliton resolution to the optimal critical regularity. Previous results had required weighted $H^{1,1}$ initial data, and in the global case the solution was additionally required to remain in $H^{1,1}$ for all time, which amounts to an extra spatial decay assumption. One key ingredient in our argument is a new modular operator-theoretic framework that separates the discrete and continuous spectral channels and identifies the radiation profile through the distorted Fourier transform. A second ingredient is the development of several independent rigidity criteria for the discrete soliton profiles, which avoids inverse scattering and handles embedded eigenvalues. Our framework also yields a unified description of both global and finite-time asymptotics, with exact mass partition and mass-defect quantization.

We study bistable pulsating waves for reaction-diffusion equations with general periodic advection in arbitrary space dimension, allowing the diffusion matrix to be nonsymmetric. Assuming that the homogenized equation admits a traveling wave with nonzero speed in a given direction, we construct moving pulsating waves for all sufficiently small spatial periods $L$ and prove their convergence to the homogenized wave as $L\to0^+$. The existence range and convergence are uniform in the propagation direction when the homogenized speeds never vanish. We also prove uniqueness of the wave speed for arbitrary periods, profile uniqueness for moving waves, and stationary-wave uniqueness in the standing case under continuity of the competing profile. For spatially homogeneous reactions, we derive the expansion $c_L=c_0+Lc_1+O(L^2)$ and an explicit formula for $c_1$. Examples with constant diffusion and zero-mean periodic advection show that $c_1$ can have either sign, demonstrating that the heterogeneity in advection may accelerate or decelerate propagation relative to the homogenized limit and that the general $O(L)$ speed estimate is sharp.

The optimal Sobolev regularity of the roots of a smooth curve of monic complex polynomials is $W^{1,q}$, for $q \in [1,\frac{d}{d-1})$, where $d$ is the degree. This result is stable in the sense that the solution map from $C^d$ coefficients to $W^{1,q}$ roots is continuous. We prove that, after a natural Lipschitz reparameterization, the roots are Lipschitz and the map from $C^d$ coefficients to reparameterization and reparameterized roots is continuous with respect to the $W^{1,q}$ topology on the target spaces, for all $q \in [1,\infty)$. The result is based on a convergence and reparameterization theorem for Almgren's $Q$-valued Sobolev functions.

We study the global dynamics of the generalized Boussinesq equation near standing solitary waves. For even-odd perturbations, Maulen [J. Math. Pures Appl. 177 (2023)] constructed an asymptotically stable center-stable manifold M near the solitary wave Q. We complete this local description by proving that any sufficiently small even-odd perturbation of Q lying outside M either scatters in the energy space as t -> +infinity or blows up in finite time. The proof relies on two main ingredients. First, we classify even-odd solutions with energy below the ground-state energy by identifying two invariant regions, associated with scattering and blow-up, respectively. Second, we establish a one-pass theorem in one dimension: a non-scattering solution cannot re-enter a suitably chosen neighborhood of the solitary wave once it has left that neighborhood. Consequently, the solution remains confined to one of the two invariant regions. Together with the classification above, this yields a dichotomy between scattering and blow-up outside M.

Naff's planarity estimate is a crucial tool for analysing singularities of quadratically pinched mean curvature flows in high codimension, showing that high-curvature regions become asymptotically codimension one. Naff established the estimate throughout the full Andrews–Baker pinching range in dimensions at least $7$, while in dimensions $5$ and $6$ a stronger pinching assumption was required. We close this remaining gap, proving the planarity estimate in dimensions $5$ and $6$ throughout the full Andrews–Baker range. The key new ingredient is a sharp Kato-type inequality relating the twisting of the principal normal to the gradient of the second fundamental form.

In this paper, we investigate a viscous regularization for the BGK model of the Boltzmann equation and its hydrodynamic limit. First, we establish rigorous convergence toward the viscous Euler equations, by exploiting bounds derived from spatial Fisher information. We then analyze the formal derivation of the Euler equations with an entropy inequality for both single-and multi-species case, comparing our results against symmetric hyperbolic thermodynamically compatible theory.

We prove interior curvature estimates for admissible graphical solutions of curvature quotient equations in the range $3\leq k<n$ on the full Gårding cone $\Gamma_k$. No convexity, semiconvexity, or $\Gamma_{k+1}$-admissibility assumption is required. The proof combines a new quantitative compression inequality, a two-surface comparison and doubling argument, and a Pogorelov estimate.

In this paper, we study nonlinear Schrödinger equations arising in the study of Markovian open quantum systems. These equations are derived from the normalization of linear stochastic Schrödinger equations. First, we prove the global well-posedness of the linear equation in the energy space, which is less regular than the usual space for this equation. Then, we show the indistinguishability of solutions to the nonlinear SPDE under a relaxation of the usual monotonicity assumption. Finally, by proving the pathwise uniqueness and normalizing the solutions to the linear equation, we are able to prove the well-posedness of solutions to the required class of nonlinear stochastic equations.

This survey presents heat kernel regularization as a method for quantitative stability of Kantorovich potentials for the quadratic transport cost. We give a complete new heat kernel proof on Heisenberg groups equipped with the Carnot–Carathéodory distance. For source densities bounded above and away from zero on bounded John domains, we obtain $L^2$ stability with rate $W_1^{1/2}$ for arbitrary targets in a fixed compact set, including atomic measures. We also establish a new $L^2$ stability estimate on finite-dimensional RCD spaces, with the optimal dimension-independent exponent $1/2$.

We study the singular sets of real-valued solutions to periodic parabolic equations with rapidly oscillating coefficients. Under a nondegeneracy condition on the corrector matrix, we obtain a local parabolic $n$-dimensional Hausdorff measure bound and a codimension-two Minkowski estimate. The Minkowski estimate holds at every scale, including scales below the oscillation scale $\varepsilon$, with a constant independent of $\varepsilon$. The bounds depend on a coarse-scale ratio of space-time $L^2$ mass to terminal-slice $L^2$ mass. Independently of corrector nondegeneracy, we prove that this ratio controls doubling at all smaller scales uniformly in $\varepsilon$. The singular-set argument combines summable errors in truncated Gaussian approximations, comparison of caloric polynomials at different time centers, and a stopping cover with microscopic singular-set estimates.

This paper studies the Cauchy problem for the stochastic Landau–Lifshitz–Baryakhtar equation on $\mathbb{R}^d$, $d=1,2,3$, subject to Stratonovich Gaussian perturbations. We establish a low-regularity well-posedness theory for initial data in $\mathbb{L}^2$, $\mathbb{H}^1$, and $\mathbb{H}^2$. For $\mathbb{L}^2$ initial data, we prove the existence and pathwise uniqueness of global very weak solutions in dimensions one and two, while in dimension three we construct martingale very weak solutions on every finite time interval. For arbitrary $\mathbb{H}^1$ initial data, we prove the existence and uniqueness of global pathwise weak solutions in all dimensions $d\leq3$. Furthermore, for $\mathbb{H}^2$ initial data, we establish the existence and uniqueness of global pathwise strong solutions in the same range of dimensions. The analysis is based on a frequency-truncation approximation scheme combined with stochastic compactness arguments in the whole-space setting. The key step in global continuation is to justify the stochastic effective-field balance at $\mathbb{H}^1$ regularity, where the drift is available only in a negative Sobolev space. Spatial mollification and passage to the limit yield the energy identity for local weak solutions. A coercive modification then gives global energy moments and excludes finite-time blow-up.

We prove quantitative homogenization estimates for linear elliptic equations in nondivergence form with stationary, symmetric coefficients, finite range of dependence, and a deterministic upper ellipticity bound. No deterministic positive lower bound is imposed. We assume that the reciprocal of the infimum of the smallest eigenvalue on a unit ball has a finite moment of order $d$, together with the common continuity condition of Armstrong and Smart. Under these assumptions, we obtain algebraic probability bounds for finite-cell errors and for Dirichlet homogenization errors with nonzero sources. We construct quadratic correctors on the whole space, modulo affine functions, and prove first- and second-order large-scale regularity and the corresponding Liouville theorems. We also identify the effective matrix through the invariant density and quantify smooth spatial averages of the density and the weighted coefficients. The proof separates a unit-trace diffusion from its physical clock. A reverse Hölder estimate for the Green function of a stopped coarse process yields the integrability gain needed to control the clock at the critical moment. Applications include weighted gradient convergence and a finite-domain approximation of the effective matrix.

In a bulk ferromagnet, magnetic energy is six orders of magnitude weaker than elastic energy. Slenderness removes this disparity. In a thin ferromagnetic rod, the two energies scale differently with aspect ratio, and an ordinary nickel wire deforms nonlinearly even under modest remote fields. We develop a three-dimensional variational theory of a ferromagnetic Kirchhoff rod under gravity and magnetic loading and study it through bifurcation analysis, computation, and experiment. A uniform field can stabilize a vertical rod against buckling under its own weight, the classical Greenhill instability. We give sharp criteria for clamped and pinned supports in terms of an Airy-type principal eigenvalue, and give a fine description of the character of the bifurcation landscape, which has qualitatively different features than its non-magnetic counterpart. When subject to the field generated by a permanent magnet, we identify a rich bifurcation landscape as the magnet is rotated quasistatically along a circle of radius slightly larger than the rest length of the rod. We prove that instability occurs only through a planar mode and that each snap is a nondegenerate saddle-node. We also locate the two folds that produce hysteresis. Experiments on $50\,\mu$m nickel wires confirm the Greenhill threshold, restabilization at ~$30$ mT for both clamped and pinned supports, as well as the salient features of the hysteresis loop such as switching angles, width, and geometry of folds accurately. Our analysis and experiments offer principled design rules for field-stabilized filaments and remotely switched bistable actuators. Our analysis also offers a well-conditioned numerical method is valid near the switching events, where the direct numerical minimization is strongly ill-conditioned.

We establish the sharp asymptotic and symmetry theory for the critical quasilinear Hartree equation \[ -\Delta_pu=\big(|x|^{-\sigma}*u^{p^*_{\sigma}}\big)u^{p^*_{\sigma}-1} \quad\text{in }\mathbb R^N, \qquad p^*_{\sigma}=\frac{p(2N-\sigma)}{2(N-p)}, \] in the complementary range $2p<\sigma<N$, beyond the quasilinear theory previously available for $\sigma\le2p$ [Math. Ann., 391(2):2653–2708, 2025]. This range is structurally different: $p^*_{\sigma}<p$, so the effective potential obtained by factoring out $u^{p-1}$ becomes singular as $u\to0$. Nevertheless, whenever $p^*_{\sigma}>1$, every nonnegative finite-energy weak solution is automatically locally bounded. Our proof provides a quasilinear regularity mechanism adapted to this complementary range, where the potential-theoretic route available for $\sigma\le2p$ breaks down. For positive finite-energy solutions with $p^*_{\sigma}>1$ we moreover uncover a sharp transition governed by the sign of $\sigma(p-2)+2p$. If this quantity is positive, the classical $p$-harmonic profile persists. At the threshold, \[ u(x)\simeq |x|^{-\gamma_h}(\log|x|)^{1/(p-p^*_{\sigma})}, \qquad \gamma_h=\frac{N-p}{p-1}, \] with the matching two-sided gradient rate. Below the threshold, \[ u(x)=A_*|x|^{-\gamma_*}+o(|x|^{-\gamma_*}), \qquad \gamma_*=\frac{2(N-p)(\sigma-p)}{p(\sigma-2p)}<\gamma_h, \] and the rescaled gradients converge uniformly to the corresponding radial profile. In all three regimes the solutions are radially symmetric about some point and strictly decreasing in the radial variable.

Let $N\ge 2$ and $1<p<N$. Let $H$ be a possibly non-reversible Minkowski norm on $\mathbb{R}^N$, with $H\in C^2(\mathbb{R}^N\setminus\{0\})$ and $D^2(H^2/2)$ uniformly positive definite away from the origin. We classify all nonnegative local weak solutions $u\in W^{1,p}_{\rm loc}(\mathbb{R}^N)$ of \[ -\operatorname{div}\big(H(\nabla u)^{p-1}\nabla H(\nabla u)\big)=u^{p^*-1},\qquad p^*=\frac{Np}{N-p}, \] without any finite-energy, integrability, or asymptotic assumption. Every nonzero solution is an anisotropic Sobolev bubble. This settles, for the full range $1<p<N$, the anisotropic analogue of Problem A of Antonini–Ciraolo–Gatti, whose Euclidean case was settled by Zhang, and removes the restriction on $p$ in the non-energy anisotropic classification of Chen–Wu–Yan–Yang. We also prove a scale-invariant Schoen-type Harnack inequality for local solutions. The proof follows the strategy of Zhang for the Euclidean $p$-Laplacian, combining rigidity of bounded blow-up profiles and a first-contact argument with the integral bootstrap of Ciraolo and Gatti. The anisotropic analysis requires two additional ingredients. First, we establish a Bochner identity for the conjugated stress tensor in distributional form on the noncritical set, using only $C^2$ regularity of $H$. Second, we prove a two-term expansion at isolated singularities of Finsler $p$-harmonic functions, including the gradient control needed by the Pohozaev invariant. For non-reversible anisotropies the argument is formulated with the reflected norm. A Morrey-type estimate proved directly for the relevant Leray–Lions operator replaces the potential-theoretic input used in the Euclidean argument.

Nonlinear parabolic partial differential equations frequently occur in the modeling of applied phenomena. They often exhibit complicated dynamics which is organized by their associated equilibrium solutions. As a function of the involved model parameters, the equilibria can be detected using a bifurcation analysis, which in turn is centered around a deeper understanding of bifurcation points. In many applied systems establishing these organizing centers rigorously is difficult to achieve using traditional mathematical techniques. In this paper, we propose a theoretical framework for computer-assisted proofs of symmetry-induced transcritical and pitchfork bifurcations in nonlinear problems. This s accomplished by reformulating the existence question via a zero finding problem for an associated extended nonlinear system. The approach is general and allows for a wide variety of underlying symmetries. We demonstrate how these results can be applied in the setting of parabolic systems and higher-order equations, on both one- and two-dimensional domains.

We establish instantaneous $H^2$ norm explosion for the two-dimensional Boussinesq equations around the linearly stable background profile $-x_2$. While previous ill-posedness results for this system relied on $L^\infty$-type spaces, this work provides the first strong ill-posedness result in the energy-critical space, where stable background stratification acts to suppress deformation growth. We adapt an idea from our previous work on the IPM equation \cite{BCMZ2025} and select an initial density perturbation vanishing near the origin to neutralize the stable stratification. Crucially, unlike the IPM setting, the evolution is here driven by the non-linear coupling between density and vorticity. Since the initial vorticity vanishes identically, the instantaneous loss of regulari\-ty is driven by density-vorticity feedback - a mechanism fundamentally distinct from both $H^1$ Euler ill-posedness of Bourgain & Li and the IPM evolution. Through an iterative scheme yielding a sequence of approximations with strong deformation at the origin, we construct a strong solution $(\rho (t), \omega (t))$ originating from data with arbitrarily small $H^2 \times (H^1\cap \dot H^{-1})$ norm, which immediately exhibits $H^2 \times H^1$ norm explosion for all $t > 0$ while maintaining uniqueness within the energy class.

We prove the existence of classical solutions for nonlinearly elastic, single-director surfaces or shells characterized by finite mid-plane strains and curvature. We employ topological methods, yielding solution continua far (in norm) from an unloaded, stress-free reference configuration, i.e., a global implicit function theorem of sorts. We first establish the existence of a local solution path, which is subsequently shown to be part of a global solution branch, viz., a maximal connected, locally compact set, emanating from the unloaded state.

We formally derive a coupled Enskog system for a gas–particle mixture from the hard-sphere BBGKY hierarchy. The derivation retains the finite separation between colliding centres, includes collisions within and between species and explicitly states the direction of spatial shifts and the mass-dependent prefactors of the four collision operators. The resulting system provides a kinetic starting point for two-phase flows for which the volume effect cannot be ignored. For unit contact factors, we derive the corresponding gas and particle balance laws, and establish an entropy inequality under suitable regularity assumptions.

This work provides a comprehensive characterization of the first dynamic transition of a general two-component reaction-diffusion system on a bounded interval in one spatial dimension. By rigorously addressing the difficulties arising from the non-linear interactions between the two components, this study significantly extends the scope of previous studies on dynamic transitions of reaction-diffusion systems, which focused on single-equation cases. The analysis is conducted by employing center manifold reduction within the standard classification of dynamic transition theory. A key contribution of this study is that the nonlinear terms are treated to involve arbitrary powers of the unknowns $u$ and $v$. Furthermore, the theoretical framework is complemented by the analysis of a representative and simplified system. As a result of this generalization, the dynamics of a broad class of fundamental reaction-diffusion systems can be systematically classified.

This paper presents approximate computational methods for modeling surface plasmon propagation on a graphene sheet with time- and space-dependent material properties. Starting from Maxwell's equations, the evolution of current density under a spatially and temporally modulated Drude weight is formulated as a partial integro-differential equation (PIDE). Assuming small-amplitude perturbations of the Drude weight, a regular perturbation expansion is developed. The solvability and Fourier invertibility of the resulting order-by-order integral equations are established in appropriate Sobolev spaces. A numerical solver combining a Volterra integral equation formulation with the fast Fourier transform (FFT) is introduced, along with pointwise error estimates and computational complexity analysis. Additionally, an analytical transform-based solution using contour integration and Laplace inversion is derived for traveling-wave Drude weight modulations. Numerical simulations demonstrate diverse wave phenomena, including traveling, standing, and exponentially growing plasmonic current densities.

In this article, we analyze plane-symmetric solutions to the relativistic Euler equations on spatially flat spacetimes with decelerated expansion. We consider a linear equation of state $p(\rho)=K\rho$ and power law inflation $a(t)=t^{\alpha}$, where $0<K<\frac{1}{3}$ and $0\leq\alpha<1$. Previous results imply nonlinear stability of the homogeneous and isotropic solution above an expansion threshold $\alpha_\text{crit}=\frac{2}{3(1-K)}$. We prove that at and below this critical expansion rate there exist arbitrarily small data that exhibit finite-time singularity formation of the shock type, i.e., gradient blowup while the solution stays bounded. For this type of data, we establish lifespan estimates that are polynomial below and exponential at the critical threshold. In addition, we prove a theorem based on energy estimates that provides lower bounds on the lifespan of plane-symmetric solutions generated from small initial data and generalizes to solutions without symmetry in $3+1$-dimensions.

We are interested in the molecule reduction algorithm used by Deng, Hani and Ma. This algorithm allows them to establish a rigidity theorem, which plays a central role in the long-time derivation of the Boltzmann equation using the hard-sphere dynamics. In the present article, we show that this algorithm is a graph traversal algorithm of Kruskal type, and we prove that it constructs a Kruskal spanning tree of the input molecule.

We give an explicit solution to the five expert prediction with expert advice partial differential equation (PDE) in the finite-time horizon setting. The solution formula establishes that the adversary's rank strategy $(1,0,1,0,0)$ is globally optimal, and the COMB strategy $(1,0,1,0,1)$ is optimal exactly on the set where $x_1=x_2$ and $x_3=x_4$. The formula is derived from the solution of the geometric-stopping problem given in our companion paper through the transform principle of Bayraktar, Ekren and Zhang, which links the two problems by a Laplace transform. Inverting the transform term by term expresses the solution through a series of Gaussian and complementary error function kernels. The optimality of $(1,0,1,0,0)$ is reduced to the signs of $41$ one-variable Gaussian series, which are certified with computer assistance by Poisson summation, first-mode domination and interval arithmetic on $1616$ rational cells. The proofs of our main theorems, certificates included, are also formalized in the Lean proof assistant.

We prove weak solvability for a two dimensional nonlinear Boussinesq type groundwaterflow problem motivated by the Moche CHAVIMOCHIC aquifer in northern Peru. The model couples state dependent storage and transmissivity with a boundary decomposition into Dirichlet, Neumann, and nonlinear Robin parts dictated by the hydrogeological geometry. In contrast with much of the groundwater Boussinesq literature, which is devoted to exact, similarity, perturbative, semi-analytical, or linearized solutions for special geometries and boundary data, our objective is a variational existence result for a genuinely two-dimensional mixed-boundary problem. Because the physical transmissivity is proportional to the saturated thickness, the equation degenerates at local depletion. We therefore isolate a physically admissible saturated range and construct uniformly positive extensions of the constitutive laws. For the resulting uniformly parabolic problem, existence of a weak solution is obtained through an implicit Euler Rothe scheme, a nonlinear energy adapted to the storage law, uniform estimates for the discrete derivative of the storage potential, discrete compactness, and strong convergence of boundary traces. We then give a precise consistency statement showing when the abstract solution solves the original Moche model and identify the fully depleted regime as a distinct porous medium type problem.

We investigate the recently introduced polygonal hypergeometric functions [arXiv:2502.12127, arXiv:2507.01904], which are conjectured to evaluate multipoint parametric one-loop conformal integrals in arbitrary dimensions. These functions can be systematically constructed using a diagrammatic algorithm that operates in terms of simple planar shapes. Given an $n$-point polygonal power series, we formulate the corresponding GKZ hypergeometric system, which admits an explicit diagrammatic interpretation. In particular, the transianic matrix, which encodes all the relevant diagrammatic data, determines the corresponding toric matrix along with a basis of the lattice of relations, thereby singling out a finite subsystem of $n(n-3)/2$ toric equations. This finite subsystem consists of second- and third-order equations, whereas the full toric family also includes higher-order ones. We illustrate the general procedure with several prominent examples of polygonal functions, including the fourth Appell and the Srivastava–Daoust functions.