Papers

For continuous viscosity solutions $u$ of fully nonlinear uniformly elliptic equations $F(D^2 u)=0$, the work of Nadirashvili–Tkachev–Vlăduţ shows that $u$ need not be $C^2$ in dimensions $n\ge 5$. It is therefore natural to ask how large the set $\operatorname{Sing}(u)$, consisting of points at which $u$ has no $C^2$ neighborhood, can be. Under an additional $C^1$ assumption on $F$, Armstrong–Silvestre–Smart proved that $\operatorname{Sing}(u)$ has Hausdorff dimension at most $n-\varepsilon$ for some $\varepsilon >0$. In this paper, we show that, in dimensions $n\ge 5$, the Hausdorff dimension of $\operatorname{Sing}(u)$ cannot be bounded away from $n$ under uniform ellipticity alone. In fact, we prove the stronger statement: $\operatorname{Sing}(u)$ can be any compact nowhere dense set in $\mathbb{R}^5$ modulo a countable set. Consequently, in every dimension $n\geq 5$, the Hausdorff dimension of $\operatorname{Sing}(u)$ can be any number in $[n-5, n]$; moreover, $\operatorname{Sing}(u)$ can even have positive Lebesgue measure. In contrast, for every dimension $n$ and every uniformly elliptic operator $F$, we prove that the set $\Sigma_{2,0}(u)$ of points at which $u$ is not twice differentiable has Hausdorff dimension at most $n-\varepsilon$ for some $\varepsilon>0$. In particular, this quantitatively strengthens Trudinger's theorem that $u$ is twice differentiable almost everywhere. More generally, we prove for every $\alpha\in [0,1)$ that the set $\Sigma_{2,\alpha}(u)$ of points at which $u$ has no $C^{2,\alpha}$ expansion has Hausdorff dimension at most $n-\varepsilon(1-\alpha)$.

We prove unique ergodicity of the projective process associated with the two dimensional Navier–Stokes equation in vorticity form, with additive diagonal noise acting on every nonzero real Fourier phase and satisfying two sided power law bounds. Consequently, the exact Furstenberg–Khasminskii formula for the top Lyapunov exponent holds. The main new ingredient is a compact dense mechanism for asymptotic generalized coupling in the absence of a Foiaş–Prodi type high-low mode decomposition for the projective dynamics. Using the dense range of the Malliavin derivative and compactness of the state derivative, we construct finite rank perturbations of the Wiener path that compensate, to first order, for perturbations of the initial condition, leaving a residual whose logarithmic growth has negative stationary mean and hence contracts locally at an exponential rate, with a cost controlled by a triangular scheme of blockwise Ramer transformations.

A well-known conjecture of Chern, do Carmo, and Kobayashi asserts that, for $n,m \geq 1$, the scalar curvature of a closed, minimally immersed $n$-submanifold of $\mathbb{S}^{n+m}$ with second fundamental form of constant length takes values in a discrete set. This property holds in every codimension when $n \in \{1,2\}$. We disprove this conjecture for all $n \geq 3$ with $m \geq 4$, and for even $n \geq 4$ with $m \geq 3$.

We show that the best constant in the weak-type $(1,1)$ bound for the vector Riesz transform on $\mathbb{R}^n$ is at most $2$, independent of the dimension $n$. The proof relies on a new decomposition of the input data involving an obstacle problem for the fractional Laplacian and an associated Lewy-Stampacchia type estimate on an unbounded domain. This settles a problem posed by E. M. Stein at the 1986 International Congress of Mathematicians.

We study the convergence rates in periodic homogenization of general nonconvex, coercive Hamilton–Jacobi equations. We show that the optimal convergence rate is $O(\varepsilon^{1/2})$ in one dimension, $O(\varepsilon^{1/3})$ in two dimensions (up to a logarithmic factor), and $O(\varepsilon^{1/3})$ in dimension three or higher.

In this work, we investigate the inverse problem of determining a semilinear term in a nonlinear parabolic equation from one single boundary flux measurement taken on an arbitrary subset of the boundary. More precisely, we address both uniqueness and stability issues of the inverse problem and establish new Hölder-type stability estimates. The Hölder exponent depends explicitly on the measurement configuration as well as on regularity properties of the semilinear term. The analysis relies on a novel approach based on the derivation of a suitable integral identity involving solutions of the associated adjoint equation. This allows reformulating the inverse problem as an inverse source problem with a sign-changing source term. The main results are obtained by combining fundamental properties of parabolic equations, including maximum principle and appropriate energy estimates. Finally, we complement the theoretical analysis with an iterative reconstruction algorithm inspired by inverse source problems, and illustrate its accuracy on several numerical experiments.

We derive duality-based $\textit{a posteriori}$ error identities for a broad class of subgradient flows induced by time-dependent convex integral functionals. Starting from the Brézis-Ekeland-Nayroles principle, we identify an unsteady primal energy functional and derive its Fenchel dual formulation, including strong duality and the corresponding optimality system under general normal-integrand assumptions. This Fenchel duality framework is used to derive $\textit{a posteriori}$ error identities for subgradient flows. In doing so, we depart from the usual duality-based $\textit{a posteriori}$ error control framework in the unsteady setting, since the Brézis-Ekeland-Nayroles formulation reveals the following unsteady feature: the minimal primal value and the maximal dual value are both prescribed by the initial datum. This allows us to pass from a combined primal-dual gap identity to separate primal and dual gap identities. These identities quantify the primal and dual errors independently and admit representations in terms of generalized Bregman divergences and, under a spatial convex conjugation formula, as non-negative time-space integral quantities suitable for localization. The abstract framework is applied to a number of variational problems of physical interest, including the unsteady heat equation, the unsteady Stokes equations, the unsteady Navier-Lamé equations, the unsteady Bingham flow through a pipe, the unsteady obstacle problem, and the unsteady elasto-plastic torsion problem.

In this paper we study the behavior of the density ratios of harmonic measure at points with vanishing density. Given an arbitrary open set $\Omega\subset\mathbb R^{n+1}$ with harmonic measure $\omega$, we show that at $\omega$-almost every point $x\in\partial\Omega$ where $\liminf_{r\to0}\frac{\omega(B(x,r))}{r^n}=0$, the density ratio $\frac{\omega(B(x,r))}{r^n}$ is not almost monotone with respect to the radius $r$, and therefore exhibits arbitrarily large oscillations at small scales.

We study the Calderón problem for the equation $-\Delta_\infty u+q(x)u=0$ in a bounded convex domain with $C^2$ boundary. We prove that a positive potential $q$ is uniquely determined and explicitly reconstructible from the measurements $\Lambda_q\bigl(t(e\cdot x+b)|_{\partial\Omega}\bigr)$, where $e\in\mathbb S^{n-1}$, the offset $b>\sup_{\overline\Omega}|x|$ is fixed, and $t\to\infty$. The first potential-dependent term in the large amplitude asymptotics determines weighted integrals of $q$ over the chords parallel to $e$. Combining the measurements in the directions $e$ and $-e$ gives the X-ray transform of the zero extension of $q$, and the Fourier slice identity yields reconstruction and uniqueness.

We study stationary periodic solutions of nonlinear Dirac equations with Soler-type nonlinearities. By splitting off a circle factor the problem is reduced to dimension two. The nonlinearity degenerates along a Lorentz null cone, causing difficulties for the variational analysis. Using a coercive perturbation we first obtain minimax perturbed solutions. The key new ingredient is a quantitative separation of the lowest positive eigenspace of the reduced Dirac operator from the null cone; together with a uniform resolvent estimate, it yields uniform bounds that allow us to remove the perturbation. This produces nontrivial periodic solutions for temporal frequencies near the corresponding positive spectral threshold, including frequencies above the mass.

This paper investigates the well-posedness and hidden regularity of boundary-degenerate hyperbolic equations in a two-dimensional setting. The shape design method is employed, which approximates the original degenerate problem by a family of uniformly elliptic problems on truncated subdomains and passes to the limit through uniform estimates. Within this framework, the existence and uniqueness of weak solutions are established in suitable weighted Sobolev spaces. A hidden regularity estimate for the conormal derivative on the nondegenerate portion of the boundary is obtained, providing a uniform bound in terms of the natural weighted energy norms of the initial data and source term. This estimate yields the boundary trace information essential for observability and controllability of degenerate hyperbolic systems.

This paper establishes a fundamental connection between quasilinear potential theory and convex geometric analysis by investigating the interplay between the quasilinear Laplace operator and quermassintegrals. We introduce a quasilinear heat dispersion law for convex conductors and prove that, among all convex conductors of a fixed mean width, the closed ball is a unique maximizer of this dispersion. By characterizing the quasilinear heat loss of a convex conductor explicitly in terms of its quermassintegrals, we demonstrate not only a formal equivalence between the isocapacitary and isoperimetric inequalities in the setting of mathematical physics but also that, among all convex conductors of a fixed mean width, the closed ball is a unique maximizer of this loss. These results provide a novel bridge between the metric properties of convex conductors and the variational analysis of quasilinear elliptic operators, offering a unified perspective on sharp geometric inequalities and their extremal cases.

In this paper, we study the Cauchy problem for the compressible Navier-Stokes system in $\mathbb{R}^3$. Suppose that the viscosity coefficients satisfy $0<\max\{\mu, \nu=\lambda+2\mu\}<1$, and set $\varepsilon=\min\{\mu, \nu=\lambda+2\mu\}$. We establish the global existence of classical solutions when the initial perturbations of the density and the curl-free part of the velocity are smaller than $\varepsilon^{\frac12+}$ (up to a logarithmic loss), while the divergence part of the initial velocity is smaller than $\varepsilon$. This improves the classical global existence result of Matsumura-Nishida \cite{MaN80}, which requires all the initial data to be smaller than $\varepsilon (<1)$. We expect that this result is representative of general Shizuta-Kawashima systems arising in physical applications. The improvement of the index from $1$ to $\frac12+$ relies on exploiting the hidden Kawashima-type dissipation for the density and controlling the spacetime trace norm of the solution at the scale $\sqrt{\varepsilon}$. These two ingredients are obtained through a weighted trace inequality and a Morawetz-type inequality for the perturbed sound speed and the divergence of the velocity.

We study an inverse problem for the heat equation in a medium that generates internal heat at a rate proportional to the heat density. The goal consists of determining the heat-generation coefficient $V$ from the knowledge of the initial-to-final map, which assigns to every initial heat density its final heat density. We state and prove a uniqueness result for the heat operator in a region modelled by $\mathbb{R}^n$ with $n \geq 2$. We assume that $V$ is bounded and decays super-exponentially. Our approach relies on constructing exponentially-growing solutions for the corresponding heat operator. A key contribution of our approach is to provide a weighted $L^2$-estimate, which follows from the moment generating function of a Gaussian random variable. This extends the initial-to-final-state inverse problem, previously studied for the Schrödinger equation, to the parabolic setting.

We study the asymptotic behavior of the initial-boundary value problem for the radially symmetric Navier–Stokes–Korteweg (NSK) equations defined on the exterior domain $\Omega = \{x\in\R^n~|~|x|> 1\}$. In particular, we consider the impermeable wall problem, where the velocity at the boundary $\{x\in\R^n~|~|x|=1\}$ is set to be zero. We show that, if the initial data is a small perturbation of the stationary solution, and the boundary data are sufficiently small, then there exists a global-in-time strong solution to the radially symmetric NSK equations, and it converges to the stationary solution time-asymptotically. Our method is based on elementary energy estimates with a combination of carefully designed energy functionals.

We study an approximate system for the compressible Navier-Stokes-Korteweg equations in a bounded domain periodically perforated by small obstacles. As the size of the obstacles decrease much faster to zero than their mutual distances, we show in spatial dimension two and three that the limiting system remains unchanged. Our result applies for a large class of monotone and non-monotone pressure functions. In particular, it holds in the physically relevant case when the approximate system is used to describe the dynamics of a compressible viscous two-phase fluid extending the corresponding homogenization results known for the compressible Navier-Stokes equations in a single-phase setting.

This paper first investigates the propagation dynamics of solutions to the Cauchy problem for a one-dimensional reaction-diffusion equation in a spatially periodic environment consisting of two distinct patch types. The novelty of this work lies in the systematic analysis of a KPP-bistable heterogeneous framework. In this setting, the respective patch lengths, the linear stability of the zero solution and the positive periodic steady state, and the magnitude of the initial data play crucial roles in the long-time dynamics. We first establish persistence properties of the species, showing that uniform persistence holds when the zero steady state of the associated periodic patch model is unstable, while local persistence is obtained under additional suitable conditions. Using a dynamical systems approach, we further establish spreading properties and demonstrate the existence of pulsating traveling waves in two different cases, depending on whether the trivial solution is unstable or stable. Finally, we present two sets of sufficient conditions characterizing species extinction.

We prove that, for a suitable small external force in $\dot H^{-2}(\mathbb R^2)$, the stationary Navier–Stokes equations on $\mathbb R^2$ admit multiple solutions in $L^2(\mathbb R^2)$.

We establish a concavity inequality for the Hessian quotient operators $\frac{\sigma_k}{\sigma_l}$ in the cases $k-l\in\{1,2\}$, and then derive the corresponding Jacobi inequality. Combining this with the framework developed by Lu and Tsai, we obtain an interior Hessian estimate for convex solutions of $\frac{\sigma_k(D^2u)}{\sigma_l(D^2u)}=f$. As an application, we prove that any entire convex solution in $\mathbb R^n$ with quadratic growth must be a quadratic polynomial.

We establish, at every pole, the asymptotic expansion of the normalized $\Gamma_2$-Green function with a finite pole set in every dimension $n\ge5$. In dimensions $5\le n\le7$ the first correction is a unique nonnegative constant, whereas dimension $8$ exhibits a Weyl-driven $\sqrt{\log}$ term. For $n>8$ we construct the finite local curvature parametrix through the first positive indicial resonance and identify the first genuinely global coefficient by a relative Newton flux.

This paper establishes the global existence of non-negative weak solutions to a two-dimensional, fourth-order nonlinear degenerate parabolic equation modeling surface-tension-driven convection in thin fluid films. First, we construct a regularized approximate problem and prove its solvability via the Galerkin method. Utilizing energy and entropy functionals alongside a singular entropy condition $1/h_0 \in L^1(\Omega)$, we secure uniform a priori bounds for higher-order spatial and time derivatives. These bounds enable the use of the Aubin-Lions lemma and Gagliardo-Nirenberg inequalities to achieve strong compactness and essential $L^6$-integrability. Furthermore, we adopt the Alber-Zhu framework to rigorously define higher-order local weak derivatives and pass to the limit. Finally, we prove the limit function is non-negative, confirming it as a global weak solution to the original problem.

We construct nonzero compactly supported stationary distributional solutions $u\in L^2(\mathbb{R}^3;\mathbb{R}^3)$ of the unforced incompressible Navier–Stokes equations, together with compactly supported pressures in $L^1$, with both norms arbitrarily small. The construction addresses the three-dimensional finite-energy endpoint at which the usual single-scale intermittent Mikado mechanism loses its algebraic gain against the Laplacian. The main new ingredient is a logarithmic Mikado profile, built from a truncated two-dimensional harmonic dipole and spread over logarithmically many transverse scales. The same perturbation yields a localized $h$-principle: nonzero stationary solutions are strongly dense in the localized $L^p$ classes for every $1\leq p<2$ and in $H^{-1}$, and weakly dense at $p=2$, while strong $L^2$ density fails because of an exact isotropic quadratic-moment constraint. We also prescribe arbitrary positive $L^2$ norms, periodize the construction to $\mathbb{T}^3$, and obtain a stationary-versus-Leray nonuniqueness mechanism for the evolutionary equations.

We consider the wave equation in 1$d$ with a spatially-localized cubic nonlinearity. We establish scattering in a weighted space and demonstrate the injectivity of the scattering map, with a corresponding stability estimate for nonlinear recovery.

For integers $n\ge 1$, consider assertions: $\mathbf{(P_n)}$: Let $\Omega \subset S^n$ be a spherical domain enclosed by totally geodesic $S^{n-1}$'s with non-obtuse dihedral angles. Then in $[0,2(n+1)]$, its Neumann spectrum can only take values among $\{0,n,2(n+1)\}$. Moreover, $n$ is a Neumann eigenvalue if and only if the corresponding eigenfunction is the restriction of a linear function in $\mathbb{R}^{n+1}$, while $2(n+2)$ is a Neumann eigenvalue if and only if the corresponding eigenfunction is restriction of a quadratic polynomial in $\mathbb{R}^{n+1}$. $\mathbf{(Q_n)}$: A weak solution $u$ to $\Delta u = f$ with the Neumann boundary condition, with $f$ Hölder continuous, in a conical polyhedral domain $\Omega$ in $\mathbb{R}^n$ with non-obtuse dihedral angles, is in $C^{2,\alpha}_{loc}(\overline{\Omega})$. We prove the implications \[\mathbf{(Q_n)} \Rightarrow \mathbf{(P_n)},\qquad \mathbf{(P_n)}\Rightarrow \mathbf{(Q_{n+1})}.\] Consequently, both assertions hold in all dimensions. These give the optimal Neumann eigenvalue lower bound and $C^2$ elliptic regularity in non-obtuse Riemannian polyhedral domains.

We prove finite-time singularity formation for the forced generalized surface quasi-geostrophic equation in the singular velocity regime $\gamma\in(0,1)$, where $\gamma=0$ corresponds to SQG. For every such $\gamma$, we construct a smooth, compactly supported initial datum and a time-dependent force $F$ for which the corresponding solution $\theta$ is classical on $[0,1)$ with finite energy for all times, but loses Sobolev regularity at $t=1$. More precisely, there exists \[ \kappa_0>2+\gamma+\frac{\gamma^2(1-\gamma)}{25(4+\gamma)}, \] such that the force satisfies $F\in L^1([0,1];H^\kappa(\mathbb{R}^2))$ for every $\kappa\in[2+\gamma,\kappa_0]$, whereas \[ \lim_{T\nearrow1}\int_0^T\|\theta(\cdot,t)\|_{H^\kappa}\,dt=\infty \] for every exponent in the same interval. At the same time, the solution remains uniformly bounded in $H^{\kappa_1}$ throughout its lifespan for any \[\kappa_1\in\left[0,2+\gamma-\frac{\gamma(1-\gamma)}{2(4+\gamma)}\right].\] Then, the singularity occurs within the Sobolev well-posedness regime and cannot be attributed to insufficient regularity of the force or the initial conditions. To the best of our knowledge, this is the first finite-time blow-up result for classical finite-energy solutions of the generalized SQG equations in a well-posedness regime.

Let $\Omega\subset\mathbb R^n$ be a bounded domain, and let $f$ be a nonnegative, nondecreasing function satisfying the Keller-Osserman condition. We study boundary blow-up solutions of $\Delta u=f(u)$ in $\Omega$. Although existence is classical, uniqueness under these assumptions is known in balls but remains open even for smooth convex domains. We identify the normalized gradient $Q_u=|\nabla u|^2/(2F(u))$, $F'=f$, as a quantity governing uniqueness. Under a structural condition on $f$, a boundary blow-up solution $u$ is unique if $\limsup_{x\to\partial\Omega}Q_u(x)\le 1$, without any regularity assumption on $\partial \Omega$. For $C^{1,1}$ domains, assuming a growth condition on $f$, we prove $Q_u(x)\to 1$ for every boundary blow-up solution and hence obtain uniqueness under the structural condition. For convex domains, we prove $Q_u\le 1$ for the minimal boundary blow-up solution and obtain uniqueness when $\sqrt F$ is eventually convex, without imposing any additional boundary regularity.

This article is motivated by a broad conjecture, formulated by the first and last authors in earlier work, asserting that one-dimensional cubic defocusing dispersive flows with small initial data have global, dispersive solutions. The conjecture was first established for a class of semilinear Schrödinger-type models at $L^2$ regularity, the classical cubic NLS among them. In a complementary direction, Harrop-Griffiths, Killip and Vişan have recently shown, using the completely integrable structure, that the cubic NLS is globally well-posed in $H^s$ for every $-\tfrac12 < s < 0$. Our aim here is to extend the reach of the global well-posedness conjecture for one dimensional cubic NLS problems to data which is small in negative Sobolev spaces, and to show that global dispersive bounds persist there. We do so for a broad class of nonlinearities which includes the cubic NLS but which in general generates flows that are not completely integrable. Our method is correspondingly robust, resting on density-flux identities, interaction Morawetz estimates and an implicit normal form transformation rather than on integrability, and it reaches all the way to the scaling-critical threshold, namely $s > -\tfrac12$. As in the earlier work, the global bounds we obtain include both $L^6_{t,x}$ Strichartz estimates and bilinear $L^2_{t,x}$ estimates; these are new even for the classical defocusing cubic NLS at negative Sobolev regularity. There, by scaling, our dispersive bounds also extend to the large data case.

Let $\mathsf{d}\geq1$, $0<\mathsf{s}<\mathsf{d}$, and $N\geq1$. We prove that the optimal fixed-particle-number constant $\Lambda_N(\mathsf{s},\mathsf{d})$ in the Riesz Lieb–Oxford inequality is attained and that these constants are strictly increasing in $N$. The proof combines grand-canonical concentration–compactness with a strict one-particle extension. After recentering, a limiting plan arising from a maximizing sequence may assign positive probability to several particle numbers and hence be grand-canonical. A strict $N$-particle completion excludes this case, while the inequalities $\Lambda_N>\Lambda_k$ for $k<N$ exclude limits with a fixed lower particle number. Once attainment at particle number $N$ is known, the compact-support theorem of Di Marino and Lelotte arXiv:2607.11440, valid for all $0<\mathsf{s}<\mathsf{d}$, permits a non-product one-particle extension and yields $\Lambda_{N+1}(\mathsf{s},\mathsf{d})>\Lambda_N(\mathsf{s},\mathsf{d})$. Together, these implications close an induction beginning at $N=1$.

We consider the sum of interaction energies of an array of dislocation loops placed on a homogenization lattice as the lattice spacing and diameter of the loops tend to zero. Under suitable well-separation and L2 -boundedness assumptions, a formula for the limit of the total interaction energy is derived in terms of the weak limit, the generated H-measure and the generated Wigner measure of the associated Burgers vectors and oriented surface areas.

We prove probabilistic pointwise convergence to the initial datum for the 2D periodic quintic NLS for data in $H^s(\mathbb T^2)$ with $s > 0$. This is an improvement with respect to the deterministic setting, in which convergence is known to fail if $s < 1/3$. The proof is based on a nonlinear maximal characterization for convergence, Bourgain's linear-nonlinear decomposition and the corresponding nonlinear smoothing for which we employ random tensor estimates.