Papers

In this article, we develop high-order, entropy-dissipative schemes for a hyperbolic system of conservation laws describing the first-order dynamics of two-layer thin-film flows of immiscible fluids with perfectly soluble solute particles. The key aspect is to construct entropy-conservative fluxes in the sense of Tadmor. Here, we consider and compare two different construction processes: an exact formulation and a commonly used simplified approximation in the flux evaluation. While the approximate approach is simpler at the level of construction and is frequently employed in practice (also in other contexts), it is less structured, leading to increased computational cost in the implementation compared to the exact formulation, which has been much more complicated to derive. We employ these fluxes within both an entropy-dissipative finite-difference framework and an entropy-dissipative discontinuous Galerkin spectral element method. Through numerical experiments using two distinct spatial discretizations, we investigate the performance and robustness of the proposed methods and demonstrate that the choice of numerical flux has a noticeable impact on the efficiency of the resulting solvers.

We prove an interior Lipschitz estimate with a sharp height threshold for continuous viscosity solutions of the three-dimensional special Lagrangian equation with subcritical phase $|\Theta|<\pi/2$. If \[ osc_{B_R}u\leq MR^2,\qquad 0\leq M<\sec\Theta, \] then $Lip_{B_{\rho R}}u\leq CR$ for some $\rho>0$ and $C>0$ depending only on $|\Theta|$ and $M$. The threshold is optimal: at $M=\sec\Theta$, there are entire real-analytic solutions whose Lipschitz seminorms are unbounded on every fixed interior ball. The proof uses a two-point maximum principle and algebraic identities specific to dimension three.

We prove everywhere regularity for Lipschitz solutions to the prescribed vertical mean curvature system in the plane, where the codimension is arbitrary and the mean curvature is in $L^r, \, r > 2$. This strengthens and extends our previous result on the Lawson-Osserman conjecture, https://arxiv.org/abs/2308.04997.

In this paper, we study the following Gierer–Meinhardt system \begin{equation*} \begin{cases} u_t=d_1\Delta u-a_1u+u^p/v^q+\delta_1(x),\quad x\in \Omega, t>0,\\ v_t=d_2\Delta v-a_2v+u^r/v^s+\delta_2(x), \quad x\in \Omega, t>0,\\ \partial_\nu u=\partial_\nu v=0,\qquad\qquad \qquad\qquad\quad x\in \partial \Omega, t>0,\\ u(x,0)=u_0(x),\quad v(x,0)=v_0(x),\quad x\in \Omega, \end{cases} \end{equation*} where $\Omega\subset \mathbb{R}^N$ is a bounded domain, $\nu$ is the outer norm vector with respect to the smooth boundary $\partial \Omega$, $d_1,d_2, p,q,r>0$, $s>-1$, $p-1<r$, $qr=(p-1)(s+1)$, $\delta_1(x)$, $\delta_2(x)$, $u_0(x)$ and $v_0(x)$ are continuous functions. For every pair $d_1,d_2>0$, we respectively establish the global existence, uniform boundedness and asymptotic behavior for Gierer–Meinhardt system with critical exponent $qr=(p-1)(s+1)$ under certain conditions.

We develop a measure-theoretic framework for small-order limits of nonlocal energies with metric-valued maps. For globally $L^p$ maps, we consider interaction measures whose marginals have uniformly bounded $L^\infty$ densities $a_\alpha$ and $b_\alpha$. If these densities converge weakly-* to $a_\infty$ and $b_\infty$, the normalized energies converge to the $L^p$ energy weighted by $a_\infty+b_\infty$ if and only if the interaction measures escape every bounded rectangle. For maps that are only locally $L^p$, we introduce target laws describing the distribution of the values sampled at infinity. Convergence of their $p$-distance profiles yields the limiting interaction energy for every $1\leq p<\infty$ and arbitrary Polish targets, with integrated $p$-Wasserstein convergence as a sufficient criterion. The arguments include the endpoint $p=1$ and require no linear structure on the target space. Applications include directional and anisotropic Maz'ya–Shaposhnikova formulas, a Bernoulli-law interpretation of fractional perimeters, and an Abelian principle for heat-semigroup energies.

We study regularised optimal transport with subquadratic or entropic regularisation. Under $C^{1,\alpha}$-assumptions on the marginals and the domains of their support, we prove global gradient-Lipschitz bounds on the potentials, uniform in the regularisation parameter. Moreover, we show that, up to the boundary, the support of the conditional supports $\text{supp}\,\pi_\varepsilon(\cdot|x)$ behaves like balls of radius $\varepsilon^\frac 4 {d(p-1)+2}$. Our work was carried out concurrently and independently of [9]. We obtain the results of [9,Theorem 1.2] under weaker assumptions and with a different proof.

Let $K\subset\mathbb{R}^n$ be a convex body, let $X$ and $Y$ be independent uniform points of $K$, and set $R_K=(|K|/\omega_n)^{1/n}$. We consider the distribution function \[ D_K(\rho)=\mathbb P\{|X-Y|/R_K<\rho\}, \] and its homogeneous counterpart \begin{equation*} \mathcal J_\rho(K)=\iint_{K\times K}\mathbf{1}_{\{|x-y|<\rho R_K\}}dxdy. \end{equation*} Thus $D_K(\rho)$ is the proportion of ordered pairs of points of $K$ whose distance is less than $\rho R_K$, or equivalently the radial distribution function of the normalized covariogram of $K$. The functional satisfies a Brunn–Minkowski inequality, with equality precisely for homothetic bodies, and its differential gives a first Minkowski inequality. We characterize the Borel measures that occur as Wulff differentials: they are exactly the nonzero finite positive measures with zero centroid which are not concentrated on a great subsphere, and the corresponding body is unique up to translation. We further prove fixed-volume and stationary rigidity of the ball in the subcritical range, together with regularity for the resulting equation. At saturation, the equation is the classical Minkowski equation.

In this paper, we consider the existence of small- and large-amplitude compressible water waves. These are solutions of the two-dimensional free boundary isentropic Euler equations subjected to a constant gravitational force and neglecting the effects of surface tension. We study periodic and solitary waves beneath vacuum as well as front-type solutions of the two-phase problem in a channel. Using local and global bifurcation techniques, we construct large families of these waves beginning at an incompressible solution and treating the Mach number (defined in a suitable sense) as the bifurcation parameter. This gives a rigorous version of the classical Janzen–Rayleigh expansion, but also furnishes solution curves that may limit to transsonic flow.

We study the focusing one-dimensional cubic nonlinear Schrödinger / Gross–Pitaevskii equation (NLS-GP), where the potential of the underlying linear Schrödinger operator, $H_V=-\partial_x^2+V(x)$, is compactly supported. In \cite{turner2026resonance} the authors proved that purely imaginary zeros (transmission resonances) or purely imaginary poles in the lower half plane (scattering resonances) of a reflection coefficient of $H_V$ seed branches of resonance-induced nonlinear bound states, and that these states bifurcate at a strictly positive $L^2(\mathbb R)$ excitation threshold. This is in contrast to nonlinear bound states which bifurcate at zero $L^2$-norm from point spectra of $H_V$ (corresponding to poles in the upper half plane). In this paper we establish precise criteria for the nonlinear orbital stability and instability of the resonance-induced states near the bifurcation point. A corollary is that resonance-induced nonlinear states are orbitally stable if i) the underlying linear (scattering or transmission) resonance frequency is sufficiently small, and ii) the corresponding linear resonance eigenmode is strictly positive. Numerical simulations are presented to explore regimes not accessible to our theory, and to explore the large time dynamics for initial conditions near stable and unstable resonance-induced nonlinear bound states.

We study a one-dimensional $2x2$ system of conservation laws with discontinuous flux. We prove the global existence of weak solutions for initial data with sufficiently small total variation. We construct explicitly the associated Riemann solver and analyze its main properties. Finally, we introduce a Glimm-type numerical scheme tailored to the exact Riemann solver and use it to reproduce the wave patterns obtained analytically.

We develop a Quantum Optimal Transport (QOT) barycenter framework for quantum states, as an analog of Wasserstein barycenters~\cite{AguCar}, and establish existence and duality results for a broad class of possibly unbounded transport costs on separable Hilbert spaces. Our framework provides, in particular, a unified treatment of $2$-quantum Wasserstein (QW) barycenters in both the quantum-state and quantum-channel formulations by specializing to the canonical quadratic cost operators associated with the $2$-quantum Wasserstein distances of Caglioti–Golse–Mouhot–Paul~\cite{caglioti2021towards,Golse-Mouhot-Paul:2016} and De Palma–Trevisan~\cite{DPaTre19}. The central results concern Gaussian input states and, in particular, Gaussian rigidity: whether Gaussian input states force the 2-QW barycenter itself to be Gaussian and uniquely determined when the minimization is taken over all quantum states. We first show that the barycenter problem admits a Gaussian minimizer and reduces to a finite-dimensional convex optimization problem over covariance matrices. The main difficulty is that uniqueness of the optimal covariance does not, in general, imply uniqueness of the underlying quantum state. We bridge this gap through a state-reconstruction principle under covariance complementary slackness that upgrades uniqueness of the optimal covariance to uniqueness of the full quantum state, and thereby prove a global rigidity theorem: if at least one Gaussian input is faithful, then the barycenter is unique among all quantum states and is necessarily Gaussian. Faithfulness is sufficient but not necessary: families of pure inputs still determine a unique barycenter, whereas partially pure nonfaithful Gaussian inputs may admit multiple barycenters.

We prove non-uniqueness for the advection-diffusion equation with a drift in a logarithmically near-critical class below $L_t^2L_x^\infty$. The first construction is a diffusion-assisted inverse cascade of scalar energy that diverges as $t \to 0$. The second uses compressed inverse mixing, with diffusion acting perturbatively, and produces a bounded parabolic solution with the energy jump at the initial time. In both constructions the drift and scalar are smooth for positive time. We also consider the drift in $L^{2,\infty}_tL^\infty_x$ and solutions satisfying the energy inequality starting from almost every time. In this setting, we prove uniqueness of solutions with bounded energy, and construct non-unique solution with energy blowing up as time goes to zero.

We investigate how root allometry and below-ground root competition due to root overlap affect the conditions for pattern formation in non-local scalar vegetation models. We start from a spatially explicit two-component vegetation-water system in which plants acquire soil water through their laterally extended root systems, with root allometry represented by an interaction kernel whose spatial extent increases with the above-ground biomass density. In the limit of fast water dynamics, we eliminate the water variable, resulting in a non-local scalar model for the vegetation dynamics. This reduction reveals that accounting for root overlap naturally leads to a double integral structure in the scalar model, in contrast to the single convolution term commonly used to model non-local competition for resources. For the resulting scalar model, we identify two distinct mechanisms that can drive Turing instabilities. The first instability mechanism requires root growth to outpace mortality, while the second mechanism relies on the development of competitive pressure zones, in which vegetation growth is impeded by the cumulative competitive pressure exerted by neighbouring areas with higher biomass density. Root allometry promotes the occurrence of both instability mechanisms, while root overlap inhibits the emergence of competitive pressure zones. Hence, our results show that both root allometry and root overlap are important features to consider when studying vegetation pattern formation.

For weak solutions to quasilinear degenerate parabolic equations of $p$-Laplace type, a central obstacle in applying the method of intrinsic scaling to prove their Hölder regularity is the derivation of forward-in-time propagation estimate for the spatial measure of level sets. In this paper, we revisit and overcome this difficulty for an inhomogeneous degenerate parabolic $p$-Laplace equation with bounded nonnegative time-independent forcing term and initial data. Our new proof is built upon the temporal Lipschitz estimate that we establish on any time interval separated from the initial time by a positive amount. The strong regularity gain furnished by this estimate allows us to derive the forward propagation of level-set measures directly, thereby bypassing the energy inequality that served as the starting point of the previous approach. Furthermore, we establish space-time $W^{1,\infty}_{x,t}$ estimates away from the initial time and construct counterexamples to demonstrate the sharpness of the positive waiting-time condition under merely bounded initial data.

In a recent paper [Kwaśnicki, Poisson kernels on the half-plane are bell-shaped, Bull. London Math. Soc. 58 (2026), no. e70303], the author proved that Poisson kernels for a class of elliptic operators in the half-plane are bell-shaped. By allowing one-sided barriers, we prove a converse result.

We prove nonlinear stability of the Guderley imploding shock under radial perturbations for the three-dimensional compressible Euler equations in the adiabatic range $1<\gamma\leq\frac{5}{3}$. The perturbed solutions develop an asymptotically self-similar implosion in finite time. Our result allows perturbations with small, nonzero constant pressure in the quiescent region ahead of the shock. The proof combines weighted $L^\infty$ estimates along characteristics with a computer-assisted proof of stability inequalities for the Guderley profile.

We construct solutions to the two-dimensional stationary Prandtl system under a constant adverse pressure gradient exhibiting a family of distinct separation laws. More precisely, for every integer $\ell\ge1$, we construct smooth initial data for which separation occurs at a finite location $x=x^\ast$, with \[ \partial_y u(x,0) =\mathbf C_\ell(x^\ast-x)^{\ell/2}\bigl(1+o(1)\bigr), \qquad x\uparrow x^\ast, \] where $\mathbf C_\ell>0$ depends on the initial data. We determine the local asymptotic structure of the solution near separation through matched inner and outer expansions based on linearization around the ground state in von Mises variables. This description addresses Open Problem~5 posed by Oleinik and Samokhin~\cite{OleinikSamok-book-99}. We also prove that the square-root separation regime, corresponding to $\ell=1$, is dynamically stable under small admissible perturbations of the initial data in a suitable topology.

This paper is concerned with establishing estimates on averaged Green's functions for a uniformly elliptic divergence form partial difference equation with random coefficients on the $d$ dimensional integer lattice $\mathbb{Z}^d$. It has previously been shown that the averaged Green's function is point-wise well approximated by the homogenized Green's function at large length scales. Corresponding results also hold for (fractional) derivatives of the averaged Green's function up to but not including the second derivative. Second derivative results have been established when the elliptic equation generates a positive semi-group, as in the diagonal case. Here the second derivative result is shown more generally by using an insight of Bourgain.

The honeycomb and Kelvin problems ask how space can be partitioned into cells of equal volume with the least surface area. We resolve this question in four dimensions for convex bodies that tile space by translations, showing that the regular $24$-cell uniquely minimizes surface area at fixed volume. We develop a framework combining an exact deficit identity, valid in every dimension, with a descent argument. As a consequence, the regular $24$-cell also minimizes surface area among such tiles containing a fixed ball.

Let $B_3=B(0,3)\subset\mathbb{R}^3$. For every sufficiently large integer $k$, we construct a nonzero real function $u_k\in C^2(\overline{B_3})$ and a real potential $V_k\in L^\infty(B_3)$ such that $\Delta u_k=V_ku_k$, $u_k|_{\partial B_3}=0$, $\mathrm{ord}_0u_k=k$, and $\|V_k\|_\infty\le Ck^{3/2}$. Consequently, for every sufficiently large $N$, there is such a Dirichlet solution with potential norm smaller than $N$ and vanishing order at least $cN^{2/3}$. The construction extends directly to higher dimensions and to spheres. Together with previous results, this example indicates that $C(1+\|V\|_\infty^{2/3})$ is the sharp bound for the vanishing order in the real-valued case. It also indicates that the bound conjectured independently by Kukavica \cite{Kukavica1998} and Kenig \cite{Kenig2006} is not attainable in general.

We study finite-time blow-up and global existence for semilinear heat equations with time-dependent reaction coefficients on metric measure spaces. For convex nonlinearities, we establish a blow-up criterion that retains the dependence on the initial datum through its linear evolution and does not require stochastic completeness. Under additional conservativity and heat kernel assumptions, we also prove blow-up at the critical Fujita exponent associated with polynomial volume growth. Our main global existence result is based on a heat-kernel-weighted functional space in which the mild solution map becomes a contraction. This yields a unified construction covering both polynomial and exponential heat kernel decay, general locally Lipschitz nonlinearities, and time-dependent reaction coefficients, without convexity or monotonicity assumptions. In particular, the global existence criterion only uses the behaviour of the nonlinearity on the range explored by the solution. Applications include Riemannian manifolds, metric graphs, and, within a separate discrete framework, weighted graphs. The arguments, especially the functional construction underlying global existence, are new and lead to new results even for the classical underlying spaces considered above.

We establish a class of sharp higher-order inequalities related to the uncertainty principle. First, by means of the expanding the squares method, we give a concise proof of sharp higher-order Heisenberg uncertainty principles. We provide explicit expressions for the optimal constants and establish the existence of extremal functions. Furthermore, we obtain sharp higher-order Heisenberg uncertainty principles for curl-free vector fields. In particular, when $N=2$, we give an answer to the higher-order version of Open Problem 9 raised by Maz'ya in [Integr. Equ. Oper. Theory (2018) 90:25]. Second, we establish sharp higher-order Heisenberg uncertainty principles that do not involve higher-order tensors. We also prove sharp higher-order Heisenberg uncertainty principles and higher-order hydrogen uncertainty principles for radially symmetric functions.

We establish a global uniqueness result at the critical Sobolev regularity for the isotropic Calderón problem in all dimensions $n\ge 4$. More precisely, for every $n\ge4$ and every bounded Lipschitz domain $\Omega\subset\mathbb{R}^n$, any real-valued, uniformly elliptic scalar conductivity $\gamma\in W^{1,n}(\Omega)$ is uniquely determined by its full Dirichlet-to-Neumann map. This settles a long-standing open problem in the field. To prove uniqueness at this endpoint, we develop an averaged trace norm method, which combines complex geometrical optics with trace norm estimates averaged over wave orientations.

We study an incompressible visco-morphoelastic system in three dimensions in which the Eulerian effective strain is transported by the Zaremba-Jaumann rate and coupled to a Navier-Stokes momentum balance through a Kelvin-Voigt stress. The strain remains symmetric and splits into a scalar transport-relaxation equation for its trace and a deviatoric equation containing the nonlinear stretching and Jaumann terms. We regularize both equations by diffusion of order $\varepsilon$, which admits a mechanical interpretation as weakly nonlocal remodeling with a characteristic length scale. For fixed $\varepsilon>0$ we construct Leray-Hopf type weak solutions by a truncated Galerkin scheme and remove the truncation a posteriori using a maximum principle for the scalar variable. As $\varepsilon\to0$, the scalar variable converges strongly in $L^p((0,T)\times\Omega)$ for every $1\le p<\infty$, by a DiPerna-Lions commutator argument on bounded Lipschitz domains, allowing the stretching term to be identified in the limit. In contrast, the Jaumann commutator is a product of two weakly convergent sequences, and its limit cannot be identified from the available estimates. We therefore obtain a defect \[\boldsymbol{\mathcal{D}}\in L^2(0,T;(H^2(\Omega;\mathbb{S}_0))^\ast)\] in the limiting deviatoric equation. We give two sufficient conditions for $\boldsymbol{\mathcal{D}}=0$. The energy-level cancellation used in related corotational viscoelastic models does not close here because the coefficient of the stretching term is itself an unknown.

We consider the semiclassical magnetic Schrödinger operator in dimensions one and two. The first result is that an even one dimensional semiclassical potential is spectrally determined by the Schrödinger operator. The second result states that a two dimensional radial magnetic field is spectrally determined by a purely magnetic Schrödinger operator. This extends the earlier work of Guillemin-Wang and answers a conjecture of theirs therein.

We study the critical points $U_\varepsilon$ of the two-dimensional Ginzburg–Landau energy on domains with corners $\Omega$ and imposed Dirichlet boundary condition which vanishes precisely at the vertices $V$. We isolate the forced scalar boundary layer $\rho_\varepsilon$ by the weighted Lassoued–Mironescu splitting. Assuming the reduced logarithmic energy bound and vertex logarithmic tightness, we prove that after passing to a subsequence, the normalized maps $v_\varepsilon = U_\varepsilon/\rho_\varepsilon$ converges in $C^k_{\mathrm{loc}}(\Omega\setminus A)$ $\forall k\ge0$ away from a finite set of interior vortices $A$. The limiting map is a canonical $S^1$-valued harmonic map $u_*$ containing integer interior vortex factors and fixed fractional corner factors. The renormalized energy is defined by removing discs around the interior vortices and sectors around the vertices and subtracting the corresponding logarithmic divergences. The interior vortex locations of the limiting map are critical points of this renormalized energy.

We construct solutions to the Allen-Cahn equation $\Delta u+u-u^3=0$ in $\mathbb{R}^{N+1}$, with $N\ge 3$, whose zero level set is asymptotic (at infinity) to a given minimal cone. Our construction is quite general, since we only use a suitable nondegeneracy assumption; neither stability nor symmetry of the underlying cone is required. Moreover, we calculate the morse index of such solutions. In particular, our main result generalizes a result by Pacard and Wei (2013) dealing with stable solutions to the Allen-Cahn equation in high dimensions. More precisely, we give a positive answer to a remark raised in the paper by Pacard and Wei about the possibility to relax the assumption for the underlying cone is area-minimizing or at least stable. As a corollary, we also obtain solutions with infinite morse index and connected zero level set.

We study an inverse boundary value problem for the Helmholtz equation on a smooth compact non-trapping Riemannian manifold with strictly convex boundary. We prove that, given two such metrics, for sufficiently large but fixed frequency $\lambda$, equality of their Dirichlet-to-Neumann maps implies equality of their lens data, up to a smooth boundary-fixing diffeomorphism. This establishes a high-frequency bridge between the Calderón problem and lens rigidity. We also prove a quantitative version of this result uniformly valid in a suitable bounded set of Riemannian metrics. Crucially, we show that Gaussian beam type solutions concentrating on maximal geodesics, split near the boundary into outgoing/incoming parts, such that the phase of the outgoing part contains information on exit point, direction, and travel time. The recovery of lens data then proceeds by a careful stationary phase analysis and a boundary integral identity.

We analyze a coupled PDE-ODE model for metastatic cancer invasion involving epithelial-like and mesenchymal-like cancer cells, cancer-associated fibroblasts (CAFs), the transforming growth factor $\beta$, matrix-degrading metalloproteinases, and extracellular matrix reconstruction by CAFs. The model incorporates bidirectional phenotypic transitions between epithelial-like and mesenchymal-like cancer cells, cancer-cell haptotaxis, negative fibroblast haptotaxis, as well as cell-specific proliferation. Under explicit parameter restrictions, we establish global classical solvability for the full model in two dimensions using an energy-based approach, and for a three-dimensional variant with constant phenotypic transition rates using coupled $L^p$-estimates. Numerical simulations illustrate primary tumor invasion and metastatic spread to secondary sites.

In this paper, we study boundary-driven behavior of the BGK model and the Boltzmann equation in a half-space with vacuum initial data. For any $1\leq\beta\leq2$ and $\alpha>0$, we construct a unique local-in-time BGK solution whose inflow boundary data have an arbitrarily small norm with the tangential exponential weight $e^{\alpha|v_h|^\beta}$, but whose norm with weight $e^{\alpha'|v_h|^\beta}$ is infinite for every $0<\alpha'\leq\alpha$ and every positive time of existence, where $v=(v_h,v_z)\in\mathbb{R}^n$ and $v_h\in\mathbb{R}^{n-1}$ denotes the tangential velocity. Thus, this BGK solution instantaneously leaves the exponentially weighted spaces associated with the inflow data, even when the exponential weight is weakened. The construction relies on sharp estimates for the density, bulk velocity, and temperature on a singular scale together with a fixed-point argument in a time-dependent anisotropic space. For comparison, we establish local existence and uniqueness for the cutoff Boltzmann equation under isotropic exponential weights $e^{\alpha|v|^\beta}$. In the Appendix, we further clarify the role of these different weights by showing that the macroscopic stability required for the BGK construction can fail even for Maxwellian inflow data and that it can be recovered by suitably weakening the normal velocity decay of the inflow data. These observations highlight a structural difference between local Maxwellian relaxation and the Boltzmann collision operator in the presence of boundary inflow and vacuum initial data. This work extends the whole-space analysis of D. Lee, S. Park, and S.-B. Yun \cite{LPY2024} to the half-space setting.