Papers

In the Drosophila melanogaster egg chamber, the collective migration of border cells toward the oocyte is guided by spatial gradients of chemoattractants. While cellular responses to these cues are well characterized, the spatial distribution of chemoattractant within the tissue remains difficult to measure experimentally due to imaging limitations and extracellular complexity. In this study, we develop a spatially resolved mathematical framework to model local chemoattractant concentrations during border cell migration. We use a phase-field approach to represent the egg chamber geometry and define a diffusion-reaction system with spatially heterogeneous diffusivity that accounts for confinement by cellular domains. This framework allows chemoattractant diffusion to be restricted to extracellular space while remaining excluded from the interiors of nurse cells, the border cell cluster, and the oocyte, similar to what we observe in vivo. We simulate secretion from the oocyte and degradation throughout the domain, showing how geometry shapes the distribution of signaling molecules. We further couple this chemical field to a mechanical model of cluster migration that includes a tangential interface migration (TIM) force, allowing the cluster to respond to both chemoattractant gradients and cell-cell contact. Our results show that signal localization and tissue geometry jointly influence directional persistence and the speed of migration. Notably, geometric bottlenecks and intersections can flatten local gradients and slow migration, consistent with experimental observations. This modeling framework offers a tool to investigate how biophysical constraints shape signaling environments and guide collective cell movement in vivo.

This paper examines the impulse controllability of degenerate singular parabolic equations through a modern framework focused on finite-time stabilization. Furthermore, we provide an explicit estimate for the exponential decay of the solution. The proof of our main result combines a logarithmic convexity estimate with specific spectral properties. Finally, we establish the existence and uniqueness of the minimal norm impulse control associated with the system.

Regularity theorems à la Avellaneda-Lin are an indispensable part of the modern quantitative theory of stochastic homogenization. While interior regularity results for random elliptic operators have been available for a while, on general smooth domains the existing theory has until recently remained limited to Lipschitz estimates. We establish $C^{1,\alpha}$ regularity results for random elliptic operators on bounded sufficiently smooth domains, as well as for scalar problems on convex polytopes. We, furthermore, prove a number of auxiliary results typically employed in the derivation of fluctuation bounds, such as a weighted Meyers estimate.

We develop reliable a posteriori error estimators for fully discrete Runge-Kutta discontinuous Galerkin approximations of nonlinear convection-diffusion systems endowed with a convex entropy in multiple spatial dimensions on the flat torus T^d, with a focus on the convection-dominated regime. In order to use the relative entropy method, we reconstruct the numerical solution via tensor-product Smoothness-Increasing Accuracy-Conserving (SIAC) filtering which has superconvergence properties. We then derive reliable a posteriori error estimators for the difference between the entropy weak solution and the reconstruction, with constants that are uniform in the vanishing viscosity limit. Our numerical experiments show that the a posteriori error bounds converge with the same order as the error of the reconstructed numerical solution.

In this paper we study Moser-Trudinger type inequalities for some nonlocal energy functionals in presence of a logarithmic convolution potential, when the domain is a ball of $\mathbb{R}^N$ with $N \geq 2$. In particular, we perform a blow-up analysis to prove existence of extremal functions in the borderline case of critical growth. Using this, we extend the results in \cite{CiWeYu} to higher dimension and sharpen \cite{CC}.

We study scattering resonances of finite one-dimensional systems of high-contrast resonators beyond the subwavelength regime. Introducing a novel tridiagonal frequency-dependent capacitance matrix, we derive quantitative asymptotic expansions of the hybridized Fabry-Pérot resonant frequencies in terms of the material contrast parameter. The leading-order shifts are governed by the eigenvalues of this matrix, while the corresponding eigenmodes are approximated, to leading order, by trigonometric functions on selected spacings between resonators. Our results extend the use of discrete approximations as a powerful tool for characterizing the resonant properties of a system of high-contrast resonators at arbitrarily high frequencies.

We derive stochastically-constrained Koiter shell models in line with the SALT (Stochastic Advection by Lie Transport) approach introduced by Holm [Proc. A. 471 (2015)]. First, we deduce the stochastic partial differential equations for the generalised nonlinear elastic and linear elastic Koiter shell models with abstract functional derivatives of their corresponding membrane and flexural energies. We then present a prototype for a stochastically-constrained (simplified) linearised Koiter shell model that captures stiffness effects arising from shell curvature, bending and membrane stresses, interior and surface forces, and, more generally, stochastic buckling. Finally, we show that if a weak pathwise solution of this prototype is parametrised by a suitably chosen family of noise coefficients, we obtain in the parameter limit, the deterministic viscoelastic shell model with viscous damping.

An old and well-known open problem in the critical point theory asks whether, for some $p \neq 2$ and some bounded domain $\Omega$, there exists a critical value of the $p$-Dirichlet energy $\|\nabla u\|_p^p$ over an $L^p(\Omega)$-sphere in $W_0^{1,p}(\Omega)$ lying outside of a Ljusternik–Schnirelman type sequence of critical values, the latter will be called LS eigenvalues of the $p$-Laplacian. In this work, we provide a positive answer by showing the existence of a non-LS eigenvalue when $p>2$ is sufficiently close to $2$ and $\Omega$ is just a planar rectangle close to the square. The arguments pursue the observation that a simple eigenvalue of the Laplacian can be a meeting point for several branches of eigenvalues of the $p$-Laplacian as $p$ varies. Since LS eigenvalues are continuous with respect to $p$ and exhaust the whole spectrum when $p=2$, we deduce that at least one of the branches must contain non-LS eigenvalues.

We study the emergence of periodic oscillations through a Hopf bifurcation in a scalar diffusion equation on the half line coupled to a dynamic boundary condition. Our results quantify the effect of delay through the buffering in the diffusive field on boundary kinetics, drawing a parallel to the emergence of oscillations in delay equations. Technically, the Hopf bifurcation occurs in the presence of essential spectrum induced by the diffusive field, preventing a simple approach via center-manifold reduction. The results are motivated by observations in biological systems where dynamic boundary conditions arise when modeling surface dynamics coupled to bulk diffusion.

We prove that the Almgren-Pitts and phase-transition half-volume spectra of a closed Riemannian manifold are equal. This confirms a conjecture of Liam Mazurowski and Xin Zhou.

We prove that for any Osgood non-Lipschitz modulus of continuity $\omega$, flow maps associated with time-periodic $\omega$-continuous velocity fields generically (in the sense of Baire) have infinite topological entropy.

We establish the linear instability of the semiclassical Einstein-Klein-Gordon system linearised about the Minkowski vacuum spacetime. The proof relies on formulating a forcing problem for both metric and state perturbations within the space of past-compact sections. This geometric framework admits a unique tensor decomposition which, in conjunction with the quantum Møller operator, enables the decoupling of the linearised system into two distinct Cauchy problems. Consequently, the metric perturbations are shown to be governed by a higher-order, nonlocal hyperbolic partial differential equation. By relegating the nonlocal contributions to subleading order, we establish the well-posedness of this forcing problem. Furthermore, we provide a rigorous asymptotic analysis for physically admissible choices of the renormalisation constants. We prove that the system exhibits a late-time linear instability: the metric perturbations grow exponentially, bounded strictly by a universal scale H, thereby indicating a quantum backreaction-driven transition toward a de Sitter cosmological spacetime. Provided the parameters governing the system are restricted to a physically relevant regime, this universal scale is compatible with the measured expansion of our universe.

We introduce a new $C^1$ algorithm for the rigorous integration of dissipative partial differential equations. The algorithm is designed for computer-assisted proofs that require rigorous control of both solutions and their derivatives with respect to initial data. As applications, we establish the existence of locally attracting periodic orbits for initial and boundary value problems for two non-autonomous dissipative PDEs: the Chafee-Infante equation and the Burgers equation with a fractional Laplacian.

In this paper, we study the optimization of the first Laplacian eigenvalue on axisymmetric doubly connected domains under positive Robin boundary conditions. Under additional geometric constraints, we prove that spherical shells maximize this eigenvalue. Our approach combines known isoperimetric inequalities for mixed Laplacian eigenvalues with a higher-dimensional extension of the effectless cut technique introduced by Hersch to study multiply connected membranes of given area fixed along their boundaries.

We investigate a fluid-structure interaction system in which the dynamics of the fluid is described by the compressible Navier-Stokes equations, while the elastic structure is modeled by a damped plate equation. The fluid evolves in a three-dimensional bounded domain, with the structure occupies a part of its boundary. Instead of standard no-slip boundary conditions, we consider the Navier-slip boundary conditions at the fluid-structure interface as well as at the fixed boundary. We establish the local-in-time existence and uniqueness of strong solutions within $L^{p}-L^{q}$ framework. The existence result is obtained for small time by decoupling the linearized system and employing a cascade strategy combined with the Tikhonov fixed point theorem, whereas the uniqueness is shown by deriving weak regularity properties for the associated linear coupled operator in a Hilbert space setting. It is the first result addressing strong solutions for a compressible fluid interacting with a damped plate under Navier-slip boundary conditions.

We investigate the relation between several generalized solution concepts for nonlinear PDE systems from fluid dynamics. More precisely, we study measure-valued solutions, dissipative weak solutions, and energy-variational solutions. For the incompressible Euler equations, we prove the equivalence of all three concepts, provided that the energy inequality is formulated in the appropriate way. For several important examples of conservation laws arising in fluid dynamics, we establish the equivalence between energy-variational and suitably refined dissipative weak solutions, where the defect measures are controlled sharply by the energy defect. These examples comprise the compressible isentropic Euler system, the Euler–Korteweg system, and the Euler–Poisson system.

In recent years, the use of entropy-regularized optimal transport with $L^p$-type entropies has become increasingly popular. In this setting, the solutions are sparse, in the sense that the support of the regularized optimal coupling, $\mathrm{supp}(\pi_\varepsilon)$, shrinks to the support of the original optimal transport problem as $\varepsilon \to 0$. The main open question concerns the rate of this convergence. In this paper, we obtain sharp local results away from the boundary. We prove that the supports $\mathrm{supp}(\pi_\varepsilon(\cdot \mid x))$ of the conditional measures, $\pi_\varepsilon(\cdot \mid x)$, behave like balls of radius $\varepsilon^\frac 1 {d(p-1)+2}$. This allows us to show that the regularized potentials are uniformly strongly convex and to derive the rate of convergence of these potentials toward their unregularized limit. Our results generalize the results of (González-Sanz and Nutz, SIAM J.~Math.~Anal.) and (Wiesel and Xu, Ibid.) to the multivariate case and beyond the case of self-transport.

We study a degenerate stochastic system inspired by bias-corrected Adam. The system arises as the continuous-time limit of an explicitly modified recursion in which the squared-gradient update is replaced by a prescribed second-moment closure. We construct an entrance solution at the singular initial time and prove convergence of the modified recursion on finite time intervals away from zero. Bias correction enters through time-dependent coefficients which converge to constants as time tends to infinity. For a smooth objective with globally Lipschitz gradient and radial dissipativity at infinity, we prove existence and uniqueness of an invariant probability measure for the autonomous system under either $b<4a$ or $\gamma L_f\le a(\sigma/\sqrt2+\varepsilon)/4$. Here $L_f$ is the Lipschitz constant of the gradient, $a,b$ are the relaxation rates, $\gamma$ is the learning-rate parameter, and $\sigma,\varepsilon$ are the noise and regularization parameters. The proof combines an exponential Lyapunov function with compact-set minorization obtained from explicit control paths and a regular endpoint map. Under an exponential Lyapunov moment condition, we obtain exponential convergence in weighted total variation and in the squared $2$-Wasserstein distance of the complete state. The invariant law has finite moments of every order, and qualitative total variation convergence holds for every initial probability law. The time-inhomogeneous process converges weakly to the same invariant law. We formulate the associated Fokker-Planck equation for probability measures and establish local hypoellipticity and smoothness of the invariant density where $\operatorname{Diag}(\nabla f)D^2f$ is invertible. A quadratic example shows that a uniform exponential convergence estimate requires a prefactor growing exponentially in the initial position along a specified family of states.

We study Serrin's overdetermined boundary value problems in bounded domains on weighted Riemannian manifolds. When the closure of the domain is compact, we establish a rigidity result that characterizes both the solution and the geometry of the ambient manifold. We further address the case of domains with non-compact closure for manifolds conformally equivalent to the Euclidean space, possibly degenerating or becoming singular at a point, where both the weight and the conformal factor are radial functions.

This paper investigates a system of nonlocal continuity equations modelling the interaction of two species coupled through Riesz-type potentials. The model incorporates self- and cross-interaction kernels of possibly different fractional orders. By exploiting optimal transportation theory and the theory of gradient flows in Wasserstein spaces, we establish the existence of weak solutions under singularity assumptions on all interaction potentials, provided the cross-interaction ones satisfy a symmetry condition. Our analysis extends previous results available for either single-species equations or multi-species systems with smoother cross-interaction kernels.

We establish the well-posedness of the Helmholtz equation with rough and compactly supported coefficients in Rd under sharp regularity assumptions. Using a paraproduct calculus in rescaled weighted Besov spaces, we rigorously define the product between the solution and the coefficient at the lowest regularity level without renormalization. A rescaled Lippmann-Schwinger formulation is shown to be equivalent to the Helmholtz equation with the Sommerfeld radiation condition. We prove existence, uniqueness, and explicit wavenumber dependent resolvent estimates in a general Lp setting, including an L2 theory relevant to scattering amplitudes. The results provide a sharp analytic foundation for wave propagation and scattering in highly irregular media.

This paper establishes a quantitative, uniform-in-time diffusion approximation for the joint law of a broad class of fully coupled multiscale stochastic systems. We derive a precise characterization of the limiting joint distribution as a specific skew-product of the conditional equilibrium of the fast process and the homogenized law of the slow component, thereby providing a rigorous uniform-in-time formulation of the adiabatic elimination principle. The convergence rate explicitly separates the initial relaxation of the fast dynamics from the long-time homogenized evolution and depends only on the regularity of the coefficients in the slow variable. As a consequence, we obtain the first quantitative identification of the limiting stationary distribution of the original multiscale system and prove the commutativity of the limits $\eps\to0$ and $t\to\infty$ for a large class of observables. Our framework accommodates unbounded and irregular coefficients, degenerate structures, and weakly mixing dynamics. We illustrate its scope with three applications: (i) a uniform-in-time averaging principle for fast-slow systems; (ii) a uniform Smoluchowski–Kramers approximation for degenerate Langevin systems, yielding convergence of the joint position-scaled velocity law and global-in-time asymptotics of key thermodynamic functionals (e.g., total energy, entropy production, free energy); and (iii) the first uniform-in-time periodic homogenization result for SDEs with distributional drifts.

We develop a unified PDE-probabilistic framework for pointwise gradient and Hessian estimates of Markov semigroups associated with stochastic differential equations with singular and unbounded coefficients. Under mild local structural assumptions on the diffusion matrix and integrability/regularity conditions on the drift, we obtain quantitative sharp short-time regularization estimates as well as long-time decay bounds (including exponential and polynomial rates) for the first and second spatial derivatives of the semigroup. A distinctive feature of our results is the explicit dependence of these estimates on local norms of the coefficients (through scale-invariant quantities), without requiring any global smoothness, boundedness or uniform ellipticity. In particular, our approach allows for degenerate or highly irregular behavior at infinity, subject to suitable local ellipticity and Lyapunov/ergodicity controls. As applications, we establish solvability and regularity results for Poisson equations on the whole space with singular coefficients, and we derive pointwise gradient estimates for SDEs with distributional drifts via a Zvonkin-type transform.

We extend the weak-strong uniqueness principle for mean-field game (MFG) systems to a broad class of second-order stationary and time-dependent problems. Under standard monotonicity, growth, and coercivity assumptions on the Hamiltonian, and relying strictly on the integrability exponents guaranteed by the existing theory for monotone MFG systems, we show that any weak solution must coincide with a given strong solution. Our analysis covers models with spatially dependent scalar diffusion coefficients, using monotonicity arguments and a coefficient-adapted mollification strategy to manage the variable diffusion terms. We extend this strategy to establish weak-strong uniqueness in the corresponding second-order, initial-terminal, time-dependent setting. Finally, to address the critical quadratic growth regime, we derive a new a priori second-order estimate for a stationary MFG system with logarithmic coupling. This estimate provides quantitative bounds on the solution in terms of the data, and yields weak-strong uniqueness in the range where the improved integrability yields $L^2$ control of the density. Since numerical and approximation methods for MFGs naturally yield weak solutions in the monotonicity sense, whereas strong solutions are known to exist in many settings, our results identify any weak limit produced by such methods with the strong solution whenever one exists.

This paper applies a discrete adjoint gradient computation method for a multi-class traffic flow model on road networks. Vehicle classes are characterized by their specific velocity functions, which depend on the total traffic density, resulting in a coupled hyperbolic system of conservation laws. The system is discretized using a Godunov-type finite volume scheme based on demand and supply functions, extended to handle complex junction coupling conditions – such as merges and diverges – and boundary conditions with buffer lengths to account for congestion spillback. The optimization of different travel-related performance metrics, including total travel time and total travel distance, is formulated as a constrained minimization problem and is accomplished through the use of an adjoint gradient approach, allowing for an efficient computation of sensitivities with respect to the chosen time-dependent control variables. Numerical simulations on a sample network demonstrate the efficiency of the proposed framework, particularly as the number of control parameters increases. This approach provides a robust and computationally efficient solution, making it suitable for large-scale traffic network optimization.

These notes give a brief introduction to differential Harnack inequalities and summarise the main results of the mini-course “Li-Yau and Harnack estimates for nonlocal diffusion problems”, presented by the author at the Seasonal School on PDEs “Oscillation Phenomena, PDEs, and Applications: A Comprehensive School in Mathematical Analysis”, held at Ghent University in October 2025.

Ensuring that a PDE model is well-posed is a necessary precursor to any form of analysis, control, or numerical simulation. Although the Lumer–Phillips theorem provides necessary and sufficient conditions for well-posedness of dissipative PDEs, these conditions must hold only on the domain of the PDE—a proper subspace of $L_{2}$—which can make them difficult to verify in practice. In this paper, we show how the Lumer–Phillips conditions for PDEs can be tested more conveniently using the equivalent Partial Integral Equation (PIE) representation. This representation introduces a fundamental state in the Hilbert space $L_{2}$ and provides a bijection between this state space and the PDE domain. Using this bijection, we reformulate the Lumer–Phillips conditions as operator inequalities on $L_{2}$. We show how these inequalities can be tested using convex optimization methods, establishing an upper bound on the exponential growth rate of solutions. We demonstrate the effectiveness of the proposed approach by verifying well-posedness for several classical examples of parabolic and hyperbolic PDEs.

We derive an explicit representation of the fundamental solution to the heat equation in a half-space of ${\mathbb R}^N$ with a diffusive dynamical boundary condition, and establish sharp pointwise upper and lower bounds. We also investigate qualitative properties of the associated solutions, including precise decay estimates. Furthermore, we analyze the diffusion limits of solutions to the initial–boundary value problem, and reveal the role of the diffusive dynamical boundary condition in the behavior of solutions.

This paper investigates the relationship between the geometric properties of a domain and the diffusion dynamics of Brownian motion, with a specific focus on the phenomenon of "trapping" in terms of the behavior of stochastic processes.

In this paper, we study the symmetric hyperbolic Schrödinger equations in the periodic setting. First, we prove full range Strichartz estimates on general tori by adapting Bourgain's major arc method. The result is sharp for rational tori. Second, on two-dimensional rational tori, we establish optimal local well-posedness for two hyperbolic nonlinear Schrödinger (HNLS) equations: the septic HNLS and the hyperbolic-elliptic Davey-Stewartson system.