Papers

We investigate self-similar blow-up solutions to the $d$-dimensional axisymmetric incompressible Euler equations without swirl for $d\ge 3$. For any $\alpha\in(0, \alpha_d)$ with $\alpha_d=1-2/d$, we construct a self-similar blow-up solution whose initial velocity field satisfies $u_0\in C^{1,\alpha}_{\rm loc}(\mathbb R^d)\cap C^\infty(\mathbb R^d\setminus\{0\})$. Our construction relies on a fixed-point argument formulated for the self-similar profile equations, which form a coupled elliptic-transport system. Specifically, the transport equation recovers the vorticity profile from given data along characteristic curves, while the elliptic equation reconstructs the velocity field via Newtonian potentials defined in an auxiliary $(d+4)$-dimensional space. The main challenge consists in choosing appropriate function spaces that remain invariant under such nonlinear compositions and that simultaneously capture the exact singular behavior near the origin and the symmetry axis. Furthermore, we establish a finite-codimensional stability result for the self-similar profiles obtained above. As a consequence, after suitable truncation and correction of finitely many unstable modes, we obtain finite-energy blow-up solutions with initial velocity in $C^{1,\alpha}(\mathbb R^d)\cap C^\infty(\mathbb R^d\setminus\{0\})\cap L^2(\mathbb R^d)$ and compactly supported initial vorticity. These solutions are asymptotically self-similar near the blow-up time.

In this paper, we explore the statistical subtleties of the nonideal Rayleigh gas, in a grand canonical mixture framework. This model allows to consider a large amount of tagged particles close to equilibrium, and their empirical measure, whose first-order convergence has been shown to converge to the solution of the linear Rayleigh-Boltzmann equation [5]. Thanks to the study of the cumulants of the system, we analyze the asymptotic behaviour of the fluctuations and large deviations of this empirical measure, hence refining the previous statistical results in the same vein as [7]. This way, we exhibit the trivial limit behaviour of the fluctuations in any overdilute regime, proving the exact relevance at any statistical scale of the low density limit. In the case of large deviations, we present the linear Boltzmann-Hamilton-Jacobi system driving their asymptotic behaviour. Eventually, we optimize the geometrical estimates on the billiards dynamics [6] to finally achieve a full convergence rate for the cumulants.

This paper introduces a grand canonical mixture model to generalize the nonideal Rayleigh gas [5] to an asymptotically infinite amount of perturbed tagged particles. This model relies precisely on grand canonical tags, to preserve symmetry in the system, contrary to [2]. We hence define and study the convergence of the correlation functions of this system in large times, linking it to the expectancy of the empirical measure of tagged and non-tagged particles, to eventually prove a law of large numbers for this dynamics. We extend the quantitative study to all the correlation functions, and not only the first one, exhibiting the resultant additional factors, and we also generalize the perturbation to the whole phase space, instead of considering a space-only initial perturbation. Eventually, we fit our adaptive time cutting [12] to the mixture system, even improving it to get better convergence rates.

In this study, we firstly establish the well-posedness of a degenerate parabolic equation under Dirichlet boundary conditions. Following this, we introduce a shape design problem, which acts as a framework for approximating the degenerate parabolic equation through a series of uniformly parabolic equations. Finally, as a tangible application of this shape design approach, we deduce a boundary observability inequality associated with the degenerate parabolic equation.

We study wrinkling patterns in a thin elastic annulus subjected to radial stretching within the framework of the Föppl–von Kármán theory. Building on the analysis of the Lamé problem in Bella and Kohn, we investigate the asymptotic regime $h\to0$ and establish a $\Gamma$-convergence result for suitably rescaled energies after subtraction of the relaxed membrane energy. The limiting functional is a scalar convex measure-valued energy coupled with a constraint on the marginal of the limiting measure, describing the distribution of wrinkle frequencies. We also prove existence and qualitative properties of minimizers of the limiting functional.

We introduce a flow that is designed to flow maps $u:\Sigma\to \mathbb{R}^n$ which map the boundary of a general domain surface $\Sigma$ into a given (not necessarily connected) submanifold $N\hookrightarrow \mathbb{R}^n$ towards a free boundary (branched) minimal immersion supported by $N$. In the case when $\Sigma$ is the unit disc $D$, this task can be achieved by means of the Plateau-flow introduced in the work [15] of the second author. When $\Sigma\neq D$, however, also the conformal type of the domain metric plays a role and it no longer suffices to deform the trace of the given map into a half-harmonic map as in [15]. In order to overcome this issue, here we combine ideas of the Plateau-flow from [15] with ideas of the Teichmüller harmonic flow from [12], in order to flow both an initial map $u_0$ with trace $u_0\colon\partial \Sigma\to N$ and an initial domain metric $g_0$ in a way that produces, as time tends to infinity, a half-harmonic map from $\partial \Sigma$ into $N$ whose harmonic extension is conformal and hence is a (branched) minimal immersion.

The rise of Physics-Informed Neural Networks (PINNs) has popularized the concept of solving differential equations via residual minimization. However, neural networks are often viewed as “black boxes" requiring significant computational overhead and stochastic optimization. Moreover, PINNs typically treat boundary conditions (BCs) as “soft constraints" within the loss function and this makes the optimization process struggling to enforce the BCs properly. This paper introduces the Spline-based Physics-Informed Numerical Scheme (SPINS), a numerical framework designed to solve both initial and boundary value problems of ordinary differential equations (ODEs). By replacing the neural network architecture of traditional PINNs with a structured spline basis, SPINS achieves high accuracy and interpretability with a minimal parameter set. In addition, the BCs are automatically satisfied from the choice of the splines architecture. Therefore, SPINS provides smooth numerical solutions for ODEs allowing analytical differentiation. Moreover, SPINS benefits from the automatic differentiation where computing the gradient of the physics-informed loss function is an easy task making the optimization process very fast using gradient-based optimizers such as the L-BFGS-B algorithm. We demonstrate the efficacy of SPINS on nonlinear second order ODEs with several choices of BCs using cubic and quintic interpolating splines and present its natural extension to high order ODEs.

Caetano et al. (Proc. R. Soc. A. 481:20230650, 2025) have proposed a formulation for sound-soft acoustic scattering by a compact scatterer O $\subset$ Rn, in which the scattered field is represented as an acoustic Newtonian potential whose density is the solution of an operator equation on a compact set $\Gamma$ $\subset$ O. In the case that $\Gamma$ is Ahlfors-David d-regular (a d-set), for some d $\in$ (n–2, n], they show, moreover, that the operator equation can be interpreted as an integral equation, the integration with respect to d-dimensional Hausdorff measure, and present a convergent Galerkin scheme for numerical computation. In this paper we make a substantial extension of these results so that they apply to more realistic fractal scatterers that are multifractal, in the sense that they have spatially varying fractal dimension. Firstly, we provide, inspired by Claret et al. (J. Math. Pures Appl. 212:103888, 2026), an interpretation of this operator equation as an equation between a trace space on $\Gamma$ and its dual, and, in many cases, relate the density to a notion of the normal derivative of the scattered field on $\Gamma$. Secondly, we show that the operator equation is equivalent to an integral equation on $\Gamma$ whenever $\Gamma$ is the support of a Radon measure $\mu$ such that: (i) the trace operator from H1(Rn) to L2($\Gamma$, $\mu$) is continuous and; (ii) certain canonical singular integrals with respect to $\mu$ are finite; and we characterise a large class of measures for which (i) and (ii) hold. Finally, we show that Galerkin methods based on finite element subspaces of L2($\Gamma$, $\mu$) are convergent if and only if, additionally, C$\infty$\_0 (Rn\$\Gamma$) is dense in the kernel of the trace operator. These results apply, in particular, if $\Gamma$ is a finite union of d-sets with different values of d. In the case that each d-set is the attractor of an iterated function system of contracting similarities, we establish rates of convergence for the Galerkin method.

We prove the existence of global solutions for some coupled systems of partially nonautonomous evolution inclusions comprised of a Cauchy problem with a compact resolvent semigroup generator and an evolution equation governed by a subdifferential of a real potential. Our system in particular includes nonautonomous generalized Schrödinger-Debye systems of inclusions with variable exponents, but extends to hyperbolic-parabolic systems of inclusions in particular to Maxwell-parabolic systems of inclusions. Methodologically, we extend an approach of Vrabie et al. to the nonautonomous case and make use of standard semigroup tools to accomodate non-parabolic behaviour of solutions paired with a new existence result for measurable selections. The combination of the latter two requires the set-valued coupling terms to be Hausdorff-continuous, to take bounded, convex and closed values, and to satisfy weak continuity with respect to one variable.

It is well known that distributions whose symmetrized gradient is a bounded Radon measure belong to the space $BD$ on bounded domains with $\mathcal{C}^1$ boundary. In this work, we extend this result to a broader class of first-order linear elliptic operators. More precisely, let $\mathcal{A}$ be a first-order linear elliptic operator satisfying the rank-one property. We prove that if a distribution defined on a Lipschitz domain has bounded $\mathcal{A}$-variation, then it belongs to the space $BV^{\mathcal{A}}$.

We consider vector-valued magnetic Schrödinger operators $-\bm \Delta_{\bm a}+V$ with magnetic potential $\bm a \in L^2_{\mathrm{loc}}(\mathbb{R}^d;\mathbb{R}^d)$ and electric potential $V$ given by a matrix-valued function whose entries belong to $L^1_{\mathrm{loc}}(\mathbb{R}^d)$. We prove maximal inequalities in $L^p(\mathbb{R}^d;\mathbb{C}^m)$, $p\in[1,\infty)$ and the boundedness of the Riesz transforms $(\nabla - i\bm a)(-\bm \Delta_{\bm a}+V)^{-\frac{1}{2}}$ and $V^{\alpha}(-\bm \Delta_{\bm a}+V)^{-\alpha}$ on $L^p(\mathbb{R}^d;\mathbb{C}^m)$ for every $p \in (1,2]$ and every $\alpha\in[0,1/p]$.

A conservation-consistent boundary condition is proposed for nonlinear models of soluble-surfactant-laden falling films, ensuring exact conservation of total surfactant mass. The formulation resolves an inconsistency in widely used reduced models, Pascal et al. (PRF, 2019), D'Alessio et al. (JFM, 2020), which exhibit a gradual drift of mass during nonlinear evolution in a closed periodic domain. We show that this originates from an inconsistency in the surface transport reduction and derive a corrected boundary condition that removes this defect. As the discrepancy appears only at the nonlinear order, linear stability results remain unaffected, explaining why the issue has remained unnoticed.

We characterize positive critical Hardy weights for general Laplacians on weighted graphs. We then apply this result to fractional Laplacians on general graphs and use the characterization to identify an optimal Hardy weight under suitable assumptions. We finally illustrate our results with examples of graphs which arise as Cayley graphs of groups, satisfy curvature assumptions or are fractal graphs.

This paper establishes the global well-posedness of the linearized regularized 13-moment (R13) equations for rarefied gas flows. We first derive an entropy inequality for the system on bounded domains subject to Onsager boundary conditions. For the steady-state problem, well-posedness is proved via the Ladyzhenskaya-Babuska-Brezzi (LBB) theorem, facilitated by novel boundary-related Korn-type inequalities. Furthermore, leveraging the Lumer-Phillips theorem, we extend these results to guarantee the global well-posedness of the time-dependent R13 equations. Our theoretical framework uniformly accommodates the models for both Maxwell and general non-Maxwell molecules.

The Fermi–Pasta–Ulam–Tsingou (FPUT) system, describing the evolution of $N$ coupled harmonic oscillators, has been the subject of much attention since the 1950's when experiments which contradicted predictions of thermalization of the system. A full explanation of this behavior is still not fully known. Here, we rigorously derive the corresponding wave kinetic equation, which provides a precise evolution of the statistics for the FPUT system and demonstrates thermalization in an appropriate regime. In particular, we justify the kinetic equation for the 4-wave $\beta$-FPUT system in the kinetic limit $N \to \infty$ and $\beta \to 0$ for weakly nonlinear scaling laws $\beta \sim N^{-\gamma}$, reaching times up to $T_{\mathrm{kin}}^{2/3}$, where $T_{\mathrm{kin}}$ represents the kinetic (thermalization) timescale. While we use a typical diagrammatic expansion to derive the kinetic equations, few works have dealt with nonlinearities with non-resonant terms, which are not part of the kinetic equation, which is the major novelty of this work. The only other such work \cite{DIP25} made use of a normal form method to push the non-resonant terms to higher order nonlinearities. Here, we directly incorporate the non-resonant terms into the diagrammatic expansion and demonstrate corresponding gains. This method can be adapted to other 4-wave non-resonant nonlinearities.

Although quantitative stability for critical points of the Sobolev and fractional Sobolev inequalities has been extensively studied, the corresponding stability theory for critical points of the Hardy–Littlewood–Sobolev (HLS) inequality remains largely unexplored. A major difficulty is that the natural stability problem for HLS critical points involves a non-Hilbertian distance, so the classical orthogonal decomposition methods used in Hilbert-space settings are no longer available. In this paper, we develop a weak-decomposition–strong-stability method tailored to the stability structure of HLS critical points and establish the corresponding stability inequality. Our approach also yields an explicit lower bound for the stability of Palais–Smale sequences of the HLS integral equation. To the best of our knowledge, this appears to be the first quantitative stability result for Palais–Smale sequences of a variational functional measured in a non-Hilbertian distance. We further introduce a duality framework connecting Struwe-type decompositions and stability inequalities for critical points of the Sobolev inequality with their HLS counterparts. As a consequence, we derive Struwe-type decomposition and stability results for critical points of the fractional Sobolev inequality for general functions, thereby removing the nonnegativity assumption imposed in [26].

We prove the weak cosmic censorship conjecture in $2+1$ spacetime dimensions for the circularly symmetric Einstein-scalar field system in the presence of a negative cosmological constant $\Lambda<0$. More precisely, we show that for any integer $k\geq2$, the maximal development of generic $C^k$ initial data does not contain naked singularities. An essential step of the proof is establishing the presence of a mass gap. In particular, this implies that all naked singularities have infinite blueshift, which represents the main instability mechanism.

We construct solutions to Monge-Ampère equations whose Monge-Ampère measures contain singular components supported on low codimensional sets. We also study the regularity of such solutions. To motivate our construction, we present examples arising from optimal transport where the potential of optimal transport maps satisfy Monge-Ampère equations similar to the ones we study.

We systematically discuss the equivalence of two variational formulations of magnetostatics, in terms of magnetization and magnetic field on the one hand and the single field formulation using only magnetic induction. To demonstrate that this link is stable also when the magnetic laws are coupled with other variational static models, elasticity is included in the models as well. Interestingly, despite the fact that the corresponding magnetoelastic energy densities in the material can be computed via Legendre-Fenchel transform in the magnetic state variables, the two formulations are not linked by standard convex duality on the level of the functionals. In addition, convexity and coercivity of the given functional are neither required for the transformation nor always preserved by it.

We prove the existence of time-periodic weak solutions for a fluid-structure interaction system coupling the incompressible Navier-Stokes equations in a three-dimensional moving domain with a nonlinear Koiter plate equation on its upper boundary. The lateral boundary is space-periodic, a natural setting for flow in pipes and channels of periodic cross-section driven by a time-periodic pressure gradient, and the fluid satisfies a no-slip coupling condition at the moving interface. The elastic energy of the plate is governed by the nonlinear Koiter model, which yields an $H^2$-coercive operator and accounts for both membrane and bending effects. To the best of our knowledge, this is the first result on time-periodic weak solutions for a fluid-structure interaction system with a nonlinear elastic energy. The main novelty, compared to our earlier works on the linear case – a linear elastic plate and a linear Koiter shell respectively – is the replacement of a two-stage fixed-point procedure – a Leray-Schauder argument at the discrete level followed by a set-valued Kakutani-Glicksberg-Fan argument at the continuous level – by a single Leray-Schauder fixed point applied directly to the fully coupled Galerkin system. This reduction is not merely a simplification: the nonlinearity of the Koiter energy destroys the convexity of the solution map on which Kakutani-Fan relies, making the two-stage approach of~\cite{Claudiu22} unavailable and the single fixed point the only viable strategy.

Penrose's original heuristic for his eponymous spacetime inequality – a conjectured lower bound on the ADM mass in terms of the area of a horizon cross-section – relies on the black hole final state conjecture. In this paper we isolate a substantially weaker but precise late-time condition, which we call the quasi final state hypothesis and prove the spacetime Penrose inequality under this hypothesis. More precisely, for an asymptotically flat globally hyperbolic spacetime with a black-hole-type apparent horizon tube ${H}_{{app}}$ satisfying the dominant energy condition and the quasi final state hypothesis, we show that every asymptotically flat initial data set whose boundary is a MOTS cross-section of ${H}_{{app}}$ satisfies the spacetime Penrose inequality. The quasi final state hypothesis requires only a late-time decay condition on the normal component of the shift and the ratio of timelike to spacelike mean curvature, together with convergence of the cross-sectional areas of ${H}_{{app}}$ to a finite limit. Our approach is new and formulated directly in spacetime. The main geometric object is what we call a tangentially maximal hypersurface, carrying a foliation by spacelike spheres whose timelike mean curvature vanishes. We show that these hypersurfaces are governed by a quasilinear inward-parabolic PDE, and we develop the corresponding a priori theory and prove global existence. On these hypersurfaces, the spacetime Hawking mass reduces to the Riemannian Hawking mass, and the dominant energy condition gives nonnegative scalar curvature. The Riemannian Penrose inequality, combined with the area laws for dynamical and isolated horizons, then yields the result.

We establish optimal $C^s$ boundary regularity for the most general class of (linear and translation invariant) nonlocal elliptic operator of order $2s$. Namely, we consider Lévy operators that are symmetric and its Fourier symbol satisfies $\mathcal{A}(\xi)\asymp |\xi|^{2s}$ in $\mathbb{R}^d$. This was only known when the kernel of the operator (or Lévy measure) is either homogeneous or comparable to that of the fractional Laplacian, with different proofs in each case. Our new proofs extend both at the same time, and work in a very general class of domains, under a $C^1$-Dini-type condition.

We study nested loops in zero sets of sums of Laplace eigenfunctions on closed surfaces. In the real-analytic category, answering a question of Logunov, we prove a uniform bound for the number of rooted double nests in terms of the surface, the root, and the spectral cutoff. We show that this analyticity hypothesis is sharp: on a smooth sphere, a linear combination of eigenfunctions with eigenvalues \(0\) and \(2\) can have infinitely many rooted double nests. We also answer a question of Logunov and Nadirashvili by constructing a planar biharmonic function whose nodal set contains a double nest, and we prove a quantitative bound for entire biharmonic functions of polynomial growth. The biharmonic construction gives a nodal-set manifestation of the failure of the Boggio–Hadamard conjecture from the 1900s.

Given any \(n\in\mathbb{N}\), we construct a real-valued biharmonic polynomial on \(\mathbb{R}^2\) whose zero set contains a nest of \(n\) smooth, disjoint topological loops, meaning that the \(k\)-th loop lies inside the domain bounded by the \((k+1)\)-st loop for \(k=1,\ldots,n-1\). The case \(n=2\), i.e., the existence of two nested loops, is related to the failure of the Boggio-Hadamard conjecture from the early 1900s.

The defocusing nonlinear Schrödinger hydrodynamics supports exact dark solitons under finite density boundary conditions. However, the dark soliton gas, an interacting ensemble of dark solitons, has not yet been studied. In this work, we introduce an arbitrary-genus potential of dark soliton gases by considering the limit of the $\mathcal{N}$-dark soliton as $\mathcal{N}\to \infty$. The large-space asymptotics and long-time evolution of this dark soliton gas potential are analytically investigated through Deift-Zhou nonlinear steepest descent approach. The genus-$N$ dark soliton gas potential approaches the genus-$N$ finite-gap solution as $x \to -\infty$ and the background $1$ as $x \to +\infty$. In the long-time evolution, as the self-similar variable $\xi=x/t$ increases, the gas configuration exhibits a cascade of behaviours, passing from unmodulated and modulated genus-$N$ regions and progressively reducing the genus down to the planar region (unmodulated genus-$0$ region). Notably, the evolution of lower-genus soliton gases can be embedded within that of higher-genus gases, exhibiting identical dynamics within specific regimes. This phenomenon is encoded by the underlying spectra. We also include numerical validations, in perfect agreement with the theoretical predictions.

We study a system of nonlinear partial differential equations modeling the electrokinetics of a nematic electrolyte material consisting of various ion species suspended in a nematic liquid crystal within a bounded domain in two or three dimensions. The system couples a Nernst-Planck model for ion concentrations with the Poisson equation for the electrostatic potential, a Navier-Stokes equation for the fluid solvent, and the Ericksen-Leslie equations with general Leslie stress for nematic liquid crystals. We consider the case of isotropic elasticity for the liquid crystal and impose a unit-length constraint on the director field. The no-flux condition for the electrochemical potential leads to a nonlinear (and nonlocal) boundary condition for the ion concentrations. Using the theory of maximal regularity, we prove existence and uniqueness of strong solutions, provide criteria for global existence, and characterize the set of equilibria.

In this work, we investigate the one-dimensional heat equation within the framework of Stieltjes calculus. We first consider the equation associated with two fixed derivators and develop a constructive approach to establish the existence of solutions. We then study the corresponding initial value problem and incorporate several types of boundary conditions. Finally, we introduce a notion of multivariable derivator, suitable for higher-dimensional settings, and obtain explicit solutions of the heat equation for relevant classes of such derivators.

We prove the Dirichlet problem for second-order iterated Vekua equations, a natural generalization of the Bitsadze equation, is well-posed when the boundary condition is defined as a product of an exponential function and a polynomial on a non-degenerate conic that is not a circumference. Also, we extend this result, as well as other related results for the Dirichlet problem for polyanalytic and generalized analytic functions from the literature, to their analogues for bicomplex differential equations.

We establish the local boundedness of (sub-)solutions to nonlinear kinetic diffusion equations with $p$-growth, where the kinetic p-Laplace equation is a prototypical example. A key ingredient is the derivation of kinetic Gagliardo-Nirenberg inequalities, where the Lebesgue norm of a function is estimated in terms of its transport and diffusive directions controlled in different Lebesgue spaces.

We establish heat-kernel bounds and regularity estimates for the transition densities of the diffusion associated with the martingale problem corresponding to the generator of a formal multidimensional Brownian SDE with singular drift. As a by-product, we also derive Schauder estimates for the associated Kolmogorov (kinetic) Cauchy problem. We consider both the cases of non-degenerate and degenerate noise (e.g. kinetic-type models), in the so-called Young regime. Namely, we consider a time inhomogeneous drift in L $\infty$ [0,T ] C $\beta$ o for some fixed time horizon T , where ), with o standing for an underlying distance, namely the usual Euclidean one in the non degenerate setting, and the scale-homogeneous one in the kinetic case. Importantly, the estimates are obtained by employing as parametrix the transition density of the SDE (with variable coefficients) without singular perturbation, as opposed to the standard Levi parametrix obtained by freezing the noise. Finally, since the noise is multiplicative, the weak well-posedness of the singular SDE is a novel result in itself, and the density estimates directly imply irreducibility and strong Feller property of its solutions.