Papers

We are concerned with the existence of normalized solutions to the $L^2$-supercritical nonlinear Schrödinger equation on a noncompact metric graph $G$, \[ \begin{cases} -u''+W(x)u+\lambda u=\chi(x)|u|^{p-2}u, & \text{on every edge } e \text{ of } G,\\[2mm] \displaystyle\sum_{e\succ v}u'_e(v)=0, & \text{at every vertex } v\in V, \end{cases} \] under the mass constraint $\int_G |u|^2\,dx=\mu>0$, where $\lambda$ arises as a Lagrange multiplier. Here $p>6$, $\chi$ is the characteristic function of the compact core $\mathcal K$, so that the nonlinearity is localized, and the potential $W$ is bounded, nonnegative and vanishing at infinity along every unbounded edge, with $\int_{1}^{\infty}xW(x)\, dx<\infty$ on at least one of them. For every $\mu>0$ we obtain a positive solution with $\lambda>0$, arising as a constrained critical point at a strictly positive energy level. Our approach combines a uniform mountain-pass geometry for a family of approximating functionals, the monotonicity trick with Morse index type information, and a blow-up analysis ruling out the divergence of the Lagrange multipliers. A key point is the strict positivity of the multiplier, for which we exhibit a potential of inverse-square decay admitting a zero-energy $L^2$ solution on a half-line. To the best of our knowledge, this is the first existence result for normalized solutions of the $L^2$-supercritical NLS equation on a noncompact metric graph in the presence of an external potential.

In this paper, only Carleman estimates are used, without energy estimates, we derive observability inequality. The main tool consists in the use of a new Carleman estimate.

We prove a resolvent estimate for the generator of the frame flow on hyperbolic manifolds away from vertical lines of resonances. A byproduct of the proof is an optimal essential spectral gap property for the generator, hence giving another proof of exponential mixing of frame flows with respect to the volume measure of the frame bundle. This extends the result of [https://arxiv.org/abs/2005.08387v2] in dimension 3 to any dimension. We make extensive use of the Borel-Weil calculus developed in [https://arxiv.org/abs/2405.14846] to overcome difficulties of this higher-dimensional case.

We investigate nonnegative mild solutions of $\partial_t u+(-\Delta)^{\ln}u=f(u)$ in $(0,T)\times\mathbb R^N$, with initial datum $u(0,\cdot)=\mu u_0$, $\mu>0$. Unlike the classical and fractional heat semigroups, the positive logarithmic heat kernel exists only for $0<t<N/2$, and the corresponding linear evolution may become singular at its terminal time, with lifespan and growth depending on the spatial decay of $u_0$. The behavior of $f$ near zero determines local solvability: if $\int_{0^+} d\sigma/f(\sigma)<\infty$, then no finite nonnegative solution exists on any positive time interval. Under suitable assumptions on $u_0$ and $f$, we establish well-posedness for the nonintegrable logarithmic heat kernel. If $f$ has at most global linear growth, the nonlinear solution attains the full linear lifespan, while the Osgood condition at infinity implies that the maximal existence time tends to zero as $\mu\to\infty$. We further distinguish slow-decay, fast-decay, and critical-tail initial data. In the noncritical regimes, a weighted Osgood tail condition yields blow-up strictly before the linear terminal time; if it fails, an amplitude threshold occurs under additional assumptions on $f$. In the critical regime, the dividing power is $3/2$: a square-root weighted Osgood condition yields premature blow-up, while its failure again leads to an amplitude threshold. Finally, we obtain terminal-time blow-up estimates and sharp rates for power nonlinearities.

Local-in-time well-posedness is established for a recently proposed diffuse interface model describing incompressible two-phase flows. The result constitutes the first analytical study of a model introduced by ten Eikelder et al. for the motion of a binary mixture of macroscopically immiscible, viscous, incompressible fluids with unmatched densities. In contrast to classic diffuse interface models based on a single mean velocity, this model is derived within the framework of mixture theory, assigning each phase its own momentum and mass balance, which results in a system of two coupled Navier–Stokes equations and two mass transport equations. The proof of the well-posedness result uses a fixed-point strategy, where the main difficulty lies in the analysis of the principal part of the associated linearized system.

We show that the assumption of a stress-free boundary condition at a finite intermediate depth, namely, at the bottom of the Ekman layer, in the analysis of wind-driven ocean flows necessarily leads to an unphysical current profile. Indeed, if the $z$-derivative of the fluid velocity vanishes at a given depth, then this depth necessarily corresponds to a minimum of the velocity profile, with the velocity increasing beneath it. Using a WKB ansatz based on the small variations of the ocean's water density at great depths, we also argue that a no-slip condition at the bottom of the ocean, if sufficiently deep, still effectively implies (up to a very small error) the orthogonality of the Ekman transport and the wind-stress.

On the $n$-dimensional torus $\mathbb{T}^n$, we consider the Kirchhoff-Poho{\v z}aev equation, which is the only Kirchhoff-type equation known to admit global solutions for small initial data in $H^s \times H^{s-1}$, $s \geq 2$. We study this equation from a dynamical perspective by means of a quasilinear normal form approach. We perform two steps of normal form reduction and show that the resulting cubic and quintic terms do not contribute to the Sobolev energy estimates. This cancellation reveals a special algebraic structure of the equation and represents a first step towards a dynamical explanation of its exceptional global well-posedness. As a further consequence, we obtain improved bounds on the growth of Sobolev norms and improved lower bounds on the existence time for initial data in $H^s \times H^{s-1}$ with $s \in [3/2, 2)$.

We consider a reaction diffusion equation, where the lower-order nonlinearity allows for a non-smooth part, described by a maximal monotone relation. We show that the global attractor has finite fractal dimension, and also obtain explicit upper estimates in terms of the data.

In this note, we show a condition on the derivatives of a multiplier $m\in L^\infty$ that guarantees that $m$ is a multiplier from $L^1(\mathbb{R}^n)$ to the Hardy Space $H^1(\mathbb{R}^n)$.

We investigate the wave interactions and stability of Riemann solutions for a nonautonomous chromatography-type system of Langmuir isotherm with time-dependent damping and flux. The system models two-component chromatographic separation with a time-dependent saturation capacity $n(t)$, leading to a nonautonomous hyperbolic system of balance laws. We consider a perturbed Riemann problem with piecewise constant initial data having two jump discontinuities at $x = \pm\epsilon$, and construct the global weak solution by analyzing all possible wave interactions, both classical (shock waves, rarefaction waves, contact discontinuities) and nonclassical (delta shock waves). We prove that as $\epsilon \to 0$, the solution of the perturbed Riemann problem converges to the solution of the corresponding Riemann problem in the space of Radon measures, establishing the stability of Riemann solutions under small perturbations of the initial data. To the best of our knowledge, this is the first instance of wave interaction and stability analysis for a nonautonomous chromatography-type system with time-dependent coefficients. Numerical experiments using a Lax-Friedrichs type scheme illustrate the wave interaction structure, the profiles at selected times, and the asymptotic convergence as $\epsilon \to 0$.

We review recent developments in the study of spreading phenomena in reaction–diffusion equations arising from ecological invasion models. Motivated by the conjecture of Shigesada and Kawasaki on staged invasions, we discuss how competition systems can lead to effective scalar models with shifting habitats. We present the Hamilton–Jacobi approach for determining spreading speeds and revisit the result of {Li–Bewick–Shang–Fagan} (2014) from this perspective. We then describe the emergence of nonlocally pulled fronts when the shifting habitat connects regions of distinct positive growth rates. Recent results including the works of Lam–Yu (2022) and Lam–Nadin–Yu (2025) are surveyed. We also discuss spreading phenomena in competition systems, predator-prey systems, and the existence of various classes of entire solutions. Finally, we discuss the logarithmic correction for invasion waves in moving environments and prove a new result.

Let $2\le k\le n$, let $\Omega\subset\mathbb{R}^n$ be open and convex, and let $u$ be a convex viscosity solution of $\sigma_k(D^2u)=1$ in $\Omega$. We prove that the set on which $u$ fails to be locally $C^2$ has vanishing $(n-1)$-dimensional Hausdorff measure. In the intermediate range $3\le k<n$, this gives a codimension-one refinement of the known almost-everywhere partial regularity, and the exponent is sharp. More generally, for a convex viscosity subsolution of $\sigma_k(D^2u)\ge\lambda>0$, we obtain Hausdorff bounds for strata defined by the affine dimension of all supporting contact sets. The proof combines a support-dependent Chou–Wang barrier argument, an estimate for the product of the smallest $k$ semiaxes of a John ellipsoid, and Mooney's convex section-covering theorem. As a direct analytical consequence, the full distributional Hessian is absolutely continuous and $u\in W^{2,1}_{\mathrm{loc}}(\Omega)$, yielding a $k$-Hessian counterpart of the $W^{2,1}$ regularity known for singular Monge–Ampère solutions. In a logically separate structural part, we characterize the distinguished number of flat directions, $n-k+1$, by an asymptotic infimum mean-value formula over affine sections, and explain how this mean-value heuristic leads to the supporting-contact geometry used in the proof.

In this paper, we study the Klein-Gordon-Schrödinger system in the nonrelativistic regime $\epsilon \to 0$, where $\epsilon$ is proportional to the inverse of the speed of light. We show that the Klein-Gordon-Schrödinger system converges to a system of decoupled linear Schrödinger equations over a long time interval of order $\epsilon^{-1}$ with error estimates of the form $(1+t)\epsilon^2$; in particular, the error estimate for the Schrödinger component is uniform in time of the form $\epsilon^{2}$ The specific forms of the error estimates coincide with the numerical results shown by Bao et a.l., and the $O(\epsilon^{2})$ convergence rates coincide with the order of initial error, and thus are optimal.

In this work, we study the nonlinear stability of linearly expanding Goldreich-Weber (GW) solutions for the gravitational Navier-Stokes-Poisson system with pressure law $p(\rho)=\rho^{\frac{4}{3}}$ and degenerate viscosity. It is well known that linearly expanding GW solutions are special solutions to the Euler-Poisson system with $\gamma=\frac{4}{3}$. With bulk viscosity equal to 0, linearly expanding GW solutions are also solutions to the Navier-Stokes-Poisson equations. Choosing shear viscosity proportional to $\rho^{\alpha}$ with $0<\alpha\le\frac{2}{3}$ and zero bulk viscosity, we prove the nonlinear stability of linearly expanding GW solutions under radial perturbations.

Let $s>1$, let $M$ be a connected, non-compact, oriented real-analytic manifold, and let $\omega_1,\ldots,\omega_m$ be real-valued closed $1$-forms of Gevrey order $s$ on $M$. We study the differential complex naturally associated with this family on $M\times\mathbb{T}^m$. We prove that its top-degree operator is globally solvable in Roumieu Gevrey ultradistributions, or equivalently that the corresponding top-degree cohomology vanishes. No global hypoellipticity assumption and no arithmetic condition on the periods of the defining forms are required. The proof is carried out in the physical variables and combines fiber translations, a local normal form, and a transport formula along paths in the base manifold. These tools yield propagation of Gevrey regularity, non-confinement of Gevrey singularities, and the support control needed to apply an abstract solvability criterion. The result highlights a sharp contrast with the compact setting, where compatibility conditions are unavoidable and solvability for compatible data may depend on exponential small-denominator conditions.

We use highly oscillatory geometric optics solutions to solve the inverse problem for the system $$ \square \begin{bmatrix} u^{(1)}\\ u^{(2)}\\ \vdots\\ u^{(n)}\end{bmatrix} = \sum_{k\geq l=1}^n \left(Q_0(u^{(k)},u^{(l)}) \begin{bmatrix} q_{1kl}\\q_{2kl}\\\vdots\\q_{nkl} \end{bmatrix} + \sum_{i>j=1}^d Q_{ij}(u^{(k)},u^{(l)})\begin{bmatrix} p_{1ijkl} \\ p_{2ijkl} \\ \vdots \\ p_{nijkl} \end{bmatrix}\right), $$ where $Q_0$ and $Q_{ij}$ are the symmetric and anti-symmetric bilinear null forms. We present solutions in both the linear and weakly nonlinear regimes. In the linear regime, we show that the coefficients of order $h^3$ determine an injective light-ray transform of a vector field which depends on the coefficients $q_{rkl}, p_{rijkl}$. In the weakly nonlinear regime, we see that the coefficients of order $h$ determine the non-abelian light ray transform for matrices associated with the coefficients. While we do not have an injectivity result for this case, we do have one if we assume the coefficients do not depend on the time variable $x_0$, as our coefficients instead determine an injective non-abelian X-ray transform.

Let $L:=-\Delta+V$ be a Schrödinger operator on the Euclidean space $\mathbb{R}^n$ with potential $V$ in the reverse Hölder class $RH^{n/2}$ satisfying some mild assumptions that $V$ neither decays too rapidly nor oscillates violently at infinity. In this paper, the authors construct a new system of dyadic cubes $\mathcal{D}^V$ that reflects the intrinsic geometry perturbed by $V$. Then using the quantitative geometric information of $\mathcal{D}^V$, the authors characterize the $L^p$ operator norm of the Riesz potential $L^{-\alpha/2}$ for all $\alpha\in (0,2]$ and $p\in (1,\infty)$. As applications, some quantitative spectral estimates for $L$ are given.

Previous works of the author and Sogge [BS17], [BS18] showed the significance of microlocal Kakeya-Nikodym averages in improving $L^p$ bounds on (approximate) eigenfunctions of the Laplacian in the high frequency limit. These averages are formed by taking the $L^2$ norm of an eigenfunction when localized in phase space to a small, frequency-dependent tube about a geodesic segment via a pseudodifferential operator. The former work showed that for values of $p$ beneath the Stein-Tomas exponent, $L^p$ norms are controlled by a supremum over these averages. The latter work then showed that when $(M,g)$ has nonpositive sectional curvatures, there is a logarithmic gain in the averages. In combination, these two works improved the $L^p$ theory for eigenfunctions over the universal bounds of Sogge in this geometric setting. In the present work, we develop sufficient conditions for improving these averages which are more general than nonpositive curvature. Instead our sufficient conditions are rooted in the dynamics of the geodesic flow on the tangent bundle, considering cases where the flow expands and contracts tangent vectors in at least some directions, e.g. partially hyperbolic flows. We make use of Gaussian wave packet (phase space) transforms on the manifold in order to fully appreciate the gain these hypotheses impart on the microlocal averages. In the process, we further develop Gaussian beam approximations to the wave equation in a coordinate invariant manner.

Several applications for systems of conservation laws of the form $U_t + (\Phi (U) U)_x =0$, $U: R_t\times R_x\rightarrow R^n$ , $n\geq 2$, with $\Phi (U) = \phi (r, \Theta): R^n\rightarrow R$, $r = |U|$, and $\Theta = U/|U|\in S^{n-1}$, are obtained by imposing structural conditions to provide a classification framework for solutions dependent on the form of $\phi(U)$. By prescribing the evolution of a particular eigenvalue, we can categorize classes of such functions, $\phi$, for which this evolution is met, into depending on either a scalar field $z=rK(\Theta)$, where $K: S^{n-1}\rightarrow R$, or on the vector field $\Theta$, and find for which the amplitude of solutions may blow up in finite time. As a consequence, solutions to the corresponding Riemann problems can be divided into those with are classical and those which involve delta-shocks and/or vacuum states.

For a bounded Lipschitz domain $\Omega\subset\mathbb R^2$, let $T(\Omega)=\int_\Omega u_\Omega\, dx$ denote its torsional rigidity, where $-\Delta u_\Omega=1$ in $\Omega$ and $u_\Omega=0$ on $\partial\Omega$. We prove a quantitative Pólya–Szegő inequality for tangential polygons. Let $N\ge3$, let $P$ be a tangential $N$-gon, set $A=|P|$, and let $R_N$ be the regular $N$-gon of area $A$. Writing $L(\cdot)$ for perimeter, we obtain the explicit deficit estimate \[ T(R_N)-T(P)\ge \frac{A^2}{8N\tan(\pi/N)} \left(1-\frac{L(R_N)^2}{L(P)^2}\right)^2.\]Thus, at fixed area, the torsional deficit is controlled from below purely by the perimeter ratio. In particular, the regular $N$-gon is the unique maximizer of torsional rigidity among tangential $N$-gons of prescribed area; for $N=3$ this gives the classical triangular Pólya–Szegő theorem with a quantitative estimate. The same perimeter estimate yields an explicit positive lower bound for $T(R_{N+1})-T(R_N)$ for equal-area regular polygons, and hence a rather short alternative proof of the strict monotonicity of torsional rigidity in $N$. Combined with the Kohler–Jobin inequality, it also gives an explicit sufficient condition for the polygonal Faber–Krahn inequality within the tangential class. Our full quantitative inequality is stronger: it contains, in addition, a nonnegative angular Jensen deficit, which yields quantitative angular stability away from degenerate configurations.

This proceedings contribution elaborates on the findings of arXiv:2605.26234v2: a joint work with Marco Usula, where we introduced a machine learning framework based on physics-informed neural networks (PINNs), aimed at constructing near-minimal discs in hyperbolic space asymptotic to a prescribed knot at infinity. We used this method to provide numerical evidence for a conjecture of Joel Fine relating minimal surfaces in $H^{4}$ to the coefficients of the HOMFLY polynomial. This is a methodological companion to that paper, based on a presentation given at the 2026 edition of the workshop "DANGER: Data, Numbers, and Geometry". Rather than reviewing the results, which are presented extensively in the preprint above, we discuss the two aspects of the framework which, in our experience, determined whether the method worked at all. First, the geometry of the problem must be encoded in the architecture of the model, so that the boundary condition and asymptotics at infinity hold exactly for every value of the learnable parameters - leaving us with a single-component loss function; second, the evaluation of the PDE residual must be engineered with care to ensure that complete trainings can be performed in a reasonable time. On the latter point, we describe two implementation techniques which are not spelled out in detail in the original paper: replacing nested reverse-mode automatic differentiation with the forward propagation of second-order jets, and compiling the computational graph of the residual once instead of rebuilding it at every optimisation step. Together, on identical hardware, these two changes reduce the cost of a training step by a factor of roughly forty to fifty. We hope these methodological discussions can be useful for researchers in differential geometry and geometric analysis who wish to deploy PINNs on problems of their own.

The paper deals with the existence and multiplicity of nontrivial solutions for the doubly elliptic problem $$\begin{cases} -\Delta u=f(u) \qquad &\text{in $\Omega$,}\\ \phantom{-}u=0 &\text{on $\Gamma_0$,}\\ -\Delta_\Gamma u +\partial_\nu u =g(u)\qquad &\text{on $\Gamma_1$,} \end{cases} $$ where $\Omega$ is a bounded open domain of $\mathbb{R}^N$ ($N\ge 2$) with $C^1$ boundary $\Gamma=\partial\Omega$, with $\Gamma=\Gamma_0\cup\Gamma_1$, $\Gamma_0\cap\Gamma_1=\emptyset$, $\Gamma_1$ being nonempty and relatively open on $\Gamma$, $\mathcal{H}^{N-1}(\Gamma_0)>0$. The terms $f$ and $g$ are subcritical with respect to Sobolev embeddings, respectively in $\Omega$ and on $\partial\Omega$. We prove that, under suitable assumptions, the problem admits nontrivial solutions at the depth of the potential well energy level, which is the minimum energy level for nontrivial solutions. We also prove that the problem has infinitely many solutions at higher energy levels.

We consider fourth-order and second-order partial differential operators localized on domains of the sphere in arbitrary dimension. These operators arise as weighted compositions of the linearized Einstein constraint operators and their adjoints, and played a key role in our resolution of the optimal localization problem in general relativity, also referred to as the gravitational shielding problem. To control the asymptotic behavior of solutions to Einstein's constraints in our companion paper (preprint arXiv:2312.17706), we introduced the notions of harmonic, radial, and shell stability. Harmonic stability controls the borderline harmonic modes, radial stability governs the radial evolution of spherical averages, and shell stability controls the coupled radial-angular evolution of solutions. In the present paper, we establish that these stability properties follow from weighted Poincaré, Korn, and Hardy inequalities. Furthermore, in arbitrary dimension, we investigate the behavior of the associated geometric constants, and conclude that the stability conditions hold for a broad class of localization functions; the theory applies to arbitrarily small localization domains, corresponding to gluing cones with arbitrarily small aperture. At the opposite extreme, our conditions also hold on the entire sphere, corresponding to the absence of localization. This completes, for gluing cones of arbitrarily small aperture in every dimension, the program initiated by A. Carlotto and R. Schoen on gravitational shielding and the construction of solutions enjoying super-harmonic decay estimates. In our proofs, we introduce Hamiltonian and momentum functionals, which we call shell functionals, and show that they enjoy monotonicity and semi-coercivity properties; their structure also suggests possible analogies with functionals arising in other curvature-related geometric problems.

We investigate the dynamics of an oscillating water column, which is a wave energy converter where water waves interact with a fixed partially immersed structure and an air chamber. Considering the shallow water regime, the fluid motion is governed by either the one-dimensional nonlinear shallow water equations or the Boussinesq-Abbott equations, while the air pressure variation in the chamber acts as a spring force on the fluid. The presence of the structure and the air chamber introduces constraints into the fluid equations. Assuming conservation of the total fluid-elastic energy in the absence of structural damping, we reformulate the constrained equations as transmission problems between the open water and the chamber and show their local well-posedness. We also consider a different modeling approach in which the upper part of the fluid in the chamber is treated as a rigid layer (water column) free to move vertically above the lower fluid. For this configuration, we reformulate the dynamics as wave-spring-mass systems, where the motion of the water column is driven by the shallow water waves outside the chamber. We establish their local well-posedness and show that the effective buoyancy period of the water column motion is determined by the competition between the added mass and the stiffness of the spring force.

We consider an elliptic equation driven by the $s$-fractional $p$-Laplacian, set in a smooth bounded domain $\Omega\subset\mathbb{R}^N$ with homogeneous nonlocal Dirichlet conditions and a reaction $f$ lying in $L^q(\Omega)$ for some $q\ge 1$. We prove that the unique solution $u$ is $\alpha$-Hölder continuous up to the boundary, for any $\alpha$ below $p'(s-N/pq)$ if $N/ps<q\le N/s$, and $\alpha=s$ if $q>N/s$. Also, we prove that if $q>N/s$ then $u/{\rm d}_\Omega^s$ admits a Hölder continuous extension to the closure of $\Omega$, where ${\rm d}_\Omega$ denotes the distance from the boundary. Our results are almost optimal and extend previous regularity theorems known in the linear case.

Motivated by marginal distribution flows of reflected diffusions in bounded domains, we investigate when a density/flux pair solving a no-flux continuity equation admits a regular Lagrangian flow that remains in the closed domain and generates the prescribed density flow. We give sufficient conditions in terms of interior bounded-variation regularity, bounded-variation control on a boundary collar, a one-sided bound on an absolutely continuous divergence, and vanishing normal trace of the velocity. The proof uses the fact that tangency removes the singular boundary contribution to the divergence of the zero extension, thereby making the extended velocity admissible for the Ambrosio-DiPerna-Lions theory. We show that these boundary assumptions cannot be jointly relaxed so as to admit a boundary current mechanism. We construct an explicit smooth density/flux pair carrying a boundary current. Its density evolution is unique in a weighted class and its characteristics are unique, confined and transport the marginals, yet it admits no regular Lagrangian flow because the compressibility bound fails arbitrarily close to the initial time. We also establish two uniqueness results for no-flux Fokker-Planck equations: a duality result for bounded measurable drifts and a weighted energy result for entrance-type drifts singular at the boundary. Our results provide a rigorous mathematical justification for using the ODE-based sampling of reflected diffusion models under minimal regularity assumptions on the coefficients, and also indicate when such ODE-based samplers may fail.

Let $N\in\{2,3\}$, $0<\mu _1\leq\mu _2$, and $0<\beta<\mu _1$. We prove that the equal-frequency two-component cubic Schrödinger system \[ -\Delta u+u=\mu _1u^3+\beta uv^2, \qquad -\Delta v+v=\mu _2v^3+\beta u^2v \quad\text{in }\mathbb{R}^N \] has exactly one positive solution in $H^1(\mathbb{R}^N)\times H^1(\mathbb{R}^N)$ modulo simultaneous translations. More precisely, every positive solution is a simultaneous translate of the synchronized state constructed from the unique positive radial solution of $-\Delta w+w=w^3$ in $\mathbb{R}^N$. This settles Sirakov's equal-frequency uniqueness conjecture throughout the weak-coupling range. The main difficulty in the proof is to exclude radial solutions for which the ratio of the normalized components is nonconstant. After normalization, the two components satisfy scalar equations with a common potential. We construct a weighted Pohozaev functional for the system together with a correction term and prove that both the corrected functional and the associated weighted functional are strictly positive. Combining these sign properties with a radial flux identity and an auxiliary quotient associated with the component ratio forces synchronization.

This paper studies the obstacle problem for first-order scalar conservation laws driven by multiplicative noise. By adapting a barrier-substitution strategy to the kinetic formulation, we permit the reflection measure to be a general Radon measure and eliminate the usual obstacle-noise compatibility condition. We establish the existence of kinetic solutions for continuous obstacles and further derive $L^1$-contraction and uniqueness under stronger spatial regularity. Moreover, we prove a Freidlin–Wentzell large deviation principle in $L^1(0,T;L^1(\mathbb{T}^N))$. Unlike existing large-deviation arguments based on control-uniform penalization of the obstacle, we work directly with the reflected skeleton equation and retain the possibly singular Radon reflection measure throughout, while a viscous approximation provides the spatial $H^1$-regularity required for compactness before passing to the inviscid limit. Our results provide a hyperbolic counterpart to the large deviation theory for parabolic reflected SPDEs developed by Matoussi, Sabbagh, and Zhang (2021).

This paper addresses the approximation of entropy solutions to a one-dimensional scalar conservation law via the so-called Follow-the-Leader particle scheme in case the velocity map is non monotone. The result is based on a discrete maximum principle, on $BV$ estimates of the approximated density, and on the characterization of the limit as entropy solution in the sense of Kru\v zkov.

We study the one-dimensional focusing mass-critical half-wave equation \[ i\partial_tu=|D|u-|u|^2u. \] For even initial data with negative energy and mass slightly above the ground-state mass, we prove finite-time blow-up and obtain the upper bound \[ \|u(t)\|_{\dot H^{1/2}} \lesssim \frac{|\log(T-t)|^{1/4}}{\sqrt{T-t}} \qquad\text{as }t\uparrow T. \] This is the first finite-time blow-up result for negative-energy solutions to the mass-critical half-wave equation in the near-ground-state regime. The proof uses the modulation analysis developed by Merle–Raphaël \cite{MerleRaphael2005AnnMath}. For the half-wave equation, the key missing ingredient is a coercivity estimate for a nonlocal quadratic form generated by the scaling direction. Unlike the local NLS, the half-wave equation does not admit the ODE methods used to establish the corresponding coercivity estimate. Instead, we prove the coercivity estimate by an analytic reduction followed by a rigorous computer-assisted proof. The spectral analysis part reduces the coercivity problem to a finite collection of spectral inequalities by combining constrained Morse index arguments with the Birman–Schwinger principle. The computer-assisted part certifies these inequalities by interval arithmetic using a validated approximation of the ground state obtained by compactifying the real line. Together with the modulation analysis, this coercivity theorem gives the finite-time blow-up result.