Papers

In this paper, we establish a sharp non-uniqueness result to the semilinear wave equation \[ \partial_{tt}u-\partial_{xx} u= \Theta_1(\partial_t u)^2 + \Theta_2(\partial_x u)^2. \] For any $\Theta_1 \Theta_2 < 0$, $\beta<1$, we construct infinitely many weak solutions $u \in C_{t,x}^{\beta}$ with zero initial data. To our knowledge, this is the first non-uniqueness result for a semilinear wave equation. At this low level of regularity $C_{t,x}^{\beta}(\beta<1)$, the quadratic term is defined by Littlewood–Paley decomposition in $C_t H_x^{-s}$. Our construction is based on a frequency-localized convex integration scheme. Together with the classical local uniqueness theory in $C_{t,x}^{1}$, this non-uniqueness result is sharp.

In this paper, we establish the validity of the Hilbert expansion for the coupled radiation-Euler model with non-relativistic source term for general initial data on the torus $\mathbb{T}^{3}$, which leads to the simplified equilibrium-diffusion limit and the initial layer corrections for the radiation intensity coupled with the temperature.

We show that the parabolic uniform rectifiability of a set can be characterized by an interior PDE property, namely Carleson measure estimates for bounded solutions to the heat equation. In particular, under very mild background hypotheses on an open set $\Omega$ and its boundary $\partial \Omega$, we show that the following are equivalent: (i) The quantity $|\nabla u(\cdot)|^2 \text{dist}(\cdot, \partial\Omega)$ is the density of a Carleson measure on $\Omega$ for all bounded solutions to the heat equation in $\Omega$. (ii) The caloric measure for $\Omega$ admits a corona decomposition. (iii) $\partial \Omega$ is parabolic uniformly rectifiable. The implication (iii) implies (i) is the main result of the cited paper [Bortz, Hoffman, Hofmann, Luna-García, Nyström, Anal. PDE 2023]. The remaining implications are new, and are direct parabolic analogues of results of Garnett, Mourgoglou and Tolsa [Duke Math. J. 2018] in the elliptic setting for the Laplacian; however, our proofs require several new ideas. In particular, we must adapt arguments to account both for time-lag, and for the fact that some characterizations of (elliptic) uniform rectifiability have either not been developed or have been shown to be untrue in the parabolic setting. Our background assumptions, as noted above, are very mild: we assume that the domain satisfies the time-symmetric capacity density condition and has interior corkscrews, and that the boundary is (non time-directed) Ahlfors-David regular. These are weaker than the background assumptions of Bortz, Hofmann, Martell and Nyström in arXiv:2510.22047, and are essentially optimal for the boundary continuity properties of caloric functions on which our arguments rely.

In 1980, Carleson posed a question about the least regularity required for initial data in a Sobolev space $H^s$ to ensure pointwise convergence of the solution to the single particle linear Schrodinger equation. In this paper, we study Carleson problem on the $d$-dimensional waveguide manifold $\mathbb R^n\times \mathbb T^m$ and initiate its analogy for the system of infinitely many orthonormal particles (arising from the transition of many-body quantum mechanics to the thermodynamic limit) on torus and waveguide manifold. For a single particle on the waveguide manifold, we establish a maximal-in-time Strichartz estimate that yields almost everywhere convergence to the initial data in $H^s$ for $s > \frac{d}{d+2}$. On the other hand, we adopt Bourgain counterexample approach to show that this convergence fails for $s< \frac{d}{2(d+1)}.$ For infinitely many particles, we generalize the classical maximal-in-time Strichartz estimate and pointwise convergence result of Compaan, Luca and Staffilani (2021) in the setting of fermionic systems on the torus. Similar result is also established for the waveguide manifold. We also establish necessary condition (in the spirit of Bourgain's counterexample) for the pointwise convergence problem for fermionic systems. These are the first results in the setting of torus and waveguide manifold, and complement the works of Bez, Lee and Nakamura (2020) and Bez, Kinoshita, Shinya and Shiraki (2024) on Euclidean space.

We study the regularity of weak solutions to doubly nonlinear orthotropic evolution equations of the form \[ \partial_{t}(\vert u\vert^{\alpha-1}u)-\sum_{i=1}^{N}\partial_{i}\left[a_{i}(x,t)\,(|\partial_{i}u|-\delta_{i})_{+}^{p-1}\frac{\partial_{i}u}{\vert\partial_{i}u\vert}\right]=f\,\,\,\,\,\,\,\,\,\,\mathrm{in}\,\,\,\Omega_{T}=\Omega\times(0,T), \] where $\Omega$ is a bounded open subset of $\mathbb{R}^{N}$ for $N\geq2$, the coefficients $a_{i}$ are measurable and bounded, $\alpha>0$, $p\in(1,\infty)$ and $\delta_{1},\ldots,\delta_{N}$ are non-negative numbers. We show that weak solutions are locally bounded due to their membership in a suitable De Giorgi-type energy class. We also obtain contractive estimates and global boundedness in space for solutions to a Cauchy problem associated with the above PDE. Our analysis extends analogous results available in the literature for diffusion equations that either do not exhibit double nonlinearity or are less degenerate than those considered here. Another main novelty of this paper is the presence of a source term $f$ on the right-hand side of the equation, for which we impose suitable integrability assumptions in the space-time variables.

We consider the Cauchy problem for the one-dimensional full compressible Navier–Stokes equations with general constitutive laws $p=p(v,\theta)$ and $e=e(v,\theta)$. The pressure and internal energy are assumed to be sufficiently smooth, thermodynamically compatible in the sense that $e_v=\theta p_\theta-p$, and to satisfy $e_\theta>0$. We first establish the local-in-time existence and uniqueness of strong solutions for large initial data without imposing monotonicity conditions on the pressure. For perturbations around a constant equilibrium $(\bar v,0,\bar\theta)$, we further assume the mechanical stability condition $p_v(\bar v,\bar\theta)<0$. By introducing a relative thermodynamic energy through the Gibbs relation, we derive a basic energy identity and prove its local quadratic coercivity near the equilibrium. Combining this estimate with higher-order a priori bounds, we obtain the global existence and uniqueness of strong solutions for sufficiently small $H^1(\mathbb R)$ perturbations. Moreover, the specific volume and temperature remain uniformly bounded away from zero and infinity for all time.

We establish sharp finite-ball shooting classifications for radial nodal Dirichlet solutions of the logarithmic equation and, within the maximal simple-zero shooting class, of the sublinear scalar-field equation. Recent whole-space uniqueness and phase-transition theorems are used as external inputs, while the finite-ball zero-curve ranges, endpoint asymptotics, and spectral consequences are proved here. The two models require different singular analyses: in the logarithmic problem the nonlinearity is not locally Lipschitz at a nodal zero and the linearized potential diverges there, whereas in the sublinear problem the limiting whole-space profile reaches a double zero at a finite support radius, beyond which continuation is nonunique. For every $R>0$ and $k\ge0$, the logarithmic problem has, up to sign, a unique radial Dirichlet solution with exactly $k$ interior zeros. Its shooting height $\beta_k^{\log}(R)$ is a strictly decreasing $C^1$ bijection from $(0,\infty)$ onto $(\alpha_k^{\log},\infty)$ and satisfies \[ \beta_k^{\log}(R)\longrightarrow\alpha_k^{\log} \quad(R\to\infty),\qquad \log\bigl(\beta_k^{\log}(R)^2\bigr) =\frac{\rho_{k+1}^2}{R^2}+\kappa_{k+1,n}+o(1) \quad(R\downarrow0), \] where $\kappa_{k+1,n}>0$ is given by an explicit Bessel integral. For the sublinear problem, the maximal simple-zero branch exists precisely for $R\in(\rho_{k+1},S_k)$, where $\rho_{k+1}$ is the $(k+1)$-st positive zero of the regular Bessel profile $\Phi_n$, and $S_k$ is the support radius of the unique compactly supported $k$-node whole-space bound state.

We introduce a fully–discrete variational approximation of Mather measures and sets for Tonelli Lagrangians on the flat torus, together with a numerical procedure for approximating the entire Mather set. The scheme is based on a fully–discrete Lax–Oleinik operator with integer winding labels. We prove an $O(\tau+h/\tau)$ error estimate for the critical value, convergence of critical solutions, and a finite-dimensional characterization of fully–discrete Mather measures. Accumulation points of the reconstructed minimizing measures are continuous Mather measures, while the supports satisfy complementary upper and lower convergence results involving the Mañé and Mather sets. To avoid the selection of only some minimizing components by exact discrete minimizers, we introduce a mass-threshold approximation based on almost-minimizing holonomic measures. This yields a finite-dimensional constrained optimization procedure designed to recover the whole Mather set.

We study nonlinear Landau damping for the Vlasov–Yukawa equation in the large-box regime, as a periodic box of length $2\pi L$ expands to the whole space. First, for $d\ge 3$, we prove global stability and scattering in an $L$-dependent Gevrey class, with estimates uniform for $L\ge L_0$. The density obeys a two-regime upper bound: the whole-space decay rate $\langle t\rangle^{-d}$ before the critical time scale $T_{\rm disp}(L)\sim L$, and a periodic phase-mixing bound in the rescaled time $t/L$ thereafter. Second, to quantify the nonlinear resonance effect, we identify an echo-Volterra operator governing the nonlinear density memory, and provide sharp estimates for its non-collinear and collinear parts. The non-collinear time-frequency interactions remain uniformly bounded, whereas the collinear interactions do not become comparable to the background until the much later time scale $T_{\rm col}(L)\sim L^{d-1}$. Third, we investigate the stability and convergence of Sobolev initial data. For data of size $\varepsilon$, polynomial Sobolev smallness persists up to the time scale $\varepsilon^{-1} T_{\rm col}(L)$. Within the shorter time scale $T_{\rm disp}(L)\sim L$, compatible periodic solutions converge locally, as $L\to\infty$, to a global solution of the whole-space Vlasov–Yukawa equation.

Exceptional points provide a mechanism for producing strong spectral sensitivity in non-Hermitian wave systems. For subwavelength resonators with highly dispersive material parameters, however, the resonance problem is not a linear eigenvalue problem in the frequency. In this paper, we use the Lippmann-Schwinger formulation and the associated volume-potential reduction to study exceptional points in dispersive resonator arrays. The reduction leads to a finite-dimensional nonlinear matrix pencil whose entries encode both the local material response and the volume-mediated interaction between the resonators. We characterize exceptional points as defective characteristic values of this pencil and derive explicit conditions for dimers and higher-order systems. We show that passive damping and detuning can generate exceptional points without imposed gain, and we formulate an inverse-design condition for placing such degeneracies at prescribed frequencies. We also prove the corresponding Puiseux splitting laws and identify how material dispersion enters the leading sensitivity prefactor. Numerical experiments, including halide-perovskite-inspired dimers and trimers, illustrate the theory and show that the exceptional-point mechanism persists for explicit highly dispersive passive material laws.

In this paper, we establish the bilinear Strichartz estimates for the following linear Schrödinger equation with repulsive inverse-square potential $i\partial_tu+\La u=0$ where $\La=-\Delta+a|x|^{-2}$ with $a\geq0$ for non-radial data. Our proof relies on the physical space method (interaction Morawetz estimate). Such type of method was first introduced in Planchon-Vega [Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 261–290] in flat case, and Planchon-Tzvetkov-Visciglia [Rev. Mat. Iberoam, 39(2023), no. 4, 1405-1436] for harmonic oscillator. Our bilinear estimate extends results obtained in Camps [Ann. Inst. H. Poincaré Anal. Non Linéaire 39 (2022), 1–58] and Pusateri-Soffer [Mem. Amer. Math. Soc. 299 (2024), v+107 pp] to the critical case. As an application, we establish the global well-posedness for cubic NLS with inverse-square potential \begin{equation*} iu_{t} +\La u = -|u|^2u,\ \ (t,x)\in \mathbb{R}\times\mathbb{R}^{3}. \end{equation*} for radial initial data $u_0\in H_a^s$ with $s>\frac{11}{13}$ and $a>0$ where $H_a^s$ is the Sobolev space adapted to the Schrödinger operator with inverse-square potential. The proof relies on Bourgain's high-low frequency decomposition argument and our bilinear estimates. This is the first result on the low regularity well-posedness for NLS with scaling-critical potential.

The discrete Calderón problem aims at recovering the conductivity on the edges of a graph from boundary measurements, encoded by the discrete Dirichlet-to-Neumann (DtN) map. The problem has been intensively studied in the linear case on square lattices since the seminal works of Curtis and Morrow. In this work, we investigate a nonlinear analogue of the discrete Calderón problem for a semilinear second-order elliptic equation on square lattices. We establish three uniqueness results for the conductivity recovery. First, we show that conductivity-dependent corner excitations allow a layer-by-layer reconstruction of the conductivity. Second, we study the linearization of the nonlinear DtN map at an arbitrary background boundary datum and prove that the conductivity and the background potential are uniquely determined by one pair of nonlinear Cauchy data and the linearized DtN map. Third, we show that the linearized data can be replaced by finitely many nonlinear measurements, which uniquely determine the conductivity.

Given $\rho>0$, we consider the problem \[ \text{find $(\lambda,u) \in \mathbb{R} \times H_0^1(B)$ such that } \begin{cases} -\Delta u+\lambda u = |u|^{p-1}u & \text{in } B \\ \int_B u^2\,dx = \rho, \end{cases} \] where $B$ is a ball in $\mathbb{R}^N$, $N \ge 1$, and $1<p<2^*-1$. Without any further restriction on $N$, $\rho$ and $p$, we prove the existence of infinitely many radial solutions, and, in dimension $N \ge 4$, of at least one non-radial solution.

This paper addresses the existence of pullback attractors for a class of nonlinear non-autonomous strongly damped plate equations with delay. The model contains the biharmonic operator, the strong damping term, a nonlinear source term and a delay operator acting on the history of the solution. The interaction between the strong damping mechanism and the hereditary delay effect gives rise to several analytical difficulties in deriving uniform estimates and proving pullback asymptotic compactness. To overcome these difficulties, we construct a modified energy functional adapted to the strong damping structure and employ the contractive function method to handle the delay term in the history phase space. By establishing the existence and uniqueness of weak solutions, uniform pullback estimates and the pullback asymptotic compactness of the associated non-autonomous process, we prove the existence of pullback attractors for the strongly damped plate equation with delay.

For the $k$-self-similar naked singularity solutions of Einstein–scalar field equations constructed by Christodoulou, we construct a family of interior gravitational perturbations leading to trapped surface formation that remain large but finite at the threshold Hölder regularity, while converging to zero in all regularities below the threshold. This extends the interior spherically symmetric result below the threshold in \cite{Li25} to non-spherically symmetric setting. We further construct a genuinely localized family of perturbations whose angular support shrinks to a single point, with smooth convergence to zero away from that point. This exploits the additional angular freedom available beyond spherical symmetry. Moreover, when measured instead in Sobolev regularity, such genuinely localized family of perturbations still converges to zero in all regularities below the same threshold.

In this paper, we study rigidity phenomena associated with a class of overdetermined two-phase problems on the $n$-dimensional sphere $\mathbb{S}^n$. Specifically, we analyze the solutions of coupled elliptic equations defined on a domain $\Omega \subset \mathbb{S}^n$ and its complement, subject to Dirichlet boundary conditions on the interface $\partial\Omega$ and a compatibility condition on the gradient. We utilize two different methods. The method of moving planes works when $\partial\Omega$ is contained inside an open hemisphere. The other method works when $\Omega$ is simply connected in $\mathbb{S}^2$. Both lead to the same conclusion. Namely, the existence of a solution to such overdetermined problems necessarily implies that $\Omega$ is a geodesic ball. In the latter, we prove this rigidity property for three distinct scenarios: a baseline problem with piecewise constant source terms, which is the two-phase version of the torsion problem; a generalization including a class of positive and regular nonlinearities; and finally an eigenvalue problem involving the first Dirichlet eigenvalue of the respective phases.

We prove the existence and uniqueness of global-in-time solutions to the Navier–Stokes–Maxwell (NSM) system for small initial data in the critical Fujita–Kato space $\dot H^{\frac{d}{2}-1}(\mathbb R^d)$, in any dimension $d\geq 3$. Under the additional assumption that the initial data belong to the Besov space $\dot B^{-\frac{d}{2}}_{2,\infty}(\mathbb R^d)$, we also establish the optimal decay rate $t^{-\frac{s}{2}-\frac{d}{4}}$ at infinity of the solution in $\dot H^s(\mathbb R^d)$, for any $s\in\left(-\frac{d}{2},\frac{d}{2}-1\right]$. This is achieved by constructing a Lyapunov functional that is equivalent to the pointwise energy on the Fourier side, and by combining it with an adaptation of the Fourier splitting method to establish its optimal decay rate. All the analysis carried out here - from the global existence theory to the study of the large-time behavior - is performed within a framework that is uniform with respect to the speed of light $c\in(0,\infty)$. In particular, in the non-relativistic limit $c\to\infty$, this allows us to recover the same results for the corresponding limiting magnetohydrodynamic system.

We prove the non-uniqueness of weak solutions to the three-dimensional hyper-viscous compressible Oldroyd-type model with general pressure laws and Newtonian viscous. Using an intermittent convex integration scheme, we construct non-unique weak solutions in $L_t^\gamma W_x^{s,p}$ with $(s,p,\gamma)$ lies in the supercritical regime relative to the Ladyzhenskaya-Prodi-Serrin criteria. In addition, the Newtonian viscosity we considered contains the fractional viscosity $(-\Delta)^\alpha$, where the viscous exponent $\alpha$ can be larger than the Lions exponent $5/4$. In the classical viscous case $\alpha=1$, our construction also yields sharp non-uniqueness in $L_t^pL_x^\infty$ for $1<p<2$. Furthermore, we obtain two singular-limit results. First, we prove the strong vanishing viscosity limit for any weak solutions in $H_{t,x}^{\widetilde{\beta}}$ to the compressible fundamental elastodynamic system. Second, $H_{t,x}^{\widetilde{\beta}}$ weak solutions to the incompressible Oldroyd-type model can be obtain as a low Mach number limit (incompressible limit) of a sequence of weak solutions to the compressible Oldroyd-type model. The main ingredient of our result is a new cancellation mechanism that overcomes the difficulty caused by the relative rigidity of the pressure. To the best of our knowledge, this is the first non-uniqueness results of weak solutions to the hyper-viscous compressible hydrodynamics equations.

This paper studies the existence of pullback $\mathcal{D}$-attractors for non-autonomous Kirchhoff wave equations with strong damping and delay effects in the phase space $\mathcal{E} = C_{H^1_0(\Omega)} \times C_{L^2(\Omega)}$. The model contains a strong damping term $-\Delta \partial_t u$, a nonlocal Kirchhoff term $\Psi(\|\nabla u\|^2)$, a state-dependent delay term $\phi(t, u_t)$, and a time-dependent external force $h(x, t)$. There appear to be no results on the pullback attractors of Kirchhoff wave equations involving both strong damping and delay effects in the literature. To this end, we establish delicate uniform estimates and employ the contraction function method to prove the pullback asymptotic compactness of the associated process, which yields the existence of pullback $\mathcal{D}$-attractors.

We study the Perron extremal solutions of the stationary one-phase Bernoulli problem. These solutions are important in several applications, but not much is known about the regularity of their free boundaries due to their non-variational construction. We show that, in fact, Perron extremal solutions retain some variational features; specifically, they are solutions in the sense of inner variations. We establish this property by showing that extremal solutions arise as infinite-time limits of monotone solutions of the parabolic Bernoulli problem. As a consequence, we are able to describe the structure of singularities for Perron extremal solutions in two dimensions: largest subsolutions have smooth free boundaries, while smallest supersolutions have free boundary points that are either smooth, or whose blow-up limits are two-plane wedges with equal slope.

Traveling waves are fundamental objects in reaction–diffusion systems; however, spatial heterogeneities can distort or inhibit their propagation. Determining whether a given heterogeneity results in propagation failure remains challenging, particularly in systems with coupled components. In this paper, we present explicit constructions of stationary coupled upper and lower solutions tailored to specific heterogeneities for a one-dimensional FitzHugh–Nagumo system. By leveraging the mixed quasimonotone structure of the system, these barriers yield sufficient conditions for the failure of propagation for families of trapped initial conditions. The stationary constructions establish directional signal blocking in FitzHugh–Nagumo neuron models with localized depressed dynamics or abrupt geometric variations, and the resulting stationary barriers persist for sufficiently small recovery diffusion. Additionally, we introduce explicit almost stationary barriers, which provide conditional bounds on the expansion of componentwise threshold regions in homogeneous media.

We propose a linearized Physics-Informed Neural Network (lPINN), a reduced-order neural basis method for forward and inverse differential equations. In an offline stage, lPINN learns operator-compatible continuous neural basis functions from an ensemble of numerical solutions. The basis functions are differentiable through automatic differentiation and are pretrained using solution data together with either derivative information or physics residuals. For each new problem instance, the basis functions are frozen and the solution is obtained by minimizing the governing-equation residual together with applicable initial, boundary, regularization, and observational terms. Unlike surrogate and operator-learning methods, the training data define the trial space offline, while the instance-specific solution is computed online by enforcing the governing physics. Relative to vanilla PINNs, lPINN pretrains the nonlinear hidden-layer representation offline and performs online inference only in the final linear layer. We evaluate lPINN on forward and inverse problems for the advection-diffusion equation, Burgers' equation, and the nonlinear pendulum equation. Compared with vanilla PINNs, lPINN achieves lower solution and parameter errors while reducing online inference times by approximately one to more than three orders of magnitude, with the largest gains generally observed for limited residual or measurement data. Cross-resolution experiments show that the learned continuous representation can be evaluated on finer meshes without retraining and with nearly unchanged accuracy.

We prove a sharp cut-off resolvent estimate for the Laplacian on non-compact Riemannian manifolds whose ends satisfy the Cardoso–Vodev assumption and whose geometry near the trapped set agrees with that of an even asymptotically hyperbolic manifold with the corresponding resolvent estimate. As applications, we show the global Strichartz and spectral projection estimates on negatively curved asymptotically hyperbolic surfaces, without assuming the evenness at infinity.

We prove global existence, both to the future and to the past, for a family of scalar quasilinear wave equations in two space dimensions for sufficiently small $C_c^\infty$ initial data. These equations have cubic leading quasilinear nonlinearities of the form $u^2\partial^2u$ and satisfy a sign condition that allows the cubic null condition to fail. To explain the sign condition, we derive the geometric reduced system for quasilinear wave equations with cubic leading nonlinearities in two space dimensions and introduce a notion of geometric weak null condition. We prove that the geometric weak null condition holds for the model scalar equation if and only if the sign condition holds. At the level of the geometric reduced system, the sign condition prevents the corresponding characteristic curves from intersecting in finite time. We also recover the geometric reduced system from the global solutions in our main theorem. If the model scalar equation satisfies the sign condition but fails the cubic null condition, and if $u$ is a nonzero global solution to the model equation, we show that, for $r=|x|$, the $L^\infty$ norm of $t^{\frac{1}{2}}\partial_r^2u$ in a region where $|r-t|\ll t$ tends to infinity as $t\to\infty$. Thus, its asymptotic behavior differs from that of a solution to the linear wave equation $\Box \psi=0$ with $C_c^\infty$ data.

In this paper, we derive an explicit solution of the stationary prediction with expert advice PDE for five experts. The formula is given in three regions. In the first two regions, it is the four-expert solution plus a single integral with an elementary positive density. In the third region, it is a finite sum of hyperbolic products whose coefficients are determined by one scalar quadrature. Our formula establishes that the direction $(1,0,1,0,0)$ is optimal throughout the ordered sector, and that the COMB strategy $(1,0,1,0,1)$ is optimal only on a lower dimensional subset of the sector (where $x_1=x_2$ and $x_3=x_4$). This disproves the COMB optimality conjecture of Gravin, Peres and Sivan. The verification of the Hamiltonian inequalities is a tedious task, part of which is completed with a computer assisted proof. The verification reduces to 21 scalar inequalities, which we prove using 147 exact rational Bernstein polynomial certificates. The exact certificates and their independent arithmetic checks are included in a supplement to this paper, and a Lean 4 formalization machine-checks the verification and both main theorems, apart from the viscosity characterization.

We prove the superlinear gradient growth for the one dimensional Vlasov-Poisson equation on the torus. We show them by first constructing the stationary solution which has a hyperbolic flow (locally). Then we perturb the solution and use the $L^\infty$ stability of the characteristic velocity field to show the superlinear growth. This proof was inspired by the work of Denisov.

We study front propagation in a time-fractional Amari neural field equation with a Caputo derivative and a Heaviside firing-rate function. For a class of nonnegative even connectivity kernels and strictly decreasing $C^1$ front-like initial data, we reduce the Cauchy problem to a nonlinear Volterra equation for the threshold location. We prove the existence and uniqueness of a global front solution, determine the short-time behavior of its threshold location, and identify its asymptotic speed. We also construct the associated asymptotic traveling-wave profile and establish uniqueness of the speed and profile, up to translation, in the nonzero-speed case. Using a rigidity argument, we prove the asymptotic linearity of the local increments of the threshold location and obtain uniform convergence to the profile in the threshold-centered frame. Finally, we characterize the kernel-dependent tail asymptotics of the wave profile.

We adapt the renormalized energy method developed in \cite{duerinckx2020mean} in order to prove the monokinetic mean field limit for relativistic Newtonian dynamics. The resulting monokinetic PDE is the relativistic Euler-Poisson equation. Since the position evolves relativistically in comparison to the momentum, the kinetic part of the modulated energy has to adjusted, leading to further obstructions that are not present in the non-relativistic settings. A weak-strong stability principle for the relativistic Euler-Poisson equation is investigated, followed by a renormalization procedure leading to the mean-field limit. Our main result constitute the first mean-field limit for relativistic Coulomb flows.

This is an expository article describing the works of Yu Deng, Zaher Hani and Xiao Ma on the long-time derivation of the fluid equations from the statistics of Newtonian particle collisions.

We consider the positive principal eigenvalue of an elliptic problem with a Robin boundary condition and bang–bang indefinite weight $\kappa\mathbf 1_D-\mathbf 1_{\Omega\setminus D}$, and ask where a favourable region $D$ of prescribed small volume $|D|=\delta$ should be located. Put $\varepsilon=\delta^{1/N}$ and $\tau_\delta=\alpha_\delta\varepsilon$. We prove that there is a finite threshold $\tau_*=\tau_*(N,\kappa)$, independent of the ambient domain, such that $\delta^{2/N}\Lambda_\delta(\alpha_\delta)\to\Lambda_{\mathbb H}(\tau)$ whenever $\tau_\delta\to\tau<\infty$. If $\tau<\tau_*$, optimal small regions concentrate at the boundary. If $\tau>\tau_*$, their concentration centres move to distances much larger than $\varepsilon$ from the boundary, and after recentring and rescaling the favourable sets converge in measure to the whole-space optimal ball. At $\tau=\tau_*$, the half-space problem admits a compact boundary optimiser as well as minimising sequences escaping to infinity. In the finer regime $\tau_\delta=\tau_*+\sigma\varepsilon+o(\varepsilon)$, we determine the first-order competition between the boundary and interior configurations. Tangential symmetry of compact threshold optimisers reduces the geometric correction to mean curvature. In particular, every fixed finite Robin coefficient is asymptotically in the boundary regime. Numerical experiments illustrate the boundary–interior transition, the three cases in the first-order selection law, and the curvature-dependent boundary location; for $N=2$ and $\kappa=1$ they suggest a transition near $\tau_*\approx3.2$.