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Wednesday, 4 November 2026: 59 new papers, 7 cross-lists and 21 replacements in Analysis of PDEs (math.AP), in arXiv's order, with full abstracts. The same papers as arXiv's own listing, set as a newspaper; every title links to arXiv.
We study the asymptotic stability and quantitative time-decay rates of rarefaction waves for one-dimensional $n\times n$ strictly hyperbolic systems of viscous conservation laws. While the nonlinear stability of rarefaction waves for $2\times 2$ systems was established by Xin \cite{Xin1988,Xin1989}, extending such results to general $n\times n$ systems presents additional difficulties arising from the possible coexistence of positive and negative transverse relative characteristic speeds and nonlinear interactions between distinct characteristic families. Moreover, establishing standalone $L^p$ energy estimates and explicit algebraic decay rates for general viscous systems without $L^1$ localization remains a significant challenge. In this paper, we establish both the global $H^2(\R)$ asymptotic stability and, for the first time, standalone physical-space $L^p$ energy estimates and explicit algebraic $L^p$ time-decay rates for weak rarefaction waves. Specifically, for every integer $p\ge 3$, we prove that the perturbation $U(t)-\bar{U}(t)$ around the smooth Burgers rarefaction profile $\bar{U}$ decays in $L^p(\R)$ at the rate \[ \|U(t)-\bar{U}(t)\|_{L^p}\le C_p\bigl(\|U_0-\bar{U}(0)\|_{H^2}^2+\delta^3\bigr)^{1/2}(1+t)^{-\frac{1}{2}(\frac{1}{2}-\frac{1}{p})}, \] where $\delta$ denotes the wave strength. Our proof combines a diagonal characteristic weight matrix that yields coercive rarefaction damping in all characteristic modes with directed wave-interaction functionals that exploit the strict separation of characteristic speeds to control nonlinear coupling between distinct characteristic families. These estimates lead to a closed $L^p$ Lyapunov inequality, from which the algebraic time-decay rates follow by the one-dimensional Nash inequality.
With the aim of extending to parabolic equations the framework developed by the first three authors in [Dipierro, Savin, and Valdinoci, Rev. Mat. Iberoam. (2019)], we introduce a notion of (viscosity) solutions for the space-fractional heat equation modulo $k$. In turn, we are able to define $s$-caloric functions with polynomial growth at infinity, showing also related Schauder estimates, a Liouville-type result, and a stability property with respect to the $L^\infty_{\rm{loc}}$-convergence.
We give new sufficient conditions for uniqueness of probability solutions to one-dimensional Fokker-Planck-Kolmogorov equations in the case where the drift coefficient depends on time.
We prove a sharp magnetic improvement of the family of weighted inequalities on the upper half-space $\mathbb{R}^{N+1}_+$ that interpolate between the Hardy inequality and the trace Hardy inequality. The magnetic potentials are of Aharonov-Bohm type, $\mathbf{A}(x)=|x|^{-1}\boldsymbol{a}(x/|x|)$, where the tangential field $\boldsymbol{a}$ may be singular on a closed subset of the sphere. We compute the sharp constant explicitly: it is given by the constant of the corresponding nonmagnetic inequality, evaluated in a virtual dimension that depends on the first eigenvalue of the magnetic Laplacian on the sphere. In particular, the magnetic trace Hardy constant strictly exceeds its Euclidean value whenever this eigenvalue is positive. The proof rests on a ground state representation of the deficit, which also shows that the sharp constant is not attained in the energy space. As a consequence, we obtain the sharp Hardy inequality for the fractional powers $(-\Delta_{\mathbf{A}})^\sigma$, $0<\sigma<1$, of the magnetic Laplacian: the constant is the same on smooth functions supported away from the singular set and on the form domain of $(-\Delta_{\mathbf{A}})^\sigma$, and it is not attained. Examples include the planar Aharonov-Bohm potential with nonintegral flux and, in dimension three, the Aharonov-Bohm potential and the Dirac monopole.
We study the concentration of logarithmically interacting defects, such as Ginzburg--Landau vortices and screw dislocations, on closed curves in two dimensions, subject to a confining field. We prove that a measure supported on a smooth closed curve solves the mean-field transport equation exactly when, in Lagrangian labels, the curve moves with the negative induced velocity and each label keeps its mass. This contour law is a gradient flow of the mean-field energy. We prove that the gradient of the energy on a curve has a leading term proportional to the operator $|D|$ with Fourier symbol $|k|$, which prevents every smooth equilibrium curve from being a local minimizer, in a gradient-flow analogue of the Kelvin--Helmholtz instability of vortex sheets. In a radially symmetric case with quadratic confinement, we find an explicit circular equilibrium whose linearization is diagonal in Fourier modes, and we show that, as $N\rightarrow \infty $, the linear spectrum of the regular $N$-gon converges, mode by mode, to that of the circle. Finally, we compute a critical point on two concentric circles with lower energy than the single circle, indicating a rich family of multi-circle equilibria.
We prove a new, holomorphic version of the Lie-Trotter product formula for nonlinear local semigroups with a parameter. We show that the product converges to the solutions of a nonlinear evolutions equation with a parameter. The solution is holomorphic in time on a centered at zero conic set in the complex plane and depends continuously on the parameter, and analytically on the initial data.
The large time dynamics of the weakly damped non-autonomous Korteweg-de Vries-Burgers equation \begin{equation*} u_t + u_{xxx}+ uu_x + \lambda u - \mu u_{xx} = g(x,t) \end{equation*} in the phase space $L^2(\mathbb R)$ is investigated. Due to the low regularity of the initial data, the framework of Bourgain spaces is employed. We first obtain the existence of a unique pullback attractor $\{\mathcal A_\mu(t)\}_{t\in\mathbb R}$ in $L^2(\mathbb R)$ for each $\mu \ge 0$. Then we show an optimal asymptotic smoothing effect of the equation, namely the pullback attractor is compact in $H^3(\mathbb R)$. These results are proved by using an energy equation method, splitting solutions into parts with low and high frequencies and exploiting a dispersive regularization. Furthermore, the attractors are shown to be upper semicontinuous as $\mu \to 0$ and have finite fractal dimensions in $H^1(\mathbb R)$.
We develop a new approach to studying discontinuity sets of measure-preserving convex gradient maps between nonconvex domains. Such maps optimize the Euclidean distance squared. Our idea is to introduce a new `phantom' optimal transport problem, that transports the missing mass from the convex hull of the target to the `excess' Monge--Amp\`ere measure of the minimal convex extension of the Brenier potential. We show that the closure of the discontinuity set of the original map contains the support of this phantom target. Moreover, inside a planar source domain these two sets actually coincide; on $n$-dimensional domains they coincide $\mathcal{H}^{n-1}$-a.e. The Hausdorff exponent $n-1$ is sharp in every dimension $n\geq3$. For planar polygonal targets, we establish a quantitative global length bound for the closure of the discontinuity set depending only on the target geometry, valid for bounded positive densities.
Let $\Omega\subset\mathbb{R}^N$ be a bounded open connected set, and let $q\in(1,2)$. We consider the sublinear $q$-eigenvalue problem for the Dirichlet Laplacian $-\Delta u=\lambda\|u\|_{L^q(\Omega)}^{2-q}|u|^{q-2}u$, $u\in H^1_0(\Omega)\setminus\{0\}$, whose first eigenvalue $\lambda_1(\Omega)$ is simple, with eigenfunctions of constant sign. The definition of a second eigenvalue is not unique. Following the approach introduced by Molle and Passaseo for the linear Laplacian, we construct the first non-trivial curve of the corresponding Fu\v{c}\'ik spectrum, and show that it has a unique diagonal point $(\gamma(\Omega),\gamma(\Omega))$. The Fu\v{c}\'ik-selected second value $\lambda_2^F(\Omega)=2^{1-\frac{2}{q}}\gamma(\Omega)$ is strictly larger than $\lambda_1(\Omega)$, admits a two-phase variational characterisation, and satisfies a sharp Hong-Krahn-Szeg\H{o} inequality: among sets of given measure it is minimised by the union of two equal balls. The selected solution at the diagonal point is a classical $(2,q)$-eigenfunction exactly when its nodal $L^q$-masses are balanced. We finally show that on balls in dimensions $N\geq2$ every Fu\v{c}\'ik-selected solution is foliated Schwarz symmetric.
We study overdetermined boundary problems for the maximal surface equation with zero Dirichlet data and constant nonzero Neumann data . For the exterior of a bounded domain with connected boundary in $\mathbb{R}^n$, $n\geq2$, we prove that such domain must be the complement of a ball and the solution is radial. In dimension two, we also show that a positive solution on a domain bounded by a properly embedded noncompact connected curve forces the domain to be a half-plane and the solution to be affine. The proofs combine the divergence structure of the equation with the induced geometry of maximal graphs.
In this paper, we construct Feynman propagators on exponentially decaying perturbations of stationary spacetime with compact Cauchy surface. The propagators are defined as parametrices of the wave operator on exponentially weighted Sobolev spaces defined by complex contours $\Gamma$. We then apply this theory to the computation of indexes of Dirac operators using a continuous Wick-like rotation to relate it to the Riemannian case.
We consider the Kuramoto-Sivashinsky equation $u_t+uu_x+u_{xx}+u_{xxxx}=0$ with $L$-periodic boundary conditions and zero spatial mean. We prove that every smooth solution satisfies $\|u(t)\|_{L^2}\leq CL^{5/6}+CL^{3/2}/t$ for $t>0$, where $C$ is a universal constant. In particular, $\|u(t)\|_{L^2}\leq CL^{5/6}$ for all $t\geq L^{2/3}$, independently of the initial norm.
We develop a decomposition technique to investigate maps from the Dbar data to the derivative NLS potentials. We define a new integral operator which is equivalent to a modified Dbar problem relevant to the derivative NLS spectral problem, including the Kaup-Newell spectral problem, the Chen-Lee-Liu spectral problem and the Gerdjikov-Ivanov spectral problem. The well-posedness for the integral operator with the kernel $\mathrm{e}^{ik^2x\sigma_3}$ is discussed via the decomposition technique. Under different constraint conditions, the derivative NLS potentials are reconstructed via the Dbar data. We discuss the symmetry condition of the Dbar data as well as the compatibility of the derivative NLS spectral problem with the modified Dbar problem. We show that the maps from the Dbar data to the derivative NLS potentials are Lipschitz continuous.
A gas giant Riemannian metric $g$ on a compact $n$-dimensional manifold-with-boundary $M$ blows up at the boundary $\partial M$ at the rate $1/x^\alpha$, for $\alpha \in (0,2)$, where $x \in C^\infty(M)$ is a boundary-defining function. Such metrics arise in the modeling of acoustic waves in gaseous planets. De Hoop-Ilmavirta-Kykk\"{a}nen-Mazzeo [arXiv:2403.05475] showed that, if $\alpha > 2/n$, the Laplace-Beltrami operator $\Delta_g$ is essentially self-adjoint and has discrete spectrum. Moreover, its spectrum is stable under truncations of $M$. The proof leverages the 0-calculus of Mazzeo-Melrose. We extend this theory by introducing a 0-transition calculus for studying families of 0-operators. We demonstrate a parametrix construction for $\Delta_g$ that is stable under truncations. Using this, we study the behavior of the resolvent of $\Delta_g$ under truncation and extend the characterization of the spectrum obtained by de Hoop-Ilmavirta-Kykk\"{a}nen-Mazzeo. In particular, we prove that (on appropriate functional-analytic domains) simple eigenvalues admit asymptotic expansions in the truncation parameter $\epsilon$ and that the corresponding eigenfunctions admit joint asymptotic expansions in $(\epsilon, x)$.
We prove local existence and uniqueness, propagation of Sobolev regularity, and continuation criteria results for the characteristic initial value problem for the 3D compressible Euler equations. This is the Cauchy problem with initial data given on a pair of transversally intersecting (acoustically) null hypersurfaces. The region of existence in our main theorem is a future-neighborhood of the union of the data null hypersurfaces, where the size of the neighborhood depends on various norms of the data as well as the structure of the data null hypersurfaces. We allow for any equation of state with positive sound speed, and we do not make any symmetry, irrotationality, or isentropicity assumptions. Our analysis relies on the geometric formulation of the flow derived in [Speck19], which decomposes the equations into a "wave-part" coupled to a "transport-div-curl" part. Compared to Rendall's celebrated solution of the characteristic initial value problem for wave equations [Rendall90], substantial new ideas are needed to handle the fact that the compressible Euler equations are more complicated than pure wave equations.
In this paper, we first establish a weighted bilinear fractional commutator estimate involving the spatial weight $\langle x \rangle^{2m}$. This estimate provides a powerful tool to resolve the difficulties arising from the nonlocality of the fractional Laplacian $(-\Delta)^{\sigma/2}$ and the interaction between the unbounded weights and the nonlocal operator at infinity. We then apply this result to investigate the Cauchy problem for the fractional wave equation with time-dependent damping and source nonlinearity: $$ \begin{cases} \partial_t^2 u(t,x) + (-\Delta)^{\sigma/2} u(t,x) + b(t) \partial_t u(t,x) = |u(t,x)|^p, & (t,x) \in (0,T) \times \mathbb{R}^N, \\ u(0,x) = \varepsilon u_0(x), \quad \partial_t u(0,x) = \varepsilon u_1(x), & x \in \mathbb{R}^N, \end{cases} $$ where $\sigma \in (0,2)$, the initial data scale with a small parameter $\varepsilon > 0$, and the time-dependent damping coefficient $b(t)$ satisfies $b(t) \approx (1+t)^{-\beta}$ for $\beta \in [-1, 1)$. Utilizing scaling variables and constructing a weighted energy framework based on Theorem 1.1, we derive a priori differential inequalities for the total energy. For the supercritical case $p > p_c := 1 + \frac{\sigma}{N}$, we prove the small-data global existence and uniqueness of strong solutions in weighted Sobolev spaces $H^{\sigma/2,m}(\mathbb{R}^N) \times H^{0,m}(\mathbb{R}^N)$ by closing a bootstrap argument. For the subcritical ($1 < p < p_c$) and critical ($p = p_c$) cases, we establish precise lower bounds for the lifespan $T_\varepsilon$ of blow-up solutions via delicate ODE analysis. Together with the corresponding upper estimates, our results fully determine the sharp lifespan estimates for all $\beta \in [-1, 1)$, confirming that $p_c = 1 + \frac{\sigma}{N}$ is indeed the critical exponent for the fractional wave equation with time-dependent damping.
We establish the \(L^p\)-boundedness, \(1<p<\infty\), of the non-centered Hardy--Littlewood maximal operator associated with \(\ell^q\) balls, \(1<q<\infty\), for the non-radial exponential measure \[ d\mu(x)=e^{-|x_1|-\cdots-|x_d|}\,dx \] in every dimension \(d\ge1\). This settles the higher-dimensional Euclidean case left open by Nowak, Sasso, Sj\"ogren and Stempak, and extends their results to the full range \(1<q<\infty\). We also prove that these operators are not of weak type \((1,1)\) when \(d\ge2\). Together with the known results for \(q=1\) and \(q=\infty\) for the exponential measure, this gives a complete description of the \(L^p\)-boundedness and weak type \((1,1)\) behavior for all \(1\le q\le\infty\). {We also extend the strong \(L^p\)-boundedness to the Laguerre-type measures \(d\mu_\alpha(x)=\prod_{j=1}^d|x_j|^{\alpha_j}e^{-|x_1|-\cdots-|x_d|}\,dx\) for all \(\alpha_j\ge0\) and \(1\le q\le\infty\), and for the full range \(\alpha_j>-1\) when \(q=\infty\).} {The main arguments rely} on geometric estimates for sections of \(\ell^q\) balls and a slicing argument.
We consider a two-well problem within geometrically linear elasticity, on a bounded Lipschitz set in dimension $n\ge 2$. We assume that the two wells are compatible through a symmetrized rank-one connection and that the displacement matches their average on the boundary, and we include a regularization based on the total surface energy. We show that there are two distinct regimes, with the energy scaling differently in terms of the surface-energy coefficient $\eps$: it is proportional to $\eps^{2/3}$ if there are two distinct compatible twin-normal directions, and to $\eps^{4/5}$ if there is only one. This matches previous findings in rectangles and other special geometries, mostly in two dimensions, and the lower bounds obtained by a Fourier method by Rai\c{t}\u{a}, R\"uland, Tissot and Tribuzio (2024). Our results are obtained by explicit constructions, which contain a self-similar refinement of oscillations close to the boundary, and by a short real-space proof of the matching lower bounds.
For every finite-dimensional simple Euclidean Jordan algebra $V$ of rank at least three, we realize the function $\operatorname{tr}(x^3)/|x|$ on its traceless subspace $V_0$ as a $C^{1,1}\setminus C^2$ viscosity solution of a uniformly elliptic equation $F(D^2 u)=0$ in $\mathbb{R}^{\operatorname{dim}V_0}$. This unifies and extends earlier constructions from Cartan isoparametric cubics, Hsiang minimal cubics, and traceless real symmetric matrices.
We investigate the Steklov--Dirichlet eigenvalues on shell-type domains in $\mathbb{R}^n$, $n\geq 2$, where the Steklov and Dirichlet conditions are imposed on the outer and the inner boundaries, respectively. We introduce the Green function associated with the problem, which is well-defined for spectral parameters outside the spectrum and exhibits blow-up behavior at eigenvalues, and establish its spectral representation. We also examine its differentiability and derive its shape derivative under translations of the inner boundary. Using these results and a layer potential approach, we propose a computational framework for optimizing the eigenvalue with respect to the position of the inner boundary, incorporating an eigenvalue estimator for which a local error bound is established. Numerical experiments for various inner and outer boundary shapes demonstrate the effectiveness of the proposed approach in identifying configurations that maximize the first eigenvalue.
For a bounded domain $\Omega\subset\mathbb R^d$, the spherical average $E_\Omega(R)=\int_{\mathbb S^{d-1}}|\widehat{\mathbb{1}_\Omega}(R\omega)|^2d\omega$ is never $O(R^{-a})$ with $a>d+1$, and for smooth boundaries it is $O(R^{-d-1})$, as it is for the ball, whether or not the boundary is curved. In 2003, Brandolini, Hofmann and Iosevich extended this to every convex domain and to every domain with $C^{1,\frac{1}{2}}$ boundary, and this regularity condition has not been improved since. How much regularity does the decay $R^{-d-1}$ require? We show that among H\"older classes the answer is $C^{1,\frac{1}{4}}$. Every bounded domain with $C^{1,\frac{1}{4}}$ boundary satisfies $E_\Omega(R)\leq C_\Omega R^{-d-1}$, while for every $0<s<\frac{1}{4}$ any smooth domain can be altered near a single boundary point to produce a $C^{1,s}$ domain for which this bound fails along a sequence of radii. More generally, when the boundary has fractional Sobolev regularity $W^{1+s,p}$, we determine the optimal decay exponent under a Dini condition on the normal, and in a large range of $s$ and $p$ without it. The upper bounds follow from $L^2$ estimates for nondegenerate oscillatory integral operators whose phases are distances between points of two graphs with Dini or fractional Sobolev gradients, and hence barely better than $C^1$. These are far too rough for the integrations by parts techniques from the smooth theory. We replace derivatives by finite differences adapted to the phase, a device that we expect to be useful in a variety of related problems.
In dimension $n\ge2$, a homeomorphism of $R^n$ preserving BMO space therein was proved to be quasiconformal, up to an orientation reversing reflection.
Let $\Omega$ be a bounded $C^1$ domain in $\mathbb{R}^d$, $d\geq2$. We establish vector-valued resolvent estimates for the Dirichlet Stokes system in $L^q(\Omega;\ell^2)$ for every $1<q<\infty$. The estimates are uniform in the number and choice of spectral parameters in any fixed sector avoiding the negative real axis. These estimates imply that the Stokes operator is $\mathcal R$-sectorial of angle zero and has maximal $L^r$-regularity for every $1<r<\infty$. Our proof builds on the perturbation approach of Geng and Shen, with a different half-space argument to handle independently varying spectral parameters.
In this paper we consider balance laws with measure-valued sources modeling commuters traveling along a road from residential to workplace locations. We introduce a notion of $(m_I, m_F)$-entropy solution, prove a compactness result for this class, and establish existence of solutions with prescribed spatial marginals for the sources representing the lodging and workplace distributions. Moreover, adopting a Lagrangian viewpoint, we describe commuting strategies as measures on Lipschitz curves with assigned entry and exit times, compatible with a given transport plan. Using the compactness result and a semi-discrete micro-macro scheme with source, based on the celebrated Follow-the-Leader model, we prove existence of admissible commuting strategies. Finally we prove the existence of optimal solutions for classical traffic cost functionals.
We study the spectral instability of small-amplitude Stokes waves in finite-depth flows with constant vorticity. A center-manifold reduction and the periodic Evans function give two explicit indices, one for long-wave instability near the origin and the other for an order-two resonance away from the origin. We prove instability when the low-frequency index is positive or the high-frequency index is nonzero, and obtain the leading expansions of the unstable spectral points. At low frequency, first- and second-order monodromy coefficients determine the leading figure-eight curve. At high frequency, we prove that the resonance exists uniquely and derive a limiting ellipse. An algebraic identity expresses the high-frequency index as a positive factor times the square of a real coupling coefficient. Thus every parameter pair with nonzero coupling gives high-frequency instability. The explicit formulas also permit a comparison of the two criteria in the parameter plane.
In the context of heat conduction, the equations of Maxwell-Cattaneo is a successful alternative to the Fourier's law. In 2009, Christov proposed a frame-independent (or objective) generalization of Maxwell-Cattaneo equation. For a more general class of objective heat conduction Maxwell-Cattaneo-Christov equations proposed by Angeles, a uniqueness result is proved, applied to an incompressible fluid.
We study positive radial singular solutions of $-\Delta u=f(u)$ in two space dimensions. Fujishima, Ioku, Ruf, and Terraneo constructed such solutions under asymptotic assumptions on $f$ and an additional convergence-rate condition, denoted by $(f_3)$. We prove their local existence result, with the same approximate solution and remainder estimate, under the original assumptions $(f_1)$ and $(f_2)$ alone. The proof combines a contraction on finite intervals with a nonlinear energy estimate uniform in the terminal point and requires only $C^2$ regularity of $f$ near infinity. We also obtain derivative asymptotics and, under additional global assumptions, a singular Dirichlet solution which solves the equation distributionally across the origin. An analytic oscillatory example satisfies our hypotheses but violates both $(f_3)$ and the derivative assumptions of Kikuchi and Kumagai. For every fixed finite iterate of the exponential of order at least two, we verify the hypotheses and quantify the remainder; condition $(f_3)$ fails for all iterates of order at least three.
We study the spectral stability of small-amplitude Stokes waves with constant vorticity in infinite depth. We describe the spectrum near the origin and near the first nonzero double eigenvalue. For nonzero vorticity $\omega<2\sqrt{g/3}$, where $g$ is the gravitational constant, the eigenvalues near the origin form a figure eight, whereas for $\omega>2\sqrt{g/3}$ the local spectrum remains purely imaginary. Away from the origin, the interaction of two Fourier modes whose indices differ by two produces an unstable isola when $\omega>-\sqrt{g/6}$, $\omega\ne0$, and the resonant wave numbers are noninteger. This curve is a real analytic oval with an explicit leading ellipse and maximal growth of quadratic order in the amplitude. We also prove instability at the integer resonances needed to cover the remaining vorticities. Together with the classical irrotational result, these two mechanisms show that sufficiently small nonzero Stokes waves are spectrally unstable for every fixed constant vorticity. The proof uses finite-dimensional spectral reductions of the full Euler linearization. Near the origin, the finite-dimensional linearized operator has a one-sided limit as the Floquet exponent vanishes, which differs from the periodic linearization. We use the Kato transformation to construct a three-dimensional reduction with biorthogonal bases and use all three eigenvectors of the reference operator to calculate the Benjamin--Feir coefficient. At the nonzero collision, a real two-dimensional reduction and an explicit second-order calculation determine the unstable curve.
Starting from a fully three dimensional model of finite viscoelasticity in the Kelvin-Voigt rheology, we derive a bending model for thin, viscoelastic rods. For this, we utilize a formulation of the model as a time-dependent metric gradient flow and pass to the limit by establishing convergence of curves of maximal slope using $\Gamma$-convergence. A posteriori, we relate the metric gradient flow limits to gradient flows in Hilbert spaces and discuss their properties. In contrast to previous results, where limit deformations are infinitesimal, in the bending regime, we deal with large rotations. This prevents us from using an additive ansatz for the construction of recovery sequences. Instead, we adapt from a previous result by the authors a multiplicative ansatz for elastoplastic Cosserat rods to the viscoelastic setting. For regularization purposes, we consider strain gradient terms in the three dimensional model that vanish in the limit.
We study the periodic homogenization of a stationary generalized Navier--Stokes system in a periodically perforated domain $\Omega\subset\mathbb R^N$, $N\in\{2,3\}$, with a heterogeneous bulk stress of general Orlicz growth and an oscillating nonlinear Navier friction law on the pore walls, both at the Darcy scale. If the lower growth exponent at infinity satisfies $p^-_\infty>3N/(N+2)$ and the inertial exponent exceeds $\kappa_c:=1+N(2/p^-_\infty-1)_+$, inertia vanishes and, under strict monotonicity of the bulk stress, the whole family converges to the nonlinear Darcy law $U=\mathcal K(f-\nabla p_0)$, $\operatorname{div}U=0$, $U\cdot\nu=0$. The permeability map $\mathcal K$ is continuous, strictly monotone, obeys two-sided Orlicz bounds and incorporates the wall slip through a nonlinear cell problem, and the extended microscopic pressure converges strongly to $p_0$. The proof relies on a uniform Orlicz--Korn inequality, a Darcy-scaled two-scale trace theorem and a coupled bulk--surface Minty argument.
We study the sharp-interface limit of the two-dimensional incompressible Navier--Stokes/Allen--Cahn system with constant mobility and prove the existence of global integral varifold solutions to the coupled Navier--Stokes/mean curvature flow system. The limiting interface measure satisfies Brakke's inequality, and its normal velocity equals the sum of its mean curvature and the normal projection of the fluid velocity.
A two-species aggregation-diffusion system on the one-dimensional torus with antisymmetric cross-interactions modeling pursuit-evasion dynamics is studied. The self-interaction strength is used as a bifurcation parameter, and the linear stability of the homogeneous state is characterized by Fourier analysis. It is shown that non-reciprocal coupling enlarges the stability region. In a symmetric diffusion setting, exponential convergence to the homogeneous state is established for sufficiently weak self-interactions by means of an entropy argument, with a decay condition independent of the cross-interaction strength. Under suitable nondegeneracy and spectral assumptions, the first loss of stability is shown to occur through a Hopf bifurcation associated with a single spatial Fourier mode. A branch of time-periodic patterns is thereby obtained, along which the two species oscillate with a fixed phase difference. An explicit formula for the cubic Hopf coefficient is derived, and an essential dependence of the direction and stability of the bifurcating branch on the second spatial harmonic is identified. In the case of matching diffusion coefficients and interaction kernels, sufficiently strong non-reciprocity is shown to transform a subcritical stationary instability of the corresponding reciprocal system into a supercritical Hopf bifurcation with stable periodic solutions. An exchange-of-stability result is proved, and the theory is illustrated for Kuramoto, two-mode cosine, and top-hat interaction kernels.
We study boundary observability for Dirichlet wave equations with symmetric, periodic coefficients in bounded $C^{2,1}$ domains of dimension $d\ge2$. We establish two-sided observability estimates, uniform in the oscillation scale $\varepsilon$, for initial data generated by eigenfunctions with eigenvalues at most $\delta\varepsilon^{-4/3}$ with some small factor $\delta $. The proof reduces wave observability to boundary gradient estimates for narrow spectral clusters. These estimates are obtained using a second-order expansion and a uniform commutator estimate for harmonic extension in boundary layers.
We establish that any nontrivial anharmonic Fourier series with exponents quadratically close to the positive integers changes sign on each interval of length 2$\pi$.
The aim of this article is to derive a stochastic interpretation of the Water Waves equations, based on the location uncertainty principle (LU). Next, these equations are used to infer simpler models from scaling limits, such as the Saint-Venant, Boussinesq and Serre-Green-Naghdi ones. Once adapted to the stochastic setting, these models have been successfully implemented numerically, and we show several experiments to assess their properties -- for example the breaking of the wave symmetry, and the influence of time correlated noises.
This paper investigates the analytical properties of the principal eigenpair for the clamped transmission eigenvalue problem, which arises in the context of biharmonic (i.e. flexural wave) scattering by a clamped planar obstacle. By deriving a suitable Rayleigh quotient, we provide rigorous proofs for several fundamental properties that were previously only observed through numerical experiments. One of the results we have shown here is that the first eigenvalue is monotone decreasing with respect to the domain. Furthermore, we prove that the corresponding eigenfunction is of a single sign, a property analogous to the principal eigenfunction of the Dirichlet eigenvalues of the Laplacian. Another key result is a Faber--Krahn type inequality, establishing that the disk minimizes the principal eigenvalue among all domains of a given measure. The theoretical results are supported by a comprehensive numerical investigation.
This paper is concerned with the inverse potential scattering problem for the three-dimensional fractional Helmholtz equation. For every $0<\alpha<1$, we establish an increasing stability of determining the potential function from multifrequency exterior measurements of the scattered field. The stability consists of a Lipschitz data discrepancy and a logarithmic remainder that decreases when the upper measured wavenumber increases. The key ingredient in the analysis is to construct appropriate geometric optics solutions, which relate the X-ray transform of the potential to the measured fields. As a consequence, the stability is derived by applying analytic continuation. Two-dimensional numerical examples with noisy near-field data illustrate the benefit of enlarging the measured frequency band for three representative fractional orders.
We study forward and inverse problems for weakly viscous nonlocal wave equations with elastic term $(-\Delta)^s u$ and dissipative term $(-\Delta)^r\partial_tu$, where $0\leq r<s$. For time-dependent potentials in suitable $L^\infty(0,T;L^p(\Omega))$ classes, we establish finite energy well-posedness, strong continuity in time, and an energy identity by parabolic regularization and time-regularization arguments. Under explicit nonlocality and nonresonance assumptions on the fractional orders, we prove Runge approximation in $L^2(0,T;\widetilde H^r(\Omega))$ and hence in $L^2(\Omega\times(0,T))$. As an application, partial exterior Dirichlet-to-Neumann measurements uniquely determine the potential in all three exponent regimes of the Runge theorem, including integer elastic orders with noninteger dissipative orders. The recovery argument combines equality of solutions with the same exterior input, unique continuation or the entanglement principle, and a single Lebesgue-space Runge approximation. It applies to the full admissible potential class, without spatial boundedness, sign, or smallness assumptions.
We study very weak solutions and Runge approximation for weakly viscous nonlocal wave equations with elastic term $(-\Delta)^su$ and dissipative term $(-\Delta)^r\partial_tu$, where $0\leq r<s$. For time-dependent potentials satisfying a smallness condition, we establish a transposition theory for sources in the dual of the natural space--time energy space. Under explicit nonlocality and nonresonance assumptions, we prove Runge approximation in $L^2(0,T;\widetilde H^s(\Omega))$ and in $L^2(0,T;\widetilde H^s(\Omega)) \cap H^1_0(0,T;\widetilde H^r(\Omega))$. As an application, partial exterior Dirichlet-to-Neumann measurements uniquely determine a bounded, nonnegative, time-dependent coefficient in the nonlinear velocity damping term $q|\partial_tu|^{p-2}\partial_tu$ for $0<r<s$ and $2<p\leq 2+4r/n$. The proof establishes the small-amplitude response by energy estimates, weighted $L^p$ monotonicity convergence, and a linear Green identity, then applies adjoint Runge approximation followed by a forward velocity approximation in $L^2$. No smallness or positive lower bound is imposed on the nonlinear damping coefficient.
We prove a space-time analogue of the HWI inequality of Otto-Villani for controlled kinetic equations with a semiconvex confining potential. It bounds the time-averaged relative entropy by spatial and conditional velocity transport distances, the velocity Fisher information, an intrinsic action of the path, and a boundary term given by the time endpoint values of the Wasserstein current. The key step is a differential comparison inequality for the Wasserstein current, which captures the transfer of dissipation from velocity to position. Along the Langevin semigroup, this inequality yields a kinetic evolution variational inequality involving the second time derivative of the squared spatial Wasserstein distance and a damping term. Under a compatible spatial Talagrand's transport inequality, we obtain a boundary-free space-time logarithmic Sobolev inequality on every interval $[0,T]$, $T>0$, with the optimal $T^{-2}$ short-time scaling of the constant. As a consequence, we also establish entropy decay for every friction coefficient and every initial density of finite entropy, with optimal cubic order at short times and exponential decay at long times.
We show that the Dirichlet-to-Neumann map of the elastic wave equation determines the density and the anisotropic stiffness tensor on the boundary for generic density and stiffness tensor fields. The proof builds on an interplay between microlocal analysis and algebraic geometry. Microlocal analysis yields a construction of special solutions, valid for many real principal type systems, that concentrate near a boundary point. Generic conditions ensure that this construction holds and allow for extraction of information about stiffness tensor and density at the boundary. We also describe the generic sets through algebraic conditions.
We study the behaviour of the solution to the Navier-Stokes equations with vanishing viscosity and a no-slip condition in a periodically perforated domain. The behaviour depends on the particle size and the viscosity of the fluid. We prove convergence to a measure-valued solution, provided that the local Reynolds number is small, in the subcritical and critical regime. In the first case, the limit solves the Euler equations, whereas in the second case it solves the Euler-Brinkman equations. This extends the results of https://doi.org/10.1088/1361-6544/acfe56 for less regular limit functions. In particular, we also prove existence of measure-valued solutions to the Euler-Brinkman equations, along with weak-strong uniqueness.
We give a penalization argument for splitting a continuous viscosity solution of two-sided Pucci inequalities into a solution of the same inequalities with zero boundary values and a function satisfying the homogeneous Pucci inequalities and with the same boundary data, settling an open question raised by Silvestre-Sirakov.
We investigate the Bloch spectrum for transverse electric (TE) modes in a 2D high-contrast dielectric metamaterial. Our analysis focuses on the high-frequency regime where the propagating wavelength is commensurate with the microstructural period $d$, and the permittivity within the inclusions scales as $d^{-2}$. Using operator factorization and Sobolev space decomposition, we demonstrate that Bloch eigenvalues $\omega^2$ can be analytically continued and represented as a power series in $d^2$. We establish a rigorous upper bound on the microstructural length scale $d$, defined by the inclusion geometry, under which this expansion remains valid. Furthermore, we provide a definitive proof for the opening of band gaps in the frequency spectrum for all periods below a critical length scale, i.e., for all $d < \overline{d}$. The threshold is explicitly characterized as a geometric function determined by the interaction between Dirichlet eigenvalues and the spectrum of the associated Neumann-Poincar\'e operator.
We study the obstacle problem for the evolution equation $\partial_t^\alpha u+(-\Delta)^s u=0$ in the whole space $\mathbb{R}^d$, where $\partial_t^\alpha$ with $\alpha\in(0,1)$ is the Caputo derivative in time and $(-\Delta)^s$ with $s\in(0,1]$ is the fractional Laplacian in space ($s<1$) or the classical Laplacian ($s=1$). We prove that the problem has at most one bounded viscosity solution, and we obtain existence as the limit of an explicit monotone finite-difference scheme, which combines the L1 discretization of the Caputo derivative with powers of the discrete Laplacian. The scheme is also used to establish regularity estimates in space and time. We then study the large-time behaviour: solutions converge, along subsequences, to stationary solutions, and we identify the limit when the initial datum coincides with the obstacle (the minimal stationary solution) or has a limit $c_*$ at infinity. In the latter case the dimension plays a role, as a consequence of the recurrence of the underlying process: the limit is the unique stationary solution with limit $c_*$ at infinity if $d>2s$, and a constant (the maximum between $c_*$ and the supremum of the obstacle) if $d\le2s$. Numerical experiments in one dimension, based on an explicit stationary solution, illustrate the convergence of the scheme and the relaxation towards the stationary profile.
Motivated by the analysis of recent models of dislocation motion in crystalline materials, this work introduces space-time varicurrents, a form of Young measures over integral currents, to describe their evolutions with rate-dependent dissipation potentials. This addresses the problem that rate-dependent versions of the (rate-independent) space-time variation suffer from a lack of lower semicontinuity with respect to the natural weak topology. This causes a dynamical relaxation to occur by means of infinitely fine oscillations. Varicurrents retain a description of this ``microstructure'' (in space-time) and, based on this theory, we are able to establish the existence of solutions to a rate-dependent gradient flow. This is in contrast to the classical formulation using integral currents, which does not have a solution in general.
Let $M$ be a closed star-shaped hypersurface in $\mathbb{R}^{n+1}$ and let $\eta=Hg-h$ be the first Newton transformation of its second fundamental form. For a fixed constant $\alpha>0$ and $2\le k\le n-1$, we study the prescribed sum Hessian curvature equation \[ \sigma_k(\lambda(\eta))+\alpha\sigma_{k-1}(\lambda(\eta))=f(X,\nu). \] Under the natural admissibility condition \[ \lambda(\eta)\in\Gamma_{k-1}\cap \{\lambda:\sigma_k(\lambda)+\alpha\sigma_{k-1}(\lambda)>0\}, \] we establish a global curvature estimate for closed star-shaped solutions with a general positive right-hand side depending on the position vector and the unit normal. A key point is that the sum Hessian operator is nonhomogeneous; the resulting lower-order term in the maximum-principle calculation is controlled by a partial curvature estimate. We also use the augmented-variable identity \[ \sigma_k(\lambda)+\alpha\sigma_{k-1}(\lambda) =\sigma_k(\lambda_1,\ldots,\lambda_n,\alpha) \] to recover the relevant ellipticity and concavity properties. As an application of the a priori estimates, we prove existence and uniqueness for the corresponding prescribed curvature problem under suitable spherical barrier and radial monotonicity assumptions.
We study homogeneous Dirichlet heat flows for real symmetric, uniformly elliptic divergence-form operators. We establish quantitative stability of singular sets near homogeneous harmonic polynomials and eigenfunctions. Based on these, in the smooth coefficient case, we prove the spatial singular set at sufficiently large time is codimension-two and has a uniform measure upper bound independent of time. However, this codimension-two property fails in the Lipschitz coefficient case. On a three-dimensional ball, we construct a fixed self-adjoint operator and a heat flow consisting of two eigenmodes whose singular sets contain two-dimensional disks at a sequence of times tending to infinity.
Modeling the nonlinear acoustic wave propagation in biological tissues needs accuracy, especially in ultrasound imaging and therapeutic applications. The 1D Westervelt equation, \( p_{xx} - p_{tt} + \gamma\, p_{ttt} + \eta\,(p^{2})_{tt}=0, \) could be used to describe a finite-amplitude ultrasound propagation in these tissues. This equation accounts for nonlinearity, \(\eta\), and attenuation, \(\gamma\). To solve this equation, we observe that traditional solvers may struggle to do so correctly. This is due to the difficulties arising from the nonlinear nature of the equation. We solve the equation by defining three synthetic cases - linear (\(\eta=0.001\), \(\gamma=10^{-6}\)), shock-dominated (\(\eta=0.02\), \(\gamma=10^{-6}\)), and dissipation-dominated (\(\eta=0.01\), \(\gamma=5\times10^{-4}\)) - and using Physics-Informed Neural Networks (PINNs) to showcase the solution. A sinusoidal activation function is used to capture the oscillatory behavior of the problem. The results of our study, with PDE residual converging to at least \(10^{-6}\), boundary condition loss to \(10^{-5}\), and the PINN loss converging to \(10^{-5}\) for each case at \(30,000\) epochs, show that \(\gamma\) scales the rate of amplitude decay, while \(\eta\) governs frequency-dependent phase speeds.
We study the symmetry, uniqueness, and nondegeneracy of positive solutions to the Choquard equation $-\Delta u+u=(I_\alpha*u^p)u^{p-1}$ in $\mathbb{R}^N$, which have long remained open problems. For $N\geq3$, $\alpha\in(0,N)$, and $\frac{N+\alpha}{N}<p<\frac{N+\alpha}{N-2}$,we prove that every positive $H^1$-solution is radially symmetric and strictly decreasing in the radial variable. The symmetry proof relies on the moving-plane method, where the sublinear singularity is controlled by analyzing the far-field behavior of the nonlocal term. Uniqueness and nondegeneracy are established through a functional framework hinging on a nonnegative joint form and a Schur complement argument.Radial nondegeneracy and Morse index one are obtained by ruling out the spectral interval $(0,p-1]$ via a strict cumulative Riesz identity, which leads to uniqueness through a Nehari two-well argument. Nondegeneracy in the full space $H^1(\mathbb{R}^N)$ follows from the spherical-harmonic decomposition of the linearized form. These results hold for the full range of $p$ when $\alpha=2$, and for $p\geq2$ when $\alpha\in(2,N)$. The cases $\alpha\in(0,2)$ and $\alpha\in(2,N)$ with $p<2$ remain open.
For the two-dimensional incompressible Navier-Stokes equations with Navier friction on a smooth, bounded, simply connected domain, we establish strong vorticity convergence along a subsequence in the vanishing viscosity limit for initial vorticity in $L^p$ with $p>1$. The friction coefficients may grow as viscosity vanishes under an explicit growth restriction depending on $p$, which includes regimes of vanishing slip length. The main challenge, even for viscosity-independent friction, is to control the vorticity generated at the boundary. We split the vorticity into contributions from the initial data and the boundary. By combining renormalization and estimates for the normal velocity near the boundary, we show that the boundary contribution vanishes in the inviscid limit. As a consequence, the limiting Euler solutions conserve kinetic energy.
We consider the inhomogeneous Landau equation and focus on solutions whose density and temperature evolve in time but remain spatially homogeneous. Such solutions are called homoenergetic. We identify one class of initial data for which homoenergetic solutions exist globally in time, and another class for which they develop finite-time hydrodynamic singularities. The initial data are spatially inhomogeneous, neither close to vacuum nor close to equilibrium; both the well-posedness and the finite-time blow-up results hold for all soft and hard potentials.
We investigate the effect of constant vorticity on the modulational spectral stability of small-amplitude periodic gravity waves for the two-dimensional finite-depth water-wave system. We derive an explicit stability criterion that agrees with the prediction obtained by Thomas, Kharif and Manna [43] through a nonlinear Schr\"odinger approximation. For waves propagating to the right, we identify a sharp range of dimensionless positive (counterclockwise) vorticity, for which modulational instability is suppressed at every depth, while at any fixed depth sufficiently strong negative (clockwise) vorticity also restores modulational stability. We give a complete description of the spectrum near the origin and identify a figure-eight spectral curve in the unstable regime. A key distinction from the irrotational case is that the usual Hamiltonian factorization of the linearized operator fails at zero Floquet exponent on the full periodic Sobolev space. We therefore carry out a spectral reduction based on reversibility, without relying on Hamiltonian factorization.
We study anisotropic Minkowski-type curvature flows associated with a perimeter defined through oscillations on Wulff neighbourhoods. The anisotropy may be non-smooth, and in particular crystalline, while the mobility may differ from the anisotropy, provided a natural compatibility condition is satisfied. We compute the first variation of the perimeter and introduce a weak formulation in terms of distance functions and measure-valued nonlocal Cahn-Hoffman fields. By means of a nonlocal minimising movement scheme and suitable estimates for the evolution of Wulff shapes, we prove existence of weak superflows and subflows starting from arbitrary closed sets. Under the additional assumption that the initial level sets have compact boundary, a comparison principle yields existence and uniqueness of the level set flow. Finally, we prove that variational weak flows converge, up to fattening, as $r\searrow 0$ to the weak flow of the local anisotropic, possibly crystalline, curvature motion. The main step in the proof consists in averaging the nonlocal Cahn-Hoffman measures to obtain vector fields converging to a local Cahn-Hoffman field.
We study a coupled elliptic-parabolic system modelling electrical discharge in a micro-electro-mechanical (MEMS) capacitor: a Poisson equation for the electrostatic potential, coupled to drift-diffusion equations for the ion and electron densities with a Townsend ionization source whose rate per electron grows linearly in the electric field. Gimperlein, He and Lacey proved local well-posedness and left open whether the densities blow up in finite time. We first prove a continuation criterion in a weak norm: a finite-time singularity forces the space charge to be unbounded in H^1_D, the dual of the space of H^1 functions vanishing on the electrodes, so blow-up cannot be quasi-neutral. We then exclude it altogether when the two diffusivities coincide and the vacuum field is bounded: the weak solution is then global in time. To our knowledge this is the first global existence result without smallness assumptions for a drift-diffusion-Poisson model of gas discharge whose ionization rate is unbounded in the field. The proof uses the electrostatic energy and the number of carriers in place of the usual free energy: the ionization source cancels in the energy balance, the Joule heating is dissipative, and the two quantities close through a linear Gronwall inequality. Numerical experiments on a discharge in a micrometre gap, with the Townsend constants and the mobilities of air, provide evidence for the extension to unequal diffusivities and bear on the existence of a breakdown threshold.
We study the hard-core variant of the Simplified Approach to the ground state of the dilute Bose gas. The hard-core potential has been the subject of renewed attention, and is physically relevant to realistic atomic Bose gases. The hard-core Simple Equation is an integro-differential equation, and we construct its solution via a monotone iteration scheme and prove existence and uniqueness (in a restricted sense) of the solution. The energy per particle $e$ and the density $\rho$ of the solution are shown to satisfy the Lee-Huang-Yang formula $e = 2\pi\rho(1 + \frac{128}{15\sqrt{\pi}}\sqrt{\rho} + o(\sqrt{\rho}))$ as $\rho \to 0$, and the condensate fraction $\eta$ predicted by the hard-core Simple Equation is shown to satisfy $\eta = 1 - \frac{8}{3\sqrt{\pi}}\sqrt{\rho} + o(\sqrt{\rho})$ as $\rho \to 0$, in agreement with Bogolyubov's theory. We thus extend results of Carlen, Jauslin, Lieb to the hard-core case. The hard-core equation is defined on its own terms, and the proofs proceed by approximating the hard-core potential with the soft-core potentials $m\,\chi_{\{|x| \le 1\}}$ of increasing height $m$: at fixed energy, we establish the $L^1$ convergence of the corresponding solutions and the convergence of their densities at an $O(m^{-1/2})$ rate, and we show that the Lee-Huang-Yang expansions of the energy and of the condensate fraction for the soft-core equations hold with errors uniform in $m$, so that they survive the hard-core limit.
Harrop-Griffiths, Killip, Ntekoume and Vi\c{s}an proved that the derivative nonlinear Schr\"odinger equation (DNLS) is globally well-posed in $L^2(\mathbb{R})$. A key ingredient of their proof is a local smoothing estimate. It holds on $L^2$-equicontinuous sets of initial data, but it cannot hold on $L^2$-balls, because rescaled stationary solitons, all of mass $2\pi$, violate it. This raises a natural question: for which masses $m$ does local smoothing hold uniformly on the ball $\{M(q_0)\le m\}$? We prove three results. First, we give a quantitative version of the local smoothing estimate in which the data enter only through the mass and an explicit frequency scale $\kappa(q_0)$. On time intervals of length $2T$ the bound grows like $1+\kappa(q_0)^2T$, and the stationary soliton shows that the power of $\kappa$ cannot be lowered below one. Second, we prove uniform local smoothing on balls of small mass, together with a scale-invariant Morrey-type form and a localized $L^4_{t,x}$ bound. Third, we analyse the soliton family from both sides. The uniform constant must blow up at least like $(2\pi-m)^{-1}$ as $m\uparrow 2\pi$, while every soliton of mass $m<2\pi$ satisfies local smoothing with a constant $O((2\pi-m)^{-1})$. These facts single out the stationary-soliton mass $2\pi$, which is also the Hayashi--Ozawa coercivity threshold, as the natural candidate for the sharp threshold. We formulate this as a conjecture, give a spectral reformulation in terms of the Kaup--Newell scattering data, and discuss the obstacles to a proof.
We study linear variational problems on bounded domains $\Omega \subset \mathbb R^d$ involving the Dirichlet biharmonic operator $\Delta^{2}$ and the Rellich potential $|x|^{-4}$, under the assumption $0\in\overline\Omega$. The case $0\in\partial\Omega$ is particularly noteworthy and subtle.
We study the classical Skyrme model. Under corotational symmetry, we prove that (i) stationary solutions, or Skyrmions, are unique in every topological degree; (ii) the Cauchy problem is globally well-posed in a nonlinear energy space; and (iii) stable soliton resolution holds: every finite-energy solution scatters to the Skyrmion of the same degree. Our key observation is that subcritical equivariant Yang--Mills dynamics emerge near the origin and suppress concentration.
In this paper, we establish the nonlinear stability of the Schwarzschild exterior for the spherically symmetric Einstein-massless Dirac system. More precisely, we consider sufficiently small perturbations of Schwarzschild characteristic initial data and prove that the corresponding maximal development remains globally controlled in the exterior region (orbital stability), and converges, in a suitable teleological double-null gauge, to a Schwarzschild spacetime with a final mass (asymptotic stability). In particular, the maximal future development contains a complete future null infinity and a regular event horizon, with a nonempty black hole region in its interior. The main difficulties in the proof arise from the delicate coupling between the Dirac operator and the geometry, compared with the scalar wave counterpart, as well as from the failure of the Dirac stress-energy tensor to satisfy a positive energy condition. To overcome these difficulties, we rely on a bootstrap argument coupling the geometry and the Dirac field. Working in suitable teleological double-null gauges adapted to the asymptotic null boundaries, we establish weighted energy estimates combining integrated local energy decay, red-shift estimates near the event horizon, and an rp-estimates towards null infinity. A core ingredient is the use of suitable reweightings of the spinor components, which allow us to close these estimates despite the nonlinear coupling to the geometry.
Self-propelled particles such as swimming micro-organisms often accumulate at boundaries. Active Brownian particles (ABPs), that is, particles moving at constant speed with an orientation undergoing rotational diffusion, provide a simple model. ABPs have a finite probability to be on the boundary. Their probability density therefore includes both regular bulk and singular boundary components. Exchanges between these represent capture at the boundary and re-entry into the bulk. We examine the distribution of ABPs in the stationary regime. We first obtain a similarity solution that describes the probability density near the turning points, where orientation is tangential to the boundary and particles re-enter the bulk. This shows that the bulk density is singular and the boundary density is non-differentiable. We then add spatial diffusion to the ABP model and investigate the limit $\eps \to 0$ of small spatial diffusivity. We use matched asymptotics to describe the regularising effect of a finite $\eps$, focusing on the vicinity of the turning points. We assess our predictions against numerical simulations using both lattice Markov chain and finite-element discretisations.
In this paper, we study Egorov's theorem for Laplacian eigenfunctions on compact hyperbolic surfaces in Zelditch's quantization. In this quantization, the remainder in Egorov's theorem for eigenfunctions is given explicitly by the second derivatives of the symbols along the weak stable foliation of the geodesic flow. Forward in time, we show that Egorov's theorem for eigenfunctions is valid to all logarithmic times, far beyond the Ehrenfest time T_E=|log h|/2, where h going to 0 is the semiclassical parameter. Backward in time, we introduce the Schrodinger group along the stable horocycles, which absorbs the exponentially growing part of the remainder. We show that Egorov's theorem for eigenfunctions is valid up to -T_E+T_h for any Th going to infinity with 0<=T_h<=T_E. Moreover, the limit of the remainder in Egorov's theorem at the Ehrenfest time -T_E is exactly the difference between the semiclassical measure and the Liouville measure, tested against the symbols propagated by the Schrodinger group. Consequently, Egorov's theorem for a sequence of eigenfunctions holds at -T_E (and in fact, at -T_E-T for any fixed T in R if and only if the sequence tends to be equidistributed in the phase space. Thus, the quantum unique ergodicity (QUE) conjecture is equivalent to this Egorov property holding for every eigenbasis. Furthermore, QUE holds with a polynomial rate if and only if Egorov's theorem holds at -T_E with a polynomial remainder. In this case, Egorov's theorem holds beyond -T_E with a polynomial remainder, and the stronger the rate is in QUE, the longer the time is in Egorov's theorem.
Self-similar solutions play an important role in the formation of singularities in the Yang--Mills equations. In five space dimensions, these equations admit infinitely many self-similar solutions with an increasing number of unstable modes. It was observed numerically that all excited profiles share the least unstable eigenvalue $\lambda=5$. We explain this remarkable spectral property by an elementary construction of the corresponding eigenfunction in terms of the profile.
In this paper, we establish a universal approximation theorem for solution operators of semilinear stochastic heat equations in spatial dimension \(d\ge2\), extending deep neural operator approximation results to SPDEs with distribution-valued Gaussian forcing. A smooth Fourier cutoff reduces the forcing to finitely many measurements, while Besov and Schauder estimates control the resulting solution error. The finite-dimensional solution map is then approximated by ReLU branch networks together with grid-based trunk functions, yielding explicit bounds on the numbers of forcing measurements and trunk functions and approximation in every finite stochastic moment. For linear equations, the reduction coincides with truncation of the first Wiener chaos. Lastly, as an application, we prove a high-probability local-in-time approximation theorem for stochastic Navier--Stokes equations, with finite measurements of both the initial condition and forcing and an explicit sensing rate.
In our previous article CMP 407 (2026) 131, we explored the rich algebraic structures arising from a specific quantum many-body Hamiltonian quartic in the bosonic creation-annihilation operators. This Hamiltonian originated from the works of G\'erard and Grellier on classical turbulent energy transfer in a Lax-integrable setting, and its quantum version respects hierarchies of polynomial conservation laws all of which possess purely integer spectra, as does the Hamiltonian itself. The latter feature, known from numerical studies, has previously remained unproved. Here, we present an elementary solution of the quantum G\'erard-Grellier Hamiltonian. The eigenstates are labelled by partitions and the corresponding eigenvalues admit simple explicit expressions in terms of the principal hook lengths. The key ingredient in this construction is a specific basis we introduce, inspired by the classical multisoliton coherent states, in which the Hamiltonian is upper-triangular. The expressions for the eigenvalues then immediately follow, while the full eigenbasis can be recovered via a Gram-Schmidt procedure.
We consider the incompressible Navier--Stokes equations in a smooth bounded domain in dimension two or three, with Dirichlet boundary conditions, when the velocity is advected by a rapidly varying stationary Markov process $\eps^{-1}m(t/\eps^2)$. The fast field may be spatially inhomogeneous. We prove that, when $\eps\to0$, the laws of the large-scale velocities are tight and that their limit points are martingale solutions of the stochastic Navier--Stokes equations with transport noise in the form of the location uncertainty framework. The limiting equation contains in particular a noise which is described in terms of the covariance of the fast advection. Under a symmetry assumption on the temporal correlations of the fast field, the noise is Stratonovich and an advection by the It\^o--Stokes drift appears. The proof relies on perturbed test functions, which we use both for the uniform energy bounds and for the identification of the limit. We have to adapt this method and add the Stokes operator to the Poisson equation associated to the fast advection in order to use fewer regularity assumptions and obtain correctors which satisfy the boundary conditions.