Wiley: Communications on Pure and Applied Mathematics: Table of Contents

Self‐Similar Blowup for the Cubic Schrödinger Equation

Roland Donninger, Birgit Schörkhuber — Tue, 09 Jun 2026 04:18:04 -0700

ABSTRACT

We give a rigorous proof for the existence of a finite-energy, self-similar solution to the focusing cubic Schrödinger equation in three spatial dimensions. The proof is computer-assisted and relies on a fixed point argument that shows the existence of a solution in the vicinity of a numerically constructed approximation. The latter is obtained by a standard pseudo-spectral method. The computer-assisted part of the rigorous proof uses nothing but fraction arithmetic in order to obtain quantitative bounds for the fixed point argument.

Invariant Measure and Universality of the 2D Yang–Mills Langevin Dynamic

Ilya Chevyrev, Hao Shen — Tue, 09 Jun 2026 04:18:04 -0700

ABSTRACT

We prove that the Yang–Mills (YM) measure for the trivial principal bundle over the two-dimensional torus, with any connected, compact structure group, is invariant for the associated renormalised Langevin dynamic. Our argument relies on a combination of regularity structures, lattice gauge-fixing and Bourgain's method for invariant measures. Several corollaries are presented including a gauge-fixed decomposition of the YM measure into a Gaussian free field and an almost Lipschitz remainder, and a proof of universality for the YM measure that we derive from a universality for the Langevin dynamic for a wide class of discrete approximations. The latter includes standard lattice gauge theories associated to Wilson, Villain and Manton actions. An important step in the argument, which is of independent interest, is a proof of uniqueness for the mass renormalisation of the gauge-covariant continuum Langevin dynamic, which allows us to identify the limit of discrete approximations. This latter result relies on Euler estimates for singular SPDEs and for Young ODEs arising from Wilson loops.

The Huang–Yang Formula for the Low‐Density Fermi Gas: Upper Bound

Emanuela L. Giacomelli, Christian Hainzl, Phan Thành Nam, Robert Seiringer — Tue, 09 Jun 2026 04:18:04 -0700

ABSTRACT

We study the ground state energy of a gas of spin 1/2$1/2$ fermions with repulsive short-range interactions. We derive an upper bound that agrees, at low density ϱ$\varrho$, with the Huang–Yang conjecture. The latter captures the first three terms in an asymptotic low-density expansion, and in particular the Huang–Yang correction term of order ϱ7/3$\varrho ^{7/3}$. Our trial state is constructed using an adaptation of the bosonic Bogoliubov theory to the Fermi system, where the correlation structure of fermionic particles is incorporated by quasi-bosonic Bogoliubov transformations. In the latter, it is important to consider a modified zero-energy scattering equation that takes into account the presence of the Fermi sea, in the spirit of the Bethe–Goldstone equation.

Issue Information ‐ TOC

Tue, 09 Jun 2026 04:18:04 -0700

Communications on Pure and Applied Mathematics, Volume 79, Issue 8, Page 1829-1830, August 2026.

Liquid Drop Model for Nuclear Matter in the Low Density Limit

Rupert L. Frank, Mathieu Lewin, Robert Seiringer — Fri, 22 May 2026 00:00:00 -0700

ABSTRACT

We consider the liquid drop model with a positive background density in the thermodynamic limit. We prove a two-term asymptotics for the ground state energy per unit volume in the dilute limit. Our proof justifies the expectation that optimal configurations consist of droplets of unit size that arrange themselves according to minimizers for the Jellium problem for point particles. In particular, we provide the first rigorous derivation of what is known as the gnocchi phase in astrophysics.

Front Propagation Through a Perforated Wall

Henri Berestycki, François Hamel, Hiroshi Matano — Thu, 07 May 2026 00:00:00 -0700

ABSTRACT

We consider a bistable reaction– diffusion equation ut=Δu+f(u)$u_t=\Delta u +f(u)$ on RN${\mathbb {R}}^N$ in the presence of an obstacle K$K$, which is a wall of infinite span with many holes. More precisely, K$K$ is a closed subset of RN${\mathbb {R}}^N$ with smooth boundary such that its projection onto the x1$x_1$-axis is bounded and that RN∖K${\mathbb {R}}^N \setminus K$ is connected. Our goal is to study what happens when a planar traveling front coming from x1=−∞$x_1 = -\infty$ meets the wall K$K$. We first show that there is clear dichotomy between “propagation”and “blocking.” In other words, the traveling front either passes through the wall and propagates toward x1=+∞$x_1=+\infty$ (propagation) or is trapped around the wall (blocking), and that there is no intermediate behavior. This dichotomy holds for any type of walls of finite thickness. Next we discuss sufficient conditions for blocking and propagation. For blocking, assuming either that K$K$ is periodic in y:=(x2,…,xN)$y:=(x_2,\ldots, x_N)$ or that the holes are localized within a bounded area, we show that blocking occurs if the holes are sufficiently narrow. For propagation, three different types of sufficient conditions for propagation will be presented, namely “walls with large holes,” “small-capacity walls,” and “parallel-blade walls.” We also discuss complete and incomplete invasions.

About Wess–Zumino–Witten Equation and Harder–Narasimhan Potentials

Siarhei Finski — Tue, 28 Apr 2026 00:57:37 -0700

ABSTRACT

For a polarized family of complex projective manifolds, we identify the algebraic obstructions that govern the existence of approximate solutions to the Wess–Zumino–Witten equation. When this is specialized to the fibration associated with a projectivization of a vector bundle, we recover a version of Kobayashi–Hitchin correspondence. More broadly, we demonstrate that a certain auxiliary Monge–Ampère type equation, generalizing the Wess–Zumino–Witten equation by taking into account the weighted Bergman kernel associated with the Harder–Narasimhan filtrations of direct image sheaves, admits approximate solutions over any polarized family. These approximate solutions are shown to be the closest counterparts to true solutions of the Wess–Zumino–Witten equation whenever the latter do not exist, as they minimize the associated Yang–Mills functional. As an application, in a fibered setting, we prove an asymptotic converse to the Andreotti–Grauert theorem conjectured by Demailly.

Stability of Viscous Three‐Dimensional Stratified Couette Flow via Dispersion and Mixing

Michele Coti Zelati, Augusto Del Zotto, Klaus Widmayer — Fri, 24 Apr 2026 02:03:07 -0700

ABSTRACT

This article explores the stability of stratified Couette flow in the viscous 3d$3d$ Boussinesq equations. In this system, mixing effects arise from the shearing background, and gravity acts as a restoring force leading to dispersive internal gravity waves. These mechanisms are of fundamentally different nature and relevant in complementary dynamical regimes. Our study combines them to establish a bound for the non-linear transition threshold, which is quantitatively larger than the inverse Reynolds number ν$\nu$, and increases with stronger stratification resp. gravity.

Global Well‐Posedness and Relaxation for Solutions of the Fokker–Planck–Alignment Equations

Roman Shvydkoy — Thu, 23 Apr 2026 08:29:34 -0700

ABSTRACT

In this paper, we prove global existence of weak solutions, their regularization, and relaxation for large data for a broad class of Fokker–Planck–Alignment models, which appear in collective dynamics. The main feature of these results, as opposed to previously known ones, is the lack of regularity or no-vacuum requirements on the initial data. With a particular application to the classical kinetic Cucker–Smale model, we demonstrate that any bounded data with finite higher moment, f0∈L1(1+|v|q)∩L∞$f_0 \in L^1(1+ |v|^q) \cap L^\infty$, q⩾n+4$q \geqslant n+4$, gives rise to a global instantly smooth solution, satisfying entropy equality and relaxing exponentially fast. The results are achieved through the use of a new thickness-based renormalization procedure, which circumvents the problem of degenerate diffusion in nonperturbative regime.

Effective Velocities in the Toda Lattice

Amol Aggarwal — Mon, 20 Apr 2026 06:52:30 -0700

ABSTRACT

In this paper, we consider the Toda lattice (p(t);q(t))$(\bm {p}(t); \bm {q}(t))$ at thermal equilibrium, meaning that its variables (pi)$(p_i)$ and (eqi−qi+1)$(e^{q_i-q_{i+1}})$ are independent Gaussian and Gamma random variables, respectively. This model can be thought of a dense collection of many “quasiparticles” that act as solitons. We establish a law of large numbers for the trajectory of these quasiparticles, showing that they travel with approximately constant velocities, which are explicit. Our proof is based on a direct analysis of the asymptotic scattering relation, an equation that approximately governs the dynamics of quasiparticles locations. This makes use of a regularization argument that essentially linearizes this relation, together with concentration estimates for the Toda lattice's (random) Lax matrix.

Optimal Convergence Rates in Multiscale Elliptic Homogenization

Weisheng Niu, Yao Xu, Jinping Zhuge — Tue, 14 Apr 2026 05:11:26 -0700

ABSTRACT

This paper is devoted to the quantitative homogenization of multiscale elliptic operator −∇·Aε∇$-\nabla \cdot A_\varepsilon \nabla$, where Aε(x)=A(x/ε1,x/ε2,…,x/εn)$A_\varepsilon (x) = A(x/\varepsilon _1, x/\varepsilon _2,\ldots, x/\varepsilon _n)$, ε=(ε1,ε2,…,εn)∈(0,1]n$\varepsilon = (\varepsilon _1, \varepsilon _2,\ldots, \varepsilon _n) \in (0,1]^n$, and εi>εi+1$\varepsilon _i > \varepsilon _{i+1}$. We assume that A(y1,y2,…,yn)$A(y_1,y_2,\ldots, y_n)$ is 1-periodic in each yi∈Rd$y_i \in \mathbb {R}^d$ and real analytic. Classically, the method of reiterated homogenization has been applied to study this multiscale elliptic operator, which leads to a convergence rate limited by the ratios max{εi+1/εi:1≤i≤n−1}$\max \lbrace \varepsilon _{i+1}/\varepsilon _i: 1\le i\le n-1\rbrace$. In the present paper, under the assumption of real analytic coefficients, we introduce the so-called multiscale correctors and more accurate effective operators, and improve the ratio part of the convergence rate to max{e−cεi/εi+1:1≤i≤n−1}$\max \lbrace e^{-c\varepsilon _{i}/\varepsilon _{i+1}}: 1\le i\le n-1 \rbrace$. This convergence rate is optimal in the sense that c>0$c>0$ cannot be replaced by a larger constant. As a byproduct, the uniform Lipschitz estimate is established under a mild double-log scale-separation condition.

The Benjamin–Ono Equation in the Zero‐Dispersion Limit for Rational Initial Data: Generation of Dispersive Shock Waves

Elliot Blackstone, Louise Gassot, Patrick Gérard, Peter D. Miller — Tue, 31 Mar 2026 00:00:00 -0700

ABSTRACT

The leading-order asymptotic behavior of the solution of the Cauchy initial-value problem for the Benjamin–Ono equation in L2(R)$L^2(\mathbb {R})$ is obtained explicitly for generic rational initial data u0$u_0$. An explicit asymptotic wave profile uZD(t,x;ε)$u^\mathrm{ZD}(t,x;\epsilon)$ is given, in terms of the branches of the multivalued solution of the inviscid Burgers equation with initial data u0$u_0$, such that the solution u(t,x;ε)$u(t,x;\epsilon)$ of the Benjamin–Ono equation with dispersion parameter ε>0$\epsilon >0$ and initial data u0$u_0$ satisfies u(t,x;ε)−uZD(t,x;ε)→0$u(t,x;\epsilon)-u^\mathrm{ZD}(t,x;\epsilon)\rightarrow 0$ in the locally uniform sense as ε→0$\epsilon \rightarrow 0$, provided a discriminant inequality holds implying that certain caustic curves in the (t,x)$(t,x)$-plane are avoided. In some cases, this convergence implies strong L2(R)$L^2(\mathbb {R})$ convergence. The asymptotic profile uZD(t,x;ε)$u^\mathrm{ZD}(t,x;\epsilon)$ is consistent with the modulated multiphase wave solutions described by Dobrokhotov and Krichever.