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View article: Linear inviscid damping for stably stratified Boussinesq flows
Linear inviscid damping for stably stratified Boussinesq flows Open
We study the linear asymptotic stability of stably stratified monotone shear flows for the Boussinesq equations in the periodic channel. By means of the limiting absorption principle, we obtain a precise description of the inviscid damping…
View article: Nonexistence of Courant-type nodal domain bounds for eigenfunctions of the Dirichlet-to-Neumann operator
Nonexistence of Courant-type nodal domain bounds for eigenfunctions of the Dirichlet-to-Neumann operator Open
Given a compact manifold \mathcal{M} with boundary of dimension n\geq 3 and any integers K and N , we show that there exists a metric on \mathcal{M} for which the first K nonconstant eigenfunctions of the Dirichlet-to-Neumann map on \parti…
View article: Low-energy dynamics in generic potential fields: Hyperbolic periodic orbits and non-ergodicity
Low-energy dynamics in generic potential fields: Hyperbolic periodic orbits and non-ergodicity Open
We prove that, on each low energy level, the natural Hamiltonian system defined by a generic smooth potential on $\mathbf{T}^2$ exhibits an arbitrarily high number of hyperbolic periodic orbits and a positive-measure set of invariant tori.…
View article: Hölder continuous dissipative solutions of ideal MHD with nonzero helicity
Hölder continuous dissipative solutions of ideal MHD with nonzero helicity Open
We prove the existence of weak solutions to the 3D ideal MHD equations, of class $C^α$ with $α=10^{-8}$, for which the total energy and the cross helicity (i.e., the so-called Elsässer energies) are not conserved. The solutions do not poss…
View article: Optimal metrics for the first curl eigenvalue on 3-manifolds
Optimal metrics for the first curl eigenvalue on 3-manifolds Open
View article: Schiffer-type problems and nonradial stationary Euler flows with compact support
Schiffer-type problems and nonradial stationary Euler flows with compact support Open
We review some recent results on the existence of nonradial stationary planar Euler flows with compact support. The approach we take relies on an elliptic overdetermined problem motivated by Schiffer’s conjecture in spectral geometry.
View article: Isolated steady solutions of the 3D Euler equations
Isolated steady solutions of the 3D Euler equations Open
We show that there exist closed three-dimensional Riemannian manifolds where the incompressible Euler equations exhibit smooth steady solutions that are isolated in the C 1 -topology. The proof of this fact combines ideas from dynamical sy…
View article: Solutions to general elliptic equations on nearly geodesically convex domains with many critical points
Solutions to general elliptic equations on nearly geodesically convex domains with many critical points Open
Consider a complete $d$-dimensional Riemannian manifold $(\mathcal M,g)$, a point $p\in\mathcal M$ and a nonlinearity $f(q,u)$ with $f(p,0)>0$. We prove that for any odd integer $N\ge3$, there exists a sequence of smooth domains $Ω_k\subse…
View article: Local limits of high energy eigenfunctions on integrable billiards
Local limits of high energy eigenfunctions on integrable billiards Open
Berry's random wave conjecture posits that high energy eigenfunctions of chaotic systems resemble random monochromatic waves at the Planck scale. One important consequence is that, at the Planck scale around "many" points in the manifold, …
View article: Steady 3d Euler flows via a topology-preserving convex integration scheme
Steady 3d Euler flows via a topology-preserving convex integration scheme Open
Given any smooth solenoidal vector field $v_0$ on $\mathbf T^3$, we show the existence of infinitely many Hölder-continuous steady Euler flows $v$ with the same topology as $v_0$, in certain weak sense. In particular, we show that $v$ poss…
View article: An extension theorem for weak solutions of the 3d incompressible Euler equations and applications to singular flows
An extension theorem for weak solutions of the 3d incompressible Euler equations and applications to singular flows Open
We prove an extension theorem for local solutions of the 3d incompressible Euler equations. More precisely, we show that if a smooth vector field satisfies the Euler equations in a spacetime region $\Omega \times (0,T)$ , one can choose an…
View article: Positive solutions to general semilinear overdetermined boundary problems
Positive solutions to general semilinear overdetermined boundary problems Open
We establish the existence of positive solutions to a general class of overdetermined semilinear elliptic boundary problems on suitable bounded open sets $Ω\subset\mathbb{R}^n$. Specifically, for $n\leq 4$ and under mild technical hypothes…
View article: Obstructions to Topological Relaxation for Generic Magnetic Fields
Obstructions to Topological Relaxation for Generic Magnetic Fields Open
View article: Isolated steady solutions of the 3D Euler equations
Isolated steady solutions of the 3D Euler equations Open
We show that there exist closed three-dimensional Riemannian manifolds where the incompressible Euler equations exhibit smooth steady solutions that are isolated in the $C^1$-topology. The proof of this fact combines ideas from dynamical s…
View article: Entanglement of Vortices in the Ginzburg–Landau Equations for Superconductors
Entanglement of Vortices in the Ginzburg–Landau Equations for Superconductors Open
In 1988, Nelson proposed that neighboring vortex lines in high-temperature superconductors may become entangled with each other. In this article we construct solutions to the Ginzburg–Landau equations which indeed have this property, as th…
View article: Smooth nonradial stationary Euler flows on the plane with compact support
Smooth nonradial stationary Euler flows on the plane with compact support Open
We prove the existence of nonradial classical solutions to the 2D incompressible Euler equations with compact support. More precisely, for any positive integer $k$, we construct compactly supported stationary Euler flows of class $C^k(\mat…
View article: Nonexistence of Courant-type nodal domain bounds for eigenfunctions of the Dirichlet-to-Neumann operator
Nonexistence of Courant-type nodal domain bounds for eigenfunctions of the Dirichlet-to-Neumann operator Open
Given a compact manifold $\mathcal M$ with boundary of dimension $n\geq 3$ and any integers $K$ and $N$, we show that there exists a metric on $\mathcal M$ for which the first $K$ nonconstant eigenfunctions of the Dirichlet-to-Neumann map …
View article: Entanglement of vortices in the Ginzburg--Landau equations for superconductors
Entanglement of vortices in the Ginzburg--Landau equations for superconductors Open
In 1988, Nelson proposed that neighboring vortex lines in high-temperature superconductors may become entangled with each other. In this article we construct solutions to the Ginzburg--Landau equations which indeed have this property, as t…
View article: MHD equilibria with nonconstant pressure in nondegenerate toroidal domains
MHD equilibria with nonconstant pressure in nondegenerate toroidal domains Open
We prove the existence of piecewise smooth MHD equilibria in three-dimensional toroidal domains of \mathbb R^3 where the pressure is constant on the boundary but not in the interior. The pressure is piecewise constant and the plasma curren…
View article: Non-integrability and chaos for natural Hamiltonian systems with a random potential
Non-integrability and chaos for natural Hamiltonian systems with a random potential Open
Consider the ensemble of Gaussian random potentials {VL(q)}L=1∞ on the d-dimensional torus where, essentially, VL(q) is a real-valued trigonometric polynomial of degree L whose coefficients are independent standard normal variables. Our ma…
View article: Critical point asymptotics for Gaussian random waves with densities of any Sobolev regularity
Critical point asymptotics for Gaussian random waves with densities of any Sobolev regularity Open
We consider Gaussian random monochromatic waves u on the plane depending on a real parameter s that is directly related to the regularity of its Fourier transform. Specifically, the Fourier transform of u is fdσ, where dσ is the Hausdorff …
View article: Overdetermined boundary problems with nonconstant Dirichlet and Neumann data
Overdetermined boundary problems with nonconstant Dirichlet and Neumann data Open
We consider the overdetermined boundary problem for a general second-order semilinear elliptic equation on bounded domains of
\nR
\nn
\n, where one prescribes both the Dirichlet and Neumann data of the solution. We are interested in the c…
View article: Local wellposedness for the free boundary incompressible Euler equations with interfaces that exhibit cusps and corners of nonconstant angle
Local wellposedness for the free boundary incompressible Euler equations with interfaces that exhibit cusps and corners of nonconstant angle Open
We prove that free boundary incompressible Euler equations are locally well posed in a class of solutions in which the interfaces can exhibit corners and cusps. Contrary to what happens in all the previously known non-C1 water waves, the a…
View article: A Schiffer-type problem for annuli with applications to stationary planar Euler flows
A Schiffer-type problem for annuli with applications to stationary planar Euler flows Open
If on a smooth bounded domain $Ω\subset\mathbb{R}^2$ there is a nonconstant Neumann eigenfunction $u$ that is locally constant on the boundary, must $Ω$ be a disk or an annulus? This question can be understood as a weaker analog of the wel…
View article: Obstructions to topological relaxation for generic magnetic fields
Obstructions to topological relaxation for generic magnetic fields Open
For any axisymmetric toroidal domain $Ω\subset \mathbf{R}^3$ we prove that there is a locally generic set of divergence-free vector fields that are not topologically equivalent to any magnetohydrostatic (MHS) equilibrium in $Ω$. Each vecto…
View article: Limiting Measures and Energy Growth for Sequences of Solutions to Taubes’s Seiberg–Witten Equations
Limiting Measures and Energy Growth for Sequences of Solutions to Taubes’s Seiberg–Witten Equations Open
We consider sequences of solutions $$(\psi _n,A_n)_{n=1}^\infty $$ to Taubes’s modified Seiberg–Witten equations, associated with a fixed volume-preserving vector field X on a 3-manifold and corresponding to arbitrarily la…
View article: Issue Information
Issue Information Open
representations) c.W. eAton (Finite groups and representation theory) b.
View article: Optimal metrics for the first curl eigenvalue on 3-manifolds
Optimal metrics for the first curl eigenvalue on 3-manifolds Open
In this article we analyze the spectral properties of the curl operator on closed Riemannian 3-manifolds. Specifically, we study metrics that are optimal in the sense that they minimize the first curl eigenvalue among any other metric of t…
View article: Reconstruction of a Lorentzian manifold from its Dirichlet-to-Neumann map
Reconstruction of a Lorentzian manifold from its Dirichlet-to-Neumann map Open
We prove that the Dirichlet-to-Neumann map of the linear wave equation determines the topological, differentiable and conformal structure of the underlying Lorentzian manifold, under mild technical assumptions. With more stringent geometri…
View article: Finite-time singularity formation for angled-crested water waves
Finite-time singularity formation for angled-crested water waves Open
We show that the water waves system is locally wellposed in weighted Sobolev spaces which allow for interfaces with corners. No symmetry assumptions are required. These singular points are not rigid: if the initial interface exhibits a cor…