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Andrea Nahmod

Publications and source records attributed to Andrea Nahmod.

11 recordsLinked to original sources
arXiv2025

Non-equilibrium steady state for a three-mode energy cascade model

Motivated by the central phenomenon of energy cascades in wave turbulence theory, we construct non-equilibrium statistical steady states (NESS), or invariant measures, for a simplified model derived from the nonlinear Schr\"odinger (NLS) equation with external forcing and dissipation. This new perspective to studying energy cascades, distinct from traditional analyses based on kinetic equations and their cascade spectra, focuses on the underlying statistical steady state that is expected to hold when the cascade spectra of wave turbulence manifest. In the full generality of the (infinite dimensional) nonlinear Schr\"odinger equation, constructing such invariant measures is more involved than the rigorous justification of the Kolmogorov-Zakharov (KZ) spectra, which itself remains an outstanding open question despite the recent progress on mathematical wave turbulence. Since such complexity remains far beyond the current knowledge (even for much simpler chain models), we confine our analysis to a three-mode reduced system that captures the resonant dynamics of the NLS equation, offering a tractable framework for constructing the NESS. For this, we introduce a novel approach based on solving an elliptic Feynman-Kac equation to construct the needed Lyapunov function.

math.PR↗
arXiv2018

An infinite sequence of conserved quantities for the cubic Gross-Pitaevskii hierarchy on $\mathbb{R}$

We consider the (de)focusing cubic Gross-Pitaevskii (GP) hierarchy on $\mathbb{R}$, which is an infinite hierarchy of coupled linear inhomogeneous PDE which appears in the derivation of the cubic nonlinear Schrödinger (NLS) equation from quantum many-particle systems. Motivated by the fact that the cubic NLS on $\mathbb{R}$ is an integrable equation which admits infinitely many conserved quantities, we exhibit an infinite sequence of operators which generate analogous conserved quantities for the GP hierarchy.

math.AP↗
arXiv2018

The Surface Quasi-Geostrophic Equation with Random Diffusion

Consider the surface quasi-geostrophic equation with random diffusion, white in time. We show global existence and uniqueness in high probability for the associated Cauchy problem satisfying a Gevrey type bound. This article is inspired by recent work of Glatt-Holtz and Vicol.

math.AP↗
arXiv2018

Almost sure boundedness of iterates for derivative nonlinear wave equations

We study nonlinear wave equations on $\mathbb R^{2+1}$ with quadratic derivative nonlinearities, which include in particular nonlinearities exhibiting a null form structure, with random initial data in $H_x^1\times L^2_x$. In contrast to the counterexamples of Zhou \cite{Zhou} and Foschi-Klainerman \cite{FK}, we obtain a uniform time interval $I$ on which the Picard iterates of all orders are almost surely bounded in $C_t(I ; \dot H_x^1)$.

math.AP↗
arXiv2017

Global Flows with Invariant Measures for the Inviscid Modified SQG Equations

We consider the family known as modified or generalized surface quasi-geostrophic equations (mSQG) consisting of the classical inviscid surface quasi-geostrophic (SQG) equation together with a family of regularized active scalars given by introducing a smoothing operator of nonzero but possibly arbitrarily small degree. This family naturally interpolates between the 2D Euler equation and the SQG equation. For this family of equations we construct an invariant measure on a rough $L^2$-based Sobolev space and establish the existence of solutions of arbitrarily large lifespan for initial data in a set of full measure in the rough Sobolev space.

math.AP↗
arXiv2014

Equivariant and self-similar standing waves for a Hamiltonian hyperbolic-hyperbolic spin-field system

In this paper we study the existence of special symmetric solutions to a Hamiltonian hyperbolic-hyperbolic coupled spin-field system, where the spins are maps from $\mathbb R^{2+1}$ into the sphere $§^2$ or the pseudo-sphere $\H^2$. This model was introduced in \cite{Martina} and it is also known as the {\it hyperbolic-hyperbolic generalized Ishimori system}. Relying on the hyperbolic coordinates introduced in \cite{KNZ11}, we prove the existence of equivariant standing waves in both regular hyperbolic coordinates as well as similarity variables, and describe their asymptotic behaviour.

math.AP↗
arXiv2010

Invariant weighted Wiener measures and almost sure global well-posedness for the periodic derivative NLS

In this paper we construct an invariant weighted Wiener measure associated to the periodic derivative nonlinear Schrödinger equation in one dimension and establish global well-posedness for data living in its support. In particular almost surely for data in a Fourier-Lebesgue space ${\mathcal F}L^{s,r}(\T)$ with $s \ge \frac{1}{2}$, $2 < r < 4$, $(s-1)r <-1$ and scaling like $H^{\frac{1}{2}-ε}(\T),$ for small $ε>0$. We also show the invariance of this measure.

math.AP↗
arXiv2007

Schrödinger Maps and their associated Frame Systems

In this paper we establish the equivalence of solutions between Schrödinger map into $\mathbb{S}^2$ or $ \mathbb{H}^2$ and their associated gauge invariant Schrödinger equations. We also establish the existence of global weak solutions into $\mathbb{H}^2$ in two space dimensions. We extend these ideas for maps into compact hermitian symmetric manifolds with trivial first cohomology.

math.AP↗
arXiv2001

On the well-posedness of the wave map problem in high dimensions

We construct a gauge theoretic change of variables for the wave map from $R \times R^n$ into a compact group or Riemannian symmetric space, prove a new multiplication theorem for mixed Lebesgue-Besov spaces, and show the global well-posedness of a modified wave map equation - $n \ge 4$ - for small critical initial data. We obtain global existence and uniqueness for the Cauchy problem of wave maps into {\it compact} Lie groups and symmetric spaces with small critical initial data and $n \ge 4$.

math.AP↗
arXiv2001

On Schrödinger maps

We study the question of well-posedness of the Cauchy problem for Schrödinger maps from $\rone \times \rtwo$ to the sphere $\stwo$ or to ${\mathbb H^2}$, the hyperbolic space. The idea is to choose an appropriate gauge change so that the derivatives of the map will satisfy a certain nonlinear Schrödinger system of equations and then study this modified Schrödinger map system (MSM). We then prove local well posedness of the Cauchy problem for the MSM with minimal regularity assumptions on the data and outline a method to derive well posedness of the Schrödinger map itself from it. In proving well posedness of the MSM, the heart of the matter is resolved by considering truly quatrilinear forms of weighted $L^2$ functions.

math.AP↗