# On the convergence of the gradient expansion in hydrodynamics

## TL;DR

This paper investigates the convergence properties of hydrodynamic gradient expansions in strongly coupled plasmas, revealing finite radii of convergence and identifying spectral obstructions through holographic duality methods.

## Contribution

It demonstrates that hydrodynamic series in strongly coupled plasmas have finite convergence radii and links convergence issues to spectral level-crossings using holography.

## Key findings

- Hydrodynamic series have finite non-zero radii of convergence.
- Convergence is obstructed by level-crossings in the quasinormal spectrum.
- Holographic duality is used to analyze spectral properties.

## Abstract

Hydrodynamic excitations corresponding to sound and shear modes in fluids are characterised by gapless dispersion relations. In the hydrodynamic gradient expansion, their frequencies are represented by power series in spatial momenta. We investigate the analytic structure and convergence properties of the hydrodynamic series by studying the associated spectral curve in the space of complexified frequency and complexified spatial momentum. For the strongly coupled ${\cal N}=4$ supersymmetric Yang-Mills plasma, we use the holographic duality methods to demonstrate that the derivative expansions have finite non-zero radii of convergence. Obstruction to the convergence of hydrodynamic series arises from level-crossings in the quasinormal spectrum at complex momenta.