Stability of nonlinear dissipative systems with applications in fluid dynamics

TL;DR

This paper develops a stability criterion for nonlinear dissipative PDEs, linking linear dissipation, nonlinearities, and forcing, with applications to fluid dynamics models like Burgers, KPP-Fisher, and Kuramoto-Sivashinsky equations.

Contribution

It introduces an explicit stability condition for nonlinear dissipative PDEs, applicable to various models, connecting stability to physical parameters like Reynolds number.

Findings

Derived a sufficient stability condition based on solution norms in Sobolev spaces.

Applied the criterion to fluid models, interpreting it in terms of Reynolds number.

Extended the analysis to multiple nonlinear PDEs, demonstrating broad applicability.

Abstract

Nonlinear partial differential equations are central to physics, engineering, and finance. Except in a limited number of integrable cases, their solution generally requires numerical methods whose cost becomes prohibitive in high-dimensional regimes or at fine resolution. Nonlinear phenomena such as turbulence are notoriously difficult to predict because of their extreme sensitivity to small variations in initial conditions, except when certain stability conditions are fulfilled. Indeed, stability allows us to achieve reliable approximate dynamics, since it determines whether small perturbations remain bounded or are amplified, potentially leading to markedly different long-term behavior. Here, we investigate the stability of dissipative partial differential equations with second-order nonlinearities. By analyzing the time evolution of solution norms in Sobolev spaces, we establish a sufficient condition for stability that links the characteristics of the linear dissipative operator, the quadratic nonlinear term, and the external forcing. The resulting criterion is expressed as an explicit inequality that guarantees stability for a wide range of initial conditions. As an illustration, we apply the framework to fluid-dynamical models governed by nonlinear partial differential equations. In particular, for the Burgers equation, the condition admits a natural interpretation in terms of the Reynolds number, thereby directly linking the stability threshold to the competition between viscous dissipation and inertial advection. We further demonstrate the scope of the approach by extending the analysis to the KPP-Fisher and Kuramoto-Sivashinsky equations.