Stability of nonlinear dissipative systems with applications in fluid dynamics

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.

Figures (3)

• Figure 1: Stability ensures that if the initial profile is slightly altered, then the resulting evolution remains close to the reference solution for all later times, guaranteeing that numerical solutions behave consistently with the true physical problem over time. Without stability, even accurate or consistent schemes can produce meaningless results.
• Figure 2: a) Long term behaviour of solutions to Riccati equation with $D>0$ and different initial conditions. Solutions in region 1 blow up in finite time, while solutions in regions 2 ($\mathcal{R}<1$) and 3 are monotonically decreasing and monotonically increasing, respectively, and both asymptotically converge to the equilibrium value $y_e^{-}$ as $t \to +\infty$. b) Long term behaviour of the Bernoulli equation when $\mathcal{R}^*=\frac{2\mu y_0}{\beta}<1$ for different initial conditions. Solutions in region 1 (too large initial perturbation) blow up in finite time, while solutions in region 2 (small enough initial perturbation $z_0<\frac{\beta-2\mu y_0}{\mu}$) are monotonically decreasing to zero.
• Figure 3: a) Solutions to Burgers equation for different values of the Reynolds number, $Re$, at $t=0.1$. b) Relative value in time of the norm of the solutions to Burgers equation with respect to the initial condition for different values of the Reynolds number, $Re$.

Theorems & Definitions (2)

• Theorem 1
• Theorem 2