Distribution of Energy and Convergence to Equilibria in Extended Dissipative Systems

We are interested in understanding the dynamics of dissipative partial differential equations on unbounded spatial domains. We consider systems for which the energy density $e\ge 0$ satisfies an evolution law of the form ${\mathrm{\partial }}_{t}e=di{v}_{x}f-d$, where $-f$ is the energy flux and $d\ge 0$ the energy dissipation rate. We also suppose that $\mathrm{\mid }f{\mathrm{\mid }}^2\le b(e)d$ for some nonnegative function $b$. Under these assumptions we establish simple and universal bounds on the time-integrated energy flux, which in turn allow us to estimate the amount of energy that is dissipated in a given domain over a long interval of time. In low space dimensions $N\le 2$, we deduce that any relatively compact trajectory converges on average to the set of equilibria, in a sense that we quantify precisely. As an application, we consider the incompressible Navier-Stokes equation in the infinite cylinder $\mathbb{R}\times \text{T}$, and for solutions that are merely bounded we prove that the vorticity converges uniformly to zero on large subdomains, if we disregard a small subset of the time interval.