Distribution of Energy and Convergence to Equilibria in Extended Dissipative Systems

Abstract: 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 $\partial_t e = div_x f - d$, where $-f$ is the energy flux and $d \ge 0$ the energy dissipation rate. We also suppose that $|f|^2 \le b(e)d$ 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 $\R \times \T$, 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.