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Infinite energy solutions of the two-dimensional Navier-Stokes equations

These notes are based on a series of lectures delivered by the author at the University of Toulouse in February 2014. They are entirely devoted to the initial value problem and the long-time behavior of solutions for the two-dimensional incompressible Navier-Stokes equations, in the particular case where the domain occupied by the fluid is the whole plane $\mathbb{R}^2$ and the velocity field is only assumed to be bounded. In this context, local well-posedness is not difficult to establish, and a priori estimates on the vorticity distribution imply that all solutions are global and grow at most exponentially in time. Moreover, as was recently shown by S. Zelik, localized energy estimates can be used to obtain a much better control on the uniformly local energy norm of the velocity field. The aim of these notes is to present, in an explanatory and self-contained way, a simplified and optimized version of Zelik's argument which, in combination with a new formulation of the Biot-Savart law for bounded vorticities, allows one to show that the uniform norm of the velocity field grows at most linearly in time. The results do not rely on the viscous dissipation, and remain therefore valid for the so-called "Serfati solutions" of the two-dimensional Euler equations. Finally, a recent work by S. Slijepcevic and the author shows that all solutions remain uniformly bounded in the viscous case if the velocity field and the pressure are periodic in one space direction.