We consider the evolution of two-dimensional incompressible flows
with variable density, only bounded and bounded away from zero.
Assuming that the initial velocity belongs to a suitable critical subspace
of , we
prove a global-in-time existence and stability result for the initial (boundary) value
problem.
Our proof relies on new time decay estimates for finite energy weak solutions and
on a “dynamic interpolation” argument. We show that the constructed solutions have a
uniformly
flow, which ensures the propagation of geometrical structures in the fluid and
guarantees that the Eulerian and Lagrangian formulations of the equations are equivalent.
By adopting this latter formulation, we establish the uniqueness of the solutions for
prescribed data and the continuity of the flow map in an energy-like functional framework.
In contrast with prior works, our results hold in the critical regularity setting
without any smallness assumption. Our approach uses only elementary tools and
applies indistinctly to the cases where the fluid domain is the whole plane, a smooth
two-dimensional bounded domain, or the torus.
Keywords
critical regularity, uniqueness, global solutions,
inhomogeneous Navier–Stokes equations, rough density
