Global well-posedness for 2D inhomogeneous viscous flows with rough data via dynamic interpolation

TL;DR

This paper proves global existence, stability, and uniqueness of solutions for 2D inhomogeneous viscous flows with rough data, using dynamic interpolation and decay estimates, without smallness assumptions, in various domains.

Contribution

It introduces a novel dynamic interpolation method and decay estimates to establish well-posedness for 2D inhomogeneous viscous flows at critical regularity.

Findings

Global-in-time existence and stability of solutions

Solutions have a uniformly C^1 flow ensuring geometrical structure propagation

Uniqueness and flow map continuity are established without smallness conditions

Abstract

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 L^2 , 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 C^1 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 true 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.