Abstract
In this article we consider the inhomogeneous incompressible Euler equations describing two fluids with different constant densities under the influence of gravity as a differential inclusion. By considering the relaxation of the constitutive laws we formulate a general criterion for the existence of infinitely many weak solutions which reflect the turbulent mixing of the two fluids. Our criterion can be verified in the case that initially the fluids are at rest and separated by a flat interface with the heavier one being above the lighter one—the classical configuration giving rise to the Rayleigh–Taylor instability. We construct specific examples when the Atwood number is in the ultra high range, for which the zone in which the mixing occurs grows quadratically in time.
Highlights
We study the mixing of two different density perfect incompressible fluids subject to gravity, when the heavier fluid is on top
In this setting an instability known as the Rayleigh–Taylor instability forms on the interface between the fluids which eventually evolves into turbulent mixing
In the spirit of the results by De Lellis and the 3rd author [16,17], for the homogeneous incompressible Euler equations, we develop a convex integration strategy for the inhomogeneous Euler system to prove the existence of weak solutions for the Cauchy problem (1.1), (1.3)
Summary
We study the mixing of two different density perfect incompressible fluids subject to gravity, when the heavier fluid is on top. While [11] shows the non-uniqueness of solutions to the incompressible porous media equation, the paper [33] provides the full relaxation of the equation allowing to establish sharp linear bounds for the growth of the mixing zone in the Muskat problem. We already mentioned the different relaxation approach for the incompressible porous media equation via gradient flow in [29], the unique solution of this relaxation approach turned out to be recovered as a subsolution in [33] Another classical instability in fluid dynamics is the Kelvin–Helmholtz instability generated by vortex sheet initial data. The paper is organized as follows: in Section 2 we present our main results, one regarding the convex integration of the inhomogeneous incompressible Euler equations regardless of initial data, and one regarding the existence of appropriate subsolutions in the case of a flat initial interface.
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