Abstract
In the present work, we consider the evolution of two fluids separated by a sharp interface in the presence of surface tension—like, for example, the evolution of oil bubbles in water. Our main result is a weak–strong uniqueness principle for the corresponding free boundary problem for the incompressible Navier–Stokes equation: as long as a strong solution exists, any varifold solution must coincide with it. In particular, in the absence of physical singularities, the concept of varifold solutions—whose global in time existence has been shown by Abels (Interfaces Free Bound 9(1):31–65, 2007) for general initial data—does not introduce a mechanism for non-uniqueness. The key ingredient of our approach is the construction of a relative entropy functional capable of controlling the interface error. If the viscosities of the two fluids do not coincide, even for classical (strong) solutions the gradient of the velocity field becomes discontinuous at the interface, introducing the need for a careful additional adaption of the relative entropy.
Highlights
In evolution equations for interfaces, topological changes and geometric singularities often occur naturally, one basic example being the pinchoff of liquid droplets
The transition from strong to weak solution concepts for PDEs is prone to incurring unphysical non-uniqueness of solutions; for example, Brakke’s concept of varifold solutions for mean curvature flow admits sudden vanishing of the evolving surface at any time [22]; for the
The main theorem of our present work provides a corresponding result for the flow of two incompressible immiscible fluids with surface tension: varifold solutions to the free boundary problem for the Navier–Stokes equation for two fluids are unique until the strong solution for the free boundary problem develops a singularity
Summary
In evolution equations for interfaces, topological changes and geometric singularities often occur naturally, one basic example being the pinchoff of liquid droplets (see Fig. 1). The transition from strong to weak solution concepts for PDEs is prone to incurring unphysical non-uniqueness of solutions; for example, Brakke’s concept of varifold solutions for mean curvature flow admits sudden vanishing of the evolving surface at any time [22]; for the. The flow of each single fluid is described by the incompressible Navier–Stokes equation, while the fluid–fluid interface evolves by pure transport along the fluid flow and a surface tension force acts at the fluid–fluid interface For this free boundary problem for the flow of two immiscible incompressible fluids with surface tension, Abels [2] has established the global existence of varifold solutions for quite general initial data. In Theorem 1, below, we prove that as long as a strong solution to this evolution problem exists, any varifold solution in the sense of Abels [4] must coincide with it
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
