Weak–strong uniqueness for the Navier–Stokes equation for two fluids with surface tension

Fischer JL, Hensel S. 2020. Weak–strong uniqueness for the Navier–Stokes equation for two fluids with surface tension. Archive for Rational Mechanics and Analysis. 236, 967–1087.

Download
OA 2020_ArchRatMechAn_Fischer.pdf 1.90 MB

Journal Article | Published | English

Scopus indexed
Department
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.
Publishing Year
Date Published
2020-05-01
Journal Title
Archive for Rational Mechanics and Analysis
Volume
236
Page
967-1087
ISSN
eISSN
IST-REx-ID

Cite this

Fischer JL, Hensel S. Weak–strong uniqueness for the Navier–Stokes equation for two fluids with surface tension. Archive for Rational Mechanics and Analysis. 2020;236:967-1087. doi:10.1007/s00205-019-01486-2
Fischer, J. L., & Hensel, S. (2020). Weak–strong uniqueness for the Navier–Stokes equation for two fluids with surface tension. Archive for Rational Mechanics and Analysis. Springer Nature. https://doi.org/10.1007/s00205-019-01486-2
Fischer, Julian L, and Sebastian Hensel. “Weak–Strong Uniqueness for the Navier–Stokes Equation for Two Fluids with Surface Tension.” Archive for Rational Mechanics and Analysis. Springer Nature, 2020. https://doi.org/10.1007/s00205-019-01486-2.
J. L. Fischer and S. Hensel, “Weak–strong uniqueness for the Navier–Stokes equation for two fluids with surface tension,” Archive for Rational Mechanics and Analysis, vol. 236. Springer Nature, pp. 967–1087, 2020.
Fischer JL, Hensel S. 2020. Weak–strong uniqueness for the Navier–Stokes equation for two fluids with surface tension. Archive for Rational Mechanics and Analysis. 236, 967–1087.
Fischer, Julian L., and Sebastian Hensel. “Weak–Strong Uniqueness for the Navier–Stokes Equation for Two Fluids with Surface Tension.” Archive for Rational Mechanics and Analysis, vol. 236, Springer Nature, 2020, pp. 967–1087, doi:10.1007/s00205-019-01486-2.
All files available under the following license(s):
Creative Commons Attribution 4.0 International Public License (CC-BY 4.0):
Main File(s)
Access Level
OA Open Access
Date Uploaded
2020-11-20
MD5 Checksum
f107e21b58f5930876f47144be37cf6c


Export

Marked Publications

Open Data IST Research Explorer

Search this title in

Google Scholar