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

J.L. Fischer, S. Hensel, Archive for Rational Mechanics and Analysis (2020).

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Journal Article | Epub ahead of print | English
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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.
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Date Published
2020-02-04
Journal Title
Archive for Rational Mechanics and Analysis
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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. 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. 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, 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, 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.
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, doi:10.1007/s00205-019-01486-2.
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