Weak-Strong Uniqueness for Navier-Stokes/Allen-Cahn system
Radim Ho\v{s}ek, V\'aclav M\'acha
TL;DR
This paper proves that in a 3D bounded domain, weak solutions to the coupled Navier-Stokes/Allen-Cahn system are unique when a strong solution exists, using a relative entropy approach.
Contribution
It establishes the weak-strong uniqueness principle for the Navier-Stokes/Allen-Cahn system in three dimensions, a result not previously known.
Findings
Weak solutions coincide with strong solutions when both exist.
The proof uses a relative entropy method.
The result applies to bounded domains in three spatial dimensions.
Abstract
The coupled Navier-Stokes/Allen-Cahn system is a simple model to describe phase separation in two-component systems interacting with an incompressible fluid flow. We demonstrate the \emph{weak-strong uniqueness} result for this system in a bounded domain in three spatial dimensions which implies that when a strong solution exists then a weak solution emanating from the same data coincides with the strong solution on its whole life-span. The proof of given assertion relies on a form of a relative entropy method.
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