1,516 research outputs found
Weak-strong uniqueness for the Navier-Stokes equation for two fluids with surface tension
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 [2] 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.Comment: 104 page
Isoperimetric inequalities for the handlebody groups
We show that the mapping class group of a handlebody of genus at least 2 has
a Dehn function of at most exponential growth type.Comment: 21 pages, 1 figur
Nielsen Realisation by Gluing: Limit Groups and Free Products
We generalise the Karrass-Pietrowski-Solitar and the Nielsen realisation
theorems from the setting of free groups to that of free products. As a result,
we obtain a fixed point theorem for finite groups of outer automorphisms acting
on the relative free splitting complex of Handel--Mosher and on the outer space
of a free product of Guirardel--Levitt, as well as a relative version of the
Nielsen realisation theorem, which in the case of free groups answers a
question of Karen Vogtmann. We also prove Nielsen realisation for limit groups,
and as a byproduct obtain a new proof that limit groups are CAT(). The
proofs rely on a new version of Stallings' theorem on groups with at least two
ends, in which some control over the behaviour of virtual free factors is
gained.Comment: 28 pages, 1 figur
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