15,141 research outputs found
Stability Conditions and Lagrangian Cobordisms
In this paper we study the interplay between Lagrangian cobordisms and
stability conditions. We show that any stability condition on the derived
Fukaya category of a symplectic manifold
induces a stability condition on the derived Fukaya category of Lagrangian
cobordisms . In addition, using stability
conditions, we provide general conditions under which the homomorphism , introduced by Biran and Cornea, is
an isomorphism. This yields a better understanding of how stability conditions
affect and it allows us to elucidate Haug's result, that the
Lagrangian cobordism group of is isomorphic to
.Comment: 53 pages, 3 figures, expansions and revisions, improvement of
expositio
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
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