15,141 research outputs found

    Stability Conditions and Lagrangian Cobordisms

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    In this paper we study the interplay between Lagrangian cobordisms and stability conditions. We show that any stability condition on the derived Fukaya category DFuk(M)D\mathcal{F}uk(M) of a symplectic manifold (M,ω)(M,\omega) induces a stability condition on the derived Fukaya category of Lagrangian cobordisms DFuk(C×M)D\mathcal{F}uk(\mathbb{C} \times M). In addition, using stability conditions, we provide general conditions under which the homomorphism Θ:ΩLag(M)→K0(DFuk(M))\Theta: \Omega_{Lag}(M)\to K_0(D\mathcal{F}uk(M)), introduced by Biran and Cornea, is an isomorphism. This yields a better understanding of how stability conditions affect Θ\Theta and it allows us to elucidate Haug's result, that the Lagrangian cobordism group of T2T^2 is isomorphic to K0(DFuk(T2))K_0(D\mathcal{F}uk(T^2)).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

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    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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