Evolution by mean curvature flow in sub-Riemannian geometries: A stochastic approach
Nicolas Dirr Federica Dragoni Max von Renesse
Communications on Pure & Applied Analysis 2010, 9(2): 307-326 doi: 10.3934/cpaa.2010.9.307
We study evolution by horizontal mean curvature flow in sub- Riemannian geometries by using stochastic approach to prove the existence of a generalized evolution in these spaces. In particular we show that the value function of suitable family of stochastic control problems solves in the viscosity sense the level set equation for the evolution by horizontal mean curvature flow.
keywords: stochastic processes and control. level set equation sub-Riemannian geometries Mean curvature flow
On the Lagrangian structure of quantum fluid models
Philipp Fuchs Ansgar Jüngel Max von Renesse
Discrete & Continuous Dynamical Systems - A 2014, 34(4): 1375-1396 doi: 10.3934/dcds.2014.34.1375
Some quantum fluid models are written as the Lagrangian flow of mass distributions and their geometric properties are explored. The first model includes magnetic effects and leads, via the Madelung transform, to the electromagnetic Schrödinger equation in the Madelung representation. It is shown that the Madelung transform is a symplectic map between Hamiltonian systems. The second model is obtained from the Euler-Lagrange equations with friction induced from a quadratic dissipative potential. This model corresponds to the quantum Navier-Stokes equations with density-dependent viscosity. The fact that this model possesses two different energy-dissipation identities is explained by the definition of the Noether currents.
keywords: magnetic Schrödinger equation symplectic form Noether theory quantum Navier-Stokes equations osmotic velocity. Geometric mechanics

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