School "Boundary Analysis for Dispersive and Viscous Fluids"

Invited Talks

                       On the effect of the Coriolis force in the double cascade of two-dimensional turbulence

      Yuri Cacchiò

       GSSI, L' Aquila

 

Abstract:

Geophysical fluid dynamics refers to the fluid dynamics of naturally occurring flows, such  as oceans and planetary atmospheres on Earth and other planets. These flows are primarily characterized by two elements: stratification and rotation. In this article we investigate the effects of rotation on the dynamics, by neglecting stratification, in a 2D model. We consider the well-known 2D β-plane Navier-Stokes equations in the statistically forced case. Our problem addresses energy-related phenomena associated with the solution of the equations. To maintain the fluid in a turbulent state, we introduce energy into the system through a stochastic force. In the 2D case, a scaling analysis argument indicates a direct cascade of enstrophy and an inverse energy cascade. Following the evolution of the so-called third-order structure function, we compare the behaviour of the direct/inverse cascade with the 2D model lacking the Coriolis force, observing that at small scales, the enstrophy flux from larger to smaller scales remains unaffected by the planetary rotation, in contrast to the large scales where the energy flux. from smaller to larger scales is dominated by the Coriolis parameter, confirming experimental and numerical observations. In fact, to the best of our knowledge this is the first mathematically rigorous study of the above equations. This talk will be based on joint work with Amirali Hannani and Gigliola Staffilani

 

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On the instability of a 1D Gross-Pitaevskii steady flow past a delta potential

Martino Caliaro 

GSSI, L' Aquila

 

Abstract:

In this talk we investigate the dynamics of a one-dimensional Gross-Pitaevskii flow in presence of a static obstacle. The flow of the quantum fluid is obtained by imposing boundary conditions on its density ρ and on its velocity v at spatial infinity. The static obstacle is modeled by a repulsive delta potential.

For subcritical values of the potential strength and for subsonic velocities, this system admits two branches of stationary states. By looking at numerical simulations, it is generally argued that the first branch is stable while the second branch is unstable. We employ the method of the Evans function to show the linear instability of the second branch, if the potential strength is small enough. This is a joint work with Paolo Antonelli (GSSI). 

 

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                                                         Introduction on quantum hydrodynamic models and some recent results

Hao Zheng

Chinese Academy of Sciences, Beijing

 

Abstract:

Quantum hydrodynamic (QHD) models study hydrodynamic phenomena where quantum effects must be taken into account, such as superfluidity, Bose-Einstein condensation, quantum plasmas, or semiconductor devices. In this talk, I will give a briefly introduction on the physical background of several QHD models, as well as some related mathematical problems and their literatures. Last, I will introduce some recent results
about the 1-dimensional collisional Quantum Hydrodynamics (QHD) system and its relaxation-time limit, which is a work in collaborating with Paolo Antonelli and Pierangelo Marcati.

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Vanishing viscosity limit for a class of hyperbolic systems in 1-d with nonlinear viscosity

Animesh Jana 

IMATI-CNR, Pavia

 

Abstract:

In this talk, we will consider a class of hyperbolic systems in one space dimension with a nonlinear viscosity matrix. First, we prove the global existence of smooth solutions to the parabolic equation for initial data with a small total variation. We show that the solution to the parabolic equation converges to a semi-group solution of the hyperbolic system as viscosity goes to zero. Furthermore, we prove that the diffusion limit coincides with the one obtained when the viscosity matrix is the identity matrix. This talk is based on a joint work with Boris Haspot.