Mathematics and Turbulence: where do we stand?

Abstract

This contribution covers the topics presented by the authors at the {\it ``Fundamental Problems of Turbulence, 50 Years after the Marseille Conference 1961"} meeting that took place in Marseille in 2011. It focuses on some of the mathematical approaches to fluid dynamics and turbulence. This contribution does not pretend to cover or answer, as the reader may discover, the fundamental questions in turbulence, however, it aims toward presenting some of the most recent advances in attacking these questions using rigorous mathematical tools. Moreover, we consider that the proofs of the mathematical statements (concerning, for instance, finite time regularity, weak solutions and vanishing viscosity) may contain information as relevant, to the understanding of the underlying problem, as the statements themselves.

References in corpus (1)

• Vanishing Viscous Limits for 3D Navier-Stokes Equations with A Navier-Slip Boundary Condition

Cited by in corpus (25)

• The instanton method and its numerical implementation in fluid mechanics
• Onsager's Conjecture for the Incompressible Euler Equations in Bounded Domains
• Onsager's Conjecture with Physical Boundaries and an Application to the Vanishing Viscosity Limit
• Towards computable flows and robust estimates for inf-sup stable FEM applied to the time-dependent incompressible Navier-Stokes equations
• Remarks on the emergence of weak Euler solutions in the vanishing viscosity limit
• The inviscid limit of Navier-Stokes for analytic data on the half-space
• Convergence of the 2D Euler-$\alpha$ to Euler equations in the Dirichlet case: indifference to boundary layers
• Remarks on high Reynolds numbers hydrodynamics and the inviscid limit
• The Euler Equations in planar nonsmooth convex domains
• Approximation of 2D Euler Equations by the Second-Grade Fluid Equations with Dirichlet Boundary Conditions
• A Characteristic Mapping Method for the two-dimensional incompressible Euler equations
• Hidden spatiotemporal symmetries and intermittency in turbulence
• Energy dissipation caused by boundary layer instability at vanishing viscosity
• On the Extension of Onsager's Conjecture for General Conservation Laws
• Vorticity measures and the inviscid limit
• On the Euler+Prandtl expansion for the Navier-Stokes equations
• The 3D incompressible Euler equations with a passive scalar: a road to blow-up?
• Boundary Vorticity Estimates for Navier-Stokes and Application to the Inviscid Limit
• Fluids, Elasticity, Geometry, and the Existence of Wrinkled Solutions
• A traveling wave bifurcation analysis of turbulent pipe flow
• Logarithmic lattice models for flows with boundaries
• Some preliminary observations on a defect Navier-Stokes system
• How to extract a spectrum from hydrodynamic equations
• Hölder regularity of the pressure for weak solutions of the 3D Euler equations in bounded domains
• Singularity formation of vortex sheets in 2D Euler equations using the characteristic mapping method