Workshop on Recent developments in incompressible fluid dynamics

For any regularity exponent β<12, we construct non-conservative weak solutions to the 3D incompressible Euler equations in the class C0t(Hβ∩L1(1−2β)).  By interpolation, such solutions belong to C0tBs3,∞ for s approaching 13 as β approaches 12.  Hence this result provides a new proof of the flexible side of the Onsager conjecture, which is independent from that of Isett.  Of equal importance is that the intermittent nature of our solutions matches that of turbulent flows, which are observed to possess an L2-based regularity index exceeding 13.  The proof employs an intermittent convex integration scheme for the 3D incompressible Euler equations.  We employ a scheme with higher-order Reynolds stresses, which are corrected via a combinatorial placement of intermittent pipe flows of optimal relative intermittency.

In his seminal work, Leray demonstrated the existence of global weak solutions, with nonincreasing energy, to the Navier-Stokes equations in three dimensions. In this talk we exhibit two distinct Leray solutions with zero initial velocity and...

Whether the 3D incompressible Euler equations can develop a finite-time singularity from smooth initial data is an outstanding open problem. The presence of vortex stretching is the primary source of a potential finite-time singularity. However, to...

The viscosity of a fluid is usually a constant, independent of the stress. There are however in nature several examples of fluids (ice, molten lava, blood, certain polymers, some salt solutions) where viscosity changes under applied forces. Such...

We discuss recent mathematical constructions of self-similar gravitational collapse for Newtonian stars governed by the Euler-Poisson system, known as Larson-Penston solutions for the isothermal stars and Yahil solutions for polytropic stars, and...

 It is known since the work of Dyachenko & Zakharov  that for the weakly nonlinear 2d infinite depth water waves, there are no 3-wave interactions and all of the 4-wave interaction coefficients vanish on the non-trivial resonant manifold. In this...

In this talk we consider the pressureless Euler system in dimension greater than or equal to two. Several works have been devoted to the search of solutions which satisfy the following adhesion or sticky particle principle: if two particles of the...

In this talk we will present a construction of global existence of small solutions of the modified SQG equations, close to the disk. The proof uses KAM theory and a Nash-Moser argument, and does not involve any external parameters. We moreover prove...

Let M be a smooth manifold in an Euclidean space; consider the motion of a material point on M in absence of friction. The D'Alembert Principle says that the acceleration vector is orthogonal to the tangent space to M, and this fact defines the...

I will discuss some questions related to turbulence.