Seminars Sorted by Series

Abstract: I will give a review of recent progress on finite-time singularity formation in incompressible fluid models.

Abstract: The quadratic Monge optimal transportation problem can be revisited in Euler's language of Hydrodynamics as was explained by Jean-David Benamou and the speaker about 20 years ago. It turns out that Einstein's theory of gravitation, at...

Abstract. I will describe a new geometric approach for the shock formation problem for the Euler equations.   A complete description of the solution along the hypersurface of first singularities or preshocks will be given.

Abstract:   I will discuss the construction of continuous solutions to the incompressible Euler equations that exhibit local dissipation of energy and the surrounding motivations.  A significant open question, which represents a strong form of the...

Abstract: The incompressible porous media (IPM) equation describes the evolution of density transported by an incompressible velocity field given by Darcy’s law. Here the velocity field is related to the density via a singular integral operator...

Abstract: We consider the density properties of divergence-free vector fields $b \in L^1([0,1],\BV([0,1]^2))$ which are ergodic/weakly mixing/strongly mixing: this means that their Regular Lagrangian Flow $X_t$ is an ergodic/weakly mixing/strongly...

Abstract: I will discuss some questions related to turbulence.

Abstract: 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...

Abstract: 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...

Abstract: 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...

Abstract:  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...

Abstract: 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...

Abstract: 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...

Abstract: 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...

Abstract: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...

For any regularity exponent $\beta<\frac 12$, we construct non-conservative weak solutions to the 3D incompressible Euler equations in the class $C^0_t (H^{\beta} \cap L^{\frac{1}{(1-2\beta)}})$.  By interpolation, such solutions belong to $C^0_tB^{s}_{3,\infty}$ for $s$ approaching $\frac 13$ as $\beta$ approaches $\frac 12$.  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 $L^2$-based regularity index exceeding $\frac 13$.  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.