It is not known if solutions to the 3D Navier-Stokes equations for incompressible Newtonian fluids flowing in a finite periodic box can become singular in finite time. Indeed, this open issue is the subject of one of the $1M Clay Prize problems. It is known that solutions remains smooth as long as the enstrophy, i.e., the mean-square vorticity in the flow filed, is finite. The instantaneous generation rate of enstrophy is given by a functional that can be bounded in terms of the enstrophy itself using elementary functional estimates. That classical analysis establishes short-time regularity of solutions starting with arbitrarily large (but finite) enstrophy but does not rule out the possibility of singularities subsequently appearing in the flow in finite time. In this work we formulate and solve the variational problem for the maximal growth rate of enstrophy and display flows that saturate the functional analytic estimates and amplify vorticity at the greatest possible rate. Implications for questions of regularity or singularity in solutions of the 3D Navier-Stokes equations are discussed.
This joint work with Lu Lu was published in Indiana University Mathematics Journal Vol. 57, pp. 2693-2727 (2008).
Many basic equations in physiscs can be written in the form of conservation law. However, as it is common in the field of PDEs and SPDEs, classical or strong solutions do not exist in general and, on the other hand, weak solutions are not unique. The notion of kinetic formulation and kinetic solution turns out to be a very convenient tool to overcome these difficulties.
In this talk, I will present a well-posedness result for degenerate parabolic stochastic conservation laws and, in the case of hyperbolic stochastic conservation laws, discuss the hydrodynamic limit of an approximation in the sense of Bhatnagar-Gross-Krook.
In order to enable all students to choose the lectures of our institute which fits their interest all lecturers will give a short overview of their courses.
