We investigate weak solutions to the Dirichlet problem for an elliptic equation with a drift b whose divergence is sign-defined. We assume b belongs to some weak Morrey class which includes in the 3D case, in particular, drifts having a singularity along the axis with the asymptotic c/r, where r is the distance to the axis. The problem under consideration is motivated by some questions arising in the theory of axially symmetric solutions to the Navier-Stokes equations. We present results on existence, uniqueness and local properties of weak solutions to this problem as well as its relation to the Navier-Stokes theory. Based on a joint work with M. Chernobai.
In general dimensions, weak solutions of the Euler Lagrange equation of the area functional are naturally constructed in the class of integral varifolds. As the regularity theory of area-stationary integral varifolds is still substantially incomplete, it is important to enlarge the related mathematical machinery, both to handle the possible singularities and to advance regularity theory. In this regard, the talk will survey the implementation of the concept of weakly differentiable functions on integral varifolds of locally bounded first variation of area. The present theory of Sobolev spaces on metric measure spaces turns out to ill-adapted for this purpose. Instead, a coherent theory can be constructed by defining a non-linear class of functions taking into account the extrinsic geometry of the varifold. The inclusion of non-vanishing first variation therein is not only theoretically preferable but is also important for the needs of problems from applied analysis.
The first connections between the research areas of convex analysis and monotone operators were noticed early in their development by authors like Kachurovskii and Rockafellar. Various other contributions in this direction followed and the rediscovery of Fitzpatrick's function in the early 2000's opened the gate towards new achievements regarding monotone operators by means of convex analysis and viceversa. In this talk we present some of the most important results obtained in one of these areas by employing tools specific to the other one, mentioning our own the contributions, too, and discuss the current tendencies and some open problems.
This talk is divided into two parts. In the first one, we begin by proving a linear uniqueness result for weak solutions of transport-diffusion equations possessing some integrability. This first theorem follows the ideas of the celebrated DiPerna-Lions theory, whose main lines will be recalled. In a second part, we use this uniqueness result - along with variations thereof - to show the smoothness of a Leray solution of the (incompressible, homogeneous) Navier-Stokes equations of which only one component is assumed to satisfy a regularity assumption at the scaling of the equations.
The topic of the talk is the Hardy space theory of compensated compactness quantities which originated in the seminal paper of Coifman, Lions, Meyer and Semmes from 1993. I concentrate on quantities relevant to fluid dynamics and discuss the relation between compensated compactness and Hardy space integrability. I also present an application to uniqueness of weak solutions with vanishing Cauchy data in 2D magnetohydrodynamics, which is joint work with Daniel Faraco.
We consider the motion of a rigid body due to the pressure of a surrounded two-dimensional irrotational perfect incompressible fluid, the whole system being confined in a bounded domain with an impermeable condition on a part of the external boundary. Thanks to an impulsive control strategy we prove that there exists an appropriate boundary condition on the remaining part of the external boundary (allowing some fluid going in and out the domain) such that the immersed rigid body is driven from some given initial position and velocity to some final position and velocity in a given positive time, without touching the external boundary. The controlled part of the external boundary is assumed to have a nonvoid interior and the final position is assumed to be in the same connected component of the set of possible positions as the initial position.