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 for solutions which satisfy the…
Recently there has been considerable research into the stability of shocks in systems of conservation laws, with stability understood in some square-integrable sense. In this talk I will give some…
Augmented Lagrangians were first employed in an algorithm for solving nonlinear programming problems with equality constraints. However, the approach was soon extended to inequality constraints and shown in the case…
We study problems of optimal transport, by embedding them in a general functional analytic framework of convex optimization. This provides a unified treatment of alarge class of related problems in…
A nonlinear model is developed for fluid-poroelastic structure interaction with quasi-Newtonian fluids that exhibit a shear-thinning property. The flow in the fluid region is described by the Stokes equations and…
In recent years, novel optimization ideas have been applied to several inverse problems in combination with machine learning approaches, to improve the inversion by optimally choosing different quantities/functions of interest. A…
Tropical convex sets arise as ``log-limits'' of parametric families of classical convex sets. The tropicalizations of polyhedra and spectrahedra are of special interest, since they can be described in terms of…
We consider the density properties of divergence-free vector fields b in L^1(,BV(^2)) which are ergodic/weakly mixing/strongly mixing: this means that their Regular Lagrangian Flow X_t is an ergodic/weakly mixing/strongly mixing measure preserving map when evaluated…
Among all drum heads of a fixed area, a circular drum head produces the vibration of lowest frequency. The general dimensional analogue of this fact is the Faber-Krahn inequality: balls…
The well-posedness of Riccati equations plays a central role in the study of the optimal control problem with quadratic functionals for linear partial differential equations (PDEs). Indeed, it allows the…
I will discuss the existence and orbital stability of standing-wave solutions (i.e., with a specific time-dependence) with minimal energy (so-called ground states) to a non-linear Schrödinger equation where the L² norm is prescribed. I will…
We study the effect of total variation regularization on PDE-constrained optimization problems, where the control input functions may only attain finitely many integer values. The regularization helps to avoid undesirable…
Control systems are a class of dynamical systems that contain forcing terms. When control systems are used in engineering applications, the forcing terms can represent forces that can be applied…
We consider a controlled reaction-diffusion equation, modeling the spreading of an invasive population. Our goal is to derive a simpler model, describing the controlled evolution of a contaminated set. We…
Classification is a fundamental task in data science and machine learning, and in the past ten years there have been significant improvements on classification tasks (e.g. via deep learning). However,…
We consider a long-time behavior of a stochastically forced nonlinear oscillator. In a long-time limit the force converges to fractional Brownian motion, a process that has memory. In contrast, we…
We will discuss some old and new results concerning the long-time behavior of solutions to the two-dimensional incompressible Euler equations. Specifically, we discuss whether steady states can be isolated, wandering…
The so-called $l_0$ pseudonorm counts the number of nonzero components of a vector. It is standard in sparse optimization problems. However, as it is a discontinuous and nonconvex function, the…
During the preparation of a foundational chapter on manifolds of mappings for a book on geometric continuum mechanics I found out that the following object behaves surprisingly well as source…