In my thesis, I consider the system of thermoelasticity endowed with polyconvex energy. After presenting the equations in their mathematical and physical context, I embed the equations of polyconvex thermoviscoelasticity into an augmented, symmetrizable, hyperbolic system which possesses a convex entropy. This allows to prove many important stability results, such as convergence from thermoviscoelasticity (with Newtonian viscosity and Fourier heat conduction) to smooth solutions of the system of adiabatic thermoelasticity, and convergence from thermoviscoelasticity to smooth solutions of thermoelasticity in the zero-viscosity limit. In addition, I establish a weak-strong uniqueness result in the class of entropy weak solutions and in a suitable class of measure-valued solutions, defined by means of generalized Young measures that describe both oscillatory and concentration effects. Also, I construct a variational scheme for isentropic processes of adiabatic polyconvex thermoelasticity: I establish existence of minimizers which converge to a measure-valued solution that dissipates the total energy, while the scheme converges when the limiting solution is smooth.

Approximating the solution to an evolutionary partial differential equation by a set of "moving particles" has several advantages. It validates the use of a continuity equation in an "individuals-based" modeling setting, it provides a link between Lagrangian and Eulerian description, and it defines a "natural" numerical approach to those equations. I will describe recent rigorous results in that context. The main one deals with one-dimensional scalar conservation laws with nonnegative initial data, for which we prove that the a suitably designed "follow-the-leader" particle scheme approximates entropy solutions in the sense of Kruzkov in the many particle limit. Said result represents a new way to solve scalar conservation laws with bounded and integrable initial data. The same method applies to second order traffic flow models, to nonlocal transport equations, and to the Hughes model for pedestrian movements.

In this talk we consider the Cauchy problem for the 2D Euler equations for incompressible inviscid fluids. It is well-known that given a smooth initial datum, the Cauchy problem is  well-posed and in particular the energy is conserved and the vorticity is transported by the flow of the velocity. When we consider weak solutions this might not be the case anymore. We will review some recent results obtained in collaboration with Gianluca Crippa and Gennaro Ciampa where we extend those properties to the case of irregular vorticities. In particular, under low integrability assumptions on the vorticity we show that certain approximations important from a physical and a numerical point of view converge to solutions satisfying those properties.

In this talk we present the derivation of a new Boussinesq-type system to describe the propagation of long waves of small amplitude in a basin with mildly varying bottom topography. We prove the existence and uniqueness of weak solutions for maximal times that do not depend on the amplitude of the waves. We then present the numerical solution of the new system using Galerkin finite element methods and prove the convergence of the semidiscrete solution to the exact solution. The system appears to describe well water waves even in benchmark experiments that involve also general bathymetries.

Abstract

 

In this course we consider multi-phase flows, i.e., flows of one substance which is present as liquid as well as vapor. We focus on models that resolve individual bubbles/droplets and that treat both phases as compressible. We will also discuss incompressible/low Mach limits, since in most applications the liquid is nearly incompressible. Understanding and simulating such small-scale models is important in order to obtain information which can be used in larger scale models for e.g. sprays which play important roles in processes of practical interest as diverse as combustion, chemical engineering, and cloud formation

In this course we consider multi-phase flows, i.e., flows of one substance which is present as liquid as well as vapor. We focus on models that resolve individual bubbles/droplets and that treat both phases as compressible. We will also discuss incompressible/low Mach limits, since in most applications the liquid is nearly incompressible. Understanding and simulating such small-scale models is important in order to obtain information which can be used in larger scale models for e.g. sprays which play important roles in processes of practical interest as diverse as combustion, chemical engineering, and cloud formation