2018, Emmrich, Etienne, Lasarzik, Robert

We study the Ericksen-Leslie system equipped with a quadratic free energy functional. The norm restriction of the director is incorporated by a standard relaxation technique using a double-well potential. We use the relative energy concept, often applied in the context of compressible Euler- or related systems of fluid dynamics, to prove weak-strong uniqueness of solutions. A main novelty, not only in the context of the Ericksen-Leslie model, is that the relative energy inequality is proved for a system with a nonconvex energy.

2020, Baňas, L'ubomír, Lasarzik, Robert, Prohl, Andreas

We consider a system of nonlinear PDEs modeling nematic electrolytes, and construct a dissipative solution with the help of its implementable, structure-inheriting space-time discretization. Computational studies are performed to study the mutual effects of electric, elastic, and viscous effects onto the molecules in a nematic electrolyte.

2019, Bayer, Christian, Rendall, Alan D., Wälde, Klaus

We study optimal savings in continuous time with exogenous transitions between employment and unemployment as the only source of uncertainty in a small open economy. We prove the existence of an optimal consumption path. We exploit that the dynamics of consumption and wealth between jumps can be expressed as a Fuchsian system. We derive conditions under which an invariant joint distribution for the state variables, i.e., wealth and labour market status, exists and is unique. We also provide conditions under which the distribution of these variables converges to the invariant distribution. Our analysis relies on the notion of T-processes and applies results on the stability of Markovian processes from Meyn and Tweedie (1993a, b,c). © 2019 The Author(s)

2021, Lasarzik, Robert

We define the concept of energy-variational solutions for the Navier--Stokes and Euler equations. This concept is shown to be equivalent to weak solutions with energy conservation. Via a standard Galerkin discretization, we prove the existence of energy-variational solutions and thus weak solutions in any space dimension for the Navier--Stokes equations. In the limit of vanishing viscosity the same assertions are deduced for the incompressible Euler system. Via the selection criterion of maximal dissipation we deduce well-posedness for these equations.

2021, Lasarzik, Robert

We introduce the new concept of maximally dissipative solutions for a general class of isothermal GENERIC systems. Under certain assumptions, we show that maximally dissipative solutions are well-posed as long as the bigger class of dissipative solutions is non-empty. Applying this result to the Navier–Stokes and Euler equations, we infer global well-posedness of maximally dissipative solutions for these systems. The concept of maximally dissipative solutions coincides with the concept of weak solutions as long as the weak solutions inherits enough regularity to be unique.

2019, Lasarzik, Robert

We introduce the new concept of maximal dissipative solutions for the Navier--Stokes and Euler equations and show that these solutions exist and the solution set is closed and convex. The concept of maximal dissipative solutions coincides with the concept of weak solutions as long as the weak solutions inherits enough regularity to be unique. A maximal dissipative solution is defined as the minimizer of a convex functional and we argue that this definition bears several advantages.