Many real-world problems involve fluids in motion. The goal of this book is to propose a new
approach to numerical analysis of the underlying nonlinear equations in the spirit of the …

We introduce a new concept of dissipative measure-valued solution to the compressible
Navier–Stokes system satisfying, in addition, a relevant form of the total energy balance …

A standard paradigm for the existence of solutions in fluid dynamics is based on the
construction of sequences of approximate solutions or approximate minimizers. This …

We give a survey of recent results on weak-strong uniqueness for compressible and
incompressible Euler and Navier-Stokes equations, and also make some new observations …

In this paper, we present convergence analysis of high-order finite element based methods,
in particular, we focus on a discontinuous Galerkin scheme using summation-by-parts …

We apply the tensor train (TT) decomposition to construct the tensor product polynomial
chaos expansion (PCE) of a random field, to solve the stochastic elliptic diffusion PDE with …

We study convergence of a mixed finite element–finite volume numerical scheme for the
isentropic Navier–Stokes system under the full range of the adiabatic exponent. We …

We introduce the concept of dissipative measure-valued solution to the complete Euler
system describing the motion of an inviscid compressible fluid. These solutions are …

We consider a class of viscous fluids with a general monotone dependence of the viscous
stress on the symmetric velocity gradient. We introduce the concept of dissipative solution to …

The Cauchy problem for the complete Euler system is in general ill-posed in the class of
admissible (entropy producing) weak solutions. This suggests that there might be …