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The book presents recent results and new trends in the theory of fluid mechanics. Each of the four chapters focuses on a different problem in fluid flow accompanied by an overview of available older results.

The chapters are extended lecture notes from the ESSAM school "Mathematical Aspects of Fluid Flows" held in Kácov (Czech Republic) in May/June 2017.

The lectures were presented by Dominic Breit (Heriot-Watt University Edinburgh), Yann Brenier (École Polytechnique, Palaiseau), Pierre-Emmanuel Jabin (University of Maryland) and Christian Rohde (Universität Stuttgart), and cover various aspects of mathematical fluid mechanics – from Euler equations, compressible Navier-Stokes equations and stochastic equations in fluid mechanics to equations describing two-phase flow; from the modeling and mathematical analysis of equations to numerical methods. Although the chapters feature relatively recent results, they are presented in a form accessible to PhD students in the field of mathematical fluid mechanics.



An Introduction to Stochastic Navier–Stokes Equations

The dynamics of liquids and gases can be modeled by the Navier–Stokes system of partial differential equations describing the balance of mass and momentum in the fluid flow. In recent years their has been an increasing interest in random influences on the fluid motion modeled via stochastic partial differential equations.
In this lecture notes we study the existence of weak martingale solutions to the stochastic Navier-Stokes equations (both incompressible and compressible). These solutions are weak in the analytical sense (derivatives exists only in the sense of distributions) and weak in the stochastic sense (the underlying probability space is not a priori given but part of the problem). In particular, we give a detailed introduction to the stochastic compactness method based on Skorokhod’s representation theorem.
Dominic Breit

Some Concepts of Generalized and Approximate Solutions in Ideal Incompressible Fluid Mechanics Related to the Least Action Principle

Various concepts of generalized and approximate solutions related to the mathematical theory of ideal incompressible fluids are discussed in relation with variational and stochastic approaches, in close connection with the least action principle.
Yann Brenier

Quantitative Regularity Estimates for Compressible Transport Equations

These notes aim at presenting some recent estimates for transport equations with rough, i.e., non-smooth, velocity fields. Our final goal is to use those estimates to obtain new results on complex systems where the transport equation is coupled to other PDE’s: A driving example being the compressible Navier–Stokes system. But for simplicity, we work in the linear setting where the velocity field is given and only briefly sketch at the end of the notes how to use the new theory for nonlinear estimates.
After reviewing some of the classical results, we focus on /quantitative/ estimates, in the absence of any bounds on the divergence of the velocity fields (or any corresponding bound on the Jacobian of the Lagrangian flow) for which a new approach is needed.
Didier Bresch, Pierre-Emmanuel Jabin

Fully Resolved Compressible Two-Phase Flow: Modelling, Analytical and Numerical Issues

Mathematical models for compressible two-phase flow of homogeneous fluids that occur in a liquid and a vapour phase can be classified as either belonging to the class of sharp interface models or to the class of diffuse interface models. Sharp interface models display the phase boundary as a sharp front separating two bulk model domains while diffuse interface models consist of a single model on the complete domain of interest such that phase boundaries are represented as transition zones. This contribution is devoted to a self-consistent introduction to both model classes.
Sharp interface models are analyzed within the theory of hyperbolic conservation laws with special focus on the Riemann problem. Based on the thermodynamically consistent solution of the Riemann problem a multidimensional finite volume method is introduced. For the associated diffuse interface ansatz the focus is on Navier–Stokes–Korteweg-type models. Several new variants are introduced which enable in particular thermodynamically consistent and asymptotically-preserving numerical discretizations. For all models it is assumed that the relevant spatial scale corresponds to fully resolved phase boundaries.
Christian Rohde
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