On the incompressible limit of a strongly stratified heat conducting fluid

A compressible, viscous and heat conducting fluid is confined between two parallel plates maintained at a constant temperature and subject to a strong stratification due to the gravitational force. We consider the asymptotic limit, where the Mach number and the Froude number are of the same order proportional to a small parameter. We show the limit problem can be identified with Majda's model of layered ``stack-of-pancake'' flow.Comment: arXiv admin note: text overlap with arXiv:2206.1404

Vanishing viscosity limit for the compressible Navier–Stokes system via measure-valued solutions

We identify a class of measure-valued solutions of the barotropic Euler system on a general (unbounded) spatial domain as a vanishing viscosity limit for the compressible Navier–Stokes system. Then we establish the weak (measure-valued)–strong uniqueness principle, and, as a corollary, we obtain strong convergence to the Euler system on the lifespan of the strong solution.TU Berlin, Open-Access-Mittel – 202

Semiflow selection for the compressible Navier–Stokes system

Although the existence of dissipative weak solutions for the compressible Navier–Stokes system has already been established for any finite energy initial data, uniqueness is still an open problem. The idea is then to select a solution satisfying the semigroup property , an important feature of systems with uniqueness. More precisely, we are going to prove the existence of a semiflow selection in terms of the three state variables: the density, the momentum, and the energy. Finally, we will show that it is possible to introduce a new selection defined only in terms of the initial density and momentum; however, the price to pay is that the semigroup property will hold almost everywhere in time.TU Berlin, Open-Access-Mittel – 202

Auswahl eines Halbflusses und Grenzwert verschwindender Viskosität in der Flüssigkeitsdynamik

Well-posedness of systems describing the motion of fluids in the class of strong and weak solutions represents one of the most challenging problems of the modern theory of partial differential equations. To overcome the problem of existence, one suitable idea is to consider a larger class of solutions. In the first part of the thesis we identify a measure-valued solution, characterized by a parametrized Young measure, of the compressible Euler system with damping on a general (unbounded) domain as a vanishing viscosity limit for the compressible Navier–Stokes system. Afterwards, we establish the weak (measure-valued)–strong uniqueness principle, and, as a consequence, we obtain convergence of the weak solutions of the Navier–Stokes equations in the zero viscosity limit to the strong solution of the Euler system, as long as the latter exists. To handle the problem of uniqueness, one possible way is to perform a semiflow selection, identifying, among all the solutions emanating from the same initial data, the one satisfying the semigroup property. In the second part of the thesis we study under which assumptions it is possible to guarantee the existence of a semiflow selection for autonomous and non-autonomous systems, choosing the Skorokhod space of càglàd functions as trajectory space. Subsequently, we adapt this abstract machinery to the compressible Navier–Stokes system, for which we will be able to prove the existence of a semiflow selection depending only on the initial density and momentum, and to models describing general non-Newtonian fluids. In this latter case, we prove the existence of dissipative solutions for a linear pressure and we will analyze under which conditions it is possible to guarantee the existence of weak solutions.Die Wohlgestelltheit von Problemen, die aus der Beschreibung der Bewegung von Fluiden stammen, in der Klasse von starken oder schwachen Lösungen zu zeigen, ist eine der herausforderndsten Aufgabenstellungen in der modernen Theorie partieller Differentialgleichungen. Eine Möglichkeit, um die Existenz von Lösungen zu zeigen, ist den Lösungsgebriff passend zu erweitern. Im ersten Teil dieser Arbeit identifizieren wir eine maßwertige Lösung der gedämpften kompressiblen Euler-Gleichungen auf einem allgemeinen (unbeschränkten) Gebiet, die durch ein parametrisiertes Young-Maß charakterisiert wird, als den Grenzwert der kompressiblen Navier-Stokes-Gleichungen bei verschwindender Viskosität. Anschließend zeigen wir ein Prinzip der schwach-beziehungsweise maßwertig-starken Einzigkeit und erhalten als Konsequenz die Konvergenz schwacher Lösungen der Navier-Stokes-Gleichungen bei verschwindender Viskosität gegen starke Lösungen der Euler-Gleichungen, falls letztere existieren. Ein Weg die Frage nach der Einzigkeit von Lösungen zu behandeln ist die Auswahl eins Halbflusses, das heißt aus der Menge aller Lösungen zu gegebenen Anfangsdaten diejenige auszuwählen, die die Halbgruppeneigenschaft besitzt. Im zweiten Teil der Arbeit untersuchen wir, unter welchen Voraussetzungen eine solche Auswahl eines Halbflusses für autonome und nicht-autonome Systeme möglich ist, wobei wir als Raum für die Trajektorien den Skorochod-Raum der Càglàd-Funktionen wählen. Anschließend adaptieren wir die abstrakten Resultate für die kompressiblen Navier-Stokes-Gleichungen, für die wir die Existenz einer Halbfluss-Auswahl zeigen können, welche nur von den Anfangswerten der Dichte und Impulsdichte abhängt, sowie für eine Klasse von Modellen zur Beschreibung nichtnewtonscher Fluide. Für Letztere zeigen wir die Existenz dissipativer Lösungen unter Annahme eines linearen Drucks und untersuchen unter welchen Bedingungen die Existenz schwacher Lösungen gesichert ist

Semiflow selection to models of general compressible viscous fluids

We prove the existence of a semiflow selection with range the space of càglàd, i.e. left-continuous and having right-hand limits functions defined on [0,\infty) and taking values in a Hilbert space. Afterwards, we apply this abstract result to the system arising from a compressible viscous fluid with a barotropic pressure of the type aϱ^γ, γ ≥ 1 , with a viscous stress tensor being a nonlinear function of the symmetric velocity gradient.TU Berlin, Open-Access-Mittel – 202

On the Incompressible Limit of a Strongly Stratified Heat Conducting Fluid

Funder: Institute of Mathematics of the Czech Academy of SciencesA compressible, viscous and heat conducting fluid is confined between two parallel plates maintained at a constant temperature and subject to a strong stratification due to the gravitational force. We consider the asymptotic limit, where the Mach number and the Froude number are of the same order proportional to a small parameter. We show the limit problem can be identified with Majda’s model of layered “stack-of-pancake” flow

Penalization method for the Navier–Stokes–Fourier system

We apply the method of penalization to the Dirichlet problem for the Navier–Stokes–Fourier system governing the motion of a general viscous compressible fluid confined to a bounded Lipschitz domain. The physical domain is embedded into a large cube on which the periodic boundary conditions are imposed. The original boundary conditions are enforced through a singular friction term in the momentum equation and a heat source/sink term in the internal energy balance. The solutions of the penalized problem are shown to converge to the solution of the limit problem. In particular, we extend the available existence theory to domains with rough (Lipschitz) boundary. Numerical experiments are performed to illustrate the efficiency of the method