7 research outputs found
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 NavierâStokes 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 NavierâStokes system. Then we establish the weak (measure-valued)âstrong 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 â 202
Semiflow selection for the compressible NavierâStokes system
Although the existence of dissipative weak solutions for the compressible NavierâStokes 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 â 202
Auswahl eines Halbflusses und Grenzwert verschwindender ViskositĂ€t in der FlĂŒssigkeitsdynamik
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 NavierâStokes system. Afterwards, we establish the weak (measure-valued)âstrong uniqueness principle, and, as a consequence, we obtain convergence of the weak solutions of the NavierâStokes 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 cĂ glĂ d functions as trajectory space. Subsequently, we adapt this abstract machinery to the compressible NavierâStokes 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 maĂwertige Lösung der gedĂ€mpften kompressiblen Euler-Gleichungen auf einem allgemeinen (unbeschrĂ€nkten) Gebiet, die durch ein parametrisiertes Young-MaĂ charakterisiert wird, als den Grenzwert der kompressiblen Navier-Stokes-Gleichungen bei verschwindender ViskositĂ€t. AnschlieĂend zeigen wir ein Prinzip der schwach-beziehungsweise maĂwertig-starken Einzigkeit und erhalten als Konsequenz die Konvergenz schwacher Lösungen der Navier-Stokes-Gleichungen bei verschwindender ViskositĂ€t 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 heiĂt aus der Menge aller Lösungen zu gegebenen Anfangsdaten diejenige auszuwĂ€hlen, die die Halbgruppeneigenschaft besitzt. Im zweiten Teil der Arbeit untersuchen wir, unter welchen Voraussetzungen eine solche Auswahl eines Halbflusses fĂŒr autonome und nicht-autonome Systeme möglich ist, wobei wir als Raum fĂŒr die Trajektorien den Skorochod-Raum der CĂ glĂ d-Funktionen wĂ€hlen. AnschlieĂend adaptieren wir die abstrakten Resultate fĂŒr die kompressiblen Navier-Stokes-Gleichungen, fĂŒr die wir die Existenz einer Halbfluss-Auswahl zeigen können, welche nur von den Anfangswerten der Dichte und Impulsdichte abhĂ€ngt, sowie fĂŒr eine Klasse von Modellen zur Beschreibung nichtnewtonscher Fluide. FĂŒr 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 cĂ glĂ d, 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^Îł, Îł â„ 1 , with a viscous stress tensor being a nonlinear function of the symmetric velocity gradient.TU Berlin, Open-Access-Mittel â 202
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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 Majdaâs model of layered âstack-of-pancakeâ flow
Penalization method for the NavierâStokesâFourier system
We apply the method of penalization to the Dirichlet problem for the NavierâStokesâFourier 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