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Subject: defect measure ×

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    Global-in-time existence for liquid mixtures subject to a generalised incompressibility constraint
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Druet, Pierre-Étienne
    We consider a system of partial differential equations describing diffusive and convective mass transport in a fluid mixture of N > 1 chemical species. A weighted sum of the partial mass densities of the chemical species is assumed to be constant, which expresses the incompressibility of the fluid, while accounting for different reference sizes of the involved molecules. This condition is different from the usual assumption of a constant total mass density, and it leads in particular to a non-solenoidal velocity field in the Navier-Stokes equations. In turn, the pressure gradient occurs in the diffusion fluxes, so that the PDE-system of mass transport equations and momentum balance is fully coupled. Another striking feature of such incompressible mixtures is the algebraic formula connecting the pressure and the densities, which can be exploited to prove a pressure bound in L1. In this paper, we consider incompressible initial states with bounded energy and show the global existence of weak solutions with defect measure.
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    Global-in-time existence of weak solutions to Kolmogorov's two-equation model of turbulence
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Mielke, Alexander; Naumann, Joachim
    We consider Kolmogorov's model for the turbulent motion of an incompressible fluid in 3. This model consists in a Navier-Stokes type system for the mean flow u and two further partial differential equations: an equation for the frequency and for the kinetic energy k each. We investigate this system of partial differential equations in a cylinder x ]0,T[ ( 3 cube, 0 < T < +∞) under spatial periodic boundary conditions on x ]0,T[ and initial conditions in x {0}. We present an existence result for a weak solution {u, , k} to the problem under consideration, with , k obeying the inequalities formula1 and formula2.
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    On the existence of global-in-time weak solutions and scaling laws for Kolmogorovs two-equation model of turbulence
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Mielke, Alexander; Naumann, Joachim
    This paper is concerned with Kolmogorov's two-equation model for free turbulence in space dimension 3, involving the mean velocity u, the pressure p, an average frequency omega, and a mean turbulent kinetic energy k. We first discuss scaling laws for a slightly more general two-equation models to highlight the special role of the model devised by Kolmogorov in 1942. The main part of the paper consists in proving the existence of weak solutions of Kolmogorov's two-equation model under space-periodic boundary conditions in cubes with positive side length l. To this end, we provide new a priori estimates and invoke existence result for pseudo-monotone operators.