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Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik × Subject: global weak solutions ×

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Now showing 1 - 2 of 2
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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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    On a higher order convective Cahn-Hilliard type equation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Korzec, Maciek Dominik; Rybka, Piotr
    A convective Cahn-Hilliard type equation of sixth order that describes the faceting of a growing surface is considered with periodic boundary conditions. By using a Galerkin approach the existence of weak solutions to this sixth order partial differential equation is established in $L^2(0,T; dot H^3_per)$. Furthermore stronger regularity results have been derived and these are used to prove uniqueness of the solutions. Additionally a numerical study shows that solutions behave similarly as for the better known convective Cahn-Hilliard equation. The transition from coarsening to roughening is analyzed, indicating that the characteristic length scale decreases logarithmically with increasing deposition rate