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Author: Druet, Pierre-Étienne × Start date: 2019 ×

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Now showing 1 - 10 of 25
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    Analysis of improved Nernst-Planck-Poisson models of compressible isothermal electrolytes. Part II: Approximation and a priori estimates
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Dreyer, Wolfgang; Druet, Pierre-Étienne; Gajewski, Paul; Guhlke, Clemens
    We consider an improved NernstPlanckPoisson model first proposed by Dreyer et al. in 2013 for compressible isothermal electrolytes in non equilibrium. The model takes into account the elastic deformation of the medium that induces an inherent coupling of mass and momentum transport. The model consists of convectiondiffusionreaction equations for the constituents of the mixture, of the Navier-Stokes equation for the barycentric velocity, and of the Poisson equation for the electrical potential. Due to the principle of mass conservation, crossdiffusion phenomena must occur and the mobility matrix (Onsager matrix) has a kernel. In this paper, which continues the investigation of [DDGG17a], we derive for thermodynamically consistent approximation schemes the natural uniform estimates associated with the dissipations. Our results essentially improve our former study [DDGG16], in particular the a priori estimates concerning the relative chemical potentials.
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    Analysis of improved Nernst-Planck-Poisson models of compressible isothermal electrolytes. Part I: Derivation of the model and survey of the results
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Dreyer, Wolfgang; Druet, Pierre-Étienne; Gajewski, Paul; Guhlke, Clemens
    We consider an improved NernstPlanckPoisson model first proposed by Dreyer et al. in 2013 for compressible isothermal electrolytes in non equilibrium. The model takes into account the elastic deformation of the medium that induces an inherent coupling of mass and momentum transport. The model consists of convectiondiffusionreaction equations for the constituents of the mixture, of the Navier-Stokes equation for the barycentric velocity, and of the Poisson equation for the electrical potential. Due to the principle of mass conservation, crossdiffusion phenomena must occur and the mobility matrix (Onsager matrix) has a kernel. In this paper we establish the existence of a globalintime weak solution for the full model, allowing for a general structure of the mobility tensor and for chemical reactions with highly non linear rates in the bulk and on the active boundary. We characterise the singular states of the system, showing that the chemical species can vanish only globally in space, and that this phenomenon must be concentrated in a compact set of measure zero in time. With respect to our former study [DDGG16], we also essentially improve the a priori estimates, in particular concerning the relative chemical potentials.
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    Local well-posedness for thermodynamically motivated quasilinear parabolic systems in divergence form
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Druet, Pierre-Étienne
    We show that fully quasilinear parabolic systems are locally well posed in the Hilbert space scala if the coefficients of the differential operator are smooth enough and the spatial domain is sufficiently regular. In the context of diffusion systems driven by entropy, the uniform parabolicity follows from the second law of thermodynamics.
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    Global-in-time solvability of thermodynamically motivated parabolic systems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Druet, Pierre-Étienne
    In this paper, doubly non linear parabolic systems in divergence form are investigated form the point of view of their global-in-time weak solvability. The non-linearity under the time derivative is given by the gradient of a strictly convex, globally Lipschitz continuous potential, multiplied by a position-dependent weight. This weight admits singular values. The flux under the spatial divergence is also of monotone gradient type, but it is not restricted to polynomial growth. It is assumed that the elliptic operator generates some equi-coercivity on the spatial derivatives of the unknowns. The paper introduces some original techniques to deal with the case of nonlinear purely Neumann boundary conditions. In this respect, it generalises or complements the results by Alt and Luckhaus (1983) and Alt (2012). A field of application of the theory are the multi species diffusion systems driven by entropy.
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    Existence of weak solutions for improved Nernst-Planck-Poisson models of compressible reacting electrolytes
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Dreyer, Wolfgang; Druet, Pierre-Étienne; Gajewski, Paul; Guhlke, Clemens
    We consider an improved Nernst-Planck-Poisson model for compressible electrolytes first proposed by Dreyer et al. in 2013. The model takes into account the elastic deformation of the medium. In particular, large pressure contributions near electrochemical interfaces induce an inherent coupling of mass and momentum transport. The model consists of convection-diffusion-reaction equations for the constituents of the mixture, of the Navier-Stokes equation for the barycentric velocity and the Poisson equation for the electrical potential. Cross-diffusion phenomena occur due to the principle of mass conservation. Moreover, the diffusion matrix (mobility matrix) has a zero eigenvalue, meaning that the system is degenerate parabolic. In this paper we establish the existence of a global-in-time weak solution for the full model, allowing for cross-diffusion and an arbitrary number of chemical reactions in the bulk and on the active boundary.
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    Incompressible limit for a fluid mixture
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Druet, Pierre-Étienne
    In this paper we discuss the incompressible limit for multicomponent fluids in the isothermal ideal case. Both a direct limit-passage in the equation of state and the low Mach-number limit in rescaled PDEs are investigated. Using the relative energy inequality, we obtain convergence results for the densities and the velocity-field under the condition that the incompressible model possesses a sufficiently smooth solution, which is granted at least for a short time. Moreover, in comparison to single-component flows, uniform estimates and the convergence of the pressure are needed in the multicomponent case because the incompressible velocity field is not divergence-free. We show that certain constellations of the mobility tensor allow to control gradients of the entropic variables and yield the convergence of the pressure in L1.
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    Higher integrability of the Lorentz force for weak solutions to Maxwell's equations in complex geometries
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Druet, Pierre-Étienne
    We consider the stationary Maxwell system in a domain filled with different materials. The magnetic permeability being only piecewise smooth, we have to take into account the natural interface conditions for the electromagnetic fields. We present two sets of hypotheses under which we can prove the existence of weak solutions to the Maxwell system such that the Lorentz force jxB is integrable to a power larger than 6/5. This property is important for the investigation of problems in magnetohydrodynamics, with many industrial applications such as crystal growth.
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    Analysis of cross-diffusion systems for fluid mixtures driven by a pressure gradient
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Druet, Pierre-Étienne; Jüngel, Ansgar
    The convective transport in a multicomponent isothermal compressible fluid subject to the mass continuity equations is considered. The velocity is proportional to the negative pressure gradient, according to Darcy?s law, and the pressure is defined by a state equation imposed by the volume extension of the mixture. These model assumptions lead to a parabolic-hyperbolic system for the mass densities. The global-in-time existence of classical and weak solutions is proved in a bounded domain with no-penetration boundary conditions. The idea is to decompose the system into a porous-medium-type equation for the volume extension and transport equations for the modified number fractions. The existence proof is based on parabolic regularity theory, the theory of renormalized solutions, and an approximation of the velocity field.
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    A curvature estimate for open surfaces subject to a general mean curvature operator and natural contact conditions at their boundary
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Druet, Pierre-Étienne
    In the seventies, L. Simon showed that the main curvatures of two-dimensional hypersurfaces obeying a general equation of mean curvature type are a priori bounded by the Hölder norm of the coefficients of the surface differential operator. This was an essentially interior estimate. In this paper, we provide a complement to the theory, proving a global curvature estimate for open surfaces that satisfy natural contact conditions at the intersection with a given boundary.
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    A theory of generalised solutions for ideal gas mixtures with Maxwell--Stefan diffusion
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Druet, Pierre-Étienne
    After the pioneering work by Giovangigli on mathematics of multicomponent flows, several attempts were made to introduce global weak solutions for the PDEs describing the dynamics of fluid mixtures. While the incompressible case with constant density was enlighted well enough due to results by Chen and Jüngel (isothermal case), or Marion and Temam, some open questions remain for the weak solution theory of gas mixtures with their corresponding equations of mixed parabolic-hyperbolic type. For instance, Mucha, Pokorny and Zatorska showed the possibility to stabilise the hyperbolic component by means of the Bresch-Desjardins technique and a regularisation of pressure preventing vacuum. The result by Dreyer, Druet, Gajewski and Guhlke avoids emphex machina stabilisations, but the mathematical assumption that the Onsager matrix is uniformly positive on certain subspaces leads, in the dilute limit, to infinite diffusion velocities which are not compatible with the Maxwell-Stefan form of diffusion fluxes. In this paper, we prove the existence of global weak solutions for isothermal and ideal compressible mixtures with natural diffusion. The main new tool is an asymptotic condition imposed at low pressure on the binary Maxwell-Stefan diffusivities, which compensates possibly extreme behaviour of weak solutions in the rarefied regime.