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- ItemAnalysis 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, ClemensWe 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.
- ItemAnalysis 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, ClemensWe 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.
- ItemLocal well-posedness for thermodynamically motivated quasilinear parabolic systems in divergence form(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Druet, Pierre-ÉtienneWe 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.
- ItemGlobal-in-time solvability of thermodynamically motivated parabolic systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Druet, Pierre-ÉtienneIn 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.
- ItemExistence 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, ClemensWe 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.
- ItemHigher 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-ÉtienneWe 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.
- ItemA 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-ÉtienneIn 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.
- ItemRegularity of second derivatives in elliptic transmission problems near an interior regular multiple line of contact(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Druet, Pierre-ÉtienneWe investigate the regularity of the weak solution to elliptic transmission problems that involve several materials intersecting at a closed interior line of contact.We prove that local weak solutions possess second order generalized derivatives up to the contact line, mainly exploiting their higher regularity in the direction tangential to the line. Moreover we are thus able to characterize the higher regularity of the gradient and the Hölder exponent by means of explicit estimates known in the literature for two dimensional problems. They show that strong regularity properties, for instance the integrability of the gradient to a power larger than the space dimension d = 3, are to expect if the oscillations of the diffusion coefficient are moderate (that is for far larger a range than what a theory of small perturbations would allow), or if the number of involved materials does not exceed three.
- ItemNoncompactness of integral operators modeling diffuse-gray radiation in polyhedral and transient settings(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Druet, Pierre-Étienne; Philip, PeterWhile it is well-known that the standard integral operator K of (stationary) diffuse-gray radiation, as it occurs in the radiosity equation, is compact if the domain of radiative interaction is sufficiently regular, we show noncompactness of the operator if the domain is polyhedral. We also show that a stationary operator is never compact when reinterpreted in a transient setting. Moreover, we provide new proofs, which do not use the compactness of K, for 1 being a simple eigenvalue of K for connected enclosures, and for I-(1-e)K being invertible, provided the emissivity e does not vanish identically
- ItemOptimal control of 3D state constrained induction heating problems with nonlocal radiation effects(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Druet, Pierre-Étienne; Klein, Olaf; Sprekels, Jürgen; Tröltzsch, Fredi; Yousept, IrwinThe paper is concerned with a class of optimal heating problems in semiconductor single crystal growth processes. To model the heating process, time-harmonic Maxwell equations are considered in the system of the state. Due to the high temperatures characterizing crystal growth, it is necessary to include nonlocal radiation boundary conditions and a temperature-dependent heat conductivity in the description of the heat transfer process. The first goal of this paper is to prove the existence and uniqueness of the solution to the state equation. The regularity analysis associated with the time harmonic Maxwell equations is also studied. In the second part of the paper, the existence and uniqueness of the solution to the corresponding linearized equation is shown. With this result at hand, the differentiability of the control-to-state mapping operator associated with the state equation is derived. Finally, based on the theoretical results, first oder necessary optimality conditions for an associated optimal control problem are established.