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Author: Druet, Pierre-Etienne × End date: 2024 × Type: Text × WGL Institute: WIAS ×

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Now showing 1 - 6 of 6
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    Well-posedness analysis of multicomponent incompressible flow models
    (Basel : Springer, 2021) Bothe, Dieter; Druet, Pierre-Etienne
    In this paper, we extend our study of mass transport in multicomponent isothermal fluids to the incompressible case. For a mixture, incompressibility is defined as the independence of average volume on pressure, and a weighted sum of the partial mass densities stays constant. In this type of models, the velocity field in the Navier–Stokes equations is not solenoidal and, due to different specific volumes of the species, the pressure remains connected to the densities by algebraic formula. By means of a change of variables in the transport problem, we equivalently reformulate the PDE system as to eliminate positivity and incompressibility constraints affecting the density, and prove two type of results: the local-in-time well-posedness in classes of strong solutions, and the global-in-time existence of solutions for initial data sufficiently close to a smooth equilibrium solution.
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    Analysis of improved Nernst-Planck-Poisson models of isothermal compressible electrolytes subject to chemical reactions: The case of a degenerate mobility matrix
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Druet, Pierre-Etienne
    We continue our investigations of the improved NernstPlanckPoisson model introduced in [DGM13]. In the paper [DDGG16] the analysis relies on the hypothesis that the mobility matrix has maximal rank under the constraint of mass conservation (rank N-1 for a mixture of N species). In this paper we allow for the case that the positive eigenvalues of the mobility matrix tend to zero along with the partial mass densities of certain species. In this approach the mobility matrix has a variable rank between zero and N-1 according to the number of locally available species. We set up a concept of weak solution able to deal with this scenario, showing in particular how to extend the fundamental notion of differences of chemical potentials that supports the modelling and the analysis in [DDGG16]. We prove the global-in-time existence in this solution class.
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    Weak solutions to a stationary heat equation with nonlocal radiation boundary condition and right-hand side in L-p with p>=1
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Druet, Pierre-Etienne
    Accurate modeling of heat transfer in high-temperatures situations requires to account for the effect of heat radiation. In complex applications such as Czochralski's method for crystal growth, in which the conduction radiation heat transfer problem couples to an induction heating problem and to the melt flow problem, we hardly can expect from the mathematical theory that the heat sources will be in a better space than L-1. In such situations, the known results on the unique solvability of the heat conduction problem with surface radiation do not apply, since a right-hand side in L-p with p < 6/5 no longer belongs to the dual of the Banach space in which coercivity is obtained. In this paper, we focus on a stationary heat equation with non-local boundary conditions and right-hand side in L-p with p>=1 arbitrary. Essentially, we construct an approximation procedure and, thanks to new coercivity results, we are able to produce energy estimates that involve only the L-p-norm of the heat-sources, and to pass to the limit.
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    On weak solutions to the stationary MHD-equations coupled to heat transfer with nonlocal radiation boundary conditions
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Druet, Pierre-Etienne
    We study the coupling of the stationary system of magnetohydrodynamics to the heat equation. Coupling occurs on the one hand from temperature-dependent coefficients and from a temperature-dependent force term in the Navier-Stokes equations. On the other hand, the heat sources are given by the dissipation of current in the electrical conductors, and of viscous stresses in the fluid. We consider a domain occupied by several different materials, and have to take into account interface conditions for the electromagnetic fields. Since we additionally want to treat high-temperatures applications, we also take into account the effect of heat radiation, which results in nonlocal boundary conditions for the heat flux. We prove the existence of weak solutions for the coupled system, under the assumption that the imposed velocity at the boundary of the fluid remains sufficiently small. We prove a uniqueness result in the case of constant coefficients and small data. Finally, we discuss the regularity issue in a simplified setting.
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    Some mathematical problems related to the 2nd order optimal shape of a crystallization interface
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Druet, Pierre-Etienne
    We consider the problem to optimize the stationary temperature distribution and the equilibrium shape of the solid-liquid interface in a two-phase system subject to a temperature gradient. The interface satisfies the minimization principle of the free energy, while the temperature is solving the heat equation with a radiation boundary conditions at the outer wall. Under the condition that the temperature gradient is uniformly negative in the direction of crystallization, the interface is expected to have a global graph representation. We reformulate this condition as a pointwise constraint on the gradient of the state, and we derive the first order optimality system for a class of objective functionals that account for the second surface derivatives, and for the surface temperature gradient.
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    Weak solutions to a time-dependent heat equation with nonlocal radiation boundary condition and right-hand side in L-p with p>=1
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Druet, Pierre-Etienne
    It is known that the time-dependent heat equation with nonlocal radiation boundary conditions possesses a unique weak solution if the heat sources are in L-2. In this paper, we generalize the known existence and uniqueness results to the case that the right-hand side belongs to an arbitrary L-p space (p >= 1). This is the continuation of the results that we recently proved for the stationary problem. The purpose of both papers is to obtain energy estimates that involve only the L-p norm of the heat sources for some exponent p close to one. Such estimates are important for the investigation of models in which the heat equation is coupled to Maxwell's equations or to the Navier-Stokes equations (dissipative heating).